## Abstract

A laser Doppler velocimeter that consists of a semiconductor laser coupled to a fiber and that uses the self-mixing effect is presented. The velocimeter can be used for solids and fluids. A theoretical model is developed to describe the self-mixing signals as a function of the amount of feedback into the laser and the distance from the laser to the moving object. Good agreement is found between this theory and measurements.

© 1992 Optical Society of America

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### Equations (27)

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(1)
$${r}_{2}(\mathrm{\nu})={r}_{2s}+(1-\mid {r}_{2s}{\mid}^{2}){r}_{2\text{ext}}\hspace{0.17em}\text{exp}(-i2\mathrm{\pi}\mathrm{\nu}{\mathrm{\tau}}_{\text{ext}}),$$
(2)
$${r}_{2}(\mathrm{\nu})=\mid {r}_{2}\mid \text{exp}(-i{\mathrm{\phi}}_{r}).$$
(3)
$$2\mathrm{\beta}L+{\mathrm{\phi}}_{r}=2\mathrm{\pi}m,$$
(4)
$$4\mathrm{\pi}{\mathrm{\mu}}_{e}\mathrm{\nu}L/c+{\mathrm{\phi}}_{r}=2\mathrm{\pi}m.$$
(5)
$${r}_{1}\mid {r}_{2}\mid \text{exp}[({g}_{c}-{\mathrm{\alpha}}_{s})L]=1,$$
(6)
$$\mid {r}_{2}\mid \hspace{0.17em}={r}_{2s}[1+\mathrm{\kappa}\hspace{0.17em}\text{cos}(2\mathrm{\pi}\mathrm{\nu}{\mathrm{\tau}}_{\text{ext}})],$$
(7)
$$\mathrm{\kappa}=\frac{{r}_{2\text{ext}}}{{r}_{2s}}(1-\mid {r}_{2s}{\mid}^{2}),\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}(0<\mathrm{\kappa}<1),$$
(8)
$${\mathrm{\phi}}_{r}=\mathrm{\kappa}\hspace{0.17em}\text{sin}(2\mathrm{\pi}\mathrm{\nu}{\mathrm{\tau}}_{\text{ext}}),$$
(9)
$${g}_{e}-{g}_{\text{th}}=-\frac{\mathrm{\kappa}}{L}\text{cos}(2\mathrm{\pi}\mathrm{\nu}{\mathrm{\tau}}_{\text{ext}}),$$
(10)
$$\mathrm{\Delta}{\mathrm{\phi}}_{1}=2\mathrm{\pi}{\mathrm{\tau}}_{L}(\mathrm{\nu}-{\mathrm{\nu}}_{\text{th}})+\mathrm{\kappa}{(1+{\mathrm{\alpha}}^{2})}^{1/2}\times \text{sin}(2\mathrm{\pi}\mathrm{\nu}{\mathrm{\tau}}_{\text{ext}}+\text{arctan}\hspace{0.17em}\mathrm{\alpha}),$$
(11)
$$C=\frac{{\mathrm{\tau}}_{\text{ext}}}{{\mathrm{\tau}}_{L}}\mathrm{\kappa}{(1+{\mathrm{\alpha}}^{2})}^{1/2}.$$
(12)
$${g}_{\text{th}}=a(n-{n}_{0}),$$
(13)
$$\mathrm{\Delta}{I}_{\text{th}}=eV(1/{T}_{s})\mathrm{\Delta}{n}_{\text{th}},$$
(14)
$$\mathrm{\Delta}g=a\mathrm{\Gamma}\mathrm{\Delta}{n}_{\text{th}},$$
(15)
$$\mathrm{\Delta}g=\frac{{T}_{s}a\mathrm{\Gamma}}{eV}\mathrm{\Delta}{I}_{\text{th}}$$
(16)
$$P=Q(I-{I}_{\text{th}}),$$
(17)
$${\mathrm{\tau}}_{\text{ext}}=2({L}_{0}+vt)/c,$$
(18)
$$\mathrm{\Delta}P~{g}_{c}-{g}_{\text{th}}=-\frac{\mathrm{\kappa}}{L}\text{cos}(4\mathrm{\pi}\mathrm{\nu}\mathrm{\tau}vt/c+4\mathrm{\pi}{L}_{0}\mathrm{\nu}/c).$$
(19)
$$\mathrm{\Delta}t\ll 1/{f}_{d}=\mathrm{\lambda}/2v$$
(20)
$$\u3008{\mathrm{\omega}}^{n}\u3009=\int {\mathrm{\omega}}^{n}S(\mathrm{\omega})\text{d}\mathrm{\omega},$$
(21)
$$E(z,t)={E}_{0}\hspace{0.17em}\text{exp}\{-i[kz+2\mathrm{\pi}\mathrm{\nu}t+\mathrm{\psi}(t)]\},$$
(22)
$${E}_{r}(t)={r}_{2s}E(t)+(1-\mid {r}_{2s}{\mid}^{2}){r}_{2\text{ext}}E(t-{\mathrm{\tau}}_{\text{ext}}).$$
(23)
$$E(t-{\mathrm{\tau}}_{\text{ext}})=E(t)\text{exp}(-i2\mathrm{\pi}\mathrm{\nu}{\mathrm{\tau}}_{\text{ext}})\text{exp}\{-i[\mathrm{\psi}(t)-\mathrm{\psi}(t-{\mathrm{\tau}}_{\text{ext}})]\}.$$
(24)
$$\u3008\text{exp}\{-i[\mathrm{\psi}(t)-\mathrm{\psi}(t-{\mathrm{\tau}}_{\text{ext}})]\}\u3009=\text{exp}(-{\mathrm{\tau}}_{\text{ext}}/{\mathrm{\tau}}_{c}),$$
(25)
$${r}_{2}(\mathrm{\nu})={r}_{2s}+(1-\mid {r}_{2s}{\mid}^{2}){r}_{2\text{ext}}\hspace{0.17em}\text{exp}(-i2\mathrm{\pi}\mathrm{\nu}{\mathrm{\tau}}_{\text{ext}})\text{exp}(-{\mathrm{\tau}}_{\text{ext}}/{\mathrm{\tau}}_{c}).$$
(26)
$${L}_{\text{ext}}<c/(2\mathrm{\pi}\mathrm{\Delta}\mathrm{\nu}).$$
(27)
$$\mathrm{\Delta}f=2vn\hspace{0.17em}\text{cos}(\mathrm{\vartheta})/\mathrm{\lambda},$$