Continuous phase amplification with a Sagnac interferometer

We describe a phase-amplification technique using a Sagnac interferometer. We monitor the relative phase between two paths of a precisely misaligned interferometer by measuring the average position of a split-Gaussian mode in the dark port. Although we monitor only the dark port, we show that the signal varies linearly with phase and that we can obtain similar sensitivity to balanced homodyne detection. We derive the source of the amplification using classical wave optics.

Figure 1

Experimental setup. A coherent light source passes through a polarizer (Pol), producing horizontally polarized light before passing through the first 50:50 beam splitter (BS). Half of the light strikes a beam block (BB) and is thrown out. The beam then enters the Sagnac interferometer via the second BS. One of the mirrors of the interferometer is controlled by a piezo actuator (PA) to precisely control the relative transverse momentum shift $k$. We use a half-wave plate (HWP) and a piezo-actuated Soleil-Babinet compensator (SBC) to produce a relative phase shift between the two light paths. During split detection, we monitor the dark port using another BS that splits the light between a quad-cell detector (QCD) and a charge-coupled-device camera (CCD). During balanced homodyne detection, the interferometer is balanced and the flip mirror (FM) is turned up, sending both bright (1) and dark (2) ports into the balance detector (BD). We use a neutral density filter (ND) in the dark port to correct for the light lost at the first BS.

Figure 2

Postselected intensity distribution. The dotted (blue) curve is the single-mode input profile of the beam in the interferometer. The solid (black) curve is the postselected split-Gaussian mode produced by the misalignment of the interferometer and a slight relative phase shift between the two light paths. The solid lines represent the tilted wave fronts in the two paths of the interferometer when they combine at the BS, one that is delayed relative to the other, producing an asymmetric split-Gaussian (shown). The dashed line represents a wave front with the same tilt but zero relative phase delay, which would result in a symmetric split-Gaussian (not shown).

Figure 3

Dependence on transverse momentum. The transverse momentum shift imparted by the piezo-actuated mirror was varied and the split-detection signal was measured using the QCD. The solid line is the theory curve from Eq. (9) using the expected phase shift.

Figure 4

Experimental comparison of split-detection to balanced homodyne. We vary the driving voltage applied to the piezo actuator and measure the SNR using balanced homodyne detection (orange triangles) and the split-detection method (blue squares). The input power to the interferometer is approximately the same for both methods. Linear fits to the data (solid lines) show that these two methods have essentially the same sensitivity. The ideal quantum limited SNR (factoring in the quantum efficiency of each detector) is plotted using the lower, dashed blue line (split-detection) or the middle, dashed orange line (balanced homodyne detection). The higher, dashed black line illustrates the theoretical ($\sqrt{N}$) improvement of the split-detection method assuming that an equal number of photons are incident on both the split-detector and the balance detector.