Dabbicco, M.

Lucia, F. De

Ottonelli, S.

Scamarcio, G.

Vietro, M. di

English

The behavior of a displacement optical sensor based on the Laser-Self-Mixing effect employing a plane mirror (PT) and a solid corner-cube (CC) as moving target is analyzed. The performances of the sensor are compared in terms of simplification of the optical setup, measurement accuracy and tolerance to angular misalignment of the target. On the basis of the innovative assumption that only the fraction of the laser beam orthogonal to the target plane gives rise to the Self – Mixing modulation, it is demonstrated that the interferometer tolerates small tilt of the plane target (up to approximately 0.7°) only when illuminated by a divergent beam, in which case the displacement measurement becomes affected by a cosine – type error. Instead the corner cube preserves the self-mixing signal over a larger angular range  (up to approximately 2°) at the same time preserving high measurement accuracy

De Lucia, F.

di Vietro, M.

JEOS:RP - Journal of the European Optical Society Rapid publications

J O U R N A L  O FTORH E  E U R O P E A N  P T I C A L  S O C I E T YA PID  PU B LIC AT IO N SJournal of the European Optical Society - Rapid Publications 4, 09036 (2009) www.jeos.orgComparison of plane mirror vs retroreflectorperformance for laser-self-mixing displacement sensorsS. Ottonellisimona.ottonelli@fisica.uniba.itCNR-INFM Regional Laboratory LIT3, Universita` degli Studi di Bari, Via Amendola 173, I-70126 Bari,Italy and Dipartimento Interateneo di Fisica “M. Merlin,” Universita` degli Studi di Bari, Via Amendola173, I-70126 Bari, ItalyF. De Lucia CNR-INFM Regional Laboratory LIT3, Universita` degli Studi di Bari, Via Amendola 173, I-70126 Bari,Italy and Dipartimento Interateneo di Fisica “M. Merlin,” Universita` degli Studi di Bari, Via Amendola173, I-70126 Bari, ItalyM. di Vietro CNR-INFM Regional Laboratory LIT3, Universita` degli Studi di Bari, Via Amendola 173, I-70126 Bari,Italy and Dipartimento Interateneo di Fisica “M. Merlin,” Universita` degli Studi di Bari, Via Amendola173, I-70126 Bari, ItalyM. Dabbicco CNR-INFM Regional Laboratory LIT3, Universita` degli Studi di Bari, Via Amendola 173, I-70126 Bari,Italy and Dipartimento Interateneo di Fisica “M. Merlin,” Universita` degli Studi di Bari, Via Amendola173, I-70126 Bari, ItalyG. Scamarcio CNR-INFM Regional Laboratory LIT3, Universita` degli Studi di Bari, Via Amendola 173, I-70126 Bari,Italy and Dipartimento Interateneo di Fisica “M. Merlin,” Universita` degli Studi di Bari, Via Amendola173, I-70126 Bari, ItalyThe behavior of a displacement optical sensor based on the Laser-Self-Mixing effect employing a plane mirror (PT) and a solid corner-cube(CC) as moving target is analyzed. The performance of the sensor is compared in terms of simplification of the optical setup, measurementaccuracy and tolerance to angular misalignment of the target. On the basis of the innovative assumption that only the fraction of the laserbeam orthogonal to the target plane gives rise to the Self – Mixing modulation, it is demonstrated that the interferometer tolerates small tiltof the plane target (up to approximately 0.7◦) only when illuminated by a divergent beam, in which case the displacement measurementbecomes affected by a cosine – type error. Instead the corner cube preserves the self-mixing signal over a larger angular range (up toapproximately 2◦) at the same time preserving high measurement accuracy. [DOI: 10.2971/jeos.2009.09036]Keywords: self-mixing, optical feedback, displacement measurement, corner-cube, target misalignment1 INTRODUCTIONLaser interferometry is one of the most disseminate and per-forming phase sensitive techniques for metrological measure-ments. Besides the traditional “external configuration” need-ing one or more external detectors, in the last two decadesa new kind of interferometry, called Laser-Self-Mixing (LSM)effect, has been established [1].The self-mixing interference occurs inside the laser cavity be-tween the standing wave inside the cavity of the laser and partof the laser beam re-entering the cavity after being coherentlyreflected or back-scattered by an external target. The result ofthe interaction between the two beams is a periodic changeof the value of some laser emission parameters, such as thewavelength, the threshold current, the carrier density and theoutput power. The most suitable solution, in order to optimizethe measuring setup, consists in the detection of the amplitudemodulation of the laser output power by means of the photo-diode integrated into the laser package for feedback control.The useful features of this scheme with respect to the tra-ditional interferometric one are the reduction of the numberof required optical elements (such as the beam splitter andthe reference mirror), a much simpler optical alignment sincethere is no external reference arm, and a reduced cost due tothe absence of the external detector. Thanks to these benefits,the self-mixing has been recognized as a useful technique ina great variety of metrological applications [2], primarily formeasuring displacement, velocity and distance.Another peculiarity of this scheme regards the shape of thedetected signal which is not simply sinusoidal, but dependson the feedback regimes, i.e. the relative amount of light cou-pled back into the laser. A useful classification of the feed-back regimes for metrological purposes can be performed byadopting the C – value as selective parameter, where C is thefeedback parameter [3] defined as follows:C = ε√R3R2(1− R2)√1 + α2Ll · ne f f (1)Eq. (1) depends on a combination of laser dependent param-eters (R2 is the output coupler reflectivity, l is the laser cav-ity length, ne f f is the effective refractive index of the activeReceived March 19, 2009; published July 06, 2009 ISSN 1990-2573Journal of the European Optical Society - Rapid Publications 4, 09036 (2009) S. Ottonelli et. al.FIG. 1 (a) Misalignment of the target Ttilt with respect to the optical axis. (b) Linear displacement ∆x in presence of a target tilted of an angle ϑ. (c) Linear displacement ∆xmeasured by two lasers: Ltilt misaligned of an angle ϕ and Laligned well- aligned along the x- axis The red line represents the fraction of the laser beam produced by the laserLD orthogonal to the target first (T) and after (T′) the displacement.medium, α is the linewidth enhancement factor) and systemadjustable parameters (R3 is the target reflectivity, L is the ex-ternal cavity length, and ε < 1 is a constant referred to as themode matching factor).By increasing the feedback power, the power modulationfringes appear sinusoidal (C  1; very weak feedbackregime), slightly asymmetric (0.1 < C < 1, weak feedbackregime) or sawtooth-like (1 < C < 4.6, moderate feedbackregime) with sharp switching every time the phase changesby 2pi. The asymmetry of the output signal in the moderatefeedback regime allows for an easy discrimination of thedirection of motion by a single quadrature reading, on thebasis of the sign of the fast slopes of the sawtooth-like signal.Thus, whereas the intrinsic λ/2 displacement measurementresolution typical of interferometric setup can be retainedby digital fringe count electronics, the discrimination ofthe direction of motion is recovered by the sign of the fastslopes of the sawtooth-like signal. By the AC derivation ofthe output signal and the algebraic sum N = N+ − N− ofthe positive (N+) / negative (N−) number of the peaks inthe derivative, the linear displacements can be obtained as∆X = N × λ/2.The target usually employed in a self – mixing interferometer,that is a single plane mirror (PT), has two drawbacks. First, theangular range of target misalignment is limited by the angulardivergence of the laser beam. Second, a small tilt of the targetresults in an inclination of the plane - mirror with respect tothe x - axis, thus introducing a “cosine - type” error in thelinear displacements along the x – axis and limiting the mea-surement accuracy. Both these limitations can be overcome byusing corner cubes (CC) in place of plane mirrors. In this pa-per, we compare the performance of a LSM interferometer forboth kinds of targets and analyze the benefits of employing acorner-cube target.2 OPTIMAL COLLIMATION CONDITIONAND MEASUREMENT ERRORSOperations in the moderate feedback regime, in a LSM appa-ratus with a plane mirror target, is usually achieved and con-trolled by means of a variable attenuator placed along the op-tical path. The attenuation of the feedback intensity is alwaysrequired in presence of reflective target and collimated semi-conductor lasers. In order to avoid the insertion of the filter,either the target should be changed or the collimating condi-tion can be properly modified. In the first case, a diffusive tar-get can be employed to reduce the back - scattered radiationre – entering the laser source; however, this solution impliesa signal fading over long-distances due to the speckle effect.The alternative choice is represented by the reduction of thedistance between the laser diode and collimating lens, in or-der to achieve a divergent beam and to obtain the followingadvantages:1. no attenuation filter is required along the measuring arm.2. the linear increase of the feedback parameter C with thetarget distance can be tailored by a careful choice of thedistance laser – target.3. since only the fraction of the laser beam orthogonal tothe target is back-reflected toward the lens following thesame optical path, a divergent laser beam will acceptlarger target misalignment than a perfectly collimatedlaser beam. This principle is schematically illustrated inFigure 1(a), where the maximum allowed beam diver-gence will be given by the minimum feedback power ra-tio required to preserve the moderate – feedback regime.The tolerance to angular misalignment described in point 3has a two-fold implication: first, it allows a simpler opti-cal alignment procedure than that required with collimatedbeams; second, it can be exploited for measuring yaw andpitch rotations of the target [4].In spite of these benefits, an angular misalignment ϑ of thenormal to the target with respect to the measurement axis, as-09036- 2Journal of the European Optical Society - Rapid Publications 4, 09036 (2009) S. Ottonelli et. al.FIG. 2 Schematics of the sensor system: LH laser head, RH reference API head, T target,RT reference API target, LS linear stage, RS rotational stage, F variable filter, A trans-Zamplifier, PC personal computer. The red (yellow) lines refer to the laser beam of thereference (proposed) system.sumed as the x – axis, affects the measurement of linear dis-placement. With reference to Figure 1(b), the LSM interferom-eter will measure a displacement ∆d along the direction or-thogonal to the target, in place of the nominal displacement∆x along the x – axis. Since ∆d = ∆x cos(ϑ), the relativeerror, given by the normalized difference between the mea-sured displacement ∆d and the expected displacement ∆x, isσ/∆x = cos(ϑ)− 1. On the other hand, any misalignment ϕof the laser diode Ltilt with respect to the x – axis (Figure 1(c))will not bring in additional measurement errors, provided ϕto be less than the useful laser divergence. If we consider amaximum linear displacement ∆x = 60 cm, the absolute erroris σ = 2 µm for ϑ = 0.1◦ and σ = 91 µm for ϑ = 1◦.From the above considerations, it can be concluded that themeasurement of linear displacement in a LSM interferometeremploying a plane reflective target is affected by a cosine –type measurement error due to angular misalignment of thetarget. This error is similar to the well – known cosine erroraffecting classical interferometric measurements, although theformer is due to a misalignment of the target whereas the lat-ter is caused by a misalignment of the laser diode, both re-ferred to the direction x of the linear motion.The requirement of a divergent laser beam for measuring thetarget tilt angle can be relaxed by using a CC in place of thePT, since in this case the backward optical path will not beaffected by the target orientation. However, the cosine – typeerror affecting the measurement of linear displacement willagain be related to misalignment of the laser as in classicalinterferometers.3 EXPERIMENTAL SETUPFigure 2 schematically illustrates the proposed sensor, com-posed of a laser head and a reflective target. The laser headis made up of a Distributed-Feedback (DFB) laser diode withnominal wavelength of 1310 nm, current threshold Ith =12 mA and biased at twice the threshold. It is equipped withmonitor photodiode and collimated lens integrated into a sin-gle collimation tube. The current signal from the photodiodeis firstly AC-coupled to a trans-impedance amplifier with gainof 105 V/A and then fed into a signal processing board in-terfaced to a computer. The moving target is made up bya squared plane mirror (PT) of side 5 cm or a retroreflectorprism (CC) with a diameter of 10 mm, rigidly fixed onto arotation stage, mounted onto a 0.6-meter long linear stage.The minimum distance between the target and the laser headwas 15 cm. A commercial 6-axis measurement system (API6D Laser), whose nominal resolution is 0.02 µm for linear dis-placements, has been used as the reference meter.4 RESULTS AND DISCUSSIONFor validating the feasibility of this system as displacementsensor by using both PT and CC target, the linear stage wasmoved from a fixed initial position in the range ±600 mm at aspeed of 10 mm/s. The measurements were performed after apreliminary optical alignment of both the sources and the tar-gets along the measurement x-axis. The focusing conditionswere fixed in the collimation for the CC, thus requiring a fil-ter along the optical path to avoid instabilities due to strongfeedback, and divergence of an angle α ≈ 0.4◦ for the PT.Results reported in Figure 3 demonstrate that both PT and CCachieve a measurement accuracy σx = ±20 µm over 600 mmof linear displacement ∆x. The linearity over the entire mea-surement range validates a useful dynamic range of at leastfive - orders of magnitude with the use of both kind of tar-gets. The longest continuous measurable displacement wasonly limited by the length of our linear stage. However, weverified the existence of the sawtooth - like signal over a con-tinuous range of 1.7 m, with no adjustment required to theoptics (the shortest distance from laser head was 0.15 m).To test the robustness of the proposed system against targetmisalignment, the laser beam was made increasingly moredivergent by slight adjustments of the distance between thelaser diode and the lens. For each divergence angle α, the max-imum angular misalignment β of the target was determinedas the greatest angle able to preserve a moderate feedbackregime during a linear displacement. The results are shown inFigure 4, where the increasingly monotone trend confirms agreater tolerance to target misalignment up to approximately0.7◦, above which the feedback power was too low for theLSM signal to reach the moderate feedback regime.On the other hand, the CC requires a small sized collimatedbeam, in order to reduce the phase distortions induced by thereflection at the faceted prism, and a filter attenuator since theCC reflects back almost 90% of the incident power. In Figure 5,the normalized number of counted fringes Nnorm, obtained fora fixed linear displacements ∆x = 600 mm in the case of atilted target (along the z – axis), is reported for both kind oftargets; the plane mirror was used with a divergent laser beam(α ≈ (0.4± 0.1)◦) with no filter, whereas the corner cube wasused with a collimated laser beam and an optical filter. Start-ing from the zero position (with the front surface of the targetorthogonal to the direction x of linear motion), it can be ob-served a decreasing number of counted fringes for the plane09036- 3Journal of the European Optical Society - Rapid Publications 4, 09036 (2009) S. Ottonelli et. al.FIG. 3 Number N of counted fringes (a) and measurement error σx (b) for increasing linear displacements, as a function of the actual displacement measured by the referencesystem. Positive (negative) ∆x values indicate backward (forward) displacements.FIG. 4 Maximum tolerated misalignment β of the plane target for increasing laser beamdivergences α.mirror, consistent with the cosine – type error introduced bythe tilt of the plane mirror.Conversely, the corner cube illuminated by a collimated beamdoes not suffer of the same limitation. It tolerates a muchlarger target rotation up to some degrees, at the same timepreserving the number of counted fringes within the experi-mental errors. This latter is mainly due to the cosine error in-duced by a slight misalignment of the laser source (angle ϕ inFigure 1(c)) and wavelength instabilities of the laser source.5 CONCLUSIONWe compared the performance of a LSM sensor for measure-ment of linear displacements with two kinds of reflective tar-gets, a plane mirror and a solid corner cube, in terms of sim-plification of the optical setup, measurement accuracy and tol-erance to angular misalignment.Two alternative solutions have been proposed for the im-provement of the robustness of the optical apparatus againstangular target misalignment: the use of a plane target witha diverging laser beam and the use of a corner cube targetwith a collimated laser beam. On the one hand, the plane mir-ror allows for the removal of the variable attenuator alongFIG. 5 Normalized number of counts Nnorm for a fixed linear displacement ∆x = 600mm, as a function of the tilt angle of the target around the vertical axis.the optical path and causes the laser to tolerate small angularmisalignment of the target within its angular divergence. Themaximum demonstrated tilt angle of the target was approx-imately 0.7◦. On the other hand, the use of a retroreflectivetarget requires the insertion of the attenuator to avoid instabil-ities caused by strong feedback power, but makes the systemcompletely insensitive to target misalignment up to at least 2◦.The full understanding of the different behavior of the sys-tem required a redefinition of what is usually referred to asthe cosine-error. In case of plane mirror, the cosine – error isdue to the target misalignment with respect to the measure-ment axis, whereas a fine alignment of the laser source alongthe longitudinal axis is not required since only the fraction ofthe laser beam orthogonal to the target contributes to the LSMeffect. In case of retroreflective target the cosine-error retrievesthe usual interpretation, as caused by a misalignment betweenthe laser source and the measurement axis.References[1] S. Donati, Electro-Optical Instrumentation Chapter 4 (Prentice Hall,Upper Saddle River, NJ, 2004) .[2] G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications” J. Opt. A: Pure Appl.Opt. 4, S283–S294 (2002).09036- 4Journal of the European Optical Society - Rapid Publications 4, 09036 (2009) S. Ottonelli et. al.[3] G. A. Acket, D. Lenstra, J. Den Boef, B. H. Verbeek, “The influenceof feedback intensity on longitudinal mode properties and opti-cal noise in index-guided semiconductor lasers” IEEE J. QuantumElectron. 20, 1163–1169 (1984).[4] S. Ottonelli, F. De Lucia, M. di Vietro, M. Dabbicco, G. Scamar-cio, and F. P. Mezzapesa, “A Compact Three Degrees-of-FreedomMotion Sensor Based on the Laser-Self-Mixing Effect” IEEE Phot.Technol. Lett. 20, 1360–1362 (2008).09036- 5

Comparison of plane mirror vs retroreflector peformance for laser-self-mixing displacement sensors

Open Access Repository

J O U R N A L  O FTORH E  E U R O P E A N  P T I C A L  S O C I E T YA PID  PU B LIC AT IO N SJournal of the European Optical Society - Rapid Publications 4, 09036 (2009) www.jeos.orgComparison of plane mirror vs retroreflectorperformance for laser-self-mixing displacement sensorsS. Ottonellisimona.ottonelli@fisica.uniba.itCNR-INFM Regional Laboratory LIT3, Università degli Studi di Bari, Via Amendola 173, I-70126 Bari,Italy and Dipartimento Interateneo di Fisica “M. Merlin,” Università degli Studi di Bari, Via Amendola173, I-70126 Bari, ItalyF. De Lucia CNR-INFM Regional Laboratory LIT3, Università degli Studi di Bari, Via Amendola 173, I-70126 Bari,Italy and Dipartimento Interateneo di Fisica “M. Merlin,” Università degli Studi di Bari, Via Amendola173, I-70126 Bari, ItalyM. di Vietro CNR-INFM Regional Laboratory LIT3, Università degli Studi di Bari, Via Amendola 173, I-70126 Bari,Italy and Dipartimento Interateneo di Fisica “M. Merlin,” Università degli Studi di Bari, Via Amendola173, I-70126 Bari, ItalyM. Dabbicco CNR-INFM Regional Laboratory LIT3, Università degli Studi di Bari, Via Amendola 173, I-70126 Bari,Italy and Dipartimento Interateneo di Fisica “M. Merlin,” Università degli Studi di Bari, Via Amendola173, I-70126 Bari, ItalyG. Scamarcio CNR-INFM Regional Laboratory LIT3, Università degli Studi di Bari, Via Amendola 173, I-70126 Bari,Italy and Dipartimento Interateneo di Fisica “M. Merlin,” Università degli Studi di Bari, Via Amendola173, I-70126 Bari, ItalyThe behavior of a displacement optical sensor based on the Laser-Self-Mixing effect employing a plane mirror (PT) and a solid corner-cube(CC) as moving target is analyzed. The performance of the sensor is compared in terms of simplification of the optical setup, measurementaccuracy and tolerance to angular misalignment of the target. On the basis of the innovative assumption that only the fraction of the laserbeam orthogonal to the target plane gives rise to the Self – Mixing modulation, it is demonstrated that the interferometer tolerates small tiltof the plane target (up to approximately 0.7◦) only when illuminated by a divergent beam, in which case the displacement measurementbecomes affected by a cosine – type error. Instead the corner cube preserves the self-mixing signal over a larger angular range (up toapproximately 2◦) at the same time preserving high measurement accuracy. [DOI: 10.2971/jeos.2009.09036]Keywords: self-mixing, optical feedback, displacement measurement, corner-cube, target misalignment1 INTRODUCTIONLaser interferometry is one of the most disseminate and per-forming phase sensitive techniques for metrological measure-ments. Besides the traditional “external configuration” need-ing one or more external detectors, in the last two decadesa new kind of interferometry, called Laser-Self-Mixing (LSM)effect, has been established [1].The self-mixing interference occurs inside the laser cavity be-tween the standing wave inside the cavity of the laser and partof the laser beam re-entering the cavity after being coherentlyreflected or back-scattered by an external target. The result ofthe interaction between the two beams is a periodic changeof the value of some laser emission parameters, such as thewavelength, the threshold current, the carrier density and theoutput power. The most suitable solution, in order to optimizethe measuring setup, consists in the detection of the amplitudemodulation of the laser output power by means of the photo-diode integrated into the laser package for feedback control.The useful features of this scheme with respect to the tra-ditional interferometric one are the reduction of the numberof required optical elements (such as the beam splitter andthe reference mirror), a much simpler optical alignment sincethere is no external reference arm, and a reduced cost due tothe absence of the external detector. Thanks to these benefits,the self-mixing has been recognized as a useful technique ina great variety of metrological applications [2], primarily formeasuring displacement, velocity and distance.Another peculiarity of this scheme regards the shape of thedetected signal which is not simply sinusoidal, but dependson the feedback regimes, i.e. the relative amount of light cou-pled back into the laser. A useful classification of the feed-back regimes for metrological purposes can be performed byadopting the C – value as selective parameter, where C is thefeedback parameter [3] defined as follows:C = ε√R3R2(1− R2)√1 + α2Ll · ne f f(1)Eq. (1) depends on a combination of laser dependent param-eters (R2 is the output coupler reflectivity, l is the laser cav-ity length, ne f f is the effective refractive index of the activeReceived March 19, 2009; published July 06, 2009 ISSN 1990-2573Journal of the European Optical Society - Rapid Publications 4, 09036 (2009) S. Ottonelli et. al.FIG. 1 (a) Misalignment of the target Ttilt with respect to the optical axis. (b) Linear displacement ∆x in presence of a target tilted of an angle ϑ. (c) Linear displacement ∆xmeasured by two lasers: Ltilt misaligned of an angle ϕ and Laligned well- aligned along the x- axis The red line represents the fraction of the laser beam produced by the laserLD orthogonal to the target first (T) and after (T′) the displacement.medium, α is the linewidth enhancement factor) and systemadjustable parameters (R3 is the target reflectivity, L is the ex-ternal cavity length, and ε < 1 is a constant referred to as themode matching factor).By increasing the feedback power, the power modulationfringes appear sinusoidal (C  1; very weak feedbackregime), slightly asymmetric (0.1 < C < 1, weak feedbackregime) or sawtooth-like (1 < C < 4.6, moderate feedbackregime) with sharp switching every time the phase changesby 2π. The asymmetry of the output signal in the moderatefeedback regime allows for an easy discrimination of thedirection of motion by a single quadrature reading, on thebasis of the sign of the fast slopes of the sawtooth-like signal.Thus, whereas the intrinsic λ/2 displacement measurementresolution typical of interferometric setup can be retainedby digital fringe count electronics, the discrimination ofthe direction of motion is recovered by the sign of the fastslopes of the sawtooth-like signal. By the AC derivation ofthe output signal and the algebraic sum N = N+ − N− ofthe positive (N+) / negative (N−) number of the peaks inthe derivative, the linear displacements can be obtained as∆X = N × λ/2.The target usually employed in a self – mixing interferometer,that is a single plane mirror (PT), has two drawbacks. First, theangular range of target misalignment is limited by the angulardivergence of the laser beam. Second, a small tilt of the targetresults in an inclination of the plane - mirror with respect tothe x - axis, thus introducing a “cosine - type” error in thelinear displacements along the x – axis and limiting the mea-surement accuracy. Both these limitations can be overcome byusing corner cubes (CC) in place of plane mirrors. In this pa-per, we compare the performance of a LSM interferometer forboth kinds of targets and analyze the benefits of employing acorner-cube target.2 OPTIMAL COLLIMATION CONDITIONAND MEASUREMENT ERRORSOperations in the moderate feedback regime, in a LSM appa-ratus with a plane mirror target, is usually achieved and con-trolled by means of a variable attenuator placed along the op-tical path. The attenuation of the feedback intensity is alwaysrequired in presence of reflective target and collimated semi-conductor lasers. In order to avoid the insertion of the filter,either the target should be changed or the collimating condi-tion can be properly modified. In the first case, a diffusive tar-get can be employed to reduce the back - scattered radiationre – entering the laser source; however, this solution impliesa signal fading over long-distances due to the speckle effect.The alternative choice is represented by the reduction of thedistance between the laser diode and collimating lens, in or-der to achieve a divergent beam and to obtain the followingadvantages:1. no attenuation filter is required along the measuring arm.2. the linear increase of the feedback parameter C with thetarget distance can be tailored by a careful choice of thedistance laser – target.3. since only the fraction of the laser beam orthogonal tothe target is back-reflected toward the lens following thesame optical path, a divergent laser beam will acceptlarger target misalignment than a perfectly collimatedlaser beam. This principle is schematically illustrated inFigure 1(a), where the maximum allowed beam diver-gence will be given by the minimum feedback power ra-tio required to preserve the moderate – feedback regime.The tolerance to angular misalignment described in point 3has a two-fold implication: first, it allows a simpler opti-cal alignment procedure than that required with collimatedbeams; second, it can be exploited for measuring yaw andpitch rotations of the target [4].In spite of these benefits, an angular misalignment ϑ of thenormal to the target with respect to the measurement axis, as-09036- 2Journal of the European Optical Society - Rapid Publications 4, 09036 (2009) S. Ottonelli et. al.FIG. 2 Schematics of the sensor system: LH laser head, RH reference API head, T target,RT reference API target, LS linear stage, RS rotational stage, F variable filter, A trans-Zamplifier, PC personal computer. The red (yellow) lines refer to the laser beam of thereference (proposed) system.sumed as the x – axis, affects the measurement of linear dis-placement. With reference to Figure 1(b), the LSM interferom-eter will measure a displacement ∆d along the direction or-thogonal to the target, in place of the nominal displacement∆x along the x – axis. Since ∆d = ∆x cos(ϑ), the relativeerror, given by the normalized difference between the mea-sured displacement ∆d and the expected displacement ∆x, isσ/∆x = cos(ϑ)− 1. On the other hand, any misalignment ϕof the laser diode Ltilt with respect to the x – axis (Figure 1(c))will not bring in additional measurement errors, provided ϕto be less than the useful laser divergence. If we consider amaximum linear displacement ∆x = 60 cm, the absolute erroris σ = 2 µm for ϑ = 0.1◦ and σ = 91 µm for ϑ = 1◦.From the above considerations, it can be concluded that themeasurement of linear displacement in a LSM interferometeremploying a plane reflective target is affected by a cosine –type measurement error due to angular misalignment of thetarget. This error is similar to the well – known cosine erroraffecting classical interferometric measurements, although theformer is due to a misalignment of the target whereas the lat-ter is caused by a misalignment of the laser diode, both re-ferred to the direction x of the linear motion.The requirement of a divergent laser beam for measuring thetarget tilt angle can be relaxed by using a CC in place of thePT, since in this case the backward optical path will not beaffected by the target orientation. However, the cosine – typeerror affecting the measurement of linear displacement willagain be related to misalignment of the laser as in classicalinterferometers.3 EXPERIMENTAL SETUPFigure 2 schematically illustrates the proposed sensor, com-posed of a laser head and a reflective target. The laser headis made up of a Distributed-Feedback (DFB) laser diode withnominal wavelength of 1310 nm, current threshold Ith =12 mA and biased at twice the threshold. It is equipped withmonitor photodiode and collimated lens integrated into a sin-gle collimation tube. The current signal from the photodiodeis firstly AC-coupled to a trans-impedance amplifier with gainof 105 V/A and then fed into a signal processing board in-terfaced to a computer. The moving target is made up bya squared plane mirror (PT) of side 5 cm or a retroreflectorprism (CC) with a diameter of 10 mm, rigidly fixed onto arotation stage, mounted onto a 0.6-meter long linear stage.The minimum distance between the target and the laser headwas 15 cm. A commercial 6-axis measurement system (API6D Laser), whose nominal resolution is 0.02 µm for linear dis-placements, has been used as the reference meter.4 RESULTS AND DISCUSSIONFor validating the feasibility of this system as displacementsensor by using both PT and CC target, the linear stage wasmoved from a fixed initial position in the range ±600 mm at aspeed of 10 mm/s. The measurements were performed after apreliminary optical alignment of both the sources and the tar-gets along the measurement x-axis. The focusing conditionswere fixed in the collimation for the CC, thus requiring a fil-ter along the optical path to avoid instabilities due to strongfeedback, and divergence of an angle α ≈ 0.4◦ for the PT.Results reported in Figure 3 demonstrate that both PT and CCachieve a measurement accuracy σx = ±20 µm over 600 mmof linear displacement ∆x. The linearity over the entire mea-surement range validates a useful dynamic range of at leastfive - orders of magnitude with the use of both kind of tar-gets. The longest continuous measurable displacement wasonly limited by the length of our linear stage. However, weverified the existence of the sawtooth - like signal over a con-tinuous range of 1.7 m, with no adjustment required to theoptics (the shortest distance from laser head was 0.15 m).To test the robustness of the proposed system against targetmisalignment, the laser beam was made increasingly moredivergent by slight adjustments of the distance between thelaser diode and the lens. For each divergence angle α, the max-imum angular misalignment β of the target was determinedas the greatest angle able to preserve a moderate feedbackregime during a linear displacement. The results are shown inFigure 4, where the increasingly monotone trend confirms agreater tolerance to target misalignment up to approximately0.7◦, above which the feedback power was too low for theLSM signal to reach the moderate feedback regime.On the other hand, the CC requires a small sized collimatedbeam, in order to reduce the phase distortions induced by thereflection at the faceted prism, and a filter attenuator since theCC reflects back almost 90% of the incident power. In Figure 5,the normalized number of counted fringes Nnorm, obtained fora fixed linear displacements ∆x = 600 mm in the case of atilted target (along the z – axis), is reported for both kind oftargets; the plane mirror was used with a divergent laser beam(α ≈ (0.4± 0.1)◦) with no filter, whereas the corner cube wasused with a collimated laser beam and an optical filter. Start-ing from the zero position (with the front surface of the targetorthogonal to the direction x of linear motion), it can be ob-served a decreasing number of counted fringes for the plane09036- 3Journal of the European Optical Society - Rapid Publications 4, 09036 (2009) S. Ottonelli et. al.FIG. 3 Number N of counted fringes (a) and measurement error σx (b) for increasing linear displacements, as a function of the actual displacement measured by the referencesystem. Positive (negative) ∆x values indicate backward (forward) displacements.FIG. 4 Maximum tolerated misalignment β of the plane target for increasing laser beamdivergences α.mirror, consistent with the cosine – type error introduced bythe tilt of the plane mirror.Conversely, the corner cube illuminated by a collimated beamdoes not suffer of the same limitation. It tolerates a muchlarger target rotation up to some degrees, at the same timepreserving the number of counted fringes within the experi-mental errors. This latter is mainly due to the cosine error in-duced by a slight misalignment of the laser source (angle ϕ inFigure 1(c)) and wavelength instabilities of the laser source.5 CONCLUSIONWe compared the performance of a LSM sensor for measure-ment of linear displacements with two kinds of reflective tar-gets, a plane mirror and a solid corner cube, in terms of sim-plification of the optical setup, measurement accuracy and tol-erance to angular misalignment.Two alternative solutions have been proposed for the im-provement of the robustness of the optical apparatus againstangular target misalignment: the use of a plane target witha diverging laser beam and the use of a corner cube targetwith a collimated laser beam. On the one hand, the plane mir-ror allows for the removal of the variable attenuator alongFIG. 5 Normalized number of counts Nnorm for a fixed linear displacement ∆x = 600mm, as a function of the tilt angle of the target around the vertical axis.the optical path and causes the laser to tolerate small angularmisalignment of the target within its angular divergence. Themaximum demonstrated tilt angle of the target was approx-imately 0.7◦. On the other hand, the use of a retroreflectivetarget requires the insertion of the attenuator to avoid instabil-ities caused by strong feedback power, but makes the systemcompletely insensitive to target misalignment up to at least 2◦.The full understanding of the different behavior of the sys-tem required a redefinition of what is usually referred to asthe cosine-error. In case of plane mirror, the cosine – error isdue to the target misalignment with respect to the measure-ment axis, whereas a fine alignment of the laser source alongthe longitudinal axis is not required since only the fraction ofthe laser beam orthogonal to the target contributes to the LSMeffect. In case of retroreflective target the cosine-error retrievesthe usual interpretation, as caused by a misalignment betweenthe laser source and the measurement axis.References[1] S. Donati, Electro-Optical Instrumentation Chapter 4 (Prentice Hall,Upper Saddle River, NJ, 2004) .[2] G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications” J. Opt. A: Pure Appl.Opt. 4, S283–S294 (2002).09036- 4Journal of the European Optical Society - Rapid Publications 4, 09036 (2009) S. Ottonelli et. al.[3] G. A. Acket, D. Lenstra, J. Den Boef, B. H. Verbeek, “The influenceof feedback intensity on longitudinal mode properties and opti-cal noise in index-guided semiconductor lasers” IEEE J. QuantumElectron. 20, 1163–1169 (1984).[4] S. Ottonelli, F. De Lucia, M. di Vietro, M. Dabbicco, G. Scamar-cio, and F. P. Mezzapesa, “A Compact Three Degrees-of-FreedomMotion Sensor Based on the Laser-Self-Mixing Effect” IEEE Phot.Technol. Lett. 20, 1360–1362 (2008).09036- 5

Comparison of plane mirror vs retroreflector performance for laser-self-mixing displacement sensors

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Comparison of plane mirror vs retroreflector performance for laser-self-mixing displacement sensors

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