In situ measurement of light polarization with ellipticity-induced nonlinear magneto-optical rotation

A precise, accurate, and relatively straightforward in situ method to measure and control the ellipticity of light resonantly interacting with an atomic vapor is described. The technique can be used to minimize vector light shifts. The method involves measurement of ellipticity-induced resonances in the magnetic-field dependence of nonlinear magneto-optical rotation of frequency-modulated light. The light propagation direction is orthogonal to the applied magnetic field $\mathbf{B}$ and the major axis of the light polarization ellipse is along $\mathbf{B}$. When the light modulation frequency matches the Larmor frequency, elliptically polarized light produces a precessing atomic spin orientation transverse to $\mathbf{B}$ via synchronous optical pumping. The precessing spin orientation causes optical rotation oscillating at the Larmor frequency by modulating the atomic vapor's circular birefringence. Based on this technique's precision, in situ nature (which avoids systematic errors arising from optical interfaces) and independent control of the most important systematic errors, it is shown that the accuracy of light ellipticity measurements achievable with this technique can exceed that of existing methods by orders of magnitude.

Figure 1

Schematic of the experimental setup. P, linear polarizer; PBS, polarizing beam splitter; $\lambda /4$, quarter-wave plate; $\lambda /2$, half-wave plate. Designation of the $\mathbf{x}, \mathbf{y}$, and $\mathbf{z}$ directions is shown in the upper-left corner. The green arrow at the center of the diagram represents the applied magnetic field $\mathbf{B}$. Assorted optics and electronics for laser control, data acquisition, and experiment control are not shown.

Figure 2

Nonlinear magneto-optical rotation $\phi$ as a function of the applied magnetic field along the light polarization axis $z$ for different values of ellipticity $\varepsilon$, where $\varepsilon$ is measured in front of the vapor cell (prior to light entering the cell). The optical rotation signal is demodulated with a lock-in amplifier at the first harmonic of the laser modulation frequency ${\mathrm{\Omega }}_m$. Left: In-phase component ${\phi }_i$. Right: Out-of-phase component ${\phi }_o$. Light power, 500 $\mu \mathrm{W}$; laser tuned to the high-frequency wing of the ${}^{85}\mathrm{Rb}F=3\to F^{\prime }$ transition; modulation frequency ${\mathrm{\Omega }}_m=2\pi \times 1$ kHz; frequency-modulation amplitude $\mathrm{\Delta }\omega =2\pi \times 65$ MHz. Magnetic resonances occur when ${\mathrm{\Omega }}_L={\mathrm{\Omega }}_m/\kappa$, where the $n=\pm 2$ resonances correspond to the $\kappa =1$ condition and the $n=\pm 1$ resonances correspond to the $\kappa =2$ condition. The zero-magnetic-field resonance, labeled $n=0$, generally has contributions from all $\kappa$. The $n=\pm 1$ resonances for the $\varepsilon =-1.0^{\circ }$ case, where the in situ ellipticity of the light is near zero, are due primarily to misalignment between the major axis of the polarization ellipse and the $z$ axis, leading to signals arising from precession of atomic alignment [54] and alignment-to-orientation conversion [50, 55]. Similar misalignment causes the appearance of $n=\pm 1$ resonances for the $\varepsilon =-10.2^{\circ }$ case as well. For these data, absolute calibration of the optical rotation angles was not carried out so the vertical axes are given in arbitrary units.

Figure 3

The upper three plots show the slope ($\partial {\phi }_o/\partial B$) of the out-of-phase component of the $n=2$ EI FM NMOR signal as a function of the laser detuning for different incident light powers (10, 100, and 1000 $\mu W$, as indicated in the upper right corners). The lower plot shows the absorption spectrum of unmodulated light transmitted through a Rb reference cell. Light ellipticity as measured in front of the vapor cell is $\varepsilon =8.3^{\circ }$; other conditions are the same as for the data shown in Fig. 2. Note the change in the scale of the vertical axis ($30\times$) between the plot for light power 10 $\mu W$ and the plots for 100 and 1000 $\mu \mathrm{W}$.

Figure 4

The magnitude of the slope ($|\partial {\phi }_o/\partial B|$) of the out-of-phase component of the $n=2$ optical rotation signal maximized with respect to laser detuning as a function of the incident light power for the two ${}^{85}\mathrm{Rb}$ hyperfine components. Conditions are the same as for the data shown in Fig. 3.

Figure 5

The magnitude of the $n=2$ in-phase component of the magneto-optical rotation resonance as a function of ellipticity $\varepsilon$, where $\varepsilon$ is measured in front of the entrance to the vapor cell. Light power, 500 $\mu \mathrm{W}$; other conditions the same as for the data shown in Fig. 2. The dotted gray line shows the linear slope $\partial {\phi }_i/\partial \varepsilon$ in the vicinity of the zero crossing for ${\phi }_i$. The thick dashed black line shows the offset of the ${\phi }_i$ zero crossing of the optical rotation from $\varepsilon =0$ is a result of the birefringence of the vapor cell walls, which induces ellipticity in nominally linearly polarized light. Zeroing the $n=2$ ellipticity-induced FM NMOR resonance can minimize the value of $\varepsilon$ for light within the cell.

Figure 6

Measurement of ellipticity $\varepsilon$ using EI FM NMOR. The ellipticity is changed by hand via rotation of a linear polarizer preceding a $\lambda /4$ plate at time $t\sim 2$ s. Light power, 100 $\mu \mathrm{W}$; other conditions the same as in Fig. 2.