Numerical explorations

3 Longitudinal foci: Meslin’s N-split lens

3.3 Numerical explorations

And hence, the disturbance at origin is

( )

with the Euler’s formula:

⎥⎦⎤

For example, if the focal length f=30000λ, a=3000λ and Δz=300λ, so that the disturbance at origin is

λ

) 150 0

( iA

U = .

3.3 Numerical explorations

In this section, we provide the result of numerical exploration. Without loss of generality, the focal length is taken as f=30,000λ, aperture radius a=3,000λ, and the separation distance along the z axis Δz=300λ or Δz=400λ. The plots of intensity distribution are normalized to 100. If with a He-Ne laser, the wavelength λ=632.8nm, then f=18.984 mm which is a typical lens, and the aperture a=1.90 mm, while Δz=0.18984 or =0.25312 mm are generally available. The observation plane is set at z=0 where the interference pattern can be clearly seen as shown in Fig. 3 where the

intensity distribution and the phase distribution for the case of N=2 (i.e., 2X1) are denoted with Fig. 3-3(1a) and Fig. 3-3(1b) respectively. The result (1a) is well known in literature [3,7]. The corresponding phase Fig. 3-3(1b) shows winding broken strips in distribution where the strip boundary indicates phase jumps as noted by the color changes in the plot. The connection between intensity and phase could be identified by the similarity sharing in the forms of distribution.

Next let us see the intensity distributions of N=6 (i.e., 2X3) and N=10 (i.e., 2X5), which are denoted by Fig. 3-3(3a) and Fig. 3-3(5a), while the corresponding phase distributions are shown in Fig. 3-3(3b) and Fig. 3-3(5b) respectively. Now, the N/2-fold rotational symmetry about the optical axis is clearly observed whereas there

is no rotational symmetry for N=2, though it is still 2/2-fold, i.e., 1-fold. On the other hand, the phase distribution displays the phase changes by π abruptly and as numerically identified that there are only two kinds of value in phase, i.e., ±

π

2. This is because the disturbance in this XY-plane is purely imaginary and this feature was also observed in the focal plane focused by a conventional focusing lens [3]. The origin is mainly the inversion symmetry in imaging for a conventional lens (singlet);

this also leads the same feature to the double-odd case.

Now we change to look on the cases of N=4 (i.e., 2X2), N=8 (i.e., 2X4), and N=12 (i.e., 2X6). The results are dramatically different and they are denoted with Fig.

3-3(2a), (4a), and (6a) for the intensity distribution respectively, while the corresponding phase distributions are labeled with Fig. 3-3(2b), (4b), and (6b) respectively. The phase has N-fold rotational anti-symmetry apart from a factor π about the optical axis in the XY-planes at z=0 as shown in Fig. 3-3 with the labels of Fig. 3-3(2b), (4b), (6b) and the intensity pattern in this plane has N-fold rotational symmetry as denoted by Fig. 3-3(2a), (4a), and (6a). The variations in phase distribution are much wild; the phase values are no more kept with only two values because of the disturbance is not purely imaginary.

In short, one could numerically identify that although the splitting operation is simply with an even number, there are two kinds of distribution and they could be further classified according the number of splitting, i.e., either double-even or double-odd.

Fig. 3-4 plots the intensity and phase distribution in the XY-plane having the same condition with Fig. 3-3 but the separation distance along the z-axis Δz now is 400λ. The embedded symmetry can still be observed but the intensity in the vicinity of the optical axis is faint. This destructive interference is caused by the nearly -180°

Gouy phase shift of between two beams focused by the “shifted” and “un-shifted”

half-lenses.

Fig. 3-5 and Fig. 3-6 show the intensity and phase distribution for N=2 in various XY-planes along the optical axis. From (a) to (i) the observation plane moves in a step

of 50λ from the first focus at F1, to the second focus, at F2. The separation distance along the z-axis Δz now is 400λ. As expected, there is no rotational symmetry property with respect to the optical axis.

The intensity and phase distribution for N=6 are plotted in Fig. 3-7 and Fig. 3-8 having the same condition with Fig. 3-5 and Fig. 3-6, respectively. The symmetry properties around the XY-plane passing through the mid-point of two foci can be readily observed from these figures. The 3-fold symmetry properties with respect to the optical axis are clearly shown in the intensity and phase distribution.

Fig. 3-9 and Fig. 3-10 show the intensity and phase distribution for N=4 and the intensity distribution has 2-fold symmetry and phase has 2-fold anti-symmetry can be readily observed. In the case of N=8, the intensity distribution has 4-fold symmetry and the phase distribution has 4-fold anti-symmetry are shown in Fig. 3-11 and Fig.

3-12, respectively.

Fig. 3-13 shows the on-axis intensity with the separation distance along the z-axis Δz=400λ. The maximum intensity for the two beams are located at F1 and F2 as

shown in Fig. 3-13 but the total maximum intensities are not located at F1 and F2

because of constructive and deconstructive interference. As expected, the intensity in the vicinity of the mid-point of two foci, z=0, are small due to the destructive interference caused by the nearly -180° Gouy phase shift.

Fig. 3-3 Normalized intensity distribution and the corresponding phase one in the XY-plane through the mid-point between two foci where the symbol (a) is for intensity and the symbol (b) is for phase, while (1) for N=2, (2) for N=4, (3) for N=6, (4) for N=8, (5) for N=10, and (6) for N=12. The separation distance along the z-axis Δz=300λ.

Fig. 3-4 Normalized intensity distribution and the corresponding phase one in the XY-plane through the mid-point between two foci where the symbol (a) is for intensity and the symbol (b) is for phase, while (1) for N=2, (2) for N=4, (3) for N=6, (4) for N=8, (5) for N=10, and (6) for N=12. The separation distance along the z-axis Δz=400λ.

Fig. 3-5 Normalized intensity distribution for N=2 at various XY-planes along z-axis.