Nonlinear modification of the laser noise power spectrum induced by frequency-shifted optical feedback

In this article, we study the nonlinear coupling between the stationary (i.e., the beating modulation signal) and transient (i.e., the laser quantum noise) dynamics of a laser subjected to frequency-shifted optical feedback. We show how the noise power spectrum and more specifically the relaxation oscillation frequency of the laser are modified under different optical feedback conditions. Specifically we study the influence of (i) the amount of light returning to the laser cavity and (ii) the initial detuning between the frequency shift and intrinsic relaxation frequency. The present work shows how the relaxation frequency is related to the strength of the beating signal, and the shape of the noise power spectrum gives an image of the transfer modulation function (i.e., of the amplification gain) of the nonlinear-laser dynamics. The theoretical predictions, confirmed by numerical resolutions, are in good agreement with the experimental data.

Figure 1

Numerical simulation. Power spectra of the laser intensity dynamics $|\mathcal{F}\left[I\left(t\right)/I_{\mathrm{sat}}\right]{|}^2$. (a) Weak feedback: $R_e={10}^{-11},F_e/F_R=1.3$; peaks: (1) $F_R$, (2) $F_e$. (b) Strong feedback: $R_e={10}^{-7},F_e/F_R=1.01$; peaks: (1) $F_e-{\tilde{F}}_R$, (2) ${\tilde{F}}_R$, (3) $F_e$, (4) $2F_e-{\tilde{F}}_R$, (5) $F_e+{\tilde{F}}_R$, (6) $2F_e$, (7) $3F_e-{\tilde{F}}_R$. Laser parameters: $I_{\mathrm{sat}}=3\times {10}^9\mathrm{photons},\eta =1.2,{\gamma }_c=1\times {10}^9{\mathrm{s}}^{-1},{\gamma }_1/{\gamma }_c=1\times {10}^{-5},F_R/{\gamma }_c=2.251\times {10}^{-4}$. The dashed curve is a fit of the laser noise power spectrum without feedback.

Figure 2

Numerical simulation. Power spectra of the laser intensity dynamics. Left column: fixed modulation frequency $\left(F_e/F_R=1.01\right)$ and an increasing amount of optical feedback $\left(R_e\right)$; (a) $R_e={10}^{-9}$; (b) $R_e={10}^{-8}$, (c) $R_e={10}^{-7}$. Right column: fixed amount of optical feedback $\left(R_e={10}^{-7}\right)$ and a modulation frequency going closer to the intrinsic relaxation frequency; (d) $F_e/F_R=1.3$; (e) $F_e/F_R=1.1$; (f) $F_e/F_R=1.01$. Laser parameters are identical to Fig. 1. The dashed curve is a fit of the laser noise power spectrum without feedback.

Figure 3

(a) Relative variation of the relaxation frequency and (b) square of the amplitude of the laser output power modulation. The lines correspond to results obtained from analytical equations [Eq. (16)], while the dots correspond to results obtained from numerical simulations. Solid lines and square dots $\left(R_e=1\times {10}^{-7}\right)$; dotted lines and circular dots $\left(R_e=1\times {10}^{-8}\right)$. Laser parameters are identical to Fig. 1.

Figure 4

Schematic diagram of the LOFI setup. $L_1,L_2$, and $L_3$: lenses; BS: beam splitter; GS: galvanometric scanner; FS: frequency shifter with a round-trip frequency shift $F_e$; PD: photodiode.

Figure 5

Experimental power spectra of the laser intensity dynamics. Left column: fixed modulation frequency $(F_e/F_R=1.075$) and an increasing amount of optical feedback $R_e$ characterized by an increase of the laser output power modulation; (a) $R_{S,\mathrm{NL}}^2=-23.1\mathrm{dB}$; (b) $R_{S,\mathrm{NL}}^2=-14.3\mathrm{dB}$; (c) $R_{S,\mathrm{NL}}^2=-9.27\mathrm{dB}$. Right column: fixed amount of optical feedback and a decreasing modulation frequency around the intrinsic relaxation frequency; (d) $F_e/F_R=1.075,R_{S,\mathrm{NL}}^2=-24.72\mathrm{dB}$; (e) $F_e/F_R\approx 1.0,R_{S,\mathrm{NL}}^2=-19.18\mathrm{dB}$; (f) $F_e/F_R=0.925,R_{S,\mathrm{NL}}^2=-12.15\mathrm{dB}$.

Figure 6

(a) Relative variation of the relaxation frequency and (b) square of the amplitude of the laser output power modulation versus the normalized scanning of the modulation frequency. The lines correspond to results obtained from analytical equations, while the dots correspond to results obtained from experimental measurement. Dashed lines and circular dots: the scanning of the modulation frequency increases. Solid lines and square dots: the scanning of the modulation frequency decreases. Estimated laser parameters: $\eta =1.2,{\gamma }_1=1\times {10}^5{\mathrm{s}}^{-1},{\gamma }_c=1\times {10}^9{\mathrm{s}}^{-1},F_R\approx 700\mathrm{kHz}$. Estimated feedback parameter: $R_e=1\times {10}^{-8}$.

Figure 7

Log-log plot of the hyperbola $g\left(x\right)=1/x$ (solid line) and three parabola $f\left(x\right)={\left(x+a\right)}^2+b^2$ (dashed, dotted, and dash-dotted lines) for a fixed value of $b=0.23$ and three values of $a$: ($a_1=-3/2^{2/3}\approx -1.89,a=-5,a_2\approx -1/b^2=-18.85$).