Simultaneous measurement of multiple target displacements by self-mixing interferometry in a single laser diode

doi:10.1364/OE.19.016160

Simultaneous measurement of multiple target displacements by self-mixing interferometry in a single laser diode

Francesco P. Mezzapesa, Lorenzo Columbo, Massimo Brambilla, Maurizio Dabbicco, Antonio Ancona, Teresa Sibillano, Francesco De Lucia, Pietro M. Lugarà, and Gaetano Scamarcio »View Author Affiliations

Optics Express, Vol. 19, Issue 17, pp. 16160-16173 (2011)
http://dx.doi.org/10.1364/OE.19.016160

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▸ Article Outline
– Abstract
1. Introduction
2. Principle of measurement
3. Theoretical description
4. Comparison between theory and experiment: discussion
5. Conclusion
– References and links

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Abstract

We demonstrate that a single all-optical sensor based on laser diode self-mixing interferometry can monitor the independent displacement of individual portions of a surface. The experimental evidence was achieved using a metallic sample in a translatory motion while partly ablated by a ps-pulsed fiber laser. A model based on the Lang-Kobayashi approach gives an excellent explanation of the experimental results.

1. Introduction

Laser self-mixing interferometry has been widely exploited in metrology and mechatronics owing to its inherently high sensitivity to the target displacement, ease of use and compactness [1

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A, Pure Appl. Opt. 4(6), S283–S294 (2002). [CrossRef]

–3

M. Norgia, S. Donati, and A. D'Alessandro, “Interferometric measurement of displacement on a diffusing target by a speckle tracking technique,” IEEE J. Quantum Electron. 37(6), 800–806 (2001). [CrossRef]

]. Applications to the real-time measurements of absolute distance [4

F. Gouaux, N. Servagent, and T. Bosch, “Absolute distance measurement with an optical feedback interferometer,” Appl. Opt. 37(28), 6684–6689 (1998). [CrossRef] [PubMed]

], vibration [5

U. Zabit, R. Atashkhooei, T. Bosch, S. Royo, F. Bony, and A. D. Rakic, “Adaptive self-mixing vibrometer based on a liquid lens,” Opt. Lett. 35(8), 1278–1280 (2010). [CrossRef] [PubMed]

], velocity [6

L. Scalise, Y. Yu, G. Giuliani, G. Plantier, and T. Bosch, “Self-mixing laser diode velocimetry: application to vibration and velocity measurement,” IEEE Trans. Instrum. Meas. 53(1), 223–232 (2004). [CrossRef]

] and flow-rate [7

R. Kliese, Y. L. Lim, T. Bosch, and A. D. Rakić, “GaN laser self-mixing velocimeter for measuring slow flows,” Opt. Lett. 35(6), 814–816 (2010). [CrossRef] [PubMed]

], have been reported. So far, the analysis of the self-mixing interferogram has been used to measure a single degree-of-freedom of a moving object, corresponding to the displacement along the optical axis. To monitor multiple points on the same target surface, simultaneous interferometric channels were needed, as already reported [8

S. Ottonelli, F. De Lucia, M. Di Vietro, M. Dabbicco, G. Scamarcio, and F. P. Mezzapesa, “A compact three degrees-of-freedom motion sensor based on the laser-self-mixing effect,” IEEE Photon. Technol. Lett. 20(16), 1360–1362 (2008). [CrossRef]

–10

Y. L. Lim, R. Kliese, K. Bertling, K. Tanimizu, P. A. Jacobs, and A. D. Rakić, “Self-mixing flow sensor using a monolithic VCSEL array with parallel readout,” Opt. Express 18(11), 11720–11727 (2010). [CrossRef] [PubMed]

]. However, the sensing system was often based on customized multisource assemblies and specific geometry configuration/layouts.

In this paper, we demonstrate a single all-optical sensor based on laser diode self-mixing interferometry which is capable to concurrently measure the independent displacement of individual sections of a single target. The working principle is inspired to the wavefront-split interferometry, also known as sub-aperture interferometry, which has been mainly employed in astronomy [11

F. Zhao, “Sub-aperture interferometers: multiple target sub-beams are derived from the same measurement beam,” NASA Tech Briefs (2010), pp. 29–30.

]. Simultaneously measuring displacements of multiple targets can be achieved by splitting the wavefront of the measurement beam into sub-beams, which are in turn aimed to different retroreflecting targets. Particularly, the diode laser wavefront of the self-mixing interferometer sensor is spatially separated in sub-beams at the target surface and the independent displacement of individual portions of the same target can be monitored by a single detector.

The experimental validation of the sensing technique has been given during ultrafast laser percussion drilling of a moving target. Recently, we have demonstrated an optical feedback interferometric sensor to monitor the dynamics of the ablation process of an otherwise static target [12

F. P. Mezzapesa, A. Ancona, T. Sibillano, F. De Lucia, M. Dabbicco, P. Mario Lugarà, and G. Scamarcio, “High-resolution monitoring of the hole depth during ultrafast laser ablation drilling by diode laser self-mixing interferometry,” Opt. Lett. 36(6), 822–824 (2011). [CrossRef] [PubMed]

]. The displacement of the ablation front was related to the self-mixing interference fringes exhibited by the output power when a fraction of light back-reflected at the bottom surface of the drilled hole was re-injected into the diode laser cavity. Here, in addition the target is translated along the optical axis during laser ablation. The target displacement is controlled using a linear stepper-motor stage. Self-mixing fringes generated by the target translation are shown to be superimposed to those relative to the ablation process for appropriate settings of the working parameters. The experimental results perfectly match the numerical simulations, based on an extended version of the Lang-Kobayashi model [13

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). [CrossRef]

] capable to account for a single optical feedback where reflection are superimposed, coming from two independent target sections.

2. Principle of measurement

2.1 Laser ablation sensor: apparatus

Figure 1 shows the schematic of the experimental setup. The source used for the ablation is a prototype ytterbium-doped fiber laser amplifier delivering 120-ps pulses at the wavelength of 1064 nm [14

A. Ancona, D. Nodop, J. Limpert, S. Nolte, and A. Tünnermann, “Microdrilling of metals with an inexpensive and compact ultra-short-pulse fiber amplified microchip laser,” Appl. Phys., A Mater. Sci. Process. 94(1), 19–24 (2009). [CrossRef]

]. The maximum average power is 10 W at 100 kHz repetition rate, corresponding to a maximum pulse energy of 100 µJ. The linear polarization of the exit beam was converted into a circular one by using a quarter-wave plate, in order to prevent anisotropic absorption inside the metal [15

S. Nolte, C. Momma, G. Kamlage, A. Ostendorf, C. Fallnich, F. von Alvensleben, and H. Welling, “Polarization effects in ultrashort-pulse laser drilling,” Appl. Phys., A Mater. Sci. Process. 68(5), 563–567 (1999). [CrossRef]

]. Percussion drilling experiments were performed in air by focusing the fiber laser beam onto the surface of a stainless steel plate having a nominal thickness of 100 μm. To drill holes of about 30 μm diameter, a plano-convex lens of nominal focal length f = 50 mm was used (L in Fig. 1). The processing time was set using a fast mechanical shutter (MS). The effective drilling-through time was measured using a high-speed photodiode (PDext), positioned in order to collect the scattered light during the hole drilling and the transmitted light after the breakthrough [12

Fig. 1 Schematic layout of the experimental setup. M: mirror. L: focusing lens. BS: dichroic beamsplitter. MS: mechanical shutter. VA: variable attenuator. L1, L2: collimating aspheric lens. LD1, LD2: laser diode. PD1, PD2: integrated monitor photodiode. The schematic at the bottom left shows the target plate and probe laser beams, LD1 and LD2 respectively.

The measuring system, based on the self-mixing interferometry, is composed of a pair of laser diodes (LD1 and LD2 in Fig. 1). A short-pass dichroic beamsplitter (BS) was used to coaxially align the fiber laser beam with that of the laser diode LD1 used in the self-mixing interferometer. The latter consisted of a laser diode (Hitachi HL8325G) emitting at a wavelength λ_LD = 823 nm, a collimating lens (L1) of nominal focal length f1 = 11 mm, and a variable attenuator (VA) which allows to adjust the fraction of light back-reflected at the bottom surface of the drilled hole into the LD1 optical cavity. This feedback radiation coherently interferes with the optical field inside the laser cavity and eventually modulates the optical power, as detected by the monitor photodiode (PD1) packaged into the laser diode case. Analogously, the laser diode LD2 (Hitachi HL8325G) was focused on the target surface by the plano-convex lens L at about 150 µm aside of the drilled-hole edge, so that was not crossing the machining area. Hence, the photodiode PD2 returns an independent measurement of the target displacement out of the percussion drilling zone. The position of the LD2 beam with respect to LD1 was carefully controlled by a CCD positioned at the target plane.

The measuring system was prealigned recording the self-mixing interferometric signal generated by the target translation along the optical axis, using a linear stepper-motor stage. In the moderate feedback regime, the characteristic asymmetric waveform assumed by the self-mixing interferogram exhibits a fast switching each time the interferometric phase is varied by 2π. Particularly, the resulting sawtooth-like fringes superimposed to the diode output signal corresponds to a change of the external optical path by multiples of λ_LD/2. From the slope sequence in the sawtooth lineshape, one can unambiguously determine the direction of the target motion [16

D. M. Kane and K. A. Shore, in Unlocking Dynamical Diversity – Optical Feedback Effects on Semiconductor Diode Lasers (John Wiley and Sons, 2005), Chap. 7.

]. Fast-slow sawtooth-like fringes were observed when the target distance was increased. The same lineshape is expected during laser ablation.

2.2 Experimental results

Figure 2 shows the photodiode signals collected during the ablation process for a fiber laser power of about 4.5 W. To facilitate the analysis, the origin of the time axis is chosen to coincide with the shutter opening (i.e. laser on). The PDext signal in Fig. 2(a) rises at Δt = 0 ms and suddenly saturates at Δt ≈18.2 ms due to the breakthrough of the target plate. At Δt ≈35 ms the shutter is closed and the PDext signal drops to zero (i.e. laser off). In the range Δt = 0-18.2 ms, i.e. during the percussion drilling process, the PD1 signal exhibits the characteristic sawtooth-like fringes of the self-mixing interferograms, as shown in the inset of Fig. 2(b). Each fringe corresponds to the displacement of the ablation front due to both ablation and translation at the hole bottom by λ_LD/2 = 0.41 µm. After the hole breakthrough, the PD1 signal drops to zero, thus confirming that the origin of the self-mixing fringes is related with the displacement of the ablation front at the hole bottom.

Fig. 2 Left: Representative oscilloscope traces showing the time dependence of the signals detected by (a) the external photodiode (PDext), (b) the integrated photodiode PD1 and (c) the integrated photodiode PD2, during the laser percussion drilling of a 100 µm-thick stainless steel plate. The target displacement is controlled by a linear stepper motor stage moving at about 0.2 mm/sec. Relative distance between the collimating lens L1 and the diode emitting window: f1. The laser fluence was ~1.35 J/cm². Inset: PD1 signal during the percussion drilling shown on an enlarged scale.

Independently, the PD2 signal in Fig. 2(c) shows the interference fringes generated by the target displacement along the laser axis. The stainless steel plate was translated at the average velocity of 0.2 mm/sec using a linear stepper-motor stage, covering a distance of 17.22 µm during the acquisition time.

In this case, the interferometric signal detected by PD1 gives the absolute displacement of the ablation front during the drilling time, as the LD1 beam was coaxially focused on the target to a spot size smaller than the ablation area [12

]. A differential reading of both PD1 and PD2 signal is required to correct the penetration depth measurement.

A single-arm feedback interferometer is however capable to simultaneously distinguish superimposed reflection coming from two target sections, simply by arranging an appropriate combination of the optical components in the setup. Indeed, we succeeded to demonstrate that PD1 signal can carry a clear signature of both target and ablation front displacement by setting the collimating lens L1 and the variable attenuator to have the LD1 beam focused on the target to a spot size comparable with the hole diameter.

In particular, the PD1 signal in Fig. 3(a) was revealed when the relative distance between the collimating lens L1 and the diode emitting window was set 35 µm apart from the focal length f1, thus causing the measured LD1 beam diameter at the waist (i.e. FWHM of the intensity distribution measured by a beam profiler analyzer) to increase from 15.4 μm to 32.5 μm. In Fig. 3(b), the time dependence of the signal detected by the integrated photodiode PD1 shows that the sawtooth-like fringes relative to the ablation process are superimposed to those generated by the target translation.

Fig. 3 Representative oscilloscope traces showing the time dependence of the signals detected by: (a) the integrated photodiode PD1 and (c) the integrated photodiode PD2. (b) Zoom of PD1 signal during the ablation time. (d) Normalized power spectrum of the PD1 trace. The dashed line envelops the low and high-frequency peaks. The target displacement is controlled by a linear stepper-motor stage moving at about 0.2 mm/sec. Relative distance between the collimating lens L1 and the diode emitting window: f1 + 35 µm. The variable attenuator filters the optical feedback of 25% less than in Fig. 2. Laser fluence for the percussion drilling ~1.35 J/cm².

The detection system provides the capability to carry simultaneous information from both the ablation front displacement and the target translation. The PD2 signal in Fig. 3(c) shows only the interference fringes generated by the target displacement along the laser axis. The velocity fluctuations were induced by the stepper-motor stage and are due to the performance of the open-loop servo motor system.

The power spectrum of the PD1 signal in Fig. 3(d) shows the corresponding low and high frequency peaks, respectively. The low-frequency peaks exhibit a maximum at about 7 kHz, which corresponds to the target velocity controlled by the stepper-motor stage. The superior harmonics are caused by the saw-tooth character of the interferometric fringes. The envelope of the high-frequency peaks is instead broader around its maximum value at about 100 kHz, showing side peaks which can be ascribed to the actual ablation rate during the acquisition time and the beating with low-frequency tones.

On the other hand, an additional reduction of the optical feedback filtered by the variable attenuator makes the PD1 insensitive to the ablation front displacement. In Fig. 4(a) , the signal detected by the photodiode PD1 reproduces very well the trace of the photodiode PD2, being the contribution from the target displacement dominant - see Fig. 4(c). Now, the power spectrum of the PD1 signal in the Fig. 4(d), displays low-frequency peaks only, as expected.

Fig. 4 Representative oscilloscope traces showing the time dependence of the signals detected by: (a) the integrated photodiode PD1 and (c) the integrated photodiode PD2. (b) Zoom of PD1 signal during the ablation time. (d) Normalized power spectrum of the PD1 trace. The target displacement is controlled by a linear stepper-motor stage moving at about 0.2 mm/sec. Relative distance between the collimating lens L1 and the diode emitting window: f1 + 35 µm. The variable attenuator filters the optical feedback of 50% less than in Fig. 2. Laser fluence for the percussion drilling ~1.35 J/cm².

In what follows, an extended version of the Lang-Kobayashi model accounting for optical feedback from two independently moving targets will be used to generalize these unique results.

3. Theoretical description

The Lang-Kobayashi model well describes the semiconductor laser dynamics in presence of optical feedback provided by a single target in terms of two coupled delayed ordinary differential equations for the slowly varying field amplitude E and the carriers density N. It holds in the hypothesis of single longitudinal and transverse mode and negligible multiple reflections in the external cavity [13

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). [CrossRef]

]. In this section, we present an extended version of the Lang-Kobayashi model to account for feedback coming from multiple parts of a single target and /or different targets.

A representative schematic of a self-mixing interferometric system monitoring two independent portions of a single target moving along the optical axis z is shown in Fig. 5 . Referring it to the experimental setup described in Fig. 1, the target 1 can be associated with the stainless steel plate, whereas the target 2 resembles the ablation front during the percussion drilling process.

Fig. 5 Gedanken experiment of a multiparametric sensor based on SMI technique.

The Lang-Kobayashi equations in the case of optical feedback from two targets placed at different distances l₁, l₂ from the laser, read:

\begin{array}{l} \frac{dE(t)}{dt} = \frac{1}{2} (1 + α) (G(N(t) - N_{0})- \frac{1}{τ_{p}}) E(t) + \frac{k_{1}}{τ_{c}} E(t - τ_{1} {)e}^{-i ω_{0} τ_{1}} + \frac{k_{2}}{τ_{c}} E(t - τ_{2} {)e}^{-i ω_{0} τ_{2}} \\ \frac{dN(t)}{dt} = μ - \frac{N(t)}{τ_{e}} - G(N(t) - N_{0}) {| E(t) |}^{2} \end{array}

(1)

where α is the Henry factor (typical value: 1-5), N₀ is the carrier density at transparency (typical value: 1.4x10²⁴ m⁻³), G is the gain coefficient (typical value: 8x10¹³ m³s⁻¹), τ_p is the photon life time in the laser cavity (typical value: 1.6 ps), τ_c is the cavity round-trip time in the laser cavity (typical value: 8 ps), l_i is the external cavity length, τ_i = l_i/c (i = 1,2) is the round-trip time in the external cavity between the laser exit facet and the two targets, ω₀ = 2πc/λ_LD is the solitary laser frequency, μ is the electrical pump power density, τ_e is the photon decay time (typical value: 1ns). The multiple reflections in the cavity formed by the two targets have been neglected.

The feedback strength coefficient k_i has the following expression:

k_{i} = ε_{i} \sqrt{\frac{R_{e x t}}{R}} (1 - R)

where R and R_ext represent the diode laser mirror and the external target reflectivity, respectively. The coupling factor ε_i depends on both the target shape and the combination of the optical components in the setup. It is defined by the following relation [17

M. Wang and G. Lai, “Self-mixing microscopic interferometer for the measurement of micro-profile,” Opt. Commun. 238(4–6), 237–244 (2004). [CrossRef]

ε_{i} = ε_{l o s s} \sqrt{\frac{P_{i, b a c k}}{P_{o u t}}}

(2)

where ε_loss accounts for the power losses through the variable attenuator, P_out is the power delivered by the laser diode and P_i,back represents the fraction of P_out reinjected into the laser diode after an external cavity round-trip.

Looking for steady-state solutions of Eq. (1):

E = E_{0} e^{i (ω_{F} - ω_{0}) t}

N (t) = N_{0}

and supposing k_i << τ_c/2τ_p (i.e. moderate feedback regime), we get after some algebra the interferometric intensity I_PD1 [16

D. M. Kane and K. A. Shore, in Unlocking Dynamical Diversity – Optical Feedback Effects on Semiconductor Diode Lasers (John Wiley and Sons, 2005), Chap. 7.

I_{PD1} {=|E}_{0} |^{2} \approx (\frac{τ_{p}}{τ_{e}}) I_{sol} (1 + m_{1} cos(ω_{F} τ_{1} {)+m}_{2} \cos (ω_{F} τ_{2}))

(3)

where ω_F is the solution of the transcendental equation:

\begin{array}{l} ω_{F} = ω_{0} - \frac{m_{1}}{2 τ_{p}} (α cos(ω_{F} τ_{1})+ \sin (ω_{F} τ_{1})) - \frac{m_{2}}{2 τ_{p}} (α cos(ω_{F} τ_{2})+ \sin (ω_{F} τ_{2})) \end{array}

(4)

I_sol represents the intensity of the solitary laser, i.e. the laser output intensity in absence of optical feedback, and

m_{i} = 2 k_{i} \frac{τ_{p}}{τ_{c}}

the feedback coupling parameters.From Eq. (3) it derives that the contribution of the two targets to the interferometric signal I_PD1 is weighted by the feedback coupling parameters m_i and periodically varying with the laser half-wavelength. I_PD1 returns the output signal experimentally measured by the photodiode PD1.

The Eqs. (3)–(4) are formally equivalent to those describing self-mixing interferometry in the presence of multiple external cavities, as derived by Wang and Lai [17

M. Wang and G. Lai, “Self-mixing microscopic interferometer for the measurement of micro-profile,” Opt. Commun. 238(4–6), 237–244 (2004). [CrossRef]

]. There, the back-reflected beams from a series of parallel mirrors were combined into the laser diode to produce interference fringes in analogy with the case of a single external cavity. The main effect consisted in modifying the amplitude and shape of the saw-tooth like fringes. Here, the feedback into the laser cavity is provided by spatially dependent portions of a single target. Each portion can move independently along the optical axis, contributing to the interferometric waveform through individual coupling parameters. The resulting interferogram bears the fingerprint of simultaneous displacement of the target portions, as shown in next section.

3.1 Feedback parameters estimate

The feedback parameters m_i appearing in Eq. (3) can be expressed in terms of the optical components in the setup, as follows:

m_{i} = 2 \frac{τ_{p}}{τ_{c}} \sqrt{\frac{R_{e x t}}{R}} (1 - R) \times ε_{l o s s} \sqrt{\frac{P_{i, b a c k}}{P_{o u t}}}

(5)

In order to calculate P_i,back and P_out , we assume that the laser diode LD1 emits a fundamental Gaussian beam, which propagates in the external cavity as the lowest order solution of the paraxial Helmholtz equation [19

A. E. Siegman, in Lasers , 3rd ed. (University Science Books, Mill Valley, 1986).

E_{g a u s s} (x, y, z, t) = A_{0} q_{(z = 0)} (\frac{1}{R (z)} + i \frac{2}{k_{0} W^{2} (z)}) \exp (i k_{0} \frac{x^{2} + y^{2}}{2 R (z)} - \frac{x^{2} + y^{2}}{W^{2} (z)})

where k₀ = ω₀/c and the complex beam parameter q(z) is given by:

\frac{1}{q (z)} = \frac{1}{R (z)} + i \frac{2}{k_{0} W^{2} (z)}

(6)

with R(z) and W(z) being the Gaussian phase front radius of curvature and the Gaussian spot size, respectively.

The propagation of the Gaussian beam through a linear optical system can be conveniently described by transforming q(z) according to the ABCD law, namely [18

F. De Lucia, M. Putignano, S. Ottonelli, M. di Vietro, M. Dabbicco, and G. Scamarcio, “Laser-self-mixing interferometry in the Gaussian beam approximation: experiments and theory,” Opt. Express 18(10), 10323–10333 (2010). [CrossRef] [PubMed]

q (z_{o u t}) = \frac{A q (z_{i n}) + B}{C q (z_{i n}) + D}

(7)

where q(z_in) and q(z_out) define the complex Gaussian parameter before and after, respectively, an optical system characterized by the unitary 2 × 2 matrix [(A, B, C, D)].

With reference to Fig. 5, where the schematic of the experimental apparatus is drawn, the ABCD matrix elements associated with a cavity half-round trip (z = l_i ) are:

\begin{array}{l} A = - \frac{1}{f_{1} f} (d f - f_{1} f + d_{T A} f - d d_{T A} + f_{1} d_{T A}) \\ B = \frac{1}{f_{1} f} (d (f_{1} - d_{l}) (f - d_{T A}) - f_{2} d_{T A} d_{l} + f_{1} (- d_{l} d_{T A} + f (d_{l} + d_{T A}))) \\ C = \frac{1}{f_{1} f} (d - f_{1} - f) \\ D = A \end{array}

(8)

and the ABCD matrix elements associated with a round trip (z = 2l_i ) are, respectively:

\begin{array}{l} A = \frac{1}{f_{1}^{2} f^{2}} ((2 (f_{1} - d_{l}) (f - d_{T A}) d^{2} - 2 ((f - d_{T A}) f_{1}^{2} + (f^{2} - 2 (d_{T A} + d_{l}) f + 2 d_{T A} d_{l}) f_{1} \\ - f d_{l} (f - 2 d_{T A})) d) + (2 f^{2} d_{T A} d_{l} + f_{1}^{2} (f^{2} - 2 (d_{T A} + d_{l}) f + 2 d_{T A} d_{l}) - 2 f_{1} f (f (d_{T A} + d_{l}) - 2 d_{T A} d_{l}))) \\ B = - \frac{1}{f_{1}^{2} f^{2}} ((2 (d (f_{1} - d_{l}) + f d_{l} + f_{1} (d_{l} - f)) (d (f_{1} - d_{l}) (f - d_{T A}) - f d_{l} d_{T A} \\ + f_{1} (f (d_{T A} + d_{l}) - d_{l} d_{T A}))) \\ C = - \frac{1}{f_{1}^{2} f^{2}} (2 (- d + f_{1} + f) (f_{1} (f - d_{T A}) - f d_{l} + d (d_{T A} - f))) \\ D = A \end{array}

(9)

where d is the distance between the lens L and L1, d_l is the distance between the lens L1 and the diode facet (z = 0), and d_TA = l_i-(d + d_d) is the relative distance between L and the target (z = l_i ).

A general feature of the Gaussian beam is that the power transmitted P_trans through an on-axis circular aperture of radius a at z = z_a depends on the incident power P_inc as follows:

\frac{P_{t r a n s} (z = z_{a})}{P_{i n c} (z = z_{a})} = \frac{\int_{0}^{a} (2 π d ρ {| E_{g} (ρ, z = z_{a}) |}^{2})}{\int_{0}^{\infty} (2 π d ρ {| E_{g} (ρ, z = z_{a}) |}^{2})} = (1 - \exp (- \frac{2 a^{2}}{W^{2} (z = z_{a})}))

According to this simple model, the feedback power ratios in Eq. (5) are given by:

\begin{array}{l} \frac{P_{1, b a c k} (2 l_{1})}{P_{o u t}} = \frac{P^{*}_{1, b a c k} (2 l_{1})}{P_{o u t}} - \frac{P_{t r a n s} (2 l_{1})}{P_{t r a n s} (l_{1})} \frac{P_{t r a n s} (l_{1})}{P_{o u t}} = (1 - \exp (- \frac{2 a_{d}^{2}}{W^{2} (2 l_{1})})) \times \exp (- \frac{2 a_{t}^{2}}{W^{2} (l_{1})}) \\ \frac{P_{2, b a c k} (2 l_{2})}{P_{o u t}} = \frac{P_{t r a n s} (2 l_{2})}{P_{t r a n s} (l_{2})} \frac{P_{t r a n s} (l_{2})}{P_{o u t}} = (1 - \exp (- \frac{2 a_{d}^{2}}{W^{2} (2 l_{2})})) \times (1 - \exp (- \frac{2 a_{t}^{2}}{W^{2} (l_{2})})) \end{array}

(10)

where a_t is the radius of the target 2 in Fig. 5, a_d is the radius of the diode facet and P*_1,back is the power back-reflected by the target 1 in Fig. 5 if the target 2 had radius of zero (a_t = 0). The Gaussian spot size W(z) in Eq. (10) can be calculated at the corresponding z-value using the Eqs. (6)–(9), in order to finally estimate the feedback parameters m_i .

As an example, we calculate the feedback parameters m_i for the optical system in Fig. 5, in which the distance d_l between the diode facet and the collimating lens is kept variable. The Gaussian beam emitted by the laser diode LD1 changes with d_l , as shown in Fig. 6 .

Fig. 6 Image plot of the Gaussian beam intensity after a cavity half-round trip. (a) d_l = 11 mm; (b) d_l = 11.035 mm. The target 2 is enclosed by the red circle (in scale). Parametric regime: beam-waist = 1.55 μm, f₁ = 11 mm, f = 57.4 mm, d = 560 mm, d_TA = 57.4 mm, R_ext = 0.97, R = 0.35, ε_loss = 0.9, a_d = 3 μm and a_t = 15 μm.

Particularly, in Fig. 6(a) the Gaussian beam intensity |E_gauss|² at z = l₁ = l₂ is computed for d_l = f1. The Gaussian beam is focused onto the target 2, as better highlighted in the picture. The feedback parameters, m₁ = 0.04 and m₂ = 0.4, are calculated using the Eqs. (5)–(10) and the experimental settings given in the caption. This configuration is expected to reproduce the results plotted in Fig. 2(b), corresponding to a ratio of m₂/m₁~10.

Figure 6(b) shows the Gaussian beam intensity |E_gauss|² at z = l₁ = l₂ when d_l = f1 + 35 µm. The Gaussian beam is more divergent, and comparable with the hole diameter. Now, the portion of the beam impinging onto the surface of the target 1 becomes no more negligible. The calculation of the feedback parameters ratio gives m₂/m₁~1 (m₁ = 0.12 and m₂ = 0.08), thus an order of magnitude smaller than the previous case. This configuration is expected to reproduce the results plotted in Fig. 3(b).

4. Comparison between theory and experiment: discussion

The feedback parameters estimated in Section 3.1 are used to numerically solve the nonlinear Eq. (4) for ω_F and compute the interferometric function I_PD1 through Eq. (3). I_PD1 corresponds to the output signal experimentally measured by the integrated photodiode PD1 in Fig. 5.

Figure 7 shows the comparison between the experimental and numerical results for different optical configurations. The relative feedback parameters ratio m₂/m₁ are chosen to reproduce the peculiar features of the multiparametric interferograms from a single laser sensor.

Fig. 7 Left: Experimental results. Representative portions of the PD1 signals of Fig. 2(b) (a), Fig. 3(b) (c), and Fig. 4(b) (e), respectively. Right: Numerical results. Interferometric signal I_PD1 for the feedback parameters: (b) m₁ = 0.04, m₂ = 0.4; (d) m₁ = 0.12, m₂ = 0.08 and (f) m₁ = 0.4, m₂ = 0.04, respectively. Target velocity: v₁ = Δl₁/Δt = 0.4 mm/s; v₂ = Δl₂/Δt = 8.5 mm/s. The other parameters are given in the text.

In Figs. 7(a) and 7(b), the zoom of the PD1 signal in Fig. 2(b) is compared with the interferometric function I_PD1 during the ablation process. The corresponding optical configuration is arranged to focus the Gaussian beam of the laser diode LD1 mainly onto the surface of target 2, as sketched in Fig. 6(a). The fast oscillations in Fig. 7(b) are associated with the target 2 displacement, corresponding to the ablation front displacement in the experiment. The slow envelope is relative to the target 1 translation, which can be ascribed to the unavoidable contribution from the tail of LD1 beam.

In Figs. 7(c) and 7(d), the zoom of the PD1 signal in Fig. 3(b) is compared with the interferometric function I_PD1 during the ablation process. The corresponding optical configuration is arranged to diverge the Gaussian beam of the laser diode LD1 so to illuminate the surface of target 1 too, as sketched in Fig. 6(b). The fast oscillations in Fig. 7(d) are associated with the target 2 displacement, as above. The slow oscillations relative to the target 1 displacement do agree with the experimental trace, confirming that a single channel can carry information about multiple independent target dynamics.

In Figs. 7(e) and 7(f), the zoom of the PD1 signal in Fig. 4(b) is compared with the interferometric function I_PD1 during the ablation process. The corresponding optical configuration is arranged to give a negligible contribution of the back-reflected radiation from the surface of target 2. Consequently, the small fringes in Fig. 7(f) are associated mainly with the target 1 translation at the actual velocity of about 0.4 mm/sec.

The power spectra of the numerical interferometric function I_PD1 during the ablation process are plotted in Fig. 8 . The signatures of the ablation front displacement (Fig. 8(a)) and the target translation (Fig. 8(c)) are both displayed in Fig. 8(b). The beating with the low-frequency tones causes side-peaks appearing around 100 kHz, giving an excellent reproduction of the experimental results shown in Fig. 3(d). Some deviations are here attributed to the intrinsic variation of the ablation rate and the translation speed during the acquisition time.

Fig. 8 Normalized power spectrum of the numerical results shown in: (a) Fig. 7(b), (b) Fig. 7(d) and (c) Fig. 7(f), respectively.

Similar results were found for drilling processes coexisting with target translation at higher velocity (i.e. v = 1 mm/sec) or opposite direction (i.e. v = −0.4 mm/sec), although a systematic investigation of the target dynamic as a function of the instantaneous ablation rate is outside the scope of the present work. More experiments have been scheduled to prove the capability of this compact multiparametric system based on a single laser feedback interferometer to monitor non-metallic as well as hybrid targets, appropriately designed to have more than two parts moving independently.

5. Conclusion

In conclusion, the first experimental evidence of a multiparametric sensing technique using a single laser diode feedback interferometer is reported. The proof of principle was given by simultaneously measuring the ablation front displacement of a moving target during laser percussion drilling micromachining. To achieve the multiparametric regime, different optical configuration and working parameters have been systematically investigated. The Lang-Kobayashi model was extended in order to interpret the experimental results. In particular, the dynamical equations were modified to account for the optical feedback coming from two independent targets. The very good agreement between calculations and experimental results paves the way towards the development of self-mixing laser ablation sensors including the real-time correction of target displacement and/or vibration.

Acknowledgments

The authors acknowledge partial financial support from Regione Puglia – Project DM01.1 related with the Apulian Technological District on Mechatronics – MEDIS, and Projects PS_046 and PS_093. The author FPM acknowledges his research contract funded by the Apulian Regional Network TRASFORMA.

References and links

1.	G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A, Pure Appl. Opt. 4(6), S283–S294 (2002). [CrossRef]
2.	S. Ottonelli, M. Dabbicco, F. De Lucia, M. di Vietro, and G. Scamarcio, “Laser-self-mixing interferometry for mechatronics applications,” Sensors (Basel Switzerland) 9(5), 3527–3548 (2009). [CrossRef]
3.	M. Norgia, S. Donati, and A. D'Alessandro, “Interferometric measurement of displacement on a diffusing target by a speckle tracking technique,” IEEE J. Quantum Electron. 37(6), 800–806 (2001). [CrossRef]
4.	F. Gouaux, N. Servagent, and T. Bosch, “Absolute distance measurement with an optical feedback interferometer,” Appl. Opt. 37(28), 6684–6689 (1998). [CrossRef] [PubMed]
5.	U. Zabit, R. Atashkhooei, T. Bosch, S. Royo, F. Bony, and A. D. Rakic, “Adaptive self-mixing vibrometer based on a liquid lens,” Opt. Lett. 35(8), 1278–1280 (2010). [CrossRef] [PubMed]
6.	L. Scalise, Y. Yu, G. Giuliani, G. Plantier, and T. Bosch, “Self-mixing laser diode velocimetry: application to vibration and velocity measurement,” IEEE Trans. Instrum. Meas. 53(1), 223–232 (2004). [CrossRef]
7.	R. Kliese, Y. L. Lim, T. Bosch, and A. D. Rakić, “GaN laser self-mixing velocimeter for measuring slow flows,” Opt. Lett. 35(6), 814–816 (2010). [CrossRef] [PubMed]
8.	S. Ottonelli, F. De Lucia, M. Di Vietro, M. Dabbicco, G. Scamarcio, and F. P. Mezzapesa, “A compact three degrees-of-freedom motion sensor based on the laser-self-mixing effect,” IEEE Photon. Technol. Lett. 20(16), 1360–1362 (2008). [CrossRef]
9.	X. Dai, M. Wang, and C. Zhou, “Multiplexing self-mixing interference in fiber ring lasers,” IEEE Photon. Technol. Lett. 22(21), 1619–1621 (2010). [CrossRef]
10.	Y. L. Lim, R. Kliese, K. Bertling, K. Tanimizu, P. A. Jacobs, and A. D. Rakić, “Self-mixing flow sensor using a monolithic VCSEL array with parallel readout,” Opt. Express 18(11), 11720–11727 (2010). [CrossRef] [PubMed]
11.	F. Zhao, “Sub-aperture interferometers: multiple target sub-beams are derived from the same measurement beam,” NASA Tech Briefs (2010), pp. 29–30.
12.	F. P. Mezzapesa, A. Ancona, T. Sibillano, F. De Lucia, M. Dabbicco, P. Mario Lugarà, and G. Scamarcio, “High-resolution monitoring of the hole depth during ultrafast laser ablation drilling by diode laser self-mixing interferometry,” Opt. Lett. 36(6), 822–824 (2011). [CrossRef] [PubMed]
13.	R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). [CrossRef]
14.	A. Ancona, D. Nodop, J. Limpert, S. Nolte, and A. Tünnermann, “Microdrilling of metals with an inexpensive and compact ultra-short-pulse fiber amplified microchip laser,” Appl. Phys., A Mater. Sci. Process. 94(1), 19–24 (2009). [CrossRef]
15.	S. Nolte, C. Momma, G. Kamlage, A. Ostendorf, C. Fallnich, F. von Alvensleben, and H. Welling, “Polarization effects in ultrashort-pulse laser drilling,” Appl. Phys., A Mater. Sci. Process. 68(5), 563–567 (1999). [CrossRef]
16.	D. M. Kane and K. A. Shore, in Unlocking Dynamical Diversity – Optical Feedback Effects on Semiconductor Diode Lasers (John Wiley and Sons, 2005), Chap. 7.
17.	M. Wang and G. Lai, “Self-mixing microscopic interferometer for the measurement of micro-profile,” Opt. Commun. 238(4–6), 237–244 (2004). [CrossRef]
18.	F. De Lucia, M. Putignano, S. Ottonelli, M. di Vietro, M. Dabbicco, and G. Scamarcio, “Laser-self-mixing interferometry in the Gaussian beam approximation: experiments and theory,” Opt. Express 18(10), 10323–10333 (2010). [CrossRef] [PubMed]
19.	A. E. Siegman, in Lasers , 3rd ed. (University Science Books, Mill Valley, 1986).

OCIS Codes
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.3930) Instrumentation, measurement, and metrology : Metrological instrumentation
(280.3420) Remote sensing and sensors : Laser sensors
(280.4788) Remote sensing and sensors : Optical sensing and sensors

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: May 3, 2011
Revised Manuscript: May 31, 2011
Manuscript Accepted: May 31, 2011
Published: August 9, 2011

Citation
Francesco P. Mezzapesa, Lorenzo Columbo, Massimo Brambilla, Maurizio Dabbicco, Antonio Ancona, Teresa Sibillano, Francesco De Lucia, Pietro M. Lugarà, and Gaetano Scamarcio, "Simultaneous measurement of multiple target displacements by self-mixing interferometry in a single laser diode," Opt. Express 19, 16160-16173 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-16160

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References

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A, Pure Appl. Opt. 4(6), S283–S294 (2002). [CrossRef]
S. Ottonelli, M. Dabbicco, F. De Lucia, M. di Vietro, and G. Scamarcio, “Laser-self-mixing interferometry for mechatronics applications,” Sensors (Basel Switzerland) 9(5), 3527–3548 (2009). [CrossRef]
M. Norgia, S. Donati, and A. D'Alessandro, “Interferometric measurement of displacement on a diffusing target by a speckle tracking technique,” IEEE J. Quantum Electron. 37(6), 800–806 (2001). [CrossRef]
F. Gouaux, N. Servagent, and T. Bosch, “Absolute distance measurement with an optical feedback interferometer,” Appl. Opt. 37(28), 6684–6689 (1998). [CrossRef] [PubMed]
U. Zabit, R. Atashkhooei, T. Bosch, S. Royo, F. Bony, and A. D. Rakic, “Adaptive self-mixing vibrometer based on a liquid lens,” Opt. Lett. 35(8), 1278–1280 (2010). [CrossRef] [PubMed]
L. Scalise, Y. Yu, G. Giuliani, G. Plantier, and T. Bosch, “Self-mixing laser diode velocimetry: application to vibration and velocity measurement,” IEEE Trans. Instrum. Meas. 53(1), 223–232 (2004). [CrossRef]
R. Kliese, Y. L. Lim, T. Bosch, and A. D. Rakić, “GaN laser self-mixing velocimeter for measuring slow flows,” Opt. Lett. 35(6), 814–816 (2010). [CrossRef] [PubMed]
S. Ottonelli, F. De Lucia, M. Di Vietro, M. Dabbicco, G. Scamarcio, and F. P. Mezzapesa, “A compact three degrees-of-freedom motion sensor based on the laser-self-mixing effect,” IEEE Photon. Technol. Lett. 20(16), 1360–1362 (2008). [CrossRef]
X. Dai, M. Wang, and C. Zhou, “Multiplexing self-mixing interference in fiber ring lasers,” IEEE Photon. Technol. Lett. 22(21), 1619–1621 (2010). [CrossRef]
Y. L. Lim, R. Kliese, K. Bertling, K. Tanimizu, P. A. Jacobs, and A. D. Rakić, “Self-mixing flow sensor using a monolithic VCSEL array with parallel readout,” Opt. Express 18(11), 11720–11727 (2010). [CrossRef] [PubMed]
F. Zhao, “Sub-aperture interferometers: multiple target sub-beams are derived from the same measurement beam,” NASA Tech Briefs (2010), pp. 29–30.
F. P. Mezzapesa, A. Ancona, T. Sibillano, F. De Lucia, M. Dabbicco, P. Mario Lugarà, and G. Scamarcio, “High-resolution monitoring of the hole depth during ultrafast laser ablation drilling by diode laser self-mixing interferometry,” Opt. Lett. 36(6), 822–824 (2011). [CrossRef] [PubMed]
R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). [CrossRef]
A. Ancona, D. Nodop, J. Limpert, S. Nolte, and A. Tünnermann, “Microdrilling of metals with an inexpensive and compact ultra-short-pulse fiber amplified microchip laser,” Appl. Phys., A Mater. Sci. Process. 94(1), 19–24 (2009). [CrossRef]
S. Nolte, C. Momma, G. Kamlage, A. Ostendorf, C. Fallnich, F. von Alvensleben, and H. Welling, “Polarization effects in ultrashort-pulse laser drilling,” Appl. Phys., A Mater. Sci. Process. 68(5), 563–567 (1999). [CrossRef]
D. M. Kane and K. A. Shore, in Unlocking Dynamical Diversity – Optical Feedback Effects on Semiconductor Diode Lasers (John Wiley and Sons, 2005), Chap. 7.
M. Wang and G. Lai, “Self-mixing microscopic interferometer for the measurement of micro-profile,” Opt. Commun. 238(4–6), 237–244 (2004). [CrossRef]
F. De Lucia, M. Putignano, S. Ottonelli, M. di Vietro, M. Dabbicco, and G. Scamarcio, “Laser-self-mixing interferometry in the Gaussian beam approximation: experiments and theory,” Opt. Express 18(10), 10323–10333 (2010). [CrossRef] [PubMed]
A. E. Siegman, in Lasers, 3rd ed. (University Science Books, Mill Valley, 1986).

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