Fourth-order coherence induced by spatial mode parity selection

Fourth-order coherence determines the visibility of interference fringes observed in two-photon correlation measurements. In some cases, one photon triggers the presence of its conjugate twin in the interferometer. Therefore, spatial or spectral filtering of the trigger photon may change the visibility of the interference fringes and the degree of fourth-order coherence as well. We show that, contrary to the intuition developed from series of previous experiments, it is possible to increase the fourth-order coherence by reducing the degree of filtering. We present a theoretical approach and experimental results demonstrating this effect and interpret it in terms of spatial mode parity selection.

Figure 1

Typical setup used to measure double-slit interference with photon pairs. When the glass slide is inserted in half of the pump beam, an approximate Hermite-Gaussian mode is generated.

Figure 2

(a) Theoretical results for $\left|\gamma \right|$ as a function of the detection position ${\rho }_2$ for HG pump beams with $n=0,1,2$. The visibility $\left|\gamma \right|$ can oscillate as a function of ${\rho }_2$. (b) Theoretical results for $\gamma$ as a function of the detection position ${\rho }_2$ for HG pump beams with $n=0,1,2$, showing that the oscillations in (a) are due to sign changes in $\gamma$. (c) Theoretical results for $\left|\gamma \right|$ as a function of the width $2d_2$ of detector $D_2$ for $n=0,1,2$. For $n=1$, the visibility is higher for a larger detection aperture, showing that the visibility can increase by relaxing the spatial filtering. In this plot detector $D_2$ is centered at position ${\rho }_{20}\simeq 0.125$ mm and $d_1=0.1$ mm. In all plots, $w=0.2$ mm, $a=0.2$ mm, $z_2=1280$ mm, $z^{\prime }=1130$ mm, and $z=1165$ mm.

Figure 3

Plot of $\gamma$ obtained from the experimental data (solid circles), as a function of the width of detector $D_2$ for a Gaussian pump beam. Slit separation $2a=0.6$ mm. The solid line corresponds to the theory given by Eq. (8), and the dashed line corresponds to the values of $\gamma$ obtained from the single photon measurements (second-order interference). The blue stripe corresponds to the error in $\gamma$ for the second-order interference.

Figure 4

Plot of $\gamma$ as a function of the position of detector $D_2$ for a Gaussian pump beam. Slit separation $2a=0.6$ mm. The solid line corresponds to the theory given by Eq. (8).

Figure 5

Plot of $\gamma$ obtained from coincidence interferograms measured when the pump beam is prepared in the HG${}_{01}$ mode and varying the width of $D_2$. Slit separation $2a=0.6$ mm. The solid line corresponds to the theory given by Eq. (7), and the dashed line corresponds to the values of $\gamma$ obtained from the single-photon measurements (second-order interference). The blue stripe corresponds to the error for $\gamma$ in the second-order interference. The inset shows the interference patterns obtained in the coincidence counts [red curve (circles)] and single counts [blue curve (squares)]. Both curves present minima at the origin, corresponding to negative $\gamma$.

Figure 6

Plot of $\gamma$ for measurements with the HG${}_{01}$ mode pumping, as a function of the position of $D_2$. Slit separation $2a=0.6$ mm. The solid line corresponds to theoretical values of $\gamma$. The insets show the interference fringes observed in the coincidence counts for three distinct positions ${\rho }_2$. It is clearly visible that the magnitude and sign of $\gamma$ change as a function of ${\rho }_2$.

Figure 7

Plot of $\gamma$ for the HG${}_{01}$ mode pumping beam and slit separation $2a=0.4$ mm. The solid line (red) corresponds to theoretical values of $\gamma$. The dashed line (black) corresponds to the theory for $\left|\gamma \right|$, and the blue dot dashed line to values of $\gamma$ for the single photon counting measurements (second-order interference). The blue stripe corresponds to the error for $\gamma$ in the second-order interference.