Laser diode (LD)-based self-mixing interferometry (SMI) is a promising technique for noncontact sensing and its instrumentation. According to the well-known Lang–Kobayashi equations, an SMI waveform is shaped by multiple parameters, including linewidth enhancement factor (denoted as α), optical feedback level factor (denoted as C), and the movement of the external cavity of an LD. This paper presents a new algorithm for simultaneously retrieving the multiple parameters from a piece of SMI signal. First, a set of linear equations is derived based on the existing SMI model. By careful selection of data samples, the linear equations can be made independent and used to determine a set of variables, and thus the values of α and C, as well as the reconstruction coefficients of the external movement. The work in this paper lifts the restrictions existing in SMI-based sensing methods, such as prerequisite knowledge of either α and C or vibration information of an external target. Simulations and experiments are conducted to verify the proposed algorithm. © 2014 Optical Society of America OCIS codes: (120.3180) Interferometry; (280.3420) Laser sensors; (140.5960) Semiconductor lasers. http://dx.doi.org/10.1364/AO.53.004256

1. Introduction

Self-mixing interferometry (SMI) has been an active area of research as an enabling technique for noncontact sensing and its instrumentation in recent decades. The basic structure of an SMI consists of a laser diode (LD), a focusing lens, and a vibrating target which forms the external cavity of the LD. When a portion of the light emitted by the LD is backscattered or reflected by the vibrating external target and reenters the laser cavity, a modulated lasing field will be formed in both amplitude and phase. The modulated laser power, detected by a photodiode packaged at the rear facet of the LD, is called an SMI signal. It carries the information of the external target vibration and the parameters related to the LD. Generally, there are two classes of applications for 1559-128X/14/194256-08$15.00/0 © 2014 Optical Society of America 4256

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SMI-based sensing: (1) estimation of parameters associated with an LD and (2) measurement of the metrological quantities of the external target. Use of SMI for measuring LD parameters mainly focuses on the linewidth enhancement factor and the optical feedback level [1–8]. The linewidth enhancement factor, also called the alpha factor (α-parameter), is one of fundamental parameters in the LDs as it characterizes the LD linewidth, the chirp, the injection lock range, and the response to optical feedback. The optical feedback level, denoted as C, describes the coupling strength of the optical feedback in the LD cavity and is utilized for classifying different optical feedback regimes, namely the weak feedback regime (where 0 0

Note that ω0 is treated as a known parameter, as it can be easily obtained by applying an autocorrelation operation on a piece of SMI signal [2]. Hence, the five quantities introduced in Eq. (16) are all constant at a certain time instance t. With the notation in Eq. (16), Eq. (13) can be written as follows:

Therefore, Eq. (10) can be written as p 8 PN g0 t ω0 k1 bk k coskω0 t−ak k sinkω0 t 1C cosarctanαgt−C sinarctanα 1−g2 t − p ; > > 2 1−g t > > > < when g0 t < 0 h pi P g0 t > N 2 t > p − ω b k coskω t−a k sinkω t 1C cosarctanαgt−C sinarctanα − 1−g > 0 k 0 k 0 k1 > − 1−g2 t > : 0 when g t > 0: 12 Rearranging Eq. (12) yields the following: 8 p > g0 t −g0 tgtu1 g0 t 1 − g2 tu2 > > > ; p P p P > < 2 N ω0 1 − g2 t N k1 ak k sinkω0 t − ω0 1 − g t k1 bk k coskω0 t p > > g0 t −g0 tgtu1 − g0 t 1 − g2 tu2 > > ; p P p P > : 2 N − ω0 1 − g2 t N k1 ak k sinkω0 t ω0 1 − g t k1 bk k coskω0 t 4258

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when g0 t < 0 ; 0

when g t > 0

13

8 P PN > > −G1 tu1 G2 tu2 N > k1 Ak tak − k1 Bk tbk G3 t > > < when g0 t < 0 : PN PN > −G1 tu1 − G2 tu2 − k1 Ak tak k1 Bk tbk G3 t > > > > : when g0 t > 0

17

Obviously, Eq. (17) is a linear equation of constant coefficients with u1 , u2 , ak , and bk as variables. As Eq. (17) has 2N 2 variables, we need the same number of independent equations. To achieve this we take 2N 2 samples from the observed gt and its corresponding derivative version g0 t, enabling us to have the following: 2

−G1 1

6 6 −G1 2 6 6 .. 6 6 . 6 6 6 −G1 N 1 6 6 −G N 2 1 6 6 .. 6 6 . 4 −G1 2N 2 2

G3 1

G2 1

A1 1

AN 1

−B1 1

G2 2 .. .

A1 2 .. .

.. .

AN 2 .. .

−B1 2 .. .

.. .

G2 N 1

A1 N 1

AN N 1

−B1 N 1

−G2 N 2 .. .

−A1 N 2 .. .

.. .

−AN N 2 .. .

B1 N 2 .. .

.. .

−G2 2N 2

−A1 2N 2 −AN 2N 2

B1 2N 2

3

32 u1 3 7 76 6 u2 7 −BN 2 7 6 76 a 7 76 1 7 7 .. 76 76 .. 7 . 76 . 7 7 7 7 −BN N 1 76 6 7 6 aN 7 7 7 BN N 2 7 76 6 b 76 1 7 7 .. 76 76 . 7 . 54 .. 7 5 BN 2N 2 bN −BN 1

7 6 7 6 G3 2 7 6 7 6 .. 7 6 7 6 . 7 6 7 6 6 G3 N 1 7: 7 6 6 G N 2 7 7 6 3 7 6 .. 7 6 7 6 . 5 4 G3 2N 2

In order to determine all variables u1, u2 , ak , and bk , we must guarantee that Eq. (18) is nontrivial by proper selection of the data samples from the SMI signal to construct the above matrix equation. Generally speaking, the larger the differences in the values and the time instances taken among the data samples, the less likely for Eq. (18) to become trivial. Hence, we try to choose the data samples carefully and the sample data will be selected based on the following: 1. The segment of SMI signal corresponding to a whole period of vibration usually consists of two regions corresponding to the increasing and decreasing parts of ϕ0 t, respectively. N 1 data samples will be taken from the g0 t < 0 region and the other N 1 samples will be from the g0 t > 0 region (shown in Fig. 1). 2. Assuming that there are M fringes in the g0 t < 0 region in one segment of SMI signal, we set N 1 equally spaced levels over the fringe peak-to-peak height, yielding N 1 sample points within each individual g0 t < 0 section.

(18)

3. Within each fringe, for the g0 t < 0 section, N 1∕M samples are selected, resulting in a total number of N 1 data samples distributed over the entire g0 t < 0 region. 4. The other N 1 samples are selected in the same way from the region where g0 t > 0. Note that for a target with reciprocating vibration, the same number of fringes will appear in the increasing and decreasing parts of ϕ0 t. However, the proposed algorithm itself does not require the same fringe number for both parts. Let us use an example to demonstrate the procedures described above. Figure 1(a) shows the signal ϕ0 t corresponding to the movement of the external target. Figures 1(c) and 1(e) show the SMI signals gt in the moderate and weak feedback regimes, respectively. Figures 1(b) and 1(d) show the derivative forms (g0 t) of the SMI signals in Figs. 1(c) and 1(e). The signals shown in Fig. 1 are generated using the model described in Eqs. (1)–(5) with parameters C∕α 3.000∕3.000 for the moderate feedback case 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS

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20 φ 0(t) 0

-20

decreasing

(a)

increasing

0

0.005

0.01

0.05

g'(t)>0

(b)

g'(t) 0 g'(t)