Self-mixing speckle interference in DFB lasers

Daofu Han; Ming Wang; Junping Zhou

doi:10.1364/OE.14.003312

1. Introduction

Laser self-mixing effects are attracting widespread attention. Speckle interference or Doppler Effect introduced the lasers cavity, which produces self-mixing effect, can be used in various industrial and scientific measurement systems. S. Shinohara, L. Scalise, S. Donati, and some other researchers, successively utilized Doppler self-mixing effect and self-mixing speckle effect to measure the velocity, vibration and length of objects [1–7]. Various lasers, such as the Laser diode, Er-Yb-doped phosphate glass fiber laser, CO2 laser and vertical-cavity surface-emitting laser (VCSEL) etc., were used. Speckle signals statistic properties analysis as well as some other analytical methods for the self-mixing signals was presented [8–10]. Self-mixing effects have been studied extensively, but all of the former researches were based on the Fabry-Perot (F-P) cavity.

Recent years, DFB lasers are found popular applications as excellent spectral selectivity effect, narrow spectral line width, and extraordinary coherent length. They can conveniently couple with the optical fiber net. Today they are playing an important role in the mass capacity single mode fiber communication system. In previous work, we have theoretically investigated self-mixing interference in theλ /4 phase-shifted DFB laser and the gain-coupled DFB laser [11–12]. The variations of emission frequency and threshold gain were systemically analyzed with external optical feedback.

In this paper, we analyze self-mixing speckle interference in theλ /4 phase-shifted DFB laser. Self-mixing speckle occurs when a portion of light emitted from the DFB laser is reflected by a rough surface and coupled into the DFB cavity. The reentered light mixes with the original light in the DFB cavity and changes the output gain and spectra of the laser. The variations of the output gain as well as their PDFs are analyzed by theoretical simulations and experiments when the rough surface is moving transversely across the laser beam.

2. Theoretic analysis

Theλ /4 phase-shifted DFB self-mixing speckle interference system is schematically shown in fig. 1. Uniform medium grating in the DFB cavity is replaced by two gratings shifted by Λ/4 from each other. M₁ and M₂, posited at Z=-L/2 and Z=L/2, are left facet and right facet of the DFB cavity respectively. The length of the DFB cavity is L. S, an external reflector with rough surface, as well as M₂, form an external cavity. L_E is the length of external cavity. We suppose that ρ_r = ρ̂_r e ^+ψr , ρ_l = ρ̂_l e ^+ψl are the reflection coefficient M₁ and M₂ respectively, and ψ_r(ψ_l) is the phase term depending on the position of the right (left) facet.

Fig. 1. the schematics of self-mixing speckle interference system

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Light output from M₂ projects onto S. The backscattered radiations from different elements of the rough surface reenter the cavity, generating random interference, which is known as speckle. Based on the theory of Fraunhofer diffraction [13], the speckles optical field at M₂can be calculated in the following.

U (X, Y) = E_{r} ∙ e^{- i ω τ} ∙ t ∙ \int A (x, y) \exp (\frac{- i 4 π h (x, y)}{λ}) \exp (\frac{- i 2 π (x ∙ X + y ∙ Y)}{λ L_{E}}) dxdy

Where E_r is the original optical field at M₂, t is the coupling coefficient of the laser coupled from laser cavity to external cavity, τ is the single trip time of the light in the external cavity, λ is the optical wavelength, (X, Y), (x, y) are the coordinates along the speckles field and the scattering area on the rough surface respectively, A(x, y) is the aperture function of scattering area, h(x, y) is the altitude function of a random surface. We can assume that the random rough surface profile function h(x, y) obeys the statistics [14, 15]:

<h(x,y)>=0,
w=[<h(x,y)h(x,y)>]^1/2, where w is root-mean-square (RMS) height,
The autocorrelation function 〈h(x,y)h(x + l,y + l)〉 = w ² exp(-l ² /T ²), here T is the correlation length along the surface, l is the distance between two successive points on the surface.

In the case of speckles coupling into the laser cavity, the optical field at M₂ is a superposed field, which can be written as

E_{r}^{'} = ρ_{r} ∙ E_{r} + ξ ∙ t ∙^{'} U (X, Y)

Where t’ is the coupling coefficient of the laser coupled from external cavity to laser cavity, ξ is the feedback ratio of external optical field coupled into laser cavity, U(X, Y) is the complex amplitude of the speckles coupled into the laser cavity.

Conveniently, we consider the external cavity as an equivalent facet of the laser cavity, and then the reflection coefficient of the equivalent facet is

ρ_{req} = ρ_{r} + \frac{ξ ∙ t' ∙ U (X, Y)}{E_{r}}

Under the assumption of weak feedback, supposed that the light back-scattered from S has an amplitude reflectivity R and phase change ϕ, we can get the equivalent reflection coefficient of the laser facet

ρ_{req} = ρ_{r} + (1 - {\hat{ρ}}_{r}^{2}) ∙ R ∙ e^{j ϕ}

Δ ρ_{r} = (1 - {\hat{ρ}}_{r}^{2}) ∙ R ∙ e^{j ϕ}

∆ρ_r is the change of ρ_r due to the feedback light from S.

Comparing the Eq. (3) and Eq. (4) and utility Eq. (1), we get

R ∙ e^{j ϕ} = \frac{ξ ∙ t' ∙ U (X, Y)}{((1 - {\hat{ρ}}^{2}) ∙ E_{r})}

In addition, from the former research [11], the eigenvalue equation of longitudinal mode in theλ /4 phase-shifted DFB is derived when the scattered light couple into the DFB cavity from the right facet.

(1 - ρ_{req} ρ) (1 - ρ_{l} ρ) = (ρ_{req} - ρ) (ρ_{l} - ρ) e^{2 γ L}

ρ = \frac{- γ + (a - j δ)}{j κ}

Here α denotes threshold loss, δ is the departure of the oscillation frequency from the Bragg frequency, κ is the couple coefficient, and γ is the complex propagation constant.

Due to change of ρ_r, we can define the deviations of α and δ in the same way as reference [16], which denote the changes of gain and phase in the DFB cavity

Δ α_{r} L - j Δ δ_{r} L = C_{r} ∙ R ∙ e^{j ϕ}

Here C_r is a complex coefficient only related to the structure of the DFB lasers.

Deduced from Eq. (7), (8), and (9), the variation of output gain is obtained

Δ G = \frac{2 c ∣ C_{r} ∣ ∣ R ∣}{η L} \cos (ϕ - \arg (C_{r}) - \arg (R))

Where c is the velocity of light, η is the equivalent refractive index of the DFB cavity.

3. Simulations and experiments

The experimental arrangement used to study self-mixing speckle effect is shown in Fig. 2. Light beam, wavelength of which is 1550nm, emitted from a λ /4 phase-shifted DFB, after coupling into a 3dB optical fiber coupler, is split in two parts. One part beam transmitting through an optical attenuator and a fiber self-focusing lens, projects onto a rough surface. Another is coupled into a PD which connected with an amplifier, then a digital storage oscilloscope. The rough surface, we selected an aluminum plate, is set on an accurate platform which is driven by a direct current motor. The backscattered light from the rotated rough surface couples into the laser cavity, and mixes with the original light, causing a fluctuant power output, self-mixing speckle signal. This signal of self-mixing speckle is detected by the PD.

Fig. 2. Schematic configuration of experimental system

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In order to understand the speckle signal waveform, the output variations of the DFB laser are obtained by numerical simulation with Eq. (10) and experiment as shown in Fig. 3, when the velocity of rough surface is 167mm/s. We notice that they are all random output power. In the simulations, the RMS height of the surface w is 1.5um, the correlation length l is 3um, the wavelength of laser λ is 1550 nm, the length of DFB cavity L is 0.04cm, the external cavity L_E 1cm, κ is 2, ρ_l is 0, and ρ_r is 0.53.

Fig. 3. The waveform of output variations from (a) simulation and (b) experiment (v= 167mm/s)

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The output variations of the DFB laser are statistically analyzed. We get the PDFs of the output variations from simulation and experiment as shown in Fig. 4. The PDFs is Gaussian-alike shape. The difference of the PDFs between simulation and experiment is small at the same velocity of surface, which proves that the result of theoretical analysis is close to that of experiment.

Fig. 4. The PDFs of output variations from experiment and simulation (v=167mm/s)

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The external cavity length is an important parameter affecting the dynamics of the DFB with optical feedback. It affects the PDF of output variation. We simulate the output variations in the external cavity lengths of 1cm, 5cm, and 10cm, and get their PDFs as shown in Fig. 5. We find that the longer the external cavity length, the smaller the output variation. In terms of this finding, the external cavity length is taken into account in subsequent experiment.

Fig. 5. Simulation results showing the effect of external cavity length on the PDF of output variations from a DFB laser

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The velocity versus the output variations in the DFB is also investigated. In experiments, we get the PDFs of output variations at velocities of 98mm/s, 167mm/s and 340mm/s. Figure 6 shows the result of experiments. It is evident that the shape of PDF becomes more Gaussian-alike with the velocity increasing.

Fig. 6. The results of experiment showing the PDFs of output variation vary with velocities of surface

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Additionally, the fluctuations frequency in the DFB output variations are studied. For different velocities of 98mm/s, 167mm/s, 257mm/s, 340mm/s, 431mm/s and 579mm/s, the output variations are experimentally obtained and processed by the method of Fast Fourier Transform (FFT). A linear relation between mean speckle frequencies and velocities of surface is shown in Fig. 7. Mean speckle frequency is defined as the ratio of number of fluctuations in the detected signal to the measurement time.

Fig. 7. The linear dependence between mean speckle frequencies and velocities of surface in experiment

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4. Conclusion

We have investigated the speckle interference in the DFB lasers. Theoretical analysis and experiments results show that the PDF of speckle signal generated in a DFB laser is Gaussian distribution. The linear relationship between the mean speckle frequency and the velocity of rough surface has been obtained. It is promising to develop a new velocity sensor based on self-mixing speckle interference of the DFB laser.

Acknowledgments

This work was supported by the National Natural Science Foundation of China and the Specialized Research Fund for the Doctoral Program of Higher Education (20050319007) and Jiangsu province high-novel technology project (BG2003024).

References and Links

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Self-mixing speckle interference in DFB lasers

Abstract

1. Introduction

2. Theoretic analysis

3. Simulations and experiments

4. Conclusion

Acknowledgments

References and Links

Cited By

Figures (7)

Equations (10)

Optics Express