Complementary weak-value amplification with concatenated postselections

We measure a transverse momentum kick in a Sagnac interferometer using weak-value amplification with two postselections. The first postselection is controlled by a polarization-dependent phase mismatch between both paths of the interferometer, and the second postselection is controlled by a polarizer at the exit port. By monitoring the dark port of the interferometer, we study the complementary amplification of the concatenated postselections, where the polarization extinction ratio is greater than the contrast of the spatial interference. In this case, we find an improvement in the amplification of the signal of interest by introducing a second postselection to the system.

Figure 1

We send antidiagonal polarized light (Pol 1) through a Sagnac interferometer. Inside the interferometer are three wave plates arranged quarter-half-quarter (QHQ) designed to control the phase for clockwise and counterclockwise propagation (see Appendix pp1). The counterclockwise-propagating beam receives a transverse momentum kick $k$ from the piezo-controlled 50:50 beam splitter. When the beam recombines, it destructively interferes at the dark port. The beam is then postselected a second time with polarizer 2 (Pol 2). The half-wave plate after the first polarizer (Pol 1) is used for independent calibration of the horizontal and vertical polarization components.

Figure 2

The average beam shift as a function of postselection angle $\mathrm{\Theta }$. The variable $\mathrm{\Theta }$ is a generalized postselection angle; for the single-postselection case, $\mathrm{\Theta }=\varphi /2$ (red circles), but for the concatenated postselection case it is the product of both postselection angles $\mathrm{\Theta }=\theta \varphi /2$ (green squares). The label PS refers to postselection. The theory (dashed blue line) is labeled $\delta x_s=4k{\sigma }^2/\mathrm{\Theta }$ as in the mean beam shift of Eq. (3) with generalized postselection angle $\mathrm{\Theta }$. Fit: SPS (solid teal line) is the fit of the single-postselection data, and Fit: CPS (solid purple line) is the fit to the concatenated postselection data as in Eqs. (5) and (9), respectively. We present here the concatenated data set from the third row in Table 1.