## Abstract

We discuss the theory and error analysis for an ultrasensitive refractive index detector based on the two-frequency Zeeman effect laser. Experimental measurements on gases agree fairly well with predictions. With a 5-cm pathlength, the typical interferometry stability is Δ*n* = 8 × 10^{−9}/h. Resolution is Δ*n* = 1 × 10^{−9}.

© 1990 Optical Society of America

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### Equations (15)

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(1)
$${\mathbf{E}}_{\mathbf{cav}}=\frac{\sqrt{2}}{2}\left\{\begin{array}{c}A\hspace{0.17em}\text{exp}[i(k-{\omega}_{1}t)]\\ -iA\hspace{0.17em}\text{exp}[i(k-{\omega}_{1}t)]\end{array}\right\}+\frac{\sqrt{2}}{2}\left\{\begin{array}{c}B\hspace{0.17em}\text{exp}[i(k-{\omega}_{2}t)]\\ iB\hspace{0.17em}\text{exp}[i(k-{\omega}_{2}t)]\end{array}\right\},$$
(2)
$$\text{QWP}=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& i\\ i& 1\end{array}\right).$$
(3)
$${\mathbf{E}}_{\mathbf{out}}=\left\{\begin{array}{c}A\hspace{0.17em}\text{exp}[i(k-{\omega}_{1}t)]\\ iB\hspace{0.17em}\text{exp}[i(k-{\omega}_{2}t)]\end{array}\right\},$$
(4)
$$\text{CBD}1=\left[\begin{array}{cc}\text{exp}(-i{\psi}_{1})& 0\\ 0& \text{exp}(-i{\psi}_{2})\end{array}\right].$$
(5)
$$\text{CELL}=\left[\begin{array}{cc}\text{exp}(-i{\varphi}_{1})& 0\\ 0& \text{exp}(-i{\varphi}_{2})\end{array}\right].$$
(6)
$$\text{CBD}2=\left[\begin{array}{cc}\text{exp}(-i{\zeta}_{1})& 0\\ 0& \text{exp}(-i{\zeta}_{2})\end{array}\right].$$
(7)
$$\text{POL}=\frac{1}{2}\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right].$$
(8)
$$I={A}^{2}+{B}^{2}+2AB\hspace{0.17em}\text{sin}(2\pi ft+\varphi +\psi +\zeta ),$$
(9)
$$\varphi =2\pi \left(\frac{L}{\mathrm{\lambda}}\right)({n}_{2}-{n}_{1}).$$
(10)
$$\mathrm{\Delta}\varphi =360\xb0\left(\frac{L}{\mathrm{\lambda}}\right)\mathrm{\Delta}n,$$
(11)
$${{\mathbf{E}}^{\prime}}_{\mathbf{out}}=\left\{\begin{array}{c}\left(1-\frac{i\delta}{2}\right)A\hspace{0.17em}\text{exp}[i(k-{\omega}_{1}t)]-\frac{\delta}{2}B\hspace{0.17em}\text{exp}[i(k-{\omega}_{2}t)]\\ \left(i+\frac{\delta}{2}\right)B\hspace{0.17em}\text{exp}[i(k-{\omega}_{2}t)]+i\frac{\delta}{2}A\hspace{0.17em}\text{exp}[i(k-{\omega}_{1}t)]\end{array}\right\}.$$
(12)
$$\text{CBD}{1}^{\prime}=\left[\begin{array}{cc}G\hspace{0.17em}\text{exp}(-i{\psi}_{1})& g\hspace{0.17em}\text{exp}(-i{\psi}_{1})\\ h\hspace{0.17em}\text{exp}(-i{\psi}_{2})& H\hspace{0.17em}\text{exp}(-i{\psi}_{2})\end{array}\right],$$
(13)
$${I}^{\prime}(t)=2\text{ABGH}\hspace{0.17em}\text{sin}(2\pi ft+\varphi +\psi +\zeta )+2\text{AB}(Gg+Hh)\hspace{0.17em}\text{sin}(2\pi ft)+\delta \text{AB}({H}^{2}-{G}^{2})\hspace{0.17em}\text{cos}(2\pi ft).$$
(14)
$$I(t)={A}^{\prime}\hspace{0.17em}\text{sin}(2\pi ft+{\varphi}^{\prime}+\psi +\zeta ),$$
(15)
$$A=\frac{RT}{P}\frac{{n}^{2}-1}{{n}^{2}+2},$$