Simultaneous measurement of linear and transverse displacements by laser self-mixing Simona Ottonelli,1,2,* Maurizio Dabbicco,1,2 Francesco De Lucia,1,2 and Gaetano Scamarcio1,2 1
CNR-INFM Laboratorio LIT3, Università degli Studi di Bari, 173 Via Amendola, Bari 70123, Italy
Dipartimento Interateneo di Fisica “M. Merlin,” Università degli Studi di Bari, 173 Via Amendola, Bari 70123, Italy
2
*Corresponding author:
[email protected] Received 20 November 2008; revised 29 January 2009; accepted 3 February 2009; posted 9 February 2009 (Doc. ID 104307); published 19 March 2009
We present a contactless optical sensor based on the laser-self-mixing effect for real-time measurement of linear and transverse displacements of a moving stage. The sensor is able to measure linear displacements of up to 400 mm along the main optical axis while simultaneously estimating straightness and flatness deviations up to 1 mm. The sensor exploits two identical coplanar nonparallel self-mixing interferometers and requires only one reference plane. The reduction in the number of optical elements allowed by the self-mixing configuration and the intrinsic stiffness of the adopted geometry result in a compact, low-cost, and easy-to-align setup. © 2009 Optical Society of America OCIS codes: 120.3180, 280.3420, 230.0250, 120.3930, 140.2020, 120.0280.
1. Introduction
A slide moving along a linear guideway is often employed in scientific and industrial environments, such as in optical translation stages or in a carriage on a lathe. Although the slide is expected to travel along a straight line, the actual path often deviates in the transverse plane due to mechanical imperfections of the guideways or thermal deformation of structural components. If the direction of the linear motion is assumed to be the x axis, deviations in the xy plane or in the xz plane are commonly referred to as straightness or flatness error, respectively, although their amplitudes are expected to be much smaller than the working range of the main linear displacement. A variety of optical sensors are already available for the purpose of measuring such deviations in real time, from knife-edge shadow projection [1] in the case of short tracks to corner cubes for longer linear displacements [2]; both require at least one or more position sensitive detectors and thus the analogic 0003-6935/09/091784-06$15.00/0 © 2009 Optical Society of America 1784
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treatment of the detected signal. A digital signal approach is typical of optical interferometers, which are themselves unable to measure transverse displacements, i.e., displacements orthogonal to the axis of the interferometer, since this motion does not induce any optical path difference with respect to the reference arm. This hindrance can be overcome by projecting a component of the transverse motion along the direction of the laser beam and this is achieved by adding some optical element along the path, such as a Wollaston prism [3], or by adopting specific geometric configurations [4,5]. Both these approaches usually require expensive prisms or target mirror assemblies and, above all, an additional reference plane besides that defined by the laser source. We report on a compact interferometric system capable of tracking the target displacement along the longitudinal axis up to a distance of 0:4 m and simultaneously measuring its transverse deviation up to 1 mm. The interferometric system is based on the laser-self-mixing (LSM) effect and is composed of three laser sources and a properly designed mirror target. The proposed technique is based on the differential measurement of linear displacement by pairs of identical self-mixing interferometers (SMI), each
formed only by a laser diode package, which includes an integrated collimated lens, a monitor photodiode, and a plane mirror target. The two SMI are coplanar with their axes forming an angle α. When the arm of one SMI is aligned along the x axis, linear displacements of the target up to 1 m can be measured and the measurement is insensitive to transverse deviations of the target motion. At the same time, the second SMI, whose arm forms a small angle α with the x axis, measures both longitudinal and transverse motion, allowing for the straightness/flatness deviation to be extracted by the differential reading of the two SMIs. The system is capable of measuring x-axis displacements with submicrometer resolution over a 0:4 m range and transverse deviations with a 20 μm resolution over about a 1 mm range. 2. Principle of Measurement A. Self-Mixing Effect
The self-mixing interference occurs inside the laser cavity between the laser radiation and the part of the laser beam reentering the cavity after being reflected or backscattered by an external target. The interference results in the modulation of both the amplitude and the frequency of the laser oscillating field [6]. This modulation carries information on the displacement of a target. The self-mixing effect presents some peculiar features. The relative amount of light coupled back into the laser directly affects the characteristics of the output signal, which changes from sinusoidal (very weak feedback regime) to slightly asymmetric (weak feedback regime) or a sawtoothlike shape (moderate feedback regime), with a slope related to the direction of motion of the external reflector [7]. Consequently, operation in the moderate feedback regime represents the most convenient solution for displacement measurements, since the module of the displacement can be obtained simply by counting the number of produced fringes, while the displacement direction can be easily recovered by the sign of the derivative of the sawtoothlike signal [8]. Besides, the resulting output optical power variation can be monitored as photocurrent fluctuations and measured by the photodiode typically integrated in the laser chip, so that the basic apparatus for linear displacements simply consists of a laser diode chip, a collimating lens, and a reflective target, as schematically described in Fig. 1.
can be obtained by the difference in the optical path length Δl1 as Δx ¼ Δl1 ¼ N 1 × λ1 =2, where N 1 and λ1 are the net number of fringes and the wavelength of the laser L1, respectively. The SMI1 thus detects pure longitudinal displacements, being totally blind to any off-axis motion. However, SMI2 , which is rigidly tilted by a small angle α with respect to the x axis in the xy plane, identifies a new coordinate system x0 y0 , correlated with the previous system xy by a rotation transformation (see the upper part of Fig. 1(a)]. SMI2 is thus sensitive to displacements along both the linear (x) and transverse (y) axes, since both cause a change Δl2 ¼ N 2 × λ2 =2 of the optical path length along x0 . By assuming that the target performed only purely linear Δx or Δy displacements, they can be measured according to
Δx ¼ Δl2 = cosðαÞ ¼ N 2 · λ22 = cosðαÞ if ΔY ¼ 0; Δy ¼ Δl2 =senðαÞ ¼ N 2 · λ22 =senðαÞ if ΔX ¼ 0: ð1Þ
A single interferometer is thus needed for measuring one single linear degree-of-freedom motion, whereas a combined linear/transverse displacement requires the comparison of the two SMIs’ readings in order to be properly measured. In fact, in this case, Δl2 results from a combination of Δx and Δy whose actual values can be recovered according to
Δx ¼ Δl1 ; Δy ¼ Δl2 = sinðαÞ − Δx= tanðαÞ
ð2Þ
B. Linear and Transverse Measurements by the Laser Self-Mixing Effect
The self-mixing interferometer SMI1, schematized in Fig. 1, is able to detect purely linear displacement Δx along the x axis. The standard signal analysis in the self-mixing approach consists of calculating the algebraic sum N ¼ N þ − N − of the positive (N þ )/negative (N − ) number of the principal peaks in the derivative of the output signal. Since each fringe corresponds to a displacement of λ=2, the linear displacement Δx
Fig. 1. (Color online) Schematic geometry of the self-mixing interferometers (SMIi ) for the (a) straightness and (b) flatness measurements. Each interferometer is made by a laser diode Li , a converging lens Fi , and a plane mirror Mi as target. The monitor photodiode, placed at the rear facet of each laser inside the laser package, is not shown. 20 March 2009 / Vol. 48, No. 9 / APPLIED OPTICS
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or, equivalently, in terms of number of interference fringes N 1 and N 2 as
Δx ¼ N 1 · λ21 ; λ2 λ1 − N 1 · 2·tanðαÞ : Δy ¼ N 2 · 2·sinðαÞ
ð3Þ
The same considerations apply for the measurement of flatness, via a third interferometer SMI3, tilted by an angle β with respect to the x axis in the xz plane [see Fig. 1(b)]. C. System Performance
Given the intrinsic linear resolution RΔx ¼ λ=2 of the self-mixing technique, the resolution for a pure straightness/flatness displacement can be derived by Eq. (3) by assuming null the counter N 1 and unitary the counter N 2=3, resulting in RΔy ¼ λ2 =½2 sinðαÞ and RΔz ¼ λ3 =½2 sinðβÞ. For example, a straightness resolution of 1 μm can be achieved with a wavelength λ of 0:8 μm and an angle α approximately equal to 25°. However, such a large angle is not practical if medium–long linear displacement (some centimeters up to some meters) are allowed along the x axis, since the drift of the spot T 2=3 of the tilted laser L2=3 in the yz plane would increase proportionally to Δx. Accordingly, the tilt angle, and the resolution, will be constrained by the dimensions of the system: given the maximum allowed size of the target (h × h) and its maximum longitudinal distance from the lasers xmax , α ≤ tan−1 ðh=xmax Þ. Our choice to limit h ≈ 50 mm and to allow for xmax ≈ 1:5 m resulted in α̣ ≈ ̣ β̣ ≈ 2° and an expected theoretical resolution of about 20 μm for a laser wavelength of 1:3 μm. The estimated measurement accuracy σ, that is, the deviation between the measured and the actual displacement, can be obtained by applying the error propagation formula to Eq. (3) in terms of number of interference fringes N 1 and N 2 as
certainty were obtained by means of a preliminary calibration against a reference meter. The result is α ¼ 2:1° with a corresponding uncertainty of σ α ≈ 0:1°. Consequently, a linear accuracy σ Δx ¼ λ ≈ 1:3 μm and a maximum transverse accuracy σ Δy ≈ 69 μm for a straightness Δy ¼ 1 mm can be achieved with a laser of wavelength λ ¼ 1:3 μm. The measurement error due to wavelength fluctuations has been neglected in the above system, since the wavelength was continuously monitored by a wavelength meter allowing a relative accuracy Δλ=λ ¼ 10−6 . D.
System Errors due to Mechanical Misalignments
Equations (2) and (3) hold in the ideal condition where the straightness and flatness deviations never occur at the same time, the target mirror is perfectly aligned in a plane orthogonal to the x axis (SMI1 ) or to the x0 axis (SMI2;3 ), and no rotation about any axis occurs during its motion. None of the above conditions is likely to be verified in practice. In the following, we discuss the possible misalignments of real systems and their effects on the precision of the measurement. As a preliminary remark, it is worth noting that a slightly divergent laser beam allows for only a small fraction of the backward beam to reenter the laser cavity: the greater the laser divergence, the smaller the feedback radiation coupled with the laser. 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, as schematically shown in Fig. 2(a). With reference to a linear displacement Δx, the following conditions may exist: 1. a misalignment angle φ between the laser beam and the x axis [Fig. 2(b)]. As long as φ lies within the beam divergence, there will be a fraction of the
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 2 > > ∂Δli ðN;λÞ > σ N 2 ¼ 2λ σ N ; > σ Δx ¼ ∂N < sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ffi > > ∂ΔyðN;λ;αÞ ∂ΔyðN;λ;αÞ > > σN 2 þ σ α 2 ≃ 12 αλ σ N 2 þ Δy σα2; : σ Δy ¼ α ∂N ∂α
where we have assumed that λ ¼ λ1 ≈ λ2 and that the small angle approximation is valid. The error in the fringe count is σ N1 ¼ σ N2 ¼ 2, since each electronic channel for the positive/negative fringe count is affected by an intrinsic uncertainty of 1 count. The assessment of the tilt angle of the target and its un1786
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ð4Þ
laser beam orthogonal to the target, independently from the direction of the optical axis, which propagates exactly along the x axis. Therefore no error occurs. 2. a misalignment angle ϑ between the normal to the target plane and the x axis [Fig. 2(c)]. The
Fig. 2. (Color online) (a) Ray-tracing analysis of the SMI with a divergent laser beam, where the peripheral rays are reflected outside of the laser whereas the rays orthogonal to the target reenter the laser. (b) Linear displacement Δx measured by two lasers: Ltilt tilted by an angle φ and Laligned well aligned to the x axis; the orthogonal beam is the same in both the cases. (c) Misalignment ϑ of the target with respect to the optical axis; only the incident rays orthogonal to the target reenter into the laser. (d) Misalignment δ of the axis of the translation motion with respect to the x axis.
displacement measured by the self-mixing technique will be Δd in the direction orthogonal to the target, in place of the nominal displacement Δx along the x axis. Since Δd ¼ Δx cosðϑÞ, the relative error, defined as the normalized difference between the measured displacement Δd and the expected displacement Δx, is ðΔd − ΔxÞ=Δx ¼ σ=Δx ¼ cosðϑÞ − 1. 3. a misalignment angle δ between the direction of the translation stage and the x axis [Fig. 2(d)]. If the target lies in the yz plane, the measured displacement will be Δd in place of the actual displacement Δx along the tilted direction of the linear motion. Since Δd ¼ Δx cosðδÞ, the relative error is again σ=Δx ¼ cosðδÞ − 1. 4. a combination of conditions 2 and 3 would result in a combined error. In conclusion, measurement errors are introduced only by a lack of parallelism between the normal to the target plane and the measurement axis. This kind of cosine error is well known in optical metrology. However, in comparison with classical interferometers, in LSM it does not arise because of a misalignment of the laser beam with respect to the axis of motion. The above analysis, when referred to the tilted direction x0 , also applies to the measurement of Δy and Δz. However, due to the short distance traveled in the transverse plane, the accuracy of the measurement is always comparable with the resolution.
threshold of I th ¼ 12 mA. In our experiments the lasers are driven by a current I ¼ 23 mA. Each laser is equipped with a collimating lens (numerical aperture NA ¼ 0:5 and nominal focal length f ¼ 8 mm) and a monitor photodiode, whose photocurrent is first AC-coupled to a transimpedance amplifier (gain ¼ 105 V=A), and then fed into the signal processing board of a computer. To avoid the presence of a variable attenuator, the moderate feedback condition has been achieved by properly defocusing the laser source. The moving target consists of three reciprocally tilted mirrors (inset in Fig. 3): M1 is a 10 mm × 10 mm squared mirror lying in the plane yz; M2 is a 40 mm × 10 mm rectangular mirror tilted at angle α ¼ ð2:1 0:1Þ° around the z axis; and M3 is a 10 mm × 40 mm rectangular mirror tilted at angle β ¼ ð1:8 0:1Þ° around the y axis. The target was mounted on a y–z translational stage mounted on
3. Experiments and Discussion A. Setup Description
The prototype of the sensor is schematically illustrated in Fig. 3. It is composed of a laser head, which consists of three laser diodes mounted side by side in an L-like configuration, and a reflective target. The sources are distributed-feedback (DFB) diode lasers with a nominal wavelength of 1310 nm and a current
Fig. 3. Schematics of the setup for a linear/transverse measuring system. 20 March 2009 / Vol. 48, No. 9 / APPLIED OPTICS
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Fig. 5. Experimental (circles) and theoretical (solid curve) straightness resolution versus the tilt angle of SMI2 .
vals were limited by the available stages and reference meter ranges. However, LSM fringes were observable over much larger displacements, up to Δx ≈ 1:8 m and Δy ðΔzÞ ≈ 5 mm. As discussed previously, the performance of the system in terms of resolution, which was experimentally measured as the minimum straightness able to produce one interference fringe, improves by increasing the SMI2 tilt angle α as reported in Fig. 5. The 15° required to achieve a resolution of 3 μm restricted the longitudinal range to Δx ≤ 150 mm for the chosen mirror size of 40 mm. Most practical applications require real-time control of a small straightness/flatness deviation of a moving target along the main x axis. The simultaneous measurement of the linear and transverse
Fig. 4. (a) Target displacement Δx measured by the linear stage interferometer SMI1 ; (b) and (c) transverse target displacement Δy ðΔzÞ along the y and z axes, measured at two distances from the laser sources, 20 (squares) and 120 cm (crosses), as a function of the reference displacements. The error bars are always smaller than the symbol size.
a 0:5 m linear stage along the x axis. The minimum distance between the target and the laser head was 15 cm. A commercial six-axis measurement system (API 6D Laser) was used as a reference meter. B. Experimental Results
Figure 4 shows the performance of the system in measuring single axis motion. SMI1 was tested for displacements Δx in the range of 10−6 –0:4 m at a speed of 10 mm=s [Fig. 4(a)], whereas SMI2 and SMI3 were tested for displacements Δy and Δz in the range of 10−5 –10−3 m at two fixed distances from the laser source [Figs. 4(b) and 4(c)]. The linearity of the response is very good over the full measurement range. In the case of both longitudinal and transverse displacements, the maximum continuous inter1788
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Fig. 6. Simultaneous measurement of two degrees of freedom as a function of the reference displacement Δxref : (a) linear displacement Δx (black squares) and straightness Δy (open circles), (b) linear displacement Δx (black squares) and flatness Δz (open circles). The error bars correspond to the calculated accuracy for a displacement Δy; Δz ¼ −0:5 mm.
motion is reported in Fig. 6. A given Δy ðΔzÞ displacement of −0:5 mm was imposed to the target during its linear translations Δx in the range of 0–400 mm. The measured deviations in the transverse plane are always within the accuracy of σ Δy ¼ 56 μm and σ Δz ¼ 65 μm as estimated by Eq. (4), that is, approximately three times the transverse resolution.
We acknowledge Costantino Florio and Roberto Martana, who were instrumental for motivating this research, and Massimiliano Putignano and Michela di Vietro for useful discussions. Financial support from Regione Puglia under projects PE093 and DM01 are also acknowledged.
4. Conclusion
References
We demonstrated the principle of operation of a novel displacement sensor based on laser self-mixing interferometry, capable of real-time monitoring small transverse (straightness and flatness) deviations of a moving target sliding along a linear track. The instrument is made of three nominally identical LSM interferometers properly tilted with respect to the track axis, resulting in a compact, low-cost, and self-aligned system. To our knowledge, this is the first time that laser-self-mixing has been exploited for the measurement of transversal degrees of freedom. The performance of the system depends on the laser wavelength and, for transverse degrees of freedom, are constrained by the dimension of the target and the length of the linear track. In the present configuration, similar to that used by optical encoders, resolution and accuracy are 0.7 and 5 μm for linear displacement in the range of 10−3 –400 mm and 20 and 48 μm for transverse displacements in the range 10−2 –5 mm.
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