Measuring the feedback parameter of a semiconductor laser with external optical feedback

doi:10.1364/OE.19.009582

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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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▸ Article Outline
– Abstract
1. Introduction
2. The proposed method
3. Verification using experimental data
4. Conclusion
– References and links

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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.

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 , 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

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

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.

Fig. 1 Schematic SMI.

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

–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]

ϕ_{F} = ϕ_{0} - C \cdot s i n [ϕ_{F} + \arctan (α)],

(1)

g (ϕ_{0}) = \cos (ϕ_{F}),

(2)

P (ϕ_{0}) = P_{0} [1 + m \times g (ϕ_{0})] .

(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)

$ϕ_{0}$	External light phase of an SL without optical feedback, $ϕ_{0} = 2 π υ_{0} τ$ .
C	Optical feedback parameter.
α	Linewidth enhancement factor of an SL.
$ϕ_{F}$	External light phase of an SL with optical feedback, $ϕ_{F} = 2 π ν_{F} τ$ .
τ	External roundtrip delay. It is determined by the external cavity length L and light speed light c as $τ = 2 L / c$ .
$υ_{0}$	Emitting laser frequency of an SL without optical feedback.
$υ_{F}$	Emitting laser frequency of an SL under optical feedback.
$g (ϕ_{0})$	Interferometric function.
m	Modulation index of the modulated SL power.
$P_{0}$	Emitting laser power of an SL without optical feedback.
$P (ϕ_{0})$	Modulated SL power due to optical feedback, it is called an SMI signal.

From Eqs. (1)–(3), we see that

ϕ_{0}

affects the emitted power

P (ϕ_{0})

, which can be easily measured with a photo detector placed on the rear mirror of the SL. As

g (ϕ_{0}) = \frac{P (ϕ_{0}) - P_{0}}{m P_{0}}

, we can obtain

g (ϕ_{0})

through data processing on

P (ϕ_{0})

. Without loss of generality, in this paper we use

g (ϕ_{0})

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 phase

ϕ_{0} = 4 π υ_{0} L / c

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

], 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

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

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

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]

], FM/AM methods [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

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

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

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

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]

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

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]

]. These different approaches were compared systematically in [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]

The feedback parameter C is also an important parameter which tells the operational modes of an SL application system. An SL is considered to operate in weak optical feedback regime when

0 < C < 1

; it runs moderate feedback regime for

1 < C < 4.6

and in strong feedback regime when

C > 4.6

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]

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]

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]

]. Nonlinear dynamics, spectral behavior and chaos in an SL are also closely related to the C value [12

,13

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

–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]

]. Therefore, the measurement of C is very important for studying the behaviors of an SL and its applications.

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

]. 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

] 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

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

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

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

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]

,12

]. An example of the hysteresis is plot in Fig. 2 using Eq. (1) and Eq. (2), where

C = 2

and

α = 5

. When

ϕ_{0}

increases,

g (ϕ_{0})

follows the path A₁-B-B₁, which is different from the path B₁-A-A₁ for decreasing

ϕ_{0}

. The area A₁-B-B₁-A is called hysteresis area with the width represented as

ϕ_{0, A B}

ϕ_{0, A B}

has been found in [12

] as:

Fig. 2 Hysteresis phenomenon of SMI signal and the measurement principle on αin [20

] with

C = 2

and

α = 5

ϕ_{0, A B} = 2 {\sqrt{C^{2} - 1} + \arccos (\frac{- 1}{C}) - π} .

(4)

ϕ_{0, A B}

can be measured by

ϕ_{0, A B} = ϕ_{0, C B} - ϕ_{0, A D}

ϕ_{0, C B}

ϕ_{0, A D}

are read from the SMI waveform [1

]. However, when C increases to a certain value, the hysteresis area will cover

ϕ_{0, C B}

and

ϕ_{0, A D}

, and thus they will disappear from the SMI waveform. In this case, the approach in [1

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

] 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

] 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

] 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

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 = η \frac{τ}{τ_{S L}} (1 - R_{2}) \sqrt{\frac{R_{3}}{R_{2}}} \sqrt{1 + α^{2}},

(5)

where R₂ , R₃ , η and

τ_{S L}

are, 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 signal

g (ϕ_{0})

. With existing time-domain approaches [1

], 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

ϕ_{0}

also periodically changing with time. Equation (2) and Eq. (1) can be rewritten as:

g (t) = \cos (ϕ_{F} (t)),

(6)

ϕ_{F} (t) = ϕ_{0} (t) - C s i n {ϕ_{F} (t) + \arctan (α)} .

(7)

Taking Fourier transform, denoted by

F {•}

, on both sides of Eq. (7), we have

Φ_{F} (f) = Φ_{0} (f) - C \cdot F {\sin [ϕ_{F} (t) + \arctan (α)]},

(8)

where

Φ_{F} (f)

and

Φ_{0} (f)

are the Fourier transforms of

ϕ_{F} (t)

and

ϕ_{0} (t)

respectively. For ease of manipulation let us introduce a function

ϕ_{1} (t) = \sin {ϕ_{F} (t) + \arctan (α)},

(9)

so that Eq. (8) becomes

Φ_{F} (f) = Φ_{0} (f) - C \cdot F {ϕ_{1} (t)} = Φ_{0} (f) - C \cdot Φ_{1} (f),

(10)

where

Φ_{1} (f)

is the Fourier transforms of

ϕ_{1} (t)

From the SMI setup, we are able to acquire the SMI signal

g (t)

. Based on our previous work [9

ϕ_{F} (t)

can be obtained from

g (t)

by applying inverse cosine operation and phase unwrapping. Thereafter, we calculate the spectrum of

ϕ_{F} (t)

, that is

Φ_{F} (f)

. Meanwhile, when

ϕ_{F} (t)

is available, we are able to work out

ϕ_{1} (t)

using Eq. (9) and compute its spectrum

Φ_{1} (f)

ϕ_{0} (t) = 4 π υ_{0} L (t) / c

ϕ_{0} (t)

also describes the moving of the target. If the external target moves in a manner close to simple harmonic vibration,

ϕ_{0} (t)

will be close to a sinusoidal, in which case it can be considered as of narrow band in frequency domain.

Given

ϕ_{0} (t)

narrow band in nature, from Eqs. (7) and (8),

ϕ_{F} (t)

should exhibit a broader spectrum than

ϕ_{0} (t)

because of the nonlinear mapping from

ϕ_{0} (t)

ϕ_{F} (t)

. In other words,

Φ_{F} (f)

and

Φ_{1} (f)

should spread over a wider frequency range than

Φ_{0} (f)

does. Now, we can divide the frequency range of non-vanishing spectrum of

Φ_{F} (f)

and

Φ_{1} (f)

into two domains, denoted by

Ω_{1}

and

Ω_{2}

respectively, in which the spectrum of

Φ_{0} (f)

is non-vanishing or vanishing, i.e.:

Ω_{1} : f \underset{}{\subset} {Φ_{F} (f) \neq 0, Φ_{1} (f) \neq 0, Φ_{0} (f) \neq 0}, and

Ω_{2} : f \underset{}{\subset} {Φ_{F} (f) \neq 0, Φ_{1} (f) \neq 0, Φ_{0} (f) = 0} .

Considering the frequency components of

Φ_{F} (f)

and

Φ_{1} (f)

Ω_{2}

, from Eq. (10) we have

Φ_{F} (f) = - C \times Φ_{1} (f), \begin{matrix} f \subset Ω_{2} \end{matrix} .

(11)

Therefore C can be calculated as:

C = \frac{| Φ_{F} (f) |}{| Φ_{1} (f) |}, \begin{matrix} f \subset Ω_{2} \end{matrix} .

(12)

The above relation is valid for all the frequency components in

Ω_{2}

. In practice, we can employ a summation over all frequency components in

Ω_{2}

to increase the accuracy of the estimate:

C = \frac{\sum_{f \subset Ω_{2}} | Φ_{F} (f) |}{\sum_{f \subset Ω_{2}} | Φ_{1} (f) |} .

(13)

To apply the method, we shall identify

Ω_{1}

and

Ω_{2}

. In the cases of harmonic vibration,

ϕ_{0} (t)

is a periodic function with a fundamental (vibration) frequency

f_{0}

. Then

Φ_{0} (f)

should exhibit a large fundamental component at

f_{0}

, and some harmonic components. The highest harmonic component will in general depend on the actual waveform, and yet in practice we consider

15 f_{0}

as the upper frequency necessary to cover well the spectrum details. This choice will be confirmed by simulations discussed in the next Section.

In terms of the upper bound of

Ω_{2}

, as

Φ_{F} (f)

and

Φ_{1} (f)

are available, we simple choose

f_{M}

as high as possible while observing

Φ_{F} (f_{M}) \neq 0

and

Φ_{1} (f) \neq 0

. In this paper we have taken

Ω_{2}

to be

{15 f_{0}, f_{M}}

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,

ϕ_{0} (t)

is given as follows:

ϕ_{0} (t) = φ_{0} + Δ φ \sin (2 π f_{0} t),

(14)

where

φ_{0}

is the light phase at the equilibrium position of a vibrating target.

Δ φ

is the maximum phase deviation caused by target vibration amplitude, and f ₀ is the vibration frequency. In the numerical simulation, we set

α = 3

C = 5

φ_{0} = 3.9 \times 10^{6} (r a d)

Δ φ = 10 π (r a d)

f_{0} = 200 H z

, and take a sampling frequency

f_{s} = 51200 H z

. Using Eqs. (14), (9) and (7), we obtain

ϕ_{0} (t)

ϕ_{1} (t)

and

ϕ_{F} (t)

as shown on the left side of Fig. 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

ϕ_{1} (t)

and

ϕ_{F} (t)

exhibit many high frequency components in contrast to signal

ϕ_{0} (t)

. This enables us to determine the two frequency ranges

Ω_{1}

and

Ω_{2}

Fig. 3 Waveforms and spectra of signals

ϕ_{0} (t)

C ϕ_{1} (t)

, and

ϕ_{F} (t)

for sinusoidal case with C=5 and α=3.

In order to demonstrate the process of calculating C, we plot details of spectra of

ϕ_{0} (t)

ϕ_{1} (t)

, and

ϕ_{F} (t)

on Fig. 4 . It is seen that

Φ_{0} (f)

is zero but

Φ_{0} (f) \neq 0

when

f > 2000

Hz . Hence we can set the lower band of

Ω_{2}

as 2000Hz, or

10 f_{0}

. Based on Fig. 4 we can also set the upper bound of

Ω_{2}

to be 25000Hz or

125 f_{0}

. By looking at the spectra at 5800Hz (which is on

Ω_{2}

) on Fig. 4, we observe

Φ_{F} (5800) = 0.3701

and

Φ_{1} (5800) = 0.0740

, and hence by Eq. (12) we can determine the C value as

C = 0.3701 / 0.0740 = 5.0001

, which is very close to the true value used in the simulations. By using Eq. (13) over

Ω_{2} : f \subset [2000 H z, 25000 H z]

, the C value is obtained as

C = 0.3701 / 0.0740 = 5.0001

, which is also very close to the true value.

Fig. 4 Detailed spectra of signals

ϕ_{0} (t)

ϕ_{1} (t)

, and

ϕ_{F} (t)

for the sinusoidal case.

Let us consider the second case, where L(t) is a triangular function with fundamental frequency of 200 Hz. Figure 5 shows

ϕ_{0} (t)

ϕ_{1} (t)

, and

ϕ_{F} (t)

in time domain and frequency domains, respectively with

C = 5

and

α = 3

. We noticed that

ϕ_{0} (t)

only exhibit spectrum over the frequency range from 0Hz to 3000Hz, and that

ϕ_{F} (t)

and

ϕ_{1} (t)

has a much broader range than

ϕ_{0} (t)

, that is, they still have frequency components in a wide range of

f \geq 3000

Hz.

Fig. 5 Waveforms and spectra of signals

ϕ_{0} (t)

C ϕ_{1} (t)

, and

ϕ_{F} (t)

for the triangular case.

We plot detailed spectra for signals

ϕ_{0} (t)

ϕ_{1} (t)

and

ϕ_{F} (t)

on Fig. 6 , based on which we can have

Ω_{2} : f \subset [3000 H z, 25000 H z]

. 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 = 5.0001

which is closer to the true value.

Fig. 6 Detailed spectra of signals

ϕ_{0} (t)

ϕ_{1} (t)

, and

ϕ_{F} (t)

for triangular case.

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 . The estimated C values using the method in [1

] 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

Table 2 Estimated C Values Using Method in [1

] and Proposed Method in this Paper

${\hat{C}}_{\sin}$ : estimated C with a sinusoidal vibrating target; ${\hat{C}}_{t r i}$ : estimated C with a triangular vibrating target; N/A: not applicable
α		$α = 1$					$α = 3$
C	${\hat{C}}_{\sin}$			${\hat{C}}_{t r i}$		${\hat{C}}_{\sin}$			${\hat{C}}_{t r i}$
C	Proposed Method		Method in [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] ]	Proposed Method	Method in [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] ]	Proposed Method		Method in [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] ]	Proposed Method	Method in [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] ]
0.5	0.4999		N/A	0.5000	N/A	0.5000		N/A	0.5000	N/A
1	0.9999		N/A	1.0001	N/A	1.0000		N/A	1.0000	N/A
1.5	1.5000		1.5769	1.5000	1.5359	1.5000		1.5290	1.5000	1.5049
2	2.0000		2.1080	2.0000	1.8396	2.0000		1.9483	2.0000	1.9965
2.5	2.5001		N/A	2.5000	N/A	2.5000		2.4225	2.5000	2.4928
3	3.0000		N/A	3.0000	N/A	3.0000		3.0170	3.0001	3.0574
3.5	3.5001		N/A	3.5001	N/A	3.5000		N/A	3.5000	N/A
4	4.0001		N/A	4.0001	N/A	4.0000		N/A	4.0003	N/A
4.5	4.5000		N/A	4.5001	N/A	4.5001		N/A	4.5001	N/A
5	4.9999		N/A	5.0002	N/A	5.0001		N/A	5.0001	N/A
10	10.0002		N/A	9.9995	N/A	9.9998		N/A	9.9998	N/A
15	15.0003		N/A	14.9996	N/A	14.9998		N/A	14.9997	N/A

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

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

Table 3 Influence of αon the Estimation of C

True C	Estimated ${\hat{C}}_{}$ using different α					Relative standard deviation
True C	α = 3.0000	α = 4.0000	α = 5.0000	α = 6.0000	α = 7.0000	Relative standard deviation
0.5000	0.4566	0.4816	0.5000	0.5073	0.5013	4.27%
1.5000	1.4261	1.4683	1.5000	1.4963	1.4805	2.47%
2.5000	2.3434	2.4336	2.5000	2.4724	2.4369	3.28%
3.5000	3.2375	3.3890	3.5000	3.4446	3.3895	3.97%
4.5000	4.0614	4.3139	4.4998	4.3900	4.3007	5.25%
5.5000	5.2851	5.4365	5.5001	5.5140	5.5125	1.83%
6.5000	6.2326	6.4433	6.5000	6.5122	6.4937	1.83%

3. Verification using experimental data

In order to test the measuring approach described in Section 2, we implemented an experimental SMI system, which is the same as the one described in our previous work [8

]. The optical set-up consists of a laser diode (LD), a focus lens and a loudspeaker as the external target. The electrical set-up consists of LD driving devices, an optical power detection circuit and a digital oscilloscope. In the experiment, the laser diode (LD) is a HL7851G laser diode emitting at

λ_{0} = 785

nm. The LD is biased with a DC current of 90 mA and operates in single longitudinal mode regime. The sinusoidal vibration is generated by a loudspeaker driven by a signal generator.

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

]. 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
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
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 $ϕ_{F} (t)$ [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 $ϕ_{1} (t)$ using Eq. (9) with assuming $α = 3$ (or obtain a α value by the methods mentioned in [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 $ϕ_{F} (t)$ and $ϕ_{1} (t)$ and obtain their spectra;
⁃ Step 7. Set $Ω_{2} \subset [15 f_{0}, f_{M}]$ , where $f_{M}$ is chosen as half of the sampling frequency. Calculate C value using Eq. (13) over all the frequency components on $Ω_{2}$ .

The waveforms of the experimental data are shown in Fig. 7 , which are taken from [8

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

] 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 . For illustration purposes, Fig. 8 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

] and the results obtained by the method in [1

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

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

] is not able to determine the value of C.

Fig. 7 Experimental SMI signals with different C values [8

Table 4 Estimated C Values Using Experimental Signals

Experimental signals shown on Fig. 7	Proposed approach	Approach in [ 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] ]	C provided in [ 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] ]
signal (a)	1.58	1.18	1.35
signal (b)	3.12	3.05	3.30
signal (c)	6.53	NA	5.57
signal (d)	5.20	NA	6.00
signal (e)	6.12	NA	6.80
signal (f)	8.70	NA	7.40

Fig. 8 Measurement process for the experimental signal shown in Fig. 7(c) using the proposed method.

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

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]
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]
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]
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]
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]
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]
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]
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]
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]
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.
R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). [CrossRef]
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]
K. Petermann, “External optical feedback phenomena in semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 1(2), 480–489 (1995). [CrossRef]
M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers–An overview,” IEEE J. Quantum Electron. 23(1), 9–29 (1987). [CrossRef]
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]
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]
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]
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]
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]
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]
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]
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]
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]
D. Fye, “Relationship between carrier-induced index change and feedback noise in diode lasers,” IEEE J. Quantum Electron. 18(10), 1675–1678 (1982). [CrossRef]
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]
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]
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]
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]
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]

Acket, G.
Annovazzi-Lodi, V.
Arakawa, Y.
Bosch, T.
Bosch, T. M.
Bouley, J. C.
Buus, J.
Chicharo, J.
Chicharo, J. F.
Chuang, S. L.
Den Boef, A.
Donati, S.
Donegan, J. F.
Fan, Y.
Feld, S.
Fordell, T.
Fye, D.
Giuliani, G.
Gouaux, F.
Halbritter, H.
Hua, L.
Jacquet, J.
Jin, X.
Kobayashi, K.
Kuksenkov, D.
Lang, R.
Leibenguth, R.
Lenstra, D.
Lindberg, A. M.
Liu, G.
Meissner, P.
Merlo, S.
Nakajima, H.
Nam, C. H.
Norgia, M.
Ohtsu, M.
Olesen, H.
Osinski, M.
Osmundsen, J.
Park, J. R.
Petermann, K.
Plantier, G.
Provost, J. G.
Riemenschneider, F.
Sagnes, I.
Scalise, L.
Schunk, N.
Scire, A.
Seo, W.-H.
Servagent, N.
Shin, C. H.
Shin, Y. S.
Swirhun, S.
Symonds, C.
Temkin, H.
Teshima, M.
Tromborg, B.
Verbeek, B.
Wilmsen, C.
Xi, J.
Yariv, A.
Ye, H.
Yoon, T. H.
Yu, Y.

Appl. Phys. Lett.

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]
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]
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]

Electron. Lett.

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]
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]
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]

IEEE J. Quantum Electron.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). [CrossRef]
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]
M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers–An overview,” IEEE J. Quantum Electron. 23(1), 9–29 (1987). [CrossRef]
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]
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]
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]
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]
D. Fye, “Relationship between carrier-induced index change and feedback noise in diode lasers,” IEEE J. Quantum Electron. 18(10), 1675–1678 (1982). [CrossRef]
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]
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]
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]
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]

IEEE J. Sel. Top. Quantum Electron.

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

IEEE Photon. Technol. Lett.

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]
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]
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]
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]

IEEE Trans. Instrum. Meas.

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]

J. Opt.

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]

J. Opt. A, Pure Appl. Opt.

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]

Opt. Commun.

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]

Proc. SPIE

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]

Other

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.

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