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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 12, NO. 9, SEPTEMBER

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Self-Mixing Interference Inside a Single-Mode Diode Laser for Optical Sensing Applications
W. M. Wang, K. T. V. Grattan, A. W. Palmer, and W. J. 0. Boyle
Abstract-This paper presents a theoretical analysis and a comparison with experimentalal results on self-mixing interference inside a single-longitudinal-modediode laser. A theoretical model, based on the steady-state equations of the lasing condition in a Fabry-Perot type laser cavity, is described, and through it a satisfactory analysis of self-mixing interference for optical sensing applications is given. In this work, the self-mixing interference produced by an external optical feedback is found to be due to the variations in the threshold gain and in the spectral distribution of the laser output. The gain variation results in an optical intensity modulation, and the spectral variation determines both the modulation waveform and the coherence properties of the interference. The theoretical analysis of the self-mixing interference is seen to yield a simulation of the laser power modulation which is in good agreement with the experiment results reported.

I. INTRODUCTION

A. Background to “Self-Mixing” Interference
NTENSITY variations in the output of a gas laser, induced by an external optical feedback, were first reported by King and Steward [l]. It was found that the intensity modulation caused by a movable external mirror was similar to that produced by a conventional optical interferometer, such as ii “fringe” shift corresponding to an optical displacement of A/2, A being the oscillation wavelength of the laser, and the intensity modulation depth was also found to be comparable to that of a conventional interferometric system [21. These two features have therefore laid the foundation of the so-called “self-mixing interference” process, which has been applied recently to optical sensing applications [3]. In the practical use of this process, a portion of light emitted from a laser source is reflected by an external reflector, such as a mirror or a diffused target, back into the laser cavity, the reflected light then mixing “actively” with the light inside the cavity, causing a modulation of
Manuscript received February 1, 1993; revised January 11, 1994. This work was supported by the Science and Engineering Research Council (SERC). W. M. Wang was supported by the British Council by way of studentship. The authors are with the Measurement and Instrumentation Centre, Department of Electrical, Electronic and Information Engineering, City University, Northhampton Square, London EClV OHB, England. IEEE Log Number 9402791.

I

the laser output power. The resulting intensity variations can be detected easily by a photodetector, at either end of the laser or by the internal photodetector accommodated inside a diode laser package [3]. In this arrangement, changes in the optical path length of the external cavity, which can be produced by longitudinal motion of the reflector [l] or by a medium of variable refractive index within the cavity [4], can be monitored and then utilized for optical sensing, for example for displacement [5]-[71, or velocity measurement, and for ranging [8]-[141. However, some characteristics observed in the self-mixing interference have been found to be dramatically different from those of a conventional interferometer. These include: (i) the interference patterns can be observed even when the optical path difference (OPD) between the laser and the target lies well beyond the coherence length of the solitary laser used [3]; (ii) the intensity modulation waveform may be sawtooth-like as well as asymmetric, relative to the movement direction of the reflector [15]; and (iii) the waveforms have a sign inversion between the two emission directions of the laser [141. In order to interpret these phenomena, which are observed in self-mixing interference, a number of theoretical formulations have been developed. The early theories [16]-[19] were modelled on a laser cavity subject to a variable loss, and a device based on the self-mixing technique was used for refractive index monitoring and termed the “laser interferometer” by Ashby and Jephcott [20]. A simple feedback amplifier model was also described by Rudd [2] for analyzing the intensity modulation in a He-Ne laser which was used not only as a light source, but also as a mixer-oscillator for velocity measurement. Churnside [8], [9] studied laser Doppler velocimetry using the self-mixing effect, with a modulated CO, laser and termed the self-mixing effect as “backscatter modulation” with diffusely reflecting targets. Using semiconductor lasers for velocity measurement, Jentink et al. [ l l l explained the intensity modulation in terms of the conventional interference between light inside and light reentering the laser cavity. Their theoretical model was challenged by de Groot et al. [14], who developed a model based on the mode structure of a three-mirror Fabry-Perot cavity to explain signal generation in selfmixing velocimetry and ranging, where the intensity mod-

0733-8724/94$04.00 0 1994 IEEE

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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 12, NO. 9, SEPTEMBER 1994

ulation was concluded to be due to the change of the carrier density inside the laser cavity. A further theroetical analysis for a multimode self-mixing laser was carried out by de Groot [21], in which the intensity modulation was explained as a spectral mode modulation. More recently, Koelink et al. 1121 presented a short outline of their theoretical basis to explain the power fluctuations in their fiber optical self-mixing laser Doppler velocimeter, and a numerical analysis has been also developed [22]. Here the coherence has still been considered as conventional “two-beam’’ interference and the spectral variations caused by the self-mixing effect have been neglected. A simplified theoretical analysis for self-mixing interference has been presented by the authors [31, which considered that the power modulation by the self-mixing was due to the reflectivity variations of the laser facet. In this analysis spectral variations were not considered. B. Spectral Effects in “Self-Mixing”Interference It was well known that optical feedback has a profound effect on the optical spectrum and the threshold gain of a diode laser [23]-[28]. The operation of a diode laser with external optical feedback has been extensively studied in the context of the external cavity locking mechanisms [231, [28] and optical noise [291, [30]. With very weak feedback the spectral linewidth may be narrowed or broadened, depending on the phase of the feedback relative to the optical field within the active laser, and this has been used for laser linewidth reduction [31], [32] and frequency tuning [33]. When the distance of an external reflector is smaller than the coherence length of the solitary laser used, the compound cavity model has been used to interpret the observed phenomena, after the approach introduced by Lang and Kobayashi [23]. However, the effects on a diode laser, when the feedback is presented, may be surprising. On the one hand, the spectral linewidth of the laser may be broadened to many times that of the solitary laser linewidth, which was termed “coherence collapse” by Lenstra and co-workers [34], [35] and Miles et al. [36], in which the feedback field changed from the coherent to the incoherent. On the other hand, the spectral linewidth may be narrowed extremely by feedback. A reduced spectral linewidth of as low as 1 Hz has been reported by Mark et al. [371 with Rayleigh backscattering reflection from a long fiber, which represents a coherence length of about 3 X lo5 km. Nevertheless, with incoherent feedback, similar results to coherent feedback have been also observed, such as a linewidth reduction [38], and “coherence collapse” [39], and even with the reflector distance as large as 7 km, using optical fibers, the laser spectrum was affected [401. In self-mixing interference, it has been also observed that the intensity modulation, induced by the feedback field, is not completely determined by the coherence length of the solitary laser used [71. In this case, it is thus not unreasonable to assume that the coherence is enhanced by the feedback because of the change of the spectral distribution. The coherence properties of a diode laser have been studied recently by

Hamel et al. [41], with feedback from a mirror which was placed at a distance exceeding the coherence length of the solitary laser. It is evident from the above investigations that the output characteristics of a semiconductor laser can be altered by external optical feedback, no matter whether the feedback is fully coherent or incoherent with the cavity radiation of the solitary laser. It is well known that the conventional temporal coherence is a measure of the spectral spread of the light source used, which is based on a stabilized spectral output of the source, and the interference patterns could not be observed if the OPD of the interfering beams were beyond the coherence length, but in self-mixing interference the intensity modulation is a consequence of the variations at the threshold gain and the spectral distribution. As the spectral output is changing with the feedback field, no longer a constant, the self-mixing interference should therefore be considered not to be dependent on the coherence length of the solitary laser but on the actual lasing spectrum in the presence of external optical feedback.

C. Purpose of this Work o n Self-Mixing Inteijierence In the present work, a self-mixing theory, based on the stationary equation of a single-longitudinal-mode diode laser, is described and theoretical models are developed from the analysis of the laser oscillation conditions. The self-mixing interference is found to be the consequence of the optical spectrum and threshold modulation produced by external optical feedback. For weak feedback and a short external cavity, the lasing spectrum can easily show a single-mode operation, and the intensity modulation is similar to that of conventional two-beam interference. The characteristics of the intensity modulation, the frequency shifts, and the spectral linewidth variations can be expressed analytically, and therefore the visibility function, the sawtooth-like waveform, and sign inversion observed can be understood easily. For large feedback and/or a distant external reflector, the laser spectrum will eventually be characterized by multiple external cavity modes. In this case, the visibility function is separated by the external cavity modes, the self-mixing interference is limited by the spectral linewidth of each of the single modes, and the larger the feedback, the more narrow the linewidth, and therefore the longer the coherence length. In a special case given later, the feedback is considered to be coherent with the optical radiation inside the cavity, no matter how far away is the external reflector. Any further increase at the feedback level may lead the laser into the the “coherence collapse” region, in which no intensity modulation can be observed [391. In this work, a brief theory of self-mixing interference inside a single-mode diode laser is presented in Section 11, where the intensity modulation, the frequency shift, the linewidth variation, and the power modulation are presented analytically. Section I11 shows a comparison of the theoretical results obtained with experimental results reported and this is discussed in Section IV.

WANG et al.: SELF-MIXING INTERFERENCE INSIDE A SINGLE-MODE DIODE LASER

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11. THEORY OF SELF-MIXING INTERFERENCE
A. Characteristics of a Diode Laser with External Optical Feedback

A schematic arrangement for a single-mode diode laser, I in the presence of external optical feedback, is shown in O d d+ L Fig. 1, where r1 and r2 denote the amplitude reflection Fig. 1. Schematic arrangement of a diode laser with external feedback. coefficients of the laser facets F, and F,, respectively, r3 F,, F,-diode laser facets, F,-external reflector, n-refractive index of represents the amplitude reflection coefficient of the ex- the active layer of the diode laser, d-diode laser cavity length, Lternal reflector or a reflecting object, F3, d , and L are the external cavity length. laser cavity length and the external cavity length, respectively, and n is the effective refractive index of the laser and cavity material. (4) ~ O ( Y ) = 5 sin 4 e x t y In general, the external reflection coefficient r3 is much less than the reflection coefficient r2 of the laser facet, where go is the linear gain in the absence of the feedback, < Ir, I, therefore the multiple reflection effect [ denotes the coupling effect from the external reflection i.e., Ir,l < within the external cavity can be neglected [44]. An initial back into the laser cavity, with electric field E,(t) inside the laser cavity undergoes a r3 roundtrip within both the cavity of F1-F2 and the ex(5) 5 = (1 - R 2 ) - , tended cavity of F,-F3 and becomes EO), which may be r 2 given by and 4ext[= 47rv(L/c)l denotes the phase of external rend flection. Eqs. (3) and (4) are generally used to describe the basic characteristics of a diode laser in the presence of an nd L + r,(l - Z?:?)r3 exp external optical feedback. A detailed theoretical backC ground may be found in the work of Petermann 1441.

1

-+

+

where Eo(t)is the initial electric field, E ( t ) is the electric field undergoing a roundtrip within the compound cavity of Fl-F2-F3, v is the optical frequency, the term (1 - R , ) accounts for the li ht transmission through the laser facet F,, with R, = 1r21 , c is the speed of light in a vacuum, g A + ( V ) = 27r( Y - v , ) T d + is the linear gain per unit length due to the stimulated emission inside the laser cavity in the presence of feed. sin [ ~ T Y + T arctan ~ ( a ) ]= 0, ( 6 ) back, and y accounts for any optical loss per unit length where A + ( v ) represents the deviation of the oscillation within the cavity. For a stationary stable laser oscillation, the amount of phase +(v) from 2 q v , a is the linewidth enhancement light amplified by the stimulated emission becomes equal factor, v0 is the laser emission frequency without feedto the total losses in the lasing system, e.g., through the back, T ~ ( = 2 n o d / c ) and T ~ ( =2 L / c ) are the roundtrip sides of the cavity, by the mirror facets, and by absorption time delays inside the laser cavity and inside the external in the semiconductor material. This results in the follow- cavity, respectively, with no representing the effective refractive index of the laser material. An important paing condition: rameter C has been defined as

F

B. Frequency Ship and Spectral Linewidth under Weak Feedback It is known that the phase condition $ 4 ~= ) 2q7r in the presence of the feedback determines the lasing frequency of the laser system [44], and this leads to

c$G

4

exp -j47rvc

{

nd

+ (g - y)d

The above condition requires both that the roundtrip amplitude of the laser is equal to unity, and that its phase 4(v) = 2 q ~where , q is an integer. By solving this basic equation for laser oscillation, the excess required gain Ag and the additional phase # > J n ( y > for the laser system in the presence of the feedback may be expressed respectively as
Ag
= g - g,, = -

I

c = ‘“[d+z;i,
7d

(7)

=

1. (2)

representing the external feedback strength [42]-[44]. In the case of weak feedback (C < 11, the phase change A4 versus the emission frequency v is a monotonic function, which results in only one solution for A + = 0 and thus in a single emission frequency. Eq. (6) may then be solved using a first-order approximation and this gives the emission frequency v as
v=v,-

- COS 4,,,
d

5

(3)

C sin [ + arctan ( a 11 2mL{1 + C cos [ + arctan ( a ) ] } ’ (8)

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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 12, NO. 9, SEPTEMBER 1994
duo=130MHz, ha=EOOnm. 1,-2crn
1.5
I

where (8) expresses the emission frequency shifts in the presence of the weak feedback and shows that the frequency shift is a periodic function with respect to the external reflection phase whose physical meaning is discussed in Section 111. For a small value of C , the variation of the frequency is of a sinusoidal form, but for larger values of C , the shifts become sawtooth-like. The are defrequency shifts versus the external phase 4ext picted as in Fig. 2, where the central line represents the lasing frequency of the solitary laser (C = 0) and the other related parameters used in the calculation are given below: (i) the central wavelength A, of the solitary diode laser is 800 nm; (ii) the corresponding spectral linewidth Sv, is 60 MHz; (iii) the external cavity length L is 2 cm; (iv) the linewidth enhancement factor a is 5. For C > 1, the value of A 4 is no longer a monotonic function but has multiple intersects with the emission frequency axis v, resulting in multiple solutions for (6). Eventually several external cavity modes may start to oscillate, and so the original single-mode laser becomes a multimode device [441. The spectral linewidth of a diode laser is also important for laser oscillation. Above threshold, both the laser linewidth and the coherence time are determined by the phase fluctuations of the laser electric field which can be described as a Brownian motion or phase diffusion [45], [46]. For simplicity, the power spectrum of a single-mode diode laser is assumed to be Lorentzian in shape [471-[491, and the spectral linewidth Sv may be thus considered to be inversely proportional to the square of the effective roundtrip delay, Teff, which is given in the presence of the feedback by
Teff

-1.5
0

n

2n

3lT

4n

External Phase (Radians)
Fig. 2. Laser frequency shifts dependence on external phase change, correspondingto different levels of the feedback strength.

rate equation [23]: d J - N = - - K~ N - G ( N ) Z , (11) dt e where J represents the injection current, e is the electric charge, K~ is the inverse spontaneous lifetime of the excited carriers, and I represents the photon density in the active layer that is normalized to make the optical intensity Z = [El2,E being the amplitude of the electric field inside the cavity. Assuming that the laser in the absence and in the presence of feedback is in a steady state and characterized by the optical intensities Io and I and carrier densities No and N , respectively, this implies that (11) meets the conditions J = K ~ N+ , G(No)Zo, (12)

1 d = --A 4 27~ dv

= T ~1 {

+ C cos [ ~ ~ T+uarctan T ~ ( a11).
(9)

J K ~+ NG ( N ) Z . (13) e By linearization of G ( N ) , centered around N o ,the following arises:
- =

In the case of weak feedback (C < 11, the lasing spectral linewidth 6v is obtained simply as

where Sv, is the linewidth of the solitary diode laser in the absence of the feedback. Obviously, the spectral linewidth Sv is determined mainly by the feedback coefficient C and the phase of external reflection field c # ~ where the emission frequency v is given by (8). C. Intensity Modulation with Weak External Feedback The optical gain G due to the stimulated emission inside the laser cavity is determined by the injection carrier density N , which can be described by the following

G ( N ) = G ( N , ) AG = G o + K ~ A N , (14) where Go = G(N,,) and AG = K , A N with K~ = d G / d N . Substituting (12) into (13) and assuming that AG < < Go, this results in a first-order expression for the intensity I which is Z = Z,(1 - K ~ A N ) , (15) where 1 G O which is a proportionality coefficient related to the gain ~ ~ , G o and the intensity I, in absence of feedback. It is also known that the amount of linear gain g provided by the active region of the laser diode is determined by the carrier density N in the active region [21], and therefore AN = q A g , (16)
K1 =

+

-(

K2

+: ) ,

~

~

WANG ef al.: SELF-MIXING INTERFERENCE INSIDE A SINGLE-MODE DIODE LASER
dvo=60MHz, X.=BOOnm.
.......... C=O.71
...._ C=O.50 . c=o.22

1581

L,,-Zcm

-.. .. . .__.. .

__ c=o.oo
. . ........ ....

--- - C=O.O7

0

n

27l

371

471

External Phase ( R a d i a n s )
Fig. 3 . Optical intensity versus external phase change for various values o f c'.

where K~ is a constant. Substituting (3) and (16) into (1.51, the following expression is obtained:
I
=

I,, 1

[

+ m cos [hV? ,

I]

(17)

where
(18)

which is termed here the modulation coefficient of the self-mixing interference. Eq. (17) is the first expression that describes theoretically the modulation of the laser optical intensity by the weak external feedback. Obviously the intensity modulation is a repetitive function with a period of 27r radians, and it is worthwhile to note that the modulation coefficient nz is not a constant as in conventional interference, but varies nearly inversely with the optical intensity I, in the absence of feedback. The intensity variation with the exkernal phase is shown graphically in Fig. 3, corresponding to different values of external feedback strength. As the intensity modulation displays some distinct features, it has been termed as "self-mixing interference" to distinguish it from that produced by conventional interference [3].
D. Visibility Function and Output Power of Self-Mixing Inteference

cavity and that from the feedback, the spectral distribution and therefore the coherence length (or time) are then changing with the feedback field, not remaining constant, by comparison to those in a conventional interferometric system. The intensity variations of self-mixing interference may be monitored by the internal photodetector inside the laser package or by an external photodetector placed at either end of the laser cavity, the process of detecting photons generates optical power over the whole band of emitting frequencies, and the total output power from the laser may thus be calculated by taking an integral of (17) over all frequencies. The output power P is then given by P
=
=

j0 h dv
K s i x p ( V ) [ l + mCoS(47TVT,)]dV, (19)

where K~ is a proportionality coefficient and p(v) represents the power spectrum of a diode laser. Expression (19) shows that the laser output power is determined not only by the self-mixing intensity variation but also by the spectral distribution of the diode laser in the presence of feedback. When the length of the external cavity is varied, the output power will vary between constructive (P,,,,,) and destructive (Pm,,>interference. In this way, the visibility function V(T,) may be defined in the usual form as
(20) + Pm,, . It can be seen that Eq. (19) is similar to that in conventional two-beam interference, and therefore the laser output power P may be expressed, with respect to the laser spectral linewidth Sv, in the case of the source spectrum being a constant Lorentzian in shape, as
V ( T L )=
Pmax
Pmax

- Pmm

The visibility function is a measure of the coherence characteristics of a light source for optical interference. For semiconductor lasers, the spatial coherence is not particularly important because of a small emitting area of the laser, and here the coherence means the temporal coherence which is determined by the source power spectral distribution. Since self-mixing interference is not a linear superposition of the electric fields but a nonlinear interfering process between the electric field inside the

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dvo=60MHz, ho=800nm, 4&2crn
1.6
I

..........

.____

- - - - c=o.22 - - - C=O.O7

C=O.71 C=0.50

- c=o.oo

e
a
h
v)

3

1.2

a
a,

W

L 1.0

0.8

2 0.6
0.4
0

a,

-1

n

2n

3n

4n

External Phase (Radians)
Fig. 4. Optical power versus external phase change for various values of C .

of the feedback. It is evident that the output power P is also a repetitive function with one period corresponding V(T~ := )m exp (22) to a A/2 optical displacement. For a small value of C the power modulation is a sinusoidal waveform, but for larger However, the complexity arises in the self-mixing intervalues of C the output waveform becomes sawtooth-like. ference since both the emission frequency v and the A typical self-mixing output waveform observed from a linewidth 6 v , in the presence of feedback, are not condiode laser is shown in Fig. 5 , with one “fringe” shift stant, but change significantly with the external cavity corresponding to a half wavelength of displacement at the will ) therefore length, and so the visibility function V ( T ~ external reflector. The upper trace shows the signal outdisplay a very complicated behavior. For example, the put of the self-mixing interference in the diode laser with linewidth may be narrowed, resulting in a coherence ena comparison of the output (lower trace) of a two-beam hancement, or a single-mode spectrum may become mulMichelson interferometer. It can be seen that the two timode, even evolving into a very broad spectrum, leading waveforms had the same number of fringes with the to the “coherence collapse.” From this point of view, it is external reflector displaced by a distance which was the considered that the self-mixing interference is not depensame in both cases, but the observed waveforms in selfdent on the coherence length of the solitary laser used. A mixing show a significant difference in shape, in which the theoretical analysis is presented in Section 111; also, delower trace (Michelson) was a sinusoidal waveform while tailed studies of coherence properties of a semiconductor the upper trace (self-mixing) was sawtooth-like. It is clear laser subjected to a distant reflector have been presented that the theoretical simulation of the self-mixing output is by Hamel et al. [411. in good correspondence to that observed in the experi111. THEORETICAL ANALYSIS WITH AN EXPERIMENTALment work.

and the visibility function T/(T[,) is then given by

{

4 . ) .

(:OMPARISON

A. Power Modulation far Self-Mixing Inteflerence

B. Asymmetric Fringe Patterns and Waveforms Sign Inversion
A n asymmetry of the sawtooth-like waveform in selfmixing interference has been observed, corresponding to a stronger feedback level. This feature can be seen easily from Fig. 6, where with C = 0.7, the sawtooth-like waveform is depicted as the upper curve. For a comparison, the experimental output signal of the self-mixing interference is shown in Fig. 7, where the lower trace is the signal with which the external reflector is driven and the upper trace is the resultant intensity modulation of the laser output. The asymmetry of the sawtooth-like waveform is

Eq. (21) represents il theoretical expression describing the self-mixing interference in a single-mode diode laser with weak optical feedback. Since both the spectral linewidth 6 v and the emission frequency v are the function of the external reflection phase &,, the relationship between the output power P and the phase 4,,, becomes very complicated. To help elucidate its physical meaning, this relationship is depicted in a graphical form as in Fig. 4. This figure shows the dependence of the power P on , , corresponding to different levels the external phase 4

WANG cr al.: SELF-MIXING INTERFERENCE INSIDE A SINGLE-MODE DIODE LASER

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Fig. 5. A typical self-mixing interference signal (upper trace) with a comparison to conventional interference output (lower trace). Vertical axis-arbitrary units; horizontal ais--0.2 ms/division.

Fig. 7. An experimental observation of a sawtooth-like waveform of self-mixing interference. Upper trace-self-mixing interference signal, lower trace-vibrational signal of external reflector. Vertical axi-arhitrary units; horizontal axis-0.2 ms/division.

2 0
~

Self-mixing interference signal f external m ~ r r o r Displacement o

1

T i m e (arb. units)
Fig. 6 . A theoretical simulation of sawtooth-like waveform of self-mixing interference. Upper trace--self-mixing interference signal, lower trace-vibrational signal of external reflector.

Conventionally, the optical intensity, I , is an even function (cosinusoidal) of the phase argument, where the emission frequency is constant; thus it does not result in the sign inversion in the interference. The theoretical simulation of the sign inversion is depicted as in Fig. 8, where the signals display two different feedback levels, C = 0.2, indicating a sinusoidal waveform, and C = 0.7, corresponding to a sawtooth-like waveform and showing an opposite inclination of fringes. The upper curve represents the output signal of the diode laser in one direction, while the middle curve gives the sign inversion in the other direction, and the lower curve is the displacement signal of the reflector over one period. Fig. 9 shows experimental results of the sign inversion observed from the two emission directions of a diode laser, and good agreement is seen between the theoretical and experimental results, the latter clearly showing the direction of inclination of the fringes. C. Coherence Length of the Self-Mixing Interference Power modulation of various lasers due to the self-mixing effect has been observed, even when the external cavity length is well outside the coherence length of the solitary laser used. This is of importance for some particular applications but shows a deviation from familiar coherence theory, where it has been seen that if the source used is ideally a single frequency, it will produce an infinite coherence length as the spectral linewidth is zero. This is in contrast to the source being “white light,” i.e., the spectrum of the source spreads over a whole range of frequencies, and the coherence length drops to zero. It has been mentioned in Section II-D that the power spectrum of a single-mode diode laser in the presence of the feedback displays a very complicated behavior, which is not like that of a conventional laser source, in which the spectrum remains nearly constant; for example, the distri-

easy to see, and when the external reflector changes its direction of movement, the sawtooth-like waveform changes its direction of the fringe inclination relative to the movement direction of the external reflector. This feature of the fringe inclination has been used for directional discrimination [71, [151. The important feature of sign inversion is seen in self-mixing interference. It has been observed experimentally by de Groot [14] and also confirmed in our experimental work that there is a phase difference of rr radians between the self-mixing signals observed in the two emission directions of the diode laser. This experimental result was not explained in the work by de Groot, but it is easily seen by applying (9) and (17) previously obtained. The sign inversion of self-mixing signals results from the emission frequency shift v of the diode laser relative to the directional movement of the external reflector, because such a shift is an odd function of the phase change.

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..........

2 5

Self-mixing interference signal C=0.7 Self-mixing interference signal C=0.2 Displacement of external mirror

1.5

4

I

will start lasing eventually [39], [411, and in this case the laser spectrum has a multiple external mode distribution, which is almost unavoidable with the use of a distant reflector (such as from the far end of an optic fiber). For simplicity, it may be assumed that the external cavity laser is operated in a multiple external cavity mode state, as shown in Fig. 10, and the mode spacing, A V , of the spectrum is given by the term c / 2 L .

D.Modulation CoefJicient Dependence on Injection Current
The modulation coefficient m defined in (18) is a measure of the available signal strength in the laser optical intensity with the self-mixing interference. The larger the modulation coefficient, the higher the signal-to-noise ratio. It is known, from the given expression, that the modulation coefficient m in the self-mixing interference is not a constant for a given external reflection coefficient r3, but varies nearly inversely with the laser intensity ll). Above threshold, the value of Z , is proportional to the injection current J , which is given by

Fig. 8. Waveform sign inversion of self-mixing interference between two emission directions under conditions of C = 0.2 and C = 0.7. Upper curves-self-mixing interference signals at one direction, middle trace-self-mixing interference signal at opposition direction, lower trace-driver signal of vibration of external reflector.

represents the optical intensity at threshold, where lth is the incremental quantum efficiency, and Jth is the threshold current. Substituting (23) into (1 81, the modulation coefficient m may be expressed as

Fig. 9. Experimental observation of waveform sign inversion of selfmixing interference. Vertical axis-arbitrary units, horizontal axis-0.2 ms /division.

bution of a Lorentzian spectrum will lead to an exponentially decaying visibility function with the change of OPD. For a small value of C (C < l), the influence of the feedback is small, the spectral linewidth may be narrowed or broadened, but the lasing spectrum still remains a single-mode distribution. A Lorentzian spectrum and its corresponding visibility function may be used approximately to describe the coherent characteristics of the self-mixing interference [101. However, for larger values of C, the visibility function may be dramatically different from that in the absence of the feedback, as this is mainly due to dramatic changes of the laser spectrum produced by strong feedback. In particular, for C > 1 corresponding to either strong feedback or the use of a longer external cavity length (greater than the coherence length of the laser) or both, the multiple external cavity modes

Above the lasing threshold, the gain G,, is clamped at the threshold value so that the value of GI, is approximately constant. This implies that the above relationship is a single function of the injection current J . The value of rn reaches the maximum value at threshold, and then decreases with the increase of the injection current. The larger the current J , the smaller the coefficient m. Fig. 1l(a) shows the tendency of the modulation coefficient variation with the injection current, which is approximately inversely proportional to the laser output power. At the threshold the coefficient reaches its maximum value, and with the increasing of the current, the coefficient is decreased and approaches a lower but more stable value. This has been found to be in good agreement with the experimental result shown in Fig. ll(b), where the laser current was varied around the threshold current value from @.8J,hto 1.6Jth with a single-mode diode laser (Sharp LT024) being used. The feedback strength at the front facet of the laser was measured to be 3%. Close to the threshold, the modulation amplitude was observed to be a maximum, and then decreasing with increasing the injection current. Unfortunately the intensity modulation around the threshold was not stable, while the modulation amplitude was very small when the current was well above threshold. In practice, the value of the laser current was chosen to be 1.1 to 1.2 times Jth; the power modulation due to the self-mixing effect here is stable and still

-

-

WANG et al.: SELF-MIXING INTERFERENCE INSlDE A SINGLE-MODE DIODE LASER

1585

1.0

Thraahold Current=4SmA
R1=0.03

Em i ti si on Fr e qu e n c y
(a)

45

Injection Current(mA)

50

55

\
J

; ;
I
t

Amplitude Data _ _*_ _* Modulation Fitting Curve Diode Laser-Shar LM24, P,,=30mW
Threshold Curren?=4@.6mA,&=0.03

,

,

Optical P a t h Difference
(b)
Fig. 10. Visibility as a function of optical path difference.

81
a
7

1

1 ,
'-++

-*. *.

s

*-.--+*-

-cc

I

r'
I I I
(

0.0--; 45

I

,
Sk

I

I

I

/

,
65

I

I

5b

Injection Current(mA)
(b)

6b

has a relatively high sensitivity, representing an acceptable compromise between sensitivity and stability.

IV. DISCUSSION
The use of the self-mixing interference, inside a singlelongitudinal-mode diode laser, has been discussed to explain results obtained in an interesting application to optical sensing technology. A theoretical model, including aspects of the variation of the emission frequency, lasing spectral linewidth, threshold gain, and output power (intensity), has been discussed, based on the assumptions of the three-mirror Fabry-Perot structure of the self-mixing system and that the power spectrum of a single-mode diode laser is Lorentzian in shape. The results obtained have been compared with those from experiment, and good agreement has been found. In the case of a weak feedback level (C < l), the power modulation by self-mixing is similar to that of a conventional two-beam interferometer and the lasing spectrum remains single-mode. For very weak feedback (C -=K 11, the output waveform of the self-mixing interference is

Fig. 11. Relationship between modulation coefficient and laser injection current. (a) Theoretical simulation of modulation coefficient versus injection current; (b) experimental measurement at different values of laser current.

sinusoidal, and an increase of the feedback level will result in sawtooth-like waveforms as well as an inclination of the fringe pattern, and thus the larger the value of C , the more sawtooth-like the output waveforms. For a longer cavity length (greater than the coherence length of the laser used), the lasing spectrum will become much more complicated and may result in a multiple external cavity mode operation. The changes of the lasing spectral distribution thus produce a number of discrete coherence regions, in which the self-mixing intensity modulation may be always observed; this indicates that the self-mixing interference is not strongly dependent on the coherence length of the solitary laser used. The asymmetry of the sawtooth-like fringes and the waveform sign inversion observed have been concluded from the theoretical model and seen to be related to the change of the phase sign in the emission frequency shifts

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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 12, NO. 9, SEPTEMBER 1994

with the relative direction change of the external reflector movement. The self-mixing modulation depth or coefficient has also been found to be inversely to . propOfiiOna1 . the laser output intensity. The optimal operating current ‘lpplications has been Observed to be for practical l.lJthto 1.25,,, where the self-mixing intensity modulation is stable and has a satisfactory signal-to-noise ratio. The observed self-mixing interference in a single-mode diode laser results from the variations of the lasing spectral distribution and of the threshold gain, which is, in principle, different from the mechanism of conventional interference. The periodical displacement of the external feedback phase causes a change in the stabilized laser oscillation condition and thus the laser output power. The comparison between the theoretical simulations and the results of experiments undertaken have clearly shown both the similarities and the differences between self-mixing interference and conventional optical interference.

-

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N. Schunk and K. Petermann, “Numerical analysis of the feedback regimes for single-mode semiconductor lasers with external feedback,‘’ IEEE J. Quantum Electron., vol. 24, pp. 1242-1247, 1988. K. Petermann, “Semiconductor lasers with optical feedback,” in Laser Diode Modulution and Noise, xxx, Ed. Norwell, MA: Kluwer Academic, Chap. 9, pp. 250--290. M. Lax, “Classic noise V: Noise in self-sustainedoscillators,” Phys. Rev., vol. 160, pp. 290-307, 1967. C. H. Henry, “Phase noise in semiconductor lasers,” J . Lightwave Technof.,vol. LT-4, pp. 298-311, 1986. M. W. Fleming and A. Mooradian, “Fundamental line broadening of single-mode GaAlAs diode lasers,” Appl. Phys. Lett., vol. 38, p. 511, 1981. S. Saito and Y. Yamamoto, 1983 “Direct observation of Lorentzian lineshape of semiconductor lasers and linewidth reduction with external grating feedback,” Electron. Lett., vol. 17, p. 325, 1983. K Vahala, Ch. Harder, and A. Yarin, “Observation of relaxation resonance effects in the field spectrum of semiconductor lasers,” Appl. Phys. Lett., vol. 42, pp. 211-213, 1983.

W.M. Wang, photograph and biography not available at time of publication.

K. T. V. Gratten, photograph and biography not available at time of publication.

A. W. Palmer, photograph and biography not available at time of publication.

W. J. 0. Boyle, photograph and biography not available at time of
publication.


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