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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 10 — May. 9, 2011
  • pp: 9582–9593
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Measuring the feedback parameter of a semiconductor laser with external optical feedback

Yanguang Yu, Jiangtao Xi, and Joe F. Chicharo  »View Author Affiliations


Optics Express, Vol. 19, Issue 10, pp. 9582-9593 (2011)
http://dx.doi.org/10.1364/OE.19.009582


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Abstract

Feedback parameter (the C factor) is an important parameter for a semiconductor laser operating in the regime of external optical feedback. Self-mixing interferometry (SMI) has been proposed for the measurement of the parameter, based on the time-domain analysis of the output power waveforms (called SMI signals) in presence of feedback. However, the existing approaches only work for a limited range of C, below about 3.5. This paper presents a new method to measure C based on analysis of the phase signal of SMI signals in the frequency domain. The proposed method covers a large range of C values, up to about 10. Simulations and experimental results are presented for verification of the proposed method.

© 2011 OSA

1. Introduction

As an active area of research, semiconductor lasers (SLs) with external optical feedback has attracted extensive theoretical and experimental work during the past decades. A significant research outcome in this area is an emerging technology for sensing and measurement called optical feedback self-mixing interferometry (SMI) technique. The basic structure of the SMI is shown in Fig. 1
Fig. 1 Schematic SMI.
, consisting of an SL, a lens and a target. The optical feedback from the external cavity modulates the SL power, which then can be used to measure the movement of the external cavity or retrieve system parameters of an SMI [1

1. Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16(4), 990–992 (2004). [CrossRef]

10

10. Y. Yu, J. Xi, and J. F. Chicharo, “Improving the Performance in an Optical feedback Self-mixing Interferometry System using Digital Signal Pre-processing,” in Proceedings of IEEE Conference on Intelligent Signal Processing (Institute of Electrical and Electronic Engineers, Alcala de Henares, 2007), pp. 1–6.

].

The scenario behind SMI sensing is the theoretical model developed from Lang and Kobayashi equations [11

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

]. The model consists of the following equations [1

1. Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16(4), 990–992 (2004). [CrossRef]

9

9. Y. Fan, Y. Yu, J. Xi, J. Chicharo, and H. Ye, “A displacement reconstruction algorithm used for optical feedback self mixing interferometry system under different feedback levels,” Proc. SPIE 7855, 78550L, 78550L-7 (2010). [CrossRef]

]:

ϕF=ϕ0Csin[ϕF+arctan(α)],
(1)
g(ϕ0)=cos(ϕF),
(2)
P(ϕ0)=P0[1+m×g(ϕ0)].
(3)

Definitions of the parameters and variables in Eqs. (1)-(3) are listed in Table 1

Table 1. Definition of the variables in Eq. (1)(3)

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.

From Eqs. (1)(3), we see that ϕ0 affects the emitted powerP(ϕ0), which can be easily measured with a photo detector placed on the rear mirror of the SL. As g(ϕ0)=P(ϕ0)P0mP0, we can obtain g(ϕ0) through data processing on P(ϕ0). Without loss of generality, in this paper we use g(ϕ0)to study the features of the SMI signal. Physical quantities associated with the external cavity (e.g. velocity, vibration or displacement, etc.) can be extracted from an SMI sensing signal through the phaseϕ0=4πυ0L/c .

It is seen that there are two parameters in Eqs. (1)(3), that is, optical feedback parameter denoted by C [12

12. G. Acket, D. Lenstra, A. Den Boef, and B. Verbeek, “The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers,” IEEE J. Quantum Electron. 20(10), 1163–1169 (1984). [CrossRef]

,13

13. K. Petermann, “External optical feedback phenomena in semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 1(2), 480–489 (1995). [CrossRef]

], and linewidth enhancement factor (or α-factor) of an SL. As a key parameter of an SL, α characterizes the linewidth, the chirp, the injection lock range in an SL, and also characterizes the response of the SL to optical feedback [14

14. M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers–An overview,” IEEE J. Quantum Electron. 23(1), 9–29 (1987). [CrossRef]

]. Measurement of α has attracted extensive research with various approaches proposed [14

14. M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers–An overview,” IEEE J. Quantum Electron. 23(1), 9–29 (1987). [CrossRef]

], such as linewidth methods [15

15. D. Kuksenkov, S. Feld, C. Wilmsen, H. Temkin, S. Swirhun, and R. Leibenguth, “Linewidth and alpha-factor in AlGaAs/GaAs vertical cavity surface emitting lasers,” Appl. Phys. Lett. 66(3), 277–279 (1995). [CrossRef]

,16

16. Y. Arakawa and A. Yariv, “Fermi energy dependence of linewidth enhancement factor of GaAlAs buried heterostructure lasers,” Appl. Phys. Lett. 47(9), 905–907 (1985). [CrossRef]

], FM/AM methods [17

17. L. Hua, “RF-modulation measurement of linewidth enhancement factor and nonlinear gain of vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 8(12), 1594–1596 (1996). [CrossRef]

19

19. H. Nakajima and J. C. Bouley, “Observation of power dependent linewidth enhancement factor in 1.55 mu m strained quantum well lasers,” Electron. Lett. 27(20), 1840–1841 (1991). [CrossRef]

], optical injection [20

20. G. Liu, X. Jin, and S. L. Chuang, “Measurement of linewidth enhancement factor of semiconductor lasers using an injection-locking technique,” IEEE Photon. Technol. Lett. 13(5), 430–432 (2001). [CrossRef]

22

22. C. H. Shin, M. Teshima, and M. Ohtsu, “Novel measurement method of linewidth enhancement factor in semiconductor lasers by optical self-locking,” Electron. Lett. 25(1), 27–28 (1989). [CrossRef]

] and optical feedback [1

1. Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16(4), 990–992 (2004). [CrossRef]

,6

6. J. Xi, Y. Yu, J. F. Chicharo, and T. Bosch, “Estimating the parameters of semiconductor lasers based on weak optical feedback self-mixing interferometry,” IEEE J. Quantum Electron. 41(8), 1058–1064 (2005). [CrossRef]

,7

7. Y. Yu, J. Xi, J. F. Chicharo, and T. Bosch, “Toward automatic measurement of the linewidth-enhancement factor using optical feedback self-mixing interferometry with weak optical feedback,” IEEE J. Quantum Electron. 43(7), 527–534 (2007). [CrossRef]

,23

23. Y. S. Shin, T. H. Yoon, J. R. Park, and C. H. Nam, “Simple methods for measuring the linewidth enhancement factor in external cavity laser diodes,” Opt. Commun. 173(1–6), 303–309 (2000). [CrossRef]

,24

24. D. Fye, “Relationship between carrier-induced index change and feedback noise in diode lasers,” IEEE J. Quantum Electron. 18(10), 1675–1678 (1982). [CrossRef]

]. These different approaches were compared systematically in [25

25. T. Fordell and A. M. Lindberg, “Experiments on the linewidth-enhancement factor of a vertical-cavity surface-emitting laser,” IEEE J. Quantum Electron. 43(1), 6–15 (2007). [CrossRef]

,26

26. T. Fordell and A. M. Lindberg, “Noise correlation, regenerative amplification, and the linewidth enhancement factor of a vertical-cavity surface-emitting laser,” IEEE Photon. Technol. Lett. 20(9), 667–669 (2008). [CrossRef]

].

In recent years, a number of SMI-based approaches are proposed to measure α, which can also yield the C value [1

1. Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16(4), 990–992 (2004). [CrossRef]

,6

6. J. Xi, Y. Yu, J. F. Chicharo, and T. Bosch, “Estimating the parameters of semiconductor lasers based on weak optical feedback self-mixing interferometry,” IEEE J. Quantum Electron. 41(8), 1058–1064 (2005). [CrossRef]

,7

7. Y. Yu, J. Xi, J. F. Chicharo, and T. Bosch, “Toward automatic measurement of the linewidth-enhancement factor using optical feedback self-mixing interferometry with weak optical feedback,” IEEE J. Quantum Electron. 43(7), 527–534 (2007). [CrossRef]

]. However, these methods are only applicable to limited range of C, as discussed below.

The first SMI based technique for α measurement was reported in [1

1. Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16(4), 990–992 (2004). [CrossRef]

] where SMI operates in moderate optical feedback regime. When C>1, the optical feedback in an SL causes hysteresis phenomenon in the laser power [2

2. S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron. 31(1), 113–119 (1995). [CrossRef]

,11

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

,12

12. G. Acket, D. Lenstra, A. Den Boef, and B. Verbeek, “The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers,” IEEE J. Quantum Electron. 20(10), 1163–1169 (1984). [CrossRef]

]. In this case, the waveform is sawtooth like [1

1. Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16(4), 990–992 (2004). [CrossRef]

,2

2. S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron. 31(1), 113–119 (1995). [CrossRef]

,8

8. Y. Yu, J. Xi, J. F. Chicharo, and T. M. Bosch, “Optical feedback self-mixing interferometry with a large feedback factor C: behavior studies,” IEEE J. Quantum Electron. 45(7), 840–848 (2009). [CrossRef]

,12

12. G. Acket, D. Lenstra, A. Den Boef, and B. Verbeek, “The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers,” IEEE J. Quantum Electron. 20(10), 1163–1169 (1984). [CrossRef]

]. An example of the hysteresis is plot in Fig. 2
Fig. 2 Hysteresis phenomenon of SMI signal and the measurement principle on αin [20] with C=2and α=5.
using Eq. (1) and Eq. (2), where C=2 and α=5. When ϕ0 increases, g(ϕ0) follows the path A1-B-B1, which is different from the path B1-A-A1 for decreasing ϕ0. The area A1-B-B1-A is called hysteresis area with the width represented as ϕ0,AB. ϕ0,AB has been found in [12

12. G. Acket, D. Lenstra, A. Den Boef, and B. Verbeek, “The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers,” IEEE J. Quantum Electron. 20(10), 1163–1169 (1984). [CrossRef]

] as:

ϕ0,AB=2{C21+arccos(1C)π}.
(4)

ϕ0,AB can be measured by ϕ0,AB=ϕ0,CBϕ0,AD. ϕ0,CB, ϕ0,AD are read from the SMI waveform [1

1. Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16(4), 990–992 (2004). [CrossRef]

]. However, when C increases to a certain value, the hysteresis area will cover ϕ0,CB and ϕ0,AD, and thus they will disappear from the SMI waveform. In this case, the approach in [1

1. Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16(4), 990–992 (2004). [CrossRef]

] is no longer valid. Our simulations show that the approach in [1

1. Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16(4), 990–992 (2004). [CrossRef]

] can only be used when 1<C<3.5.

An SMI signal at weak feedback regime is similar to a traditional interference signal containing sinusoidal fringe pattern. In this case a data fitting algorithm and its improved version were reported in [6

6. J. Xi, Y. Yu, J. F. Chicharo, and T. Bosch, “Estimating the parameters of semiconductor lasers based on weak optical feedback self-mixing interferometry,” IEEE J. Quantum Electron. 41(8), 1058–1064 (2005). [CrossRef]

,7

7. Y. Yu, J. Xi, J. F. Chicharo, and T. Bosch, “Toward automatic measurement of the linewidth-enhancement factor using optical feedback self-mixing interferometry with weak optical feedback,” IEEE J. Quantum Electron. 43(7), 527–534 (2007). [CrossRef]

] for the measurement of α and C . The algorithm utilizes the SMI model incorporates an optimal estimate of α and C to yield the best match for the observed signals. Obviously, the approaches presented in [6

6. J. Xi, Y. Yu, J. F. Chicharo, and T. Bosch, “Estimating the parameters of semiconductor lasers based on weak optical feedback self-mixing interferometry,” IEEE J. Quantum Electron. 41(8), 1058–1064 (2005). [CrossRef]

,7

7. Y. Yu, J. Xi, J. F. Chicharo, and T. Bosch, “Toward automatic measurement of the linewidth-enhancement factor using optical feedback self-mixing interferometry with weak optical feedback,” IEEE J. Quantum Electron. 43(7), 527–534 (2007). [CrossRef]

] are only valid for the case of C<1. Hence the existing SMI methods for C measurement can only be used for the cases of C<3.5.

For an SL with external optical feedback, the value of C is determined by the following [2

2. S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron. 31(1), 113–119 (1995). [CrossRef]

]:
C=ηττSL(1R2)R3R21+α2,
(5)
where R2, R3, η and τSLare, respectively, the power reflectivity of the SL cavity, the power reflectivity of the external cavity, the coupling coefficient of the feedback power, and the round trip time in the SL cavity. Note that η and τ are not fixed values when a moving target is used to form the external cavity of an SL. It is obvious that the value of C varies in different SL systems and the value can be greater than 3.5. Hence measurement of C over a large range is required.

In this paper, we propose to estimate C over an extend range of its value (up to 10 and more). Compared to existing techniques which are based on waveform analysis in time domain, the proposed technique is in the frequency domain where the C value is determined by looking at the spectrum of the phase signals derived from the SMI signalg(ϕ0). With existing time-domain approaches [1

1. Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16(4), 990–992 (2004). [CrossRef]

,6

6. J. Xi, Y. Yu, J. F. Chicharo, and T. Bosch, “Estimating the parameters of semiconductor lasers based on weak optical feedback self-mixing interferometry,” IEEE J. Quantum Electron. 41(8), 1058–1064 (2005). [CrossRef]

,7

7. Y. Yu, J. Xi, J. F. Chicharo, and T. Bosch, “Toward automatic measurement of the linewidth-enhancement factor using optical feedback self-mixing interferometry with weak optical feedback,” IEEE J. Quantum Electron. 43(7), 527–534 (2007). [CrossRef]

], different algorithms must be employed in different regimes of optical feedback. In contrast, the proposed approach employs the same algorithm to determine C, which covers different optical feedback regimes.

2. The proposed method

2.1 Operation principle

Let us consider the case of an external target moving periodically, resulting in the external cavity length L(t)and thus ϕ0 also periodically changing with time. Equation (2) and Eq. (1) can be rewritten as:

g(t)=cos(ϕF(t)),
(6)
ϕF(t)=ϕ0(t)Csin{ϕF(t)+arctan(α)}.
(7)

Taking Fourier transform, denoted by F{}, on both sides of Eq. (7), we have
ΦF(f)=Φ0(f)CF{sin[ϕF(t)+arctan(α)]},
(8)
where ΦF(f)and Φ0(f)are the Fourier transforms of ϕF(t)and ϕ0(t)respectively. For ease of manipulation let us introduce a function
ϕ1(t)=sin{ϕF(t)+arctan(α)},
(9)
so that Eq. (8) becomes
ΦF(f)=Φ0(f)CF{ϕ1(t)}=Φ0(f)CΦ1(f),
(10)
where Φ1(f) is the Fourier transforms of ϕ1(t).

As ϕ0(t)=4πυ0L(t)/c, ϕ0(t) also describes the moving of the target. If the external target moves in a manner close to simple harmonic vibration, ϕ0(t) will be close to a sinusoidal, in which case it can be considered as of narrow band in frequency domain.

Given ϕ0(t)narrow band in nature, from Eqs. (7) and (8), ϕF(t) should exhibit a broader spectrum than ϕ0(t) because of the nonlinear mapping from ϕ0(t) to ϕF(t). In other words, ΦF(f) and Φ1(f) should spread over a wider frequency range than Φ0(f) does. Now, we can divide the frequency range of non-vanishing spectrum of ΦF(f) and Φ1(f) into two domains, denoted by Ω1 and Ω2respectively, in which the spectrum of Φ0(f) is non-vanishing or vanishing, i.e.:

Ω1:f{ΦF(f)0,Φ1(f)0,Φ0(f)0},  and
Ω2:f{ΦF(f)0,Φ1(f)0,Φ0(f)=0}.

Considering the frequency components of ΦF(f) and Φ1(f) in Ω2, from Eq. (10) we have

ΦF(f)=C×Φ1(f),fΩ2.
(11)

Therefore C can be calculated as:

C=|ΦF(f)||Φ1(f)|,fΩ2.
(12)

The above relation is valid for all the frequency components in Ω2. In practice, we can employ a summation over all frequency components in Ω2to increase the accuracy of the estimate:

C=fΩ2|ΦF(f)|fΩ2|Φ1(f)|.
(13)

In terms of the upper bound of Ω2, as ΦF(f) and Φ1(f) are available, we simple choose fM as high as possible while observing ΦF(fM)0 and Φ1(f)0. In this paper we have taken Ω2 to be {15f0,fM}.

2.2 Verification of the proposed method by simulation

In order to verify the effectiveness of the formula presented in Eq. (12) and Eq. (13), let us first consider that the external target vibrates in two possible modes, simple harmonic vibration where L(t) is a sinusoidal function, and a more complicated case where L(t) is a triangular function. In this Section we present the results of computer simulation on these two cases.

In the first case, ϕ0(t) is given as follows:
ϕ0(t)=φ0+Δφsin(2πf0t),
(14)
where φ0 is the light phase at the equilibrium position of a vibrating target. Δφ is the maximum phase deviation caused by target vibration amplitude, and f 0 is the vibration frequency. In the numerical simulation, we setα=3, C=5,φ0=3.9×106(rad), Δφ=10π(rad), f0=200Hz, and take a sampling frequency fs=51200Hz. Using Eqs. (14), (9) and (7), we obtain ϕ0(t), ϕ1(t) andϕF(t)as shown on the left side of Fig. 3
Fig. 3 Waveforms and spectra of signals ϕ0(t), Cϕ1(t), and ϕF(t) for sinusoidal case with C=5 and α=3.
. Applying Discrete Fourier Transform (DFT) on these three signals, we obtain their spectra (in modulus) as shown in the right side of Fig. 3. We can see that both ϕ1(t) and ϕF(t)exhibit many high frequency components in contrast to signal ϕ0(t). This enables us to determine the two frequency ranges Ω1 and Ω2.

In order to demonstrate the process of calculating C, we plot details of spectra ofϕ0(t), ϕ1(t), andϕF(t)on Fig. 4
Fig. 4 Detailed spectra of signals ϕ0(t), ϕ1(t), and ϕF(t) for the sinusoidal case.
. It is seen that Φ0(f)is zero but Φ0(f)0 when f>2000Hz . Hence we can set the lower band of Ω2as 2000Hz, or 10f0. Based on Fig. 4 we can also set the upper bound of Ω2 to be 25000Hz or 125f0. By looking at the spectra at 5800Hz (which is on Ω2) on Fig. 4, we observeΦF(5800)=0.3701 and Φ1(5800)=0.0740, and hence by Eq. (12) we can determine the C value asC=0.3701/0.0740=5.0001, which is very close to the true value used in the simulations. By using Eq. (13) over Ω2:f[2000Hz,25000Hz], the C value is obtained as C=0.3701/0.0740=5.0001, which is also very close to the true value.

Let us consider the second case, where L(t) is a triangular function with fundamental frequency of 200 Hz. Figure 5
Fig. 5 Waveforms and spectra of signals ϕ0(t), Cϕ1(t), and ϕF(t) for the triangular case.
shows ϕ0(t), ϕ1(t), and ϕF(t)in time domain and frequency domains, respectively with C=5 and α=3. We noticed that ϕ0(t) only exhibit spectrum over the frequency range from 0Hz to 3000Hz, and that ϕF(t) and ϕ1(t) has a much broader range than ϕ0(t), that is, they still have frequency components in a wide range of f3000Hz.

We plot detailed spectra for signalsϕ0(t), ϕ1(t) and ϕF(t) on Fig. 6
Fig. 6 Detailed spectra of signals ϕ0(t), ϕ1(t), and ϕF(t) for triangular case.
, based on which we can haveΩ2:f[3000Hz,25000Hz] . For example, using Eq. (12) at the frequency of 3800Hz we are able to obtain 5.0060 for C. Applying Eq. (13) to all the frequency components over [3000Hz, 25000Hz], we get C=5.0001which is closer to the true value.

Extensive simulations have been done for verification of the proposed method. We generated SMI data using different sets of C and α shown in Table 2

Table 2. Estimated C Values Using Method in [1] and Proposed Method in this Paper

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. The estimated C values using the method in [1

1. Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16(4), 990–992 (2004). [CrossRef]

] and the proposed method in this paper are shown in Table 2. It is seen that the proposed method is more accurate than the one in [1

1. Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16(4), 990–992 (2004). [CrossRef]

].

Note that the results shown in Table 2 are obtained using an accurate α value. In practice, it is hard to obtain accurate α for an SL. In fact α is such a complex parameter in that for the same SL, different α values can be obtained by using different measurement technique [26

26. T. Fordell and A. M. Lindberg, “Noise correlation, regenerative amplification, and the linewidth enhancement factor of a vertical-cavity surface-emitting laser,” IEEE Photon. Technol. Lett. 20(9), 667–669 (2008). [CrossRef]

]. However, α is usually considered between 3 and 7 [2

2. S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron. 31(1), 113–119 (1995). [CrossRef]

]. Hence, we may simply choose an α value within the possible range to measure C using the proposed approach. This requires us to investigate the influence of using such an inaccurate α value on the performance of the proposed approach. To this end, we carried out a set of simulations with the results shown in Table 3

Table 3. Influence of αon the Estimation of C

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. In the simulations, the true value of α is assumed to be 5 and C takes different values from 0.5 to 6.5. For each C values, we choose different α values (that is, 3, 4, 5, 6 and 7) respectively to estimate C using Eq. (13). It is seen that the estimated C values are always close to the true values as shown by the small relative standard deviations in Table 3.

3. Verification using experimental data

For the purpose of comparison, we use the same experimental data presented in [8

8. Y. Yu, J. Xi, J. F. Chicharo, and T. M. Bosch, “Optical feedback self-mixing interferometry with a large feedback factor C: behavior studies,” IEEE J. Quantum Electron. 45(7), 840–848 (2009). [CrossRef]

]. We also preprocessed the signal in order to remove unwanted noise and slow time-varying fluctuation in its amplitude. The measurement procedure is summarized below:

  • Step 1. Preprocess an SMI signal by means of digital filtering for spike-like noise removal [10

    10. Y. Yu, J. Xi, and J. F. Chicharo, “Improving the Performance in an Optical feedback Self-mixing Interferometry System using Digital Signal Pre-processing,” in Proceedings of IEEE Conference on Intelligent Signal Processing (Institute of Electrical and Electronic Engineers, Alcala de Henares, 2007), pp. 1–6.

    ];
  • Step 2. Normalize the SMI signal to get g(t);
  • Step 3. Determine the vibration frequency f 0 by auto-correlation [7

    7. Y. Yu, J. Xi, J. F. Chicharo, and T. Bosch, “Toward automatic measurement of the linewidth-enhancement factor using optical feedback self-mixing interferometry with weak optical feedback,” IEEE J. Quantum Electron. 43(7), 527–534 (2007). [CrossRef]

    ];
  • Step 4. Carry out inverse cosine function operation on g(t) and then phase-unwrapping to obtain ϕF(t) [9

    9. Y. Fan, Y. Yu, J. Xi, J. Chicharo, and H. Ye, “A displacement reconstruction algorithm used for optical feedback self mixing interferometry system under different feedback levels,” Proc. SPIE 7855, 78550L, 78550L-7 (2010). [CrossRef]

    ];
  • Step 5. Calculate ϕ1(t) using Eq. (9) with assuming α=3 (or obtain a α value by the methods mentioned in [26

    26. T. Fordell and A. M. Lindberg, “Noise correlation, regenerative amplification, and the linewidth enhancement factor of a vertical-cavity surface-emitting laser,” IEEE Photon. Technol. Lett. 20(9), 667–669 (2008). [CrossRef]

    ]).
  • Step 6. Apply DFT on ϕF(t) and ϕ1(t) and obtain their spectra;
  • Step 7. Set Ω2[15f0,fM], where fMis chosen as half of the sampling frequency. Calculate C value using Eq. (13) over all the frequency components onΩ2.

The waveforms of the experimental data are shown in Fig. 7
Fig. 7 Experimental SMI signals with different C values [8].
, which are taken from [8

8. Y. Yu, J. Xi, J. F. Chicharo, and T. M. Bosch, “Optical feedback self-mixing interferometry with a large feedback factor C: behavior studies,” IEEE J. Quantum Electron. 45(7), 840–848 (2009). [CrossRef]

]. Each C value shown on Fig. 7 was obtained in [8

8. Y. Yu, J. Xi, J. F. Chicharo, and T. M. Bosch, “Optical feedback self-mixing interferometry with a large feedback factor C: behavior studies,” IEEE J. Quantum Electron. 45(7), 840–848 (2009). [CrossRef]

] by simulating waveforms with different C values to match the experimental ones (which is a tedious task). We applied the above procedure to the data and obtained C values for each of the waveforms depicted in Fig. 7, and the results are shown in Table 4

Table 4. Estimated C Values Using Experimental Signals

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. For illustration purposes, Fig. 8
Fig. 8 Measurement process for the experimental signal shown in Fig. 7(c) using the proposed method.
shows the details of applying the proposed method to the waveform in Fig. 7. (c), yielding C=0.2901/0.0444=6.534. For comparison, we also show the results of the C values provided by [8

8. Y. Yu, J. Xi, J. F. Chicharo, and T. M. Bosch, “Optical feedback self-mixing interferometry with a large feedback factor C: behavior studies,” IEEE J. Quantum Electron. 45(7), 840–848 (2009). [CrossRef]

] and the results obtained by the method in [1

1. Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16(4), 990–992 (2004). [CrossRef]

]. It is seen that the C values obtained by the proposed method are consistent to the values provided in [8

8. Y. Yu, J. Xi, J. F. Chicharo, and T. M. Bosch, “Optical feedback self-mixing interferometry with a large feedback factor C: behavior studies,” IEEE J. Quantum Electron. 45(7), 840–848 (2009). [CrossRef]

]. However, for most cases, the method in [1

1. Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16(4), 990–992 (2004). [CrossRef]

] is not able to determine the value of C.

4. Conclusion

The paper describes an SMI-based method for measuring feedback factor C in a semiconductor laser with external optical feedback. In contrast to existing methods, which are usually based on time-domain analysis of SMI signals, the proposed one is of frequency domain in nature, which is based on analysis of the magnitude spectra of two phase signals derived from an SMI signal. The proposed approach is advantageous in that it can be used for any optical feedback level, and hence lifts the limitation in existing SMI based methods for measuring C. Effectiveness of the proposed method has been confirmed by both simulations and experimental verifications.

References and links

1.

Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16(4), 990–992 (2004). [CrossRef]

2.

S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron. 31(1), 113–119 (1995). [CrossRef]

3.

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]

4.

N. Servagent, F. Gouaux, and T. Bosch, “Measurements of displacement using the self-mixing interference in a laser diode,” J. Opt. 29(3), 168–173 (1998). [CrossRef]

5.

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]

6.

J. Xi, Y. Yu, J. F. Chicharo, and T. Bosch, “Estimating the parameters of semiconductor lasers based on weak optical feedback self-mixing interferometry,” IEEE J. Quantum Electron. 41(8), 1058–1064 (2005). [CrossRef]

7.

Y. Yu, J. Xi, J. F. Chicharo, and T. Bosch, “Toward automatic measurement of the linewidth-enhancement factor using optical feedback self-mixing interferometry with weak optical feedback,” IEEE J. Quantum Electron. 43(7), 527–534 (2007). [CrossRef]

8.

Y. Yu, J. Xi, J. F. Chicharo, and T. M. Bosch, “Optical feedback self-mixing interferometry with a large feedback factor C: behavior studies,” IEEE J. Quantum Electron. 45(7), 840–848 (2009). [CrossRef]

9.

Y. Fan, Y. Yu, J. Xi, J. Chicharo, and H. Ye, “A displacement reconstruction algorithm used for optical feedback self mixing interferometry system under different feedback levels,” Proc. SPIE 7855, 78550L, 78550L-7 (2010). [CrossRef]

10.

Y. Yu, J. Xi, and J. F. Chicharo, “Improving the Performance in an Optical feedback Self-mixing Interferometry System using Digital Signal Pre-processing,” in Proceedings of IEEE Conference on Intelligent Signal Processing (Institute of Electrical and Electronic Engineers, Alcala de Henares, 2007), pp. 1–6.

11.

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

12.

G. Acket, D. Lenstra, A. Den Boef, and B. Verbeek, “The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers,” IEEE J. Quantum Electron. 20(10), 1163–1169 (1984). [CrossRef]

13.

K. Petermann, “External optical feedback phenomena in semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 1(2), 480–489 (1995). [CrossRef]

14.

M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers–An overview,” IEEE J. Quantum Electron. 23(1), 9–29 (1987). [CrossRef]

15.

D. Kuksenkov, S. Feld, C. Wilmsen, H. Temkin, S. Swirhun, and R. Leibenguth, “Linewidth and alpha-factor in AlGaAs/GaAs vertical cavity surface emitting lasers,” Appl. Phys. Lett. 66(3), 277–279 (1995). [CrossRef]

16.

Y. Arakawa and A. Yariv, “Fermi energy dependence of linewidth enhancement factor of GaAlAs buried heterostructure lasers,” Appl. Phys. Lett. 47(9), 905–907 (1985). [CrossRef]

17.

L. Hua, “RF-modulation measurement of linewidth enhancement factor and nonlinear gain of vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 8(12), 1594–1596 (1996). [CrossRef]

18.

H. Halbritter, F. Riemenschneider, J. Jacquet, J. G. Provost, C. Symonds, I. Sagnes, and P. Meissner, “Chirp and linewidth enhancement factor of tunable, optically-pumped long wavelength VCSEL,” Electron. Lett. 40(4), 242–244 (2004). [CrossRef]

19.

H. Nakajima and J. C. Bouley, “Observation of power dependent linewidth enhancement factor in 1.55 mu m strained quantum well lasers,” Electron. Lett. 27(20), 1840–1841 (1991). [CrossRef]

20.

G. Liu, X. Jin, and S. L. Chuang, “Measurement of linewidth enhancement factor of semiconductor lasers using an injection-locking technique,” IEEE Photon. Technol. Lett. 13(5), 430–432 (2001). [CrossRef]

21.

W.-H. Seo and J. F. Donegan, “Linewidth enhancement factor of lattice-matched InGaNAs/GaAs quantum wells,” Appl. Phys. Lett. 82(4), 505–507 (2003). [CrossRef]

22.

C. H. Shin, M. Teshima, and M. Ohtsu, “Novel measurement method of linewidth enhancement factor in semiconductor lasers by optical self-locking,” Electron. Lett. 25(1), 27–28 (1989). [CrossRef]

23.

Y. S. Shin, T. H. Yoon, J. R. Park, and C. H. Nam, “Simple methods for measuring the linewidth enhancement factor in external cavity laser diodes,” Opt. Commun. 173(1–6), 303–309 (2000). [CrossRef]

24.

D. Fye, “Relationship between carrier-induced index change and feedback noise in diode lasers,” IEEE J. Quantum Electron. 18(10), 1675–1678 (1982). [CrossRef]

25.

T. Fordell and A. M. Lindberg, “Experiments on the linewidth-enhancement factor of a vertical-cavity surface-emitting laser,” IEEE J. Quantum Electron. 43(1), 6–15 (2007). [CrossRef]

26.

T. Fordell and A. M. Lindberg, “Noise correlation, regenerative amplification, and the linewidth enhancement factor of a vertical-cavity surface-emitting laser,” IEEE Photon. Technol. Lett. 20(9), 667–669 (2008). [CrossRef]

27.

N. Schunk and K. Petermann, “Numerical analysis of the feedback regimes for a single-mode semiconductor laser with external feedback,” IEEE J. Quantum Electron. 24(7), 1242–1247 (1988). [CrossRef]

28.

H. Olesen, J. Osmundsen, and B. Tromborg, “Nonlinear dynamics and spectral behavior for an external cavity laser,” IEEE J. Quantum Electron. 22(6), 762–773 (1986). [CrossRef]

29.

V. Annovazzi-Lodi, S. Merlo, M. Norgia, and A. Scire, “Characterization of a chaotic telecommunication laser for different fiber cavity lengths,” IEEE J. Quantum Electron. 38(9), 1171–1177 (2002). [CrossRef]

OCIS Codes
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(140.5960) Lasers and laser optics : Semiconductor lasers
(280.3420) Remote sensing and sensors : Laser sensors

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: March 18, 2011
Revised Manuscript: April 24, 2011
Manuscript Accepted: April 26, 2011
Published: May 2, 2011

Citation
Yanguang Yu, Jiangtao Xi, and Joe F. Chicharo, "Measuring the feedback parameter of a semiconductor laser with external optical feedback," Opt. Express 19, 9582-9593 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-10-9582


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References

  1. Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16(4), 990–992 (2004). [CrossRef]
  2. S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron. 31(1), 113–119 (1995). [CrossRef]
  3. 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]
  4. N. Servagent, F. Gouaux, and T. Bosch, “Measurements of displacement using the self-mixing interference in a laser diode,” J. Opt. 29(3), 168–173 (1998). [CrossRef]
  5. 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]
  6. J. Xi, Y. Yu, J. F. Chicharo, and T. Bosch, “Estimating the parameters of semiconductor lasers based on weak optical feedback self-mixing interferometry,” IEEE J. Quantum Electron. 41(8), 1058–1064 (2005). [CrossRef]
  7. Y. Yu, J. Xi, J. F. Chicharo, and T. Bosch, “Toward automatic measurement of the linewidth-enhancement factor using optical feedback self-mixing interferometry with weak optical feedback,” IEEE J. Quantum Electron. 43(7), 527–534 (2007). [CrossRef]
  8. Y. Yu, J. Xi, J. F. Chicharo, and T. M. Bosch, “Optical feedback self-mixing interferometry with a large feedback factor C: behavior studies,” IEEE J. Quantum Electron. 45(7), 840–848 (2009). [CrossRef]
  9. Y. Fan, Y. Yu, J. Xi, J. Chicharo, and H. Ye, “A displacement reconstruction algorithm used for optical feedback self mixing interferometry system under different feedback levels,” Proc. SPIE 7855, 78550L, 78550L-7 (2010). [CrossRef]
  10. Y. Yu, J. Xi, and J. F. Chicharo, “Improving the Performance in an Optical feedback Self-mixing Interferometry System using Digital Signal Pre-processing,” in Proceedings of IEEE Conference on Intelligent Signal Processing (Institute of Electrical and Electronic Engineers, Alcala de Henares, 2007), pp. 1–6.
  11. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). [CrossRef]
  12. G. Acket, D. Lenstra, A. Den Boef, and B. Verbeek, “The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers,” IEEE J. Quantum Electron. 20(10), 1163–1169 (1984). [CrossRef]
  13. K. Petermann, “External optical feedback phenomena in semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 1(2), 480–489 (1995). [CrossRef]
  14. M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers–An overview,” IEEE J. Quantum Electron. 23(1), 9–29 (1987). [CrossRef]
  15. D. Kuksenkov, S. Feld, C. Wilmsen, H. Temkin, S. Swirhun, and R. Leibenguth, “Linewidth and alpha-factor in AlGaAs/GaAs vertical cavity surface emitting lasers,” Appl. Phys. Lett. 66(3), 277–279 (1995). [CrossRef]
  16. Y. Arakawa and A. Yariv, “Fermi energy dependence of linewidth enhancement factor of GaAlAs buried heterostructure lasers,” Appl. Phys. Lett. 47(9), 905–907 (1985). [CrossRef]
  17. L. Hua, “RF-modulation measurement of linewidth enhancement factor and nonlinear gain of vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 8(12), 1594–1596 (1996). [CrossRef]
  18. H. Halbritter, F. Riemenschneider, J. Jacquet, J. G. Provost, C. Symonds, I. Sagnes, and P. Meissner, “Chirp and linewidth enhancement factor of tunable, optically-pumped long wavelength VCSEL,” Electron. Lett. 40(4), 242–244 (2004). [CrossRef]
  19. H. Nakajima and J. C. Bouley, “Observation of power dependent linewidth enhancement factor in 1.55 mu m strained quantum well lasers,” Electron. Lett. 27(20), 1840–1841 (1991). [CrossRef]
  20. G. Liu, X. Jin, and S. L. Chuang, “Measurement of linewidth enhancement factor of semiconductor lasers using an injection-locking technique,” IEEE Photon. Technol. Lett. 13(5), 430–432 (2001). [CrossRef]
  21. W.-H. Seo and J. F. Donegan, “Linewidth enhancement factor of lattice-matched InGaNAs/GaAs quantum wells,” Appl. Phys. Lett. 82(4), 505–507 (2003). [CrossRef]
  22. C. H. Shin, M. Teshima, and M. Ohtsu, “Novel measurement method of linewidth enhancement factor in semiconductor lasers by optical self-locking,” Electron. Lett. 25(1), 27–28 (1989). [CrossRef]
  23. Y. S. Shin, T. H. Yoon, J. R. Park, and C. H. Nam, “Simple methods for measuring the linewidth enhancement factor in external cavity laser diodes,” Opt. Commun. 173(1–6), 303–309 (2000). [CrossRef]
  24. D. Fye, “Relationship between carrier-induced index change and feedback noise in diode lasers,” IEEE J. Quantum Electron. 18(10), 1675–1678 (1982). [CrossRef]
  25. T. Fordell and A. M. Lindberg, “Experiments on the linewidth-enhancement factor of a vertical-cavity surface-emitting laser,” IEEE J. Quantum Electron. 43(1), 6–15 (2007). [CrossRef]
  26. T. Fordell and A. M. Lindberg, “Noise correlation, regenerative amplification, and the linewidth enhancement factor of a vertical-cavity surface-emitting laser,” IEEE Photon. Technol. Lett. 20(9), 667–669 (2008). [CrossRef]
  27. N. Schunk and K. Petermann, “Numerical analysis of the feedback regimes for a single-mode semiconductor laser with external feedback,” IEEE J. Quantum Electron. 24(7), 1242–1247 (1988). [CrossRef]
  28. H. Olesen, J. Osmundsen, and B. Tromborg, “Nonlinear dynamics and spectral behavior for an external cavity laser,” IEEE J. Quantum Electron. 22(6), 762–773 (1986). [CrossRef]
  29. V. Annovazzi-Lodi, S. Merlo, M. Norgia, and A. Scire, “Characterization of a chaotic telecommunication laser for different fiber cavity lengths,” IEEE J. Quantum Electron. 38(9), 1171–1177 (2002). [CrossRef]

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