Measuring angular rotation via the rotatory dispersion effect

We propose a scheme for an ultrasensitive weak measurement system with an ultrasmall rotatory dispersion effect to measure angular rotation. Estimation of angular rotation is obtained by the shift of the center wavelength. In our scheme, we obtain a continuously adjustable coupling strength and show that our system provides a 13 times higher sensitivity of 38 840 nm/rad compared with previous work. Here we demonstrate that our scheme may offer an optimal weak value $A_w$ in practice and outperforms conventional measurements in the presence of technical noise. The precision achieved by optimized weak value amplification (WVA) is about three times higher than that of a polarimeter. Our work extends the working regime of WVA. Its simplicity and robustness clear the way for widespread use of WVA involving measurements with commercial cameras and small signals.

Figure 1

Mach-Zehnder interferometer for small time delay $\tau$ between two arms. The incoming light is linear polarized and then split by PBS into two arms with two linear polarizations $|H\rangle$ and $|V\rangle$. Transformation between {$|{\psi }_r^{+}\rangle , |{\psi }_r^{-}\rangle$} and {$|H\rangle , |V\rangle$} can be realized through a half-wave plate (HWP) and a quarter-wave plate (QWP). The light is recombined at the second PBS, and the postselection procedure is realized through the latter polarizer.

Figure 2

Experimental configuration of optical rotation measurement. The signal beam is focused inside the optical crystal with high collimation. To set the preselection and postselection, polarizers (Thorlabs Inc., LPVIS050-MP, extinction ratio of 100 000:1) are put before and after the optical crystal and are almost orthogonal. The second polarizer is fixed in a rotation platform. The quartz rotators act as weak interaction between polarization and momentum of photons, and the rotation angle is nearly linear to the momentum of photons. Left-handed and right-handed quartz rotators are both applied in our system to achieve a tiny rotatory dispersion effect. A light source (SLD, IPSDD0804, 830 nm, 5 mW, Inphenix) and spectrograph are used in our experiment for spectral analysis.

Figure 3

Center wavelength shifts varying $\varepsilon$. The theoretical and experimental results are shown for four different values of effective thickness of quartz crystal. The dashed curve is the corrected curve while taking the zero point offset into account for $l$ =0.1 mm. The sensitivity to measure $\varepsilon$ can be characterized by the dashed black lines at zero point, and the solid black line (lowest) is the center wavelength shift in Li's scheme [19].

Figure 4

Comparison between WVA with different $\mathrm{Im}{A}_w$. (a) Precision obtained from CCD with saturation threshold $k_s$ = 16 384 and readout noise parameters $\mu =$ 6, ${\sigma }_1=10$. The center wavelength of the input Gaussian beam is 840 nm, and the FWHM is 25 nm, carrying ${10}^{10}$ photons per exposure in average. The results show that the precision of the system is significantly reduced for $\mathrm{Im}{A}_w$ = 5.7. (b) The average intensity distribution of the spectrum received by the CCD for various $\mathrm{Im}{A}_w$ in (a). The plot shows that saturation effects are obvious for $\mathrm{Im}{A}_w$ = 5.7.

Figure 5

Comparison between the precision of a polarimeter and that of WVA with $\mathrm{Im}{A}_w$ = 14.3 for three kinds of light source. The plot shows that the performance of a polarimeter is greatly affected by light intensity jitter. Although a polarimeter outperforms WVA for a classical thermal light, the output signal is similar to noisy light due to air disturbance, vibration, and so on. In this situation, WVA can outperform a polarimeter.

Figure 6

Experimental results of a polarimeter and WVA with various $\mathrm{Im}{A}_w$. The experiments are performed based on the above setup. (a) The integration time of CCD is $4\times {10}^{-3}$ s. As the inserted picture shows, saturation is negligible for all $\mathrm{Im}{A}_w$. The plot shows that WVA obtains similar precision for high $\overline{n}$. For low $\overline{n}$, intrinsic readout noise of CCD restricts the performance of the system. (b) The integration time of CCD is $1.5\times {10}^{-2}$ s. As the inserted picture shows, saturation is non-negligible for $\mathrm{Im}{A}_w$ = 8.1 and $\mathrm{Im}{A}_w$ = 5.7. As $\overline{n}$ increases, saturations restrict further improvements of precision.