1. Introduction

It has been known now for some time that the phase of a wavefield can exhibit unusual behavior in the neighborhood of points where the amplitude of the field is zero. This unusual behavior is connected to the fact that the phase of the field is undefined when the amplitude vanishes. The systematic study of such singular points, initiated by Nye and Berry [1], has developed into a vibrant field of optics, now usually called singular optics [2, 3]. The phase of the field in the neighborhood of such singular points can exhibit a rich variety of behaviors, such as vortices and dislocations. Furthermore, it is well-known that these singular points possess certain conserved quantities, such as topological charge, and may only be created and annihilated in ways which satisfy certain conservation laws.

A seemingly unrelated field of research that has been investigated in recent years is the study of anomalous light transmission through sub-wavelength sized apertures in thin plates. Ebbesen et al. demonstrated experimentally [4] that certain arrays of cylindrical cavities in metal plates allow much more light transmission than predicted by the standard theory of aperture diffraction. These authors suggested that this enhanced transmission was generated by the coupling of the light to surface plasmons in the metal. More recently, it was demonstrated [5] that enhanced transmission can be achieved even with the use of a single aperture. A good understanding of the causes and requirements for such enhanced transmission could lead to, among other things, novel near-field optical measurement devices as well as optical storage devices with a density not restricted by the diffraction limit.

In this article we describe two-dimensional numerical simulations we have undertaken of the electromagnetic field in the neighborhood of a sub-wavelength sized slit in a thin metal plate. In particular, the behavior of the time-averaged Poynting vector is analyzed using a rigorous integral equation method. It is found that the field of power flow in the vicinity of the slit typically possesses numerous phase singularities. More importantly, it is found that the annihilation of phase singularities coincides with the onset of extraordinary light transmission through the slit. Furthermore, the incident field is taken to be TE polarized, which does not allow for the excitation of surface plasmons; this demonstrates that enhanced transmission can occur even in their absence. Our analysis suggests that a good understanding of enhanced transmission requires that the behavior of the phase singularities of the field be taken into account.

2. Singular optics of electromagnetic fields

In our analysis, we are interested in the singular optics of a real-valued vector field, namely the two-dimensional time-averaged Poynting vector field,

where Ê and Ĥ are the amplitudes of a complex monochromatic electromagnetic field (of time dependence exp[-iωt]).

The phase Φ_S of the Poynting vector is given by the pair of relations

where |S| is the modulus of S. It follows from these equations that Φ _S (x, z)is singular whenever S = 0. Electromagnetic systems which exhibit singularities of power flow have been known of for some time [6, 7], and the creation and annihilation of singularities in such systems has been studied [8].

For a field which is TE polarized, i.e. the only component of the electric field is Ê _y , it can be readily shown using Eq. (1) and Maxwell’s equations that

We may express Ê _y as an amplitude and phase,

On substituting from Eq. (4)in to Eq. (3), one immediately finds that the Poynting vector may be expressed in the form

This equation suggests that the singular points of S may generally be divided into two categories: those which are related to the singular points of the phase of Ê _y (when |Ê _y (x, z)| = 0)a nd those which are related to the stationary points of the phase of Ê _y (for which ∇ΦE(x, z)= 0). Because these topological features of Ê _y are directly related to the singular points of S, we will briefly review some properties of the singular and stationary points of Ê _y and their relation to the singular points of S.

The phase around a singular point of a complex scalar field is well-known to possess a vortex-like structure (as described for instance in [3]). The phase increases or decreases as one moves about the singular point in a counterclockwise direction; the vortex is then referred to as positive or negative, respectively (see Fig. 1(a)). The phase vortices of Ê _y correspond to vortices (also referred to as centers)of the power flow S, around which the power flow circulates (see Fig. 1(b)). A center is referred to as right-handed (left-handed) if it is counterclockwise with respect to the positive (negative) y-axis. It is to be noted that in all the figures in this article the y-axis points into the page; a left-handed or right-handed center therefore corresponds to a positive or negative phase vortex, respectively.

At stationary points of a complex scalar field, the phase is well-defined but its gradient vanishes. Stationary points includes both minima and maxima as well as saddles, to be referred to as phase saddles (Fig. 1(c)). Phase saddles of Ê _y correspond to saddle points of the power flow, as illustrated in Fig. 1(d). A phase maximum (Fig. 1(e)) of Ê _y corresponds to a sink of power flow (Fig. 1(f)), and a phase minimum corresponds to a source of power flow; it is to be noted that sinks and sources do not occur in free space.

Both phase singularities and stationary points are topological features of the complex field Ê _y , and several conserved quantities can be associated with each topological feature. The first of these is the so-called topological charge s_E of the field, defined as the integral of ∇Φ _E around a closed loop enclosing the feature such that

where C is a closed counterclockwise path of winding number 1. It can be shown that the topological charge of a given phase singularity takes on a unique positive or negative integer value, independent of the choice of the enclosing path C. Likewise, the topological charge of a phase saddle, maximum or minimum is always zero.

Another quantity of interest is the topological index t_E , which is defined as the topological charge of the phase singularities of the vector field ∇Φ _E . It can be shown that for a positive or negative vortex t_E = +1, while for a phase saddle t_E = -1. The topological index of a phase maximum or minimum is t_E = +1.

 

Fig. 1. Illustrating the relation between phase singularities (a), stationary points of the phase (c,e), and the corresponding singularities of the power flow (b,d,f). The arrows in the left-hand column indicate the direction of increasing phase Φ _E .

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A topological charge s_S and index t_S can also be associated with the phase Φ _S of the power flow. It follows directly from Eq. (5)t hat the topological charge of S for a given feature is equal to the topological index of Ê _y . The topological charge of a vortex of power flow is therefore s_S = +1 regardless of whether it is a positive or negative vortex of Ê _y . Similarly, the topological charge of a saddle point of power flow is s_S = -1, and the topological charge of a source or sink is s_S = +1. A topological index may be defined for the singularities of power flow, but is not necessary for our interests and will not be considered here.

Both topological charge and index are quantities which are conserved under smooth variations of the configuration parameters, and as such can only appear or disappear via creation and annihilation of multiple stationary and/or singular points. The most common process, which we will be primarily concerned with, involves the creation (annihilation) of a positive vortex (s_E = +1, t_E = +1), a negative vortex (s_E = -1, t_E = +1), and two phase saddles (s_E = 0, t_E = -1 for each). This event may also be described in terms of the field of power flow as the creation (annihiliation)o f two centers of opposite direction (s_S = +1 for each) with two saddle points (s_S = -1). Other, more complex, events are possible, but are not typical.

3. Integral equation solution for the electromagnetic field near a slit

The configuration under consideration is illustrated in Fig. 2. An monochromatic electromagnetic wave is normally incident upon a metal plate of thickness d and permittivity ε_plate from the negative z-direction. A single slit of width w, infinitely long in the y-direction, is present in the plate. Because the system is invariant with respect to y-translations, we may treat the problem as two-dimensional, with relevant coordinates x and z.

 

Fig. 2. Illustrating the notation relating to transmission through a slit.

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The total electric field Ê may be written as the sum of two parts, namely the incident field, Ê^inc, and the scattered field, Ê^scatt. The incident field is here taken to be the field that would occur in the absence of the slit in the plate; it can readily be calculated analytically by use of the electromagnetic boundary conditions.

It can then be shown [9] that the ith component of the total field, denoted by Ê _i(x, z), satisfies an integral equation of the form

where Δε= ε0 - ε_plate is the difference between the vacuum permittivity and the permittivity of the metal plate, ${Ĝ}_{\mathit{\text{ij}}}^E$ is the electric Green’s tensor pertaining to the plate without the slit, and the integration is over the domain of the slit D. For points which lie within the slit, Eq. (7) is a Fredholm equation of the second kind for Ê, which can be solved numerically by the collocation method with piecewise-constant basis functions. The electric field outside the domain of the slit may then be calculated by substituting this solution back into Eq. (7). With the electric field determined everywhere, the magnetic field everywhere follows directly from Maxwell’s equations. The Poynting vector may then be calculated using Eq. (1).

The definition of the transmission coefficient of the slit consists of two parts: the first is the integral of the normal component of the actual time-averaged Poynting vector S over the slit, and the second is the difference of the normal components of the actual time-averaged Poynting vector and that of the Poynting vector in the absence of the slit, S ^inc, integrated over the dark side of the plate (not the region of the slit). The result is normalized by the normal component of S ⁽⁰⁾, the Poynting vector of the field emitted by the source and impinging on the slit, i.e.

The subtraction in the second integral in the numerator corrects for the small part of the incident field which may flow through the plate itself.

With calculations of the power flow in the neighborhood of the slit and of the transmission coefficient, we are now able to demonstrate the relation between phase singularities and enhanced transmission (see also Ref. [10]).

4. Phase singularities near sub-wavelength slits

An example of the field of power flow (i.e., the time-averaged Poynting vector) near a narrow slit in a thin plate of evaporated silver is shown in Fig. 3. In this example the incident field is taken to be TE polarized (i.e., with the Ê field parallel to the slit). It follows from the structure of ${Ĝ}_{\mathit{\text{ij}}}^E$ that the scattered field is then also TE polarized implying that the analysis of Section 2 applies. It can be seen from Fig. 3 that the field exhibits several phase singualarities, namely vortices and saddles. In addition, the aperture is seen to have a funnel-like effect on the field, corresponding to a transmission coefficient T = 1.11. When the slit width is increased in a continuous manner, the four singularities below the slit (in the region indicated by the box)mo ve together and finally annihilate each other. After the annihilation takes place, a smoother field of power flow results, corresponding to a greater transmission coefficient. This process is shown in Fig. 4. It can be seen that the transmission takes on a maximum value of T = 1.33 when the slit width is w = 0.5λ.

Figure 5 shows the location of different kinds of phase singularities for the same configuration on a larger scale. In Fig. 6 the creation, movement, and subsequent annihilation of the phase singularities of the field of power flow are shown when the slit width is gradually increased. The slit width w and the corresponding power transmission coefficient T are also shown. Three groups of singularities obstruct the power flow directly in front of the slit; these groups each annihilate at w = 0.42λ, w = 0.43λ, and w = 0.46λ. Only when they have disappeared does the transmission show anomalously high behavior.

The two movies presented here clearly show that anomalous light transmission through a sub-wavelength slit is accompanied by annihilation of singularities of the field of power flow in the immediate neighborhood of the slit. It is to be noted that, for the choice of polarization that was analyzed, no surface plasmons are generated in the metal plate. Our study demonstrates that anomalous light transmission can even occur in their absence.

 

Fig. 3. Illustration of the power flow in the neighborhood of a 200 nm wide slit in a 100 nm thick plate of evaporated silver, with wavelength λ = 500 nm and n = 0.05+i2.87 (the value of the refractive index wastak en from [11]). Features (a) and (d) are left-handed centers, (b) and (c) are right-handed centers, and (e) and (f) are saddles. For this example the transmission coefficient T = 1.11. The color coding indicatesthe modulus of the (normalized) Poynting vector. The dashed box indicatest he region illustrated in Movie 1.

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Fig. 4. (186 KB) The field of power flow asa function of the slit width w in the region indicated in Fig. 3. The four phase singularities move together as the slit width is increased, and finally annihilate, leaving a smoother field of power flow corresponding to a higher transmission coefficient T.

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Fig. 5. Array of singularities in the field of power flow in the neighborhood of the slit. All parametersar e as given in Fig. 3. Left-handed vortices(cen ters) and righthanded vortices (centers) are denoted by LV and RV, respectively, and saddles are denoted by S.

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Fig. 6. (212 KB) Schematic of the position and type of singularities of power flow near a sub-wavelength slit. It is to be observed that multiple creation and annihilation events occur as the slit width is gradually increased. Left-handed vortices (centers) and right-handed vortices (centers) are denoted by LV and RV, respectively, and saddlesar e denoted by S. U denotesa vortex (center) very close to a saddle point which cannot be spatially resolved by the particular grid used for these calculations; it can be seen that eventually the singularities separate sufficiently to be distinguishable.

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