Nearfield properties of light emanated from a subwavelength double slit of finite length in a thin metal film, which is essential for understanding fundamental physical mechanisms for nearfield optical beam manipulations and various potential nanophotonic device applications, is investigated by using a threedimensional finitedifference timedomain method. Nearfield intensity distribution along the propagation direction of light after passing through the slit has been obtained from the phase relation of transverse electric and magnetic fields and the wave impedance. It is found that the near field of emerged light from the both slits is evanescent, that is consistent with conventional surface plasmon localization near the metal surface. Due to the finite of the slit, the amplitude of this evanescent field does not monotonically approach to than of the infinite slit as the slit length increases,
i.e.
the nearfield of the longer slit along the center line can be weaker than that of the shorter one.
I. INTRODUCTION
The famous experiment on the interference of light diffracted by a double slit was performed by Thomas Young, who found that unique interference patterns were related to light sources at a specific separation distance between two slits
[1
,
2]
. This experiment also inspired the development of the particlewave duality
[3]
, which became again a trigger for building up the concepts in quantum mechanics.Although this experiment was initially performed more than two centuries ago, even nowadays, intensive research activity on double slit interference is still in progress to explore some novel physical phenomena, including for instance, the light interference characteristics due to a coherence degree of two light sources
[2
,
4
8]
, analyses of geometrical effects leading to enhanced light transmission and propagation[
9

11
]. Recent significant attention to the surface plasmons
[12]
has been also related to the double slit interference. The interference pattern produced by the double slit in a plasmonic metal film is quite different from the traditional ones due to the surface plasmon generation and localization near the metal surface at optical wavelengths
[2
,
4

7
,
13

15]
. While the plasmonic double slits have also been utilized for novel subwavelength optics applications such as unidirectional surface plasmon excitation
[16
,
17]
, optical beam manipulations
[18
,
19]
, and subdiffraction limited optical spot generation in the intermediate field region
[20
,
21]
, the studies on the a subwavelength double slit are still inactive compared with those on the single slit structures. Furthermore, most work done on subwavelength slit structures has been limited to twodimensional approximations or to very long slits compared with operating wavelengths, though finite slit effect is critical to the light interference in practical structures which is essential in the performance of the subwavelength slit based optical devices.
In this work, fundamental nearfield behavior of light diffracted by a subwavelength double slit of finite length in a a thin metal film investigated using an intensive threedimensional finitedifference timedomain (3D FDTD) method
[22]
; in particular, the nearfield intensity distributions are obtained from phase relation and field peak ratio between the the electric and magnetic fields transverse to propagation direction. Analysis of the finite slit length effect on the nearfield intensity distributions and the comparison of those with twodimensional infinite slit length case
[23]
allow us to gain further insights into the physics of the subwavelength plasmonic double slit, which can be potentially useful for various optical devices dealing with subwavelength optical beam manipulation or light focusing in the nearfield region.
II. subWAVELENGTH PLASMONIC DOUBLE SLIT OF FINITE LENGTH AND 3D FDTD SIMULATION SETUP
Figure 1
shows a schematic of the subwavelength double slit with finite slit length, denoted as
sl
, perforated in a thin metal film placed in air. Without loss of generality,the
x
polarized monochromatic plane wave of
λ
=633 nm wavelength is normally incident upon a thin metal film in the 
y
direction from the upper region. Individual slit width is 80 nm which is much smaller than the incident wavelength. The metal used in this numerical experiment is silver (Ag) with the dielectric constant of
ε_{r}
= 15.91 +
i
1.07 at 633nm
[24]
. Due to this negative value of the real part of the dielectric constant, surface plasmons can
Schematic of the subwavelength double slit of finite length in thin metal film. The observation plane (the yzplane at x=0) in which electromagnetic field distributions are going to be obtained is shown below the metal film. The propagation distance r originates from the bottom surface of the metal film.
be supported at the interface between the bottom surface of the silver film and air, which is one of the reasons that the nearfield characteristics are different from those of conventional double slit in a perfect electric conductor(PEC). Thickness of the metal film is 200nm, which is thick enough to block the light passing directly through the metal film and at the same time it is also less than halfthewavelength to eliminate the FabryPerot resonance in the longitudinal
y
direction inside the slits
[25]
. The slit separation is 320nm, which is about halfthewavelength,where optical transmission is known to be stronger than those for other separation distances between the slits
[26
,
27]
. After passing through each slit, the light emerges at the center position between slits,
i.e.
at the observation plane in
Fig. 1
, due to the interference of light emanated from each slit, which is a merit of using double slit over the single slit in terms of light focusing or manipulating the light on subwavelength scale. The interference pattern formed by this subwavelength double slit can be influenced by the finite length of the slit. So, as one can expect, the nearfield distribution can be changed with respect to the slit length, which is the main focus of this work.
In order to investigate the nearfield characteristics of interfered light after passing through the slits in the metal film, the 3D FDTD simulations were performed using a freely available software package [28], in which material dispersion for real metal has been implemented. Entire computation domain has 1400×1400×400 cells in
x
,
z
,and
y
directions, respectively, with the Yee’s spatial unit cell of 5 nm. The perfectly matched layers have been set up at all boundaries for the fullwave simulations.
III. NUMERICAL RESULTS AND DISCUSSION
Figure 2
shows snapshots of the xcomponent of the electric field
E_{x}
and zcomponent of the magnetic field
H_{z}
for three different slit length cases, in the observation plane shown in
Fig. 1
. Those are taken when the Hz field is close enough to zero at the bottom surface of the silver film, as indicated in
Fig. 2
(a), (b), and (c). Here, both the electric and magnetic fields of the incident plane wave have been normalized to unity. The positions of the slits with different lengths have been denoted as solid lines in the upper region. The numbers on the ordinate are the distances originated from the bottom surface of the silver film,
i.e
. the distance
r
shown in the observation plane in
Fig. 1
. Even when the magnetic field is zero at the bottom surface of the silver film, the electric field is rather far from zero (this will be briefly discussed below), causing the nonzero value of the nearfield intensity. Comparing the cases of
sl
=600 nm and
sl
=1200 nm in
Fig. 2
(d) and
Fig. 2
(e), the intensity of the first focused spot near
r
=300 nm and the field near the bottom surface of the silver film are seen to be stronger for the longer slit case.
Snapshots of the electric and magnetic fields when the magnetic fields at the exit of the metal film are close to zero for different slit lengths of sl=600, 1200, and 1800 nm in the observation plane as shown in Fig. 1, i.e., horizontal and vertical axes are z and ypositions, respectively. [(a)(b)(c): the zcomponent of the magnetic field (H_{z}), (d)(e)(f): the xcomponent of the electric field (E_{x}),and (a)(d): sl=600 nm, (b)(e): sl=1200 nm, (c)(f): sl=1800 nm].
However, the focused spot for the
sl
=1800 nm case splits as shown in
Fig. 2
(f), and the field intensity along the central line becomes smaller than that in previous (shorter length) cases. This peak splitting is thought to be a result of nonuniform field distribution along the slit direction which in turn can result from the excitation of higherorder TEm0 eigenmodes of the cavity. In other words, the splitting can be considered as resulting from the interaction of fields emanated from each slit, while these fields are associated with corresponding eigenmodes and hence should depend on the slit length. More detailed discussion on this splitting for the finite length slit and its physical mechanism will be reported elsewhere. Obviously, a correct treatment of the problem should include such phenomena as nearfield interaction between modes and mode mixing.
Figure 3
shows the intensities of the
E
_{x}
and
ZH
_{z}
fields versus the normalized propagation distance
kr
, where
Z
is the complex wave impedance and
k
is the free space wave
The E_{x}, H_{z}, and nearfield intensities along the propagation direction at the center position (z=0) in the observation plane (shown in Fig. 1) for (a) sl=600 nm, (b) sl=1200 nm, and (c) sl=1800 nm. The nearfield intensity distributions along the propagation direction extracted from the subtraction between the E_{x} and ZHz fields.
number, at the center position (
z
=0) in the observation plane shown in
Fig. 1
and also denoted as dotted lines in
Fig. 2
. In the nearfield region, the electric and magnetic fields are out of phase. However, beyond the nearfield region, the ratio and the locations of their peaks allow us to determine the impedance
Z
, since the focused
E
_{x}
field has similar distributions with the
H
_{z}
field and they are expected to be almost in phase
[29]
. Because the instants when the
H
_{z}
fields are close to zero at the bottom surface of the silver film have been taken as shown in
Fig. 2
, the
ZH
_{z}
field can be used to estimate the Ex field distributions in the near zone without the nearfield contributions (the real Ex field includes inherent nearfield contributions from the surface plasmon supported by the silver / air interface).Thereby, the evanescent intensity distributions along the normalized propagation distance kr can be extracted from the difference between the
E
_{x}
and
ZH
_{z}
fields as shown in
Fig. 3
. According to this procedure, the nonzero evanescent fields have been obtained. That is a major difference from the case of the PEC film,
i.e
. the evanescent field intensity in the case of the PEC film is nearly negligible
[23]
.While the
E
_{x}
and
ZH
_{z}
field profiles in
Fig. 3
(a) and
Fig.3
(b) above the peak positions are almost the same as expected, a deviation between both fields for the
sl
=1800 nm case is seen about
kr
>3.0 as shown in
Fig. 3
(c). This is probably because the peaks are split and the focusing at the center position is not very strong as compared with the previous two shorter slit cases as shown in
Fig. 2
(c) and 2 (f).
For the twodimensional case,
i.e.
infinite slit length, the electromagnetic field distributions are supposed to be uniform at the same lateral position because there is no dependence along the zdirection. However, in the current threedimensional case of finite length slit, the electromagnetic field distributions are seen to be changed due to the finite slit length even though measured at the same position,
e.g.
, here at the central line in the observation plane. It is probably because the surface charge and surface current distributions at the bottom surface of the silver film are not uniform any more along the slit length direction due to the finite length of the slit, which eventually makes the light interference more complex than that in the twodimensional infinite slit length case. The obtained nearfield intensities decrease evanescently as the normalized propagation distance
kr
increases and become negligible above certain normalized propagation distance kr, that is consistent with conventional nearfield localization of the surface plasmons. For
sl
=600 nm (about 1
λ
),the nearfield intensity become negligible at about kr>1.5.Meanwhile, for
sl
=1200 nm (about
2λ
), the range of the negligible nearfield intensity shifts to about
kr
>2.8,meaning stronger surface plasmons are being supported for this slit length. However, for
sl
=1800 nm (about 3
λ
), the nearfield intensity becomes considerably reduced to be even weaker than that for
sl
=600 nm case and the boundary of the negligible nearfield intensity range is at
kr
>1.1,which means that the nearfield characteristics of the longer but finite slit may not be closer to those of the infinite slit length case. This is because the peaks of the fields at z=0 in the observation plane split and shift towards both
The evanescentfield intensity obtained as in Fig. 3along the propagation direction at the center position in theobservation plane.
directions along the zaxis due to an oscillation of the fields along zdirection as shown in
Fig. 2
(c) and
Fig. 2
(f), resulting in even weaker focused fields near the center position than those in the previous two cases as mentioned.
Figure 4
shows the evanescent field intensities, though each of them is already shown separately in
Fig. 3
, for a clearer comparison with each case. The evanescent field intensities at the bottom surface of the silver film,
i.e.
, at
r
=0, vary with the slit length and are also stretching with the distance
r
to the propagation direction but decrease evanescently when
kr
increases as mentioned above. The evanescent field intensity for
sl
=1200 nm (about 2
λ
) is stronger than for the other two cases and is stretched further also as compared with the
sl
=600nm and 1800nm cases; it is also closer to that of the twodimensional infinite slit length case [
23
] as compared with the other two cases.
Such a behavior of the evanescent field intensity can be a result of the contribution of higherorder eigenmodes. In fact, as the slit length increases, more and more higher order modes can occur in the cavity. Obviously, the odd modes are favorable for increasing the evanescent field at the center, while the even modes are favorable to decrease it. As the slit length approaches infinity, the evanescent field intensity should naturally approach to that of twodimensional case. However, this approaching is oscillating rather than monotonic because it involves more and more higher order eigenmodes.
IV. CONCLUSION
In conclusion, the nearfield characteristics of light from a subwavelength double slit of finite length in a thin metal film have been investigated by using 3D FDTD method.Nearfield intensity along the symmetry axis at the center position of the slit direction has been obtained with phase relation of the transverse electric and magnetic fields and the ratio between them. It is found that the evanescent field distribution is strongly dependent on the slit length.As is shown, the evanescent field intensity distribution along the propagation direction may not necessarily increase toward that of the ideal twodimensional infinite slit length case as the slit length increases. That might be significant in terms of designing practical subwavelength metal slit based plasmonic devices and the fundamental physics behind it as well. This effect may be attributed to the contribution of higherorder eigenmodes of the cavity, which are capable of critically changing the field distribution in the near zone.
Acknowledgements
This work was supported by the Ministry of Education,Taiwan, R.O.C., under the NCKU Project of Promoting Academic Excellence & Developing World Class Research Centers and the National Science Council. We are grateful to the National Center for HighPerformance Computing for the computer time and facilities.
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