Analysis of multipath interference in three-slit experiments

It is demonstrated that the three-slit interference, as obtained from explicit solutions of Maxwell's equations for realistic models of three-slit devices, including an idealized version of the three-slit device used in a recent three-slit experiment with light [U. Sinha et al., Science 329, 418 (2010)], is nonzero. The hypothesis that the three-slit interference should be zero is the result of dropping the one-to-one correspondence between the symbols in the mathematical theory and the different experimental configurations, opening the route to conclusions that cannot be derived from the theory proper. It is also shown that under certain experimental conditions, this hypothesis is a good approximation.

Figure 1

Amplitudes of the (a) $E_x$ and (b) $E_z$ components of the electric fields as obtained from a FDTD solution of Maxwell's equation for light incident on a metallic plate with three slits. The incident wave is monochromatic and has a wavelength of $\lambda =500$ nm. The slits are $\lambda$ wide, their centers being separated by $3\lambda$. The index of refraction of the $4\lambda$-thick metallic plate (colored black) is $2.29+2.61i$. In the FDTD simulations, the material (steel) is represented by a Drude model [3].

Figure 2

(a) Normalized angular distribution of light $I(\theta ;OOO)/I(\theta =0;OOO)$ transmitted by $N=3$ slits (see Fig. 1) as obtained from the FDTD simulation (solid circles) and Fraunhofer theory (solid line) $I(\theta ,s=1,d=2,N=3)$ [see Eq. (6)] where $s$ and $d$ are the dimensionless slit width and slit separation, respectively [4]. (b) Normalized difference $\Sigma \left(\theta \right)=\left[I\right(\theta ;OOO)-I(\theta ;COO)-I(\theta ;OCO)-I(\theta ;OOC)+I(\theta ;CCO)+I(\theta ;COC)+I(\theta ;OCC\left)\right]/I(\theta =0;OOO)$ as a function of $\theta$. According to the WH of Refs. [1, 2], this difference should be zero.

Figure 3

Amplitudes of the (a) and (c) $E_x$ and (b) and (d) $E_z$ components of the electric fields as obtained from a FDTD solution of Maxwell's equation for light incident on a metallic plate with two slits and a hole between the two slits. The incident wave is monochromatic and has a wavelength of $\lambda =500$ nm. The slits are $\lambda$ wide, their centers being separated by $3\lambda$. The index of refraction of the $4\lambda$-thick metallic plate (steel, colored black) is $2.29+2.61i$. The holes are $\lambda$ wide and $2\lambda$ deep. (e) Normalized angular distribution of light transmitted by the devices extracted from the FDTD simulation data. The closed circles denote simulation results for $I\left(\theta \right)/I(\theta =0)$ for the slit in the center filled with material half-way from the bottom [see (c) and (d)], the crosses denote simulation results for $I^{\prime }\left(\theta \right)/I^{\prime }(\theta =0)$ for the slit in the center filled with material halfway from the top [see (a) and (b)], and the dashed line is a guide to the eye. On the scale used, the two angular distributions cannot be distinguished. (f) Normalized difference between the angular distributions of the device with the hole in the bottom [(a) and (b)] and the top [(c) and (d)]: $[I\left(\theta \right)-I^{\prime }\left(\theta \right)]/\max [I(\theta =0),I^{\prime }(\theta =0)]$. According to the WH of Refs. [1, 2], this difference should be zero.

Figure 4

Two-dimensional representation of the experiment reported in Ref. [2]. The three slits at the top are $30\mu$m wide, their centers being separated by $100\mu$m. The blocking mask at the bottom can have one, two, or three slits, each slit being $60\mu$m wide with its center aligned with one of the slits in the top plate [2]. In the example shown, the middle slit of the blocking mask is closed (corresponding to the case $OCO$). The separation between the top plate and blocking mask is $50\mu$m. The index of refraction of the $25-\mu$m-thick material (colored black) is $2.29+2.61i$ (index of refraction of steel at 405 nm). The wavelength of the incident light is 405 nm. Also shown are the amplitudes of the (a) $E_x$ and (b) $E_z$ components of the electric fields as obtained from a FDTD solution of Maxwell's equation for a monochromatic light source (not shown) illuminating the blocking mask. Note that the $E_x$ and $E_z$ components propagate in a very different manner.

Figure 5

(a) Angular distribution of light transmitted by the system shown in Fig. 4 for the cases in which all slits are open ($OOO$, solid line), one slit is closed ($COO$ and $OOC$, dashed line; $OCO$, dash-dotted line), and two slits are closed ($CCO$, $COC$, and $OCC$, double-dotted line), as obtained from FDTD simulations. (b) $\kappa$ as a function of $\theta$, as defined by Eq. (12). According to the WH [1, 2], this difference should be zero.