1. Introduction

Compared with conventional interference, self-mixing interference has similar phase sensitivity and modulation depth ratio, and the movement direction can be discriminated through the self-mixing interference signal. These factors indicate that self-mixing interference has good prospects for optical sensing applications. Along with further development of self-mixing interference theory and research of a number of new types of semiconductors, more and more researchers are considering using new types of semiconductor lasers for optical sensing applications of self-mixing interference and look forward to obtaining better performance. Mourat et al. [1] reported researching absolute distance measurement utilizing a trielectrode distributed Bragg reflector laser in 2000. Porta et al. [2] reported studying Doppler velocimetry using vertical-cavity surface-emitting lasers in 2002.

We have systemically studied the self-mixing interference effect of λ/4 phase-shifted DFB lasers for what we believe to be the first time [3]. In this paper, we focus on the self-mixing interference effect of GC DFB lasers based on coupled-wave theory. The general expression of the threshold gain and frequency are deduced, and we conclude from numerical simulation that the GC DFB laser is more sensitive to external optical feedback than the Fabry-Perot (F-P) laser or the λ/4 phase-shifted DFB laser for specific coupling length coefficient κL. Therefore self-mixing interference on GC DFB lasers can be applied to high-accuracy sensors.

2. Analysis

2.1 Condition for oscillation of the GC DFB laser

According to the Maxwell equation, we get the following coupled-wave equations of GC DFB lasers [4]:

where E is the wave amplitude along the laser cavity; E⁺, E^- are amplitudes of the forward and backward components; κ_i, κ_g are the index-coupling and gain-coupling coefficient, respectively; g_th is the threshold gain for the traveling wave; and δ is the deviation of the wave number β from the Bragg wave number β_B. The solutions of Eq. (1) can be written as

where a_1,2 and b_1,2 are constants to be evaluated from boundary conditions. The complex propagation constant γ satisfies the dispersion relation:

The boundary conditions at z=0, z=L for the wave amplitudes are expressed as

where r_l, r_r are the reflectivity of the left- and right-hand laser facet, respectively.

Using the boundary conditions, we get the condition for oscillation of the GC DFB semiconductor laser given by Eq. (5):

2.2 Influences of self-mixing interference

Under the condition of weak feedback, provided a reflector with the reflectivity r on the right-hand side, the equivalent reflectivity of the right-hand facet can be written as

Here ω is the emission frequency and τ=2L_ext/c is the external round-trip time, where L_ext is the external cavity length and c is the velocity of light in vacuum. The complex feedback sensitivity defined in Ref. [5] is

Δδ can be written as below, and n is the effective refractive index:

From Eqs. (7) and (8),

We now define the material linewidth enhancement factor in terms of the parameters describing propagation of a wave in the laser structure as α_m=(ω/c)Δn/Δ(g_th/2). Inserting α_m into Eq. (9), we obtain the emission frequency and threshold gain variations:

order:

(1). For X<1, the self-mixing interference signal is an inclined sine waveform with linear variation of the external cavity length, Eq. (10) is only one-valued for certain parameter values, and the system is monostable.

(2). For 1≤X≤4.6034, the operating modes that satisfy Eq. (10) can reach three, the system becomes bistable, and the self-mixing interference signal appears sawtooth-like.

(3). For X>4.6034, resulting in more than five solutions for Eq. (10), several external cavity modes may start to oscillate and the system becomes multistable, but this condition does not exist for the self-mixing interference system, so the interference phenomenon cannot be detected.

When there are two residual facets, the laser is less sensitive to external feedback than the laser with only one residual facet. For the latter condition, r_l=|r_l|e^-jφ1, r_r=0, the oscillation condition for complex propagation constant γ can be written as

We know that C_r is a complex feedback sensitivity that depends only on the proper laser parameters in the limit of a weak feedback level, and C_r can be written as follows when the feedback is produced on the antireflection- (AR-) coated facet side [6]:

where qL=(g_th/2)L-iδL, κL=(κ_i+jκ_g)L, for a pure GC DFB laser: κL=jκ_gL.

Figure 1(a) shows the relation of the complex feedback sensitivity |C_r| of a pure GC DFB laser versus |κL|. |C_r| decreases when |κL| increases. Figure 1(b) shows a comparison of |C_r| under the condition of having the same parameters: the blue curve denotes a pure GC (κ_iL=0, r_l=0.56, r_r=0) DFB laser with one AR-coated facet; the purple curve denotes a pure index-coupled (κ_gL=0, r_l=r_r=0.56) DFB laser; and the red dot denotes the F-P laser (|κL|=0, r_l=r_r=0.56). We can find, for |κL|<0.92 approximately, that the GC DFB laser is more sensitive to the external feedback than the λ/4 phase-shifted DFB laser; otherwise, the latter is more sensitive than the former. For |κL|<0.96 approximately, the GC DFB laser is more sensitive to the external feedback than the F-P laser; otherwise, the latter is more sensitive than the former.

 

Fig. 1. Curves showing |Cr| versus |κL|. (a) Gain-coupled DFB laser for different values of r_l. (b) Gain-coupled DFB laser compared with λ/4 phase-shifted DFB laser and F-P laser.

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3. Numerical analysis

The self-mixing interference system can operate stably under the condition of X≤4.6034. Through choosing a suitable parameter, we obtain the self-mixing interference signal with the following parameters: n=3.2, Λ=0.244µm, r_l=0.56, L_int=400µm, and initial external cavity length L_ext0=0.1m.

3.1 Self-mixing interference at different |κL| values

Figure 2 shows self-mixing interference for a pure GC DFB semiconductor laser at different |κL| values with parameters r=0.1, α_m=π/4, κ_gL=0.995, 0.997, and 0.999. When κ_gL<0.98, resulting in X>4.6034, the interference phenomenon cannot be detected. Figure 2(a) shows the numerical solution of the frequency, and the signal tends to be bistable rather than monostable when κgL decreases. Figure 2(b) shows the numerical solution of the output signal with linear variation of the external cavity length, the signal tends to have a sawtooth-like waveform from the sine waveform when κ_gL decreases, and the amplitude of the signal increases when κ_gL decreases. Figure 2(c) shows the output signal with cosine variation of external cavity length. The movement direction of the external reflector can be discriminated through change in the inclination direction of the sawtooth-like waveform.

3.2 Self-mixing interference at different αm values

Figure 3 shows self-mixing interference at different αm values with parameters r=0.3, κ_gL=0.999, α_m=π/4, π/2, 3π/4, and π. When α_m>5.82, resulting in X>4.6034, the interference phenomenon cannot be detected. The frequency signal [similar to Fig. 2(a)] tends to be bistable rather than monostable when α_m increases. Figure 3(a) shows the numerical solution of the output signal, which is different from Fig. 2(a) in that the amplitude of the signal does not change when α_m increases, but the inclination degree of the waveform changes. So with increase in α_m, discrimination of the movement direction becomes easier. Figure 3(b) shows the output signal with cosine variation of the external cavity length.

3.3 Self-mixing interference at different r values

Two parameters mentioned above influence the signal of self-mixing interference, but they are both laser internal parameters that depend only on laser modal characteristics. After the laser model is selected, we consider the influence of the external parameters.

Figure 4 shows self-mixing interference at different r values, and the parameters are κ_gL=0.999, α_m=π/2. r=0.1, 0.2, and 0.3. When r>0.54, resulting in X>4.6034, the interference phenomenon cannot be detected. The frequency signal [similar to Fig. 2(a)] tends to be bistable rather than monostable when r increases. Figure 4 is similar to Fig. 2; the feedback level becomes deeper when r increases, and the amplitude of the signal increases.

Furthermore, the external cavity length should satisfy the relevant conditions. With a short external cavity, the system becomes stable more easily; with a long external cavity, possibly resulting in X>4.6034, the interference phenomenon cannot be detected.

 

Fig. 2. Self-mixing interference at different |κL| values. (a) Numerical solution of the frequency. (b) Numerical solution of the output signal. (c) Output signal accompanying cosine variation of the external cavity length.

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Fig. 3. Self-mixing interference at different α_m values. (a) Numerical solution of the output signal. (b) Output signal accompanying cosine variation of the external cavity length.

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3.4 Comparison between gain-coupled and λ/4 phase-shifted DFB lasers

Self-mixing interference of λ/4 phase-shifted DFB lasers was analyzed in Ref. [3], where we compare the results of the F-P laser and the λ/4 phase-shifted DFB laser with those of the GC DFB laser. Figure 5(a) shows self-mixing interference of the F-P, GC, and λ/4 phase-shifted DFB lasers at |κL|=0.85, r=0.1, and α_m=π/4. From the figure we see that the GC DFB laser is more sensitive than the λ/4 phase-shifted DFB laser and the F-P laser, and the interference signal could be detected more easily. In addition, the numerical solution of the opposite instance (|κL|=0.99) is shown in Fig. 5(b); the GC DFB laser is less sensitive than the λ/4 phase-shifted DFB laser and the F-P laser.

 

Fig. 4. Self-mixing interference at different r values. (a) Numerical solution of the output signal. (b) Output signal accompany cosine variation of the external cavity length.

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Fig. 5. Self-mixing interference of gain-coupled, λ/4 phase-shifted DFB and F-P laser. (a) |κL|=0.85. (b) |κL|=0.99.

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4. Conclusion

In this paper we have analyzed the effects of self-mixing interference on GC DFB lasers based on the single-mode oscillation equation. From coupled-wave equations, the oscillation frequency, threshold gain variations, and the complex feedback sensitivity C_r on the AR-coated facet side were theoretically deduced. Considering the influences on self-mixing from the coupling coefficient, the linewidth-enhancement factor of the GC DFB laser, and the reflection coefficient of the external reflector, we discussed the results using numerical simulations and compared the effects of self-mixing interference with those of λ/4 phase-shifted DFB lasers and F-P lasers. In conclusion, for |κL|<0.92 approximately, the GC DFB laser is more sensitive to external feedback than λ/4 phase-shifted DFB lasers and F-P lasers. Therefore, self-mixing interference on GC DFB lasers can be applied to high-accuracy sensors.