## Abstract

A computer program based on the finite element method is used
to study variations in pit visibility for a pit structure that is similar to those
used in TwoDOS systems. It is concluded that pit visibility is best enhanced
by making the pit width larger, and that destructive interference by making
pit depth d =λ_{Poly}/4 (where λ_{Poly} is the wavelength in Polycarbonate) does not play a major role. Also, pit visibility depends strongly on the thickness of the Al layer. The simulations are compared with experiments and with a scalar model.

© 2007 Optical Society of America

## 1. Introduction

Nowadays, in many opto-electronic devices, structures are used that are of the order of the wavelength of light. In this case, theoretical calculations on these structures using geometrical optics do not yield valid results anymore. Instead, rigorous solutions of Maxwell’s equations are required to obtain results that can be compared with experiments. Recently, at Philips Research Laboratories, a computer program was constructed, that can perform calculations for problems that involve the diffraction of light by structures that are of the order of the wavelength of the light that is used [1]. These structures can be layers with various indices of refraction, with scatterers in them, that can be non-periodic, or 1D-, 2D-, or 3D-periodic. Thus it is more powerful than the previously used program [2], which, although the electromagnetic fields can be 3-dimensional, can only handle scattering structures that are periodic in the *x*-direction, and uniform in the *y*-direction. The calculations in the program are performed with the aid of the finite element method [3]. A brief discussion of the method that is used is given in Section 4.1. The new program has already been used for calculations on optical recording [1] and lithography [4]. In this paper we will concentrate on the scattering of light by a structure of very small pits that are covered by a layer of Aluminum (Al). Such a structure is used, e.g., in the so-called TwoDOS (Two-Dimensional Optical Storage) systems [5]–[7]. These systems use a parallel reading of data. They allow larger storage densities by a factor of 2 and, in particular, much larger bit rates that the present systems, that use sequential reading and writing. We will give a description of the characteristics of the TwoDOS system in Section 2.

## 2. TwoDOS characteristics

In the TwoDOS setup, storage is, as is usual in optical storage, provided by a structure of “pits” and “lands” (i.e. the absence of a pit). Usually, the pits are embossed into a structure of polycarbonate (Poly) with the aid of injection molding. The pits and lands are then covered with a layer of Al and, on top of this, a coverlayer of a UV curable material is put. This material is chosen in such a way that the index of refraction is almost equal to that of Poly. Therefore, in the rest of our article, we will assume that the material above and below the Al layer is Poly. The thickness of the Al layer has, so far, not been optimized for optical contrast. The thickness of the cover layer can be in the order of 100 μm. In our model we assume that the layer is thick enough to be regarded as being infinite in the near-field calculations, but so thin that our lens (with focal distance *f* = 100λ, where λ is the wavelength of the light) can be placed above it in vacuum. (Thus we can show the far field figures in vacuum).

In the TwoDOS concept a number of methods is used to increase storage capacity:

- • The wavelength λ of the light of the incoming spot is chosen small. Typically, we have
This is also the wavelength that is used in a blue ray disc (BD) system.

- • The lens that is used to focus the incoming light has a high numerical aperture. Typically,
- • In practice, circularly polarized light is used, which is a combination of two directions of linear polarization. In our simulations, to be able to discern the effects of these two directions of polarization, we used linearly polarized light. We define a coordinate system in such a way, that the
*z*-axis is the optical axis of the system, and that the*x*- and*y*-axes are perpendicular to it. When an*x*-polarized plane wave is focussed by the lens, a spot occurs in the focal plane, that is predominantly polarized parallel to the*x*-direction, and has the shape of an Airy function. This type of spot will henceforth be called called “*x*-polarized Airy spot” for short. The spot has a full width half maximumLikewise we can create a “

*y*-polarized Airy spot”. The effects of a circularly polarized spot can be derived from those of the linearly polarized spots. Because in the case of one circular spot, the effects of*x*- and*y*-polarized spots should be similar, we can in this case compare the resulting far field intensities that are computed, directly with the measurements. - • The pits and “land” structures are organized into a hexagonal 2D lattice, which is a close packed structure in 2 dimensions for circular structures (hence the name Two Dimensional Optical Storage). This is different from the previously used method of optical storage, in which single (one-dimensional) tracks of pits were used. In current practice, the 2D lattice of the TwoDOS system is organized into a “supertrack” of 7–11 single tracks wide. These tracks will be read simultaneously by a number of laser beams, which increases the read-out rate. Typically, the hexagon separation has a value [6], [7]
which corresponds to a factor of 2 relative to a BD system. Ideally, the pits are circular with a width that should fit into the hexagonal lattice. The width should be chosen in such a way, that the contrast between a pit and a land structure is as high as possible. This means that the pit width cannot be too large, because otherwise the signal emanating from a number of adjacent pits will become the same as that of a land structure [6]. On the other hand, the pit width should be large enough for the pit to still be visible. In practice the pits are often elliptical with a width varying between

*b*= 80 nm and*b*= 120 nm. Thus, there is roughly 50% “pit area” and 50% “land area” in a “pit bit structure” (and of course 100% “land area” in a “land bit structure”. - • By lack of rigorous analysis so far, it was assumed (see [6], [7]) that, to obtain maximum modulation, the pit depth should ideally be$$d=\frac{\lambda}{4{n}_{\mathit{Ploy}}}=\frac{405}{4\cdot 1.619}\approx 62.5\phantom{\rule{.2em}{0ex}}\mathrm{nm},$$
so as to make the destructive interference between the light that is reflected from the bottom of the pit, and that which is reflected from the land, as large as possible. (

*n*= 1.619 is the index of refraction of the Poly). However, because the pit diameter is smaller than the cut-off value for the guiding of waves in cylindrical waveguides with a perfectly conducting wall [8],_{Poly}$${b}_{\mathit{min}}=\frac{\lambda}{2{n}_{\mathit{Poly}}}\approx 125\phantom{\rule{.2em}{0ex}}\mathrm{nm}$$this assumption could be incorrect. In fact, this seems to be confirmed by our simulations (see below).

- • The thickness of the Al layer should be chosen in such a way, that the difference in contrast between a land and a pit structure is high, but is still not too sensitive to variations in the thickness of the Al. Typically, the layer thickness of a disc can be chosen to lie between
*t*= 5 nm and*t*= 25 nm, with a variation of 1 nm around the chosen thickness. Thus, the optimal thickness will be a compromise between visibility and precision of deposition.

In practice, the pit wall usually has a slope of around 80*°*, so that the bottom width of a pit is smaller than the top width. This could decrease the visibility of the pit somewhat. As we will see in Section 4.3.3, however, the decrease in visibility is not so large.

## 3. Measurements performed so far; scalar theory

Measurements for TwoDOS systems have been performed on a few types of pit structures [5], [7]. In these references, the intensity *P* of the reflected signal was measured as function of the average value of the amount of hexagons “filled” in a structure of a central hexagon and its nearest neighbor hexagons. Each hexagon could be “filled” with either a “pit” or a “land”. *P* was then divided by the intensity *P*
_{0} of an “all-land” structure. In [5] the pit diameter was about 70 nm. Here it is shown that for one central pit and no pits in the nearest neighbor hexagons, *P*/*P*
_{0} ≈ 80–90%, while for a two-pit structure it is approximately 80%.

In ref. [6] and [7] a “scalar model” is presented, which determines *P*/*P*
_{0} through the phase differences of the plane waves that are reflected by the bottom of the pits and those that are reflected by the “land” structures (hence it is based on geometrical optics). By comparison of Fig. 8 in ref. [5] with Fig. 7 in ref. [7] it appears that the ratio *P*/*P*
_{0} of the scalar model is somewhat lower than that of the measurements. Because, strictly speaking, geometrical optics could very well be not reliable for length scales of the order of the wavelength, it is useful to check what results are generated by a rigorous solution of the Maxwell equations by using computer simulations. As we will see below, it follows from our simulations that variation of the pit depth has only a small influence on the value of *P*/*P*
_{0}, so that it appears that the mechanism of the variation of the pit depth, as suggested by the scalar theory, does not play a major role in determining *P*/*P*
_{0}.

## 4. Computer simulations of TwoDOS pit structure

#### 4.1. Finite element program

In this Section we give a brief description of the methods used by the finite element program. For a more extensive treatment the reader is referred to [1]. Throughout this paper we assume that the time dependence of the fields is given by the factor exp(-*iωt*). We omit this factor from all equations. The program is made to solve the Maxwell equations for the total electric (**E**) and magnetic (**H**) fields:

Here *ϵ _{r}*͇ and

*μ*͇ are the relative permittivity tensor, and relative permeability tensor, respectively. (We will take

_{r}*μ*͇ = 1 in our simulations).

_{r}**J**and ρ are the known external current and charge density, respectively. From equations (7) and (8) the vector Helmholtz equations can be derived:

The boundary conditions can be derived from the fact that the difference of the total and the known incident field satisfies Sommerfeld’s outgoing radiation conditions [13]. These equations, together with the boundary conditions, are solved with the aid of the finite element method. First, a computational domain Ω is defined. The computational domain should be chosen such, that the illuminated pits are contained in it. In particular, if there is only one pit, for accuracy Ω should be so large that it contains the pit but may otherwise be as small as possible. In our case we chose Ω to be a rectangular block. This domain is surrounded by a so-called Perfectly Matched Layer (PML) [9]. In the PML the field equations are modified so that at the boundary of Ω and the PML no reflection occurs, and so that the scattered field that is transmitted into the PML is absorbed and falls off to zero. On the outer boundary of the PML the fields are set equal to zero. In this way, the computed total fields inside Ω are the same as when the total computational domain would extend to infinity. The use of a PML also allows us to perform calculations on computational domains that are smaller than the size of an incoming spot [1]: First the solution is calculated for the case of a spot that is incident on a geometry consisting only of the multilayer of the system, and without the scatterers. This is called the “zero-field” **E**
_{0}. Once this field is known, the “scattered field” **E**
* _{s}* is defined to be

**E**
* _{s}* now has to satisfy a set of equations that are similar to the Maxwell equations, but which contain a source term that is contained within Ω. Therefore, the calculation of the solution of

**E**

*can be restricted to Ω and the PML. Once*

_{s}**E**

*is known, the solution from the total field within Ω follows from Eq.(13). In order to calculate the solution of the fields, the computational domain is discretized into a tetrahedral grid. For this, an automatic grid generator, belonging to the package*

_{s}*SEPRAN*[10] is used. The elements we use are lowest order curl conforming tetrahedral Nédélec elements of the first type [11]. The number of elements is typically of the order of 10

^{5}. Therefore, an iterative solver must be used. We use BCGSTAB, and ILUTP [12] as a preconditioner. Because the pit structures (including the Al layer) are quite complex, and because the finite element grid has to be fine enough to yield reliable results, only structures with one or two pits could thus far be simulated. However, the “structural parameters” (pit width, pit depth and thickness of the Al layer), could be changed without great effort, and this could lead to useful information about the visibility of the pits as a function of these parameters. In the finite element method, materials with permittivity with negative real parts, (i.e. metals), cause no problems.

#### 4.2. Macroscopic set-up

By performing the computer calculations we wanted to investigate what the effect was on the “visibility” of a pit structure, when the parameters mentioned in Section 2 were varied. To this end, we devised the following set-up for the computer simulations: an *x*-polarized or *y*-polarized plane wave passes through a lens with *NA* = 0.85 and focal distance *f* = 100λ. The resulting Airy spot is predominantly polarized along the *x*- or *y*-direction, and will be called the “*x*-polarized Airy spot” or “*y*-polarized Airy spot” for short. This spot is focussed on top of the sample where the light is refracted. Subsequently the “total near field” is calculated in the vicinity of the sample. In addition the reflected “far field” is calculated with the use of the Fraunhofer approximation [13]. The focussing by the lens of the polarized plane wave was simulated with the aid of the program *DIFFRACT* [14], using the model of Ignokovski and Richards and Wolf. The rotation of the polarization by the lens is hereby taken into account.

Within *DIFFRACT*, the beam is confined within the lens pupil, i.e. a circular aperture of radius

and the maximum absolute value of *E _{x}* (i.e. the absolute value at (

*x*,

*y*) = (0,0) given by:

Our finite element program is able to calculate all components of the total electric and magnetic field, as well as the time average of the Poynting vector of the scattered far field, which is defined by:

where ∗ means complex conjugation. This vector yields the time averaged energy flow as a function of **r**. In appendix A an expression for **S**(**r**) is given in Eq.(35). As can be seen from Eq.(35), the quantity *r*
^{2}∣**S**(**r**)∣ is independent of *r*, and depends only on the direction *r*̂ of the vector **r**, as should be in the Fraunhofer approximation. Therefore, in our figures of the far field, we show *r*
^{2}∣**S**(**r**)∣ (we will denote this as “the flux”). Usually, in an experimental set-up, the integrated energy flow of the far field (for *z* > 0) is measured. This quantity is defined by:

Here *r*, *θ* and *ϕ* are spherical coordinates. In Appendix A an expression of *P* as a function of **E**(**r**) and **H**(**r**) in the Fraunhofer approximation is given. In our discussion of the simulations we will mainly focus on the dominant component of the electric near field (*E _{x}*(

**r**) or

**E**(

_{y}**r**)), and on the fraction of the power

*P*in the far field, that will pass through the lens again. For this we will need that in the integral in Eq. (17), only those (plane wave) components will be taken into account, for which

As we will see in Figs. 4, 8, 11 and 17, hardly any reflected light occurs outside the lens pupil, when there is no pit present, or when the pits are much smaller than the wavelength of the light. When the spot width becomes of the order of the wavelength of light, however, scattering to the exterior of the lens pupil becomes more prominent, as can be seen in Fig. 13. In this case, integrating over all space to determine P would not be correct anymore.

### 4.3. Simulations

### 4.3.1. Parameters used

We simulated a number of structures, consisting of Poly and Aluminum (Al). For comparison, in the case of a layered structure, we also made a simulation with a perfect conductor. The index of refraction of these materials is given in Table 1. In the cases where a flat layer or a single pit was considered, the incident field consisted of the “*x*-polarized Airy spot” that was mentioned above. (Wavelength λ = 405 nm; the numerical aperture *NA* = 0.85). In the cases where two pits were considered, the spot could be either *x*- or *y*-polarized. We defined both spots on a grid of (*N _{x}*,

*N*) = (512,512) points, with a width of 51.46 wavelengths. A plot of ∣

_{y}*E*∣ and ∣

_{x}*E*∣ for the case of an

_{y}*x*-polarized Airy spot is given in Figs. 1 and 2, respectively. The computational domain Ω had a length as well as a width of 70 nm in the case of a flat layer or a single pit structure, and a lengtgih of 290 nm and a width of 70 nm in the case of two pits. The PML had a width of 50 nm in all cases. (Note that the spot width (= 238 nm) is larger than the computational domain Ω. We stress that nevertheless, as explained in section 4.1, the incident field in the complement of Ω is taken into account). Hence, as long as the pits are contained in Ω, the accuracy is not improved when Ω is increased. The number of Nédélec elements per vacuum wavelength was

*N*= 30 for the Poly, and

_{elements}*N*= 120 for the Al in the case of a layered structure or a 1-pit structure, and

_{elements}*N*= 12 for the Poly, and

_{elements}*N*= 80 for the Al, in the case of a two-pit structure. The number of elements per wavelength in the PML was

_{elements}*N*= 5 in all cases, and the complex stretching parameter [1] was chosen to be ζ= 15+15

_{elements}*i*. This resulted in a matrix with 1.5∙10

^{5}to 3.5∙10

^{5}unknowns. The memory needed was of the order of 20GB. Solving the system required computer times of between 3000 and 5000 seconds on a

*LINUX HP-DL585*machine with 48 GB memory and 2.4 GHz processors.

### 4.3.2. Layered structure

In order to get a first impression of the amount of light that is reflected by the Al layer, we first simulated the reflection of the Airy spot by a structure consisting of a flat Al layer of variable thickness, that was situated in between two half-infinite layers of Poly. We also made a computation for the case that the layer consisted of a perfect conductor. In this case the reflection is 100%, of course. An example of the x-component of the electric near-field *E _{x}* in the

*xz*-plane for an Al layer of 10 nm is given in Fig.3.

In Fig.3 the field in the PML, surrounding the computational domain Ω is also shown. As can be seen from the figure, the field in the PML decreases to 0 rapidly. Because we want to concentrate on the physical aspects, the field in the PML is not shown in the subsequent figures. For a structure consisting of three layers, such as the one used here, i.e. a material with index of refraction *n*
_{2}, inserted between two layers with index of refraction *n*
_{1}, the amplitudes of the electric field can be calculated by a straightforward expansion of plane waves. When the incoming field is a perpendicularly incoming plane wave with wave vector $k\phantom{\rule{.2em}{0ex}}=\phantom{\rule{.2em}{0ex}}\frac{2\pi}{\mathrm{\pi \lambda}}$ we have:

Here *E ^{i}* is the (scalar) amplitude of the incoming field, and

*E*and

^{r}*E*are the amplitudes of the reflected and transmitted field respectively. The coefficients

^{t}*R*and

*T*are given by:

where

and *t* is the thickness of the layer with index of refraction *n*
_{2}. When we now consider the Poynting vector (see Eq.(16)) we have that, in the far field:

We could also calculate the power *P* (Eq.(17)). In case of a plane wave the total power is infinite. The intensity *I* can be calculated, however. In this case we have here too:

An example of the absolute value of the flux *r*
^{2}∣*S*∣ for a layer with *t* = 15 nm is given in Fig.4. In this figure (and in similar figures below), *k*
_{0} = 2*π*/λ is the wave vector in vacuum, and *k _{x}* and

*k*are the

_{y}*x*- and

*y*-coordinates of the wave vector, respectively. The coordinates in the plane of the detector are related to

*k*and

_{x}*k*by

_{y}where *z* = -*f* is the focal plane of the lens. As we can see from figure 4, the energy flux within the exit pupil of the lens is not uniform, as one would expect when a scalar theory is used. Instead, the field has a minimum for (*k _{x}*/

*k*

_{0},

*k*/

_{y}*k*

_{0}) = (±0.85,0) and a maximum for (

*k*/

_{x}*k*

_{0},

*k*/

_{y}*k*

_{0}) = (0, ±0.85). The reason for this is that in the case of an

*x*-polarized spot, the

*P*-polarized plane waves (hence, those in the

*xz*-plane) approach the Brewster angle, while the S-polarized plane waves do not approach such an angle.

We now compared the numerically calculated values of *P* for the case of an incident Airy spot with the theoretical values of ∣*R*∣^{2}, for various values of the thickness *t* of the Al layer. (Note that the calculations for *t* = ∞ could be performed by setting the material of the lower part of
the PML to Al). The results are given in Table 2. Strictly speaking the theoretical results hold only for a single perpendicular plane wave, but comparing them with the computer calculations can still give a qualitative impression of the correctness of the calculations. The skin depth δ of Al at λ = 405 nm is

Therefore, the reflection drops quickly for thicknesses *t* smaller than 15 nm. As we can see when we compare the 2^{nd} and 3^{rd} columns of the Table, the agreement between the theory for an perpendicularly incident plane wave and the simulations for the incident Airy spot is rather good, even when, as stated above, the distribution of the field along the exit pupil is not uniform.

### 4.3.3. Single “realistic” pit; comparison with “straight” pit

We performed calculations on a pit that is chosen as closely as possible to a pit structure that is used in practice. Such a pit has a circular “top” shape, and edges that have an approximate 80*°* slope. The pit shape that we used is shown in Fig.5 and Fig.6. The opening of the pit could not be chosen exactly circular, because the program used allows only for polyhedrally shaped objects. Instead a regular octagon is used. The pit had a top width of *b* = 80 nm, and a bottom width of *a* = 60 nm. The height of the pit was *d* = 80 nm. The Al layer had a thickness *t* = 15 nm. The absolute value of the field *E _{x}* in the

*xz*-plane (i.e. the plane that goes “through the center” of the pit) is given in Fig.7. The absolute value of the flux

*r*

^{2}∣

*S*∣ in the Fraunhofer field is given in Fig.8. The value of the total reflected power is given in Table 3.

The first thing that comes to mind, when the values of *P* of Table 3 are compared with those of Table 2, is that apparently the effect of altering the thickness of the Al layer has a greater effect on the amount of light that is reflected than the presence or absence of a pit. In a real storage disc, however, fluctuations in the thickness of the layer will be of a long-range character, while the presence of a pit is a local phenomenon. Therefore these two effects can still be discerned.

At the bottom and top edges of the pit the field becomes singular. This is due to the fact that the pits have an angle there: the parts of the field that should be continuous along both edges, i.e. the field parallel to the edges, should be parallel to both edges at the position of the angle, which is not possible. This is in accordance with the established theory of field components near edges [15].

Another thing that should be considered is the effect of the sloped walls of the pit. To this end, we performed a simulation with an “idealized” pit, in which the walls of the pit are “straight”, and hence the bottom width *a* is equal to the top width *b* (see Fig.9). Pictures of the near field *E _{x}* in the

*xz*-plane, and the far field ∣

*S*∣, are given in Figs.10 and 11 respectively. The value of

*P*is given in Table 3. When the data of the straight pit are compared with those of the sloped pit, we observe that they are very similar. Particularly, the difference in reflected power

*P*is not so large. We therefore decided to use straight pits in our further calculations, also because variations in the parameters such as width

*b*, and layer thickness t are more easily defined.

### 4.3.4. Single straight pit: varying the pit ptructure

We now consider a single straight pit as in Fig.9. In this pit we varied the thickness *t*, the width *b*, and the depth *d*, and we studied the effects on the “visibility” of the pit, compared to a “land” structure of the same thickness *t* of the Al layer. The results are given in Tables 4–6. As can be seen from these Tables, the ratio *P*/*P _{layer}* is about 75–90% for almost all of these simulations, which is in agreement with the results of Fig.8 of ref. [5]. (The exceptions are the simulations
with pits of large diameter (see table 5, where the contrast between pit and land structure is considerably larger).

When the thickness *t* is varied (Table 4), the visibility of the pit varies considerably. This does not agree with the predictions of the scalar model, which predicts that *P*/*P _{layer}* should not be dependent on

*t*. For small t the visibility is quite good. However, in practice, a very thin layer with constant thickness might not be easily realized. For

*t*= 5 nm, the near field in the

*xz*-plane is shown in Fig.12. As we can see, a significant amount of the field in the pit “leaks away” at the sides of the pit. Therefore, for this thickness the assumption of the scalar model that no leakage occurs at the sides (instead only refraction at the bottom of the pit is considered), is not valid.

When the width *b* is varied (Table 5), a considerable change in visibility is seen. When the pitsize becomes more of the order of the wavelength of light, more scattering to the exterior of the lens pupil appears, as can be seen from Fig.13. It appears that the width for a pit, to be well visible, should be 80 nm or larger.

When the pit depth is varied (Table 6), visibility changes somewhat, but not much. The visibility increases as the pit depth is increased. The absolute value of *E _{x}* in the

*xz*-plane is shown for

*d*= 60 nm in Fig.14, and for

*d*= 120 nm in Fig. 15. The wavelength in Poly is

Therefore, λ_{Poly}/4 ≈ 62.5 nm and λ_{Poly}/2 ≈ 125 nm. When we look at Fig.14, however, we notice that there is no destructive interference at the top of the pit. (Note that the maximum value ∣*E _{x}*∣

*= 0.803 V/m for the incoming Airy spot). Likewise, for Fig.15, there does not seem to be extra strong interference. Instead, both figures look very similar qualitatively, and the electric field does not seem to penetrate the pit very deeply. The field amplitude at the bottom of the pit is quite low, except at the re-entrant side of the wedges, where the*

_{max}*E*-component is singular [15]. Therefore reflection from the bottom of the pit does not seem to play a major role. Thus, the assumptions of the scalar model [6], [7] that the contrast between a pit and a land structure is mainly due to destructive interference of the waves emanating from the bottom of the pit and those of the land structure, is incorrect. Indeed, the cut-off width for a pit to serve as a waveguide is [8]

_{x}and this is larger than all pit widths that were used in both the experiments and the simulations. Instead the modulation of the near field seems to be caused mainly by induced currents in the Al Layer at the top and at the sides of the pit.

### 4.3.5. Two pits

We also performed a number of simulations for a two-pit structure. Both pits had a width *b* = 80 nm, a depth *d* = 80 nm, and a thickness of the Al-Layer *t* = 10 nm.

### Convergence of “large-box” calculations

As mentioned in Section 4.3.1, the size of the computational boxes that are used here are larger than those that are used in the previous calculations, and therefore, because the available computer memory is limited, the number of elements per wavelength had to be chosen smaller. The question is then, whether the outcomes of the simulations are still convergent. In order to investigate this, we performed a simulation on a “single-pit structure” that was situated in the “two-pit box”: the box had *x*-coordinates -220 < *x* < 70 (with the unit of length being nm), and the pit center was situated at (*x*,*y*) = (0,0). The Airy spot was then focussed on the center of the pit. Subsequently, The result was compared with the simulation of the corresponding single-pit simulation of Section 4.3.4. In particular, we compared the total power *P* of both calculations. For the large box, we found *P*= 1.858∙10^{-15} which is nearly equal to *P* = 1.839∙10^{-15} found previously. The results of the “two-pit” calculations can therefore be well compared with the “single-pit” calculations.

### Results of calculations

We performed a number of calculations on a two-pit structure with pit centers at (*x*,*y*) = (0,0) and (*x*,*y*) = (- 138,0). The spots we used were both the *x*- and *y*-polarized Airy spot. Both spots were focussed either at (*x*,*y*) = (0,0), or at (*x*,*y*) = (-69,0) (i.e. in the middle between the two pits). The amplitude *E _{x}* of the electric near field for an

*x*-polarized Airy spot that is focussed at (

*x*,

*y*) = (0,0) is given in Fig. 16, and the length of the flux ∣

**S**∣ of the Fraunhofer field is given in Fig. 17. The results for

*P*are given for all two-pit calculations in Table 7. In Fig. 18, we also show the amplitude of

*E*in the focal plane (i.e. the plane parallel to the

_{x}*xy*-plane and situated at the top of the pit (i.e. at

*z*= 90 nm)) for the case of the “

*x*-polarized Airy spot”.

As we can see from Table 7, the presence of a second pit reduces the reflected power somewhat, but not significantly. The field is even more reduced when the spot goes to the middle of the two pits. It therefore seems that the position of the pit or land structures must be known beforehand if these structures are to be discerned. Note also that the ratio *P*/*P _{layer}* is approximately 80%, which is in agreement with the experiments of Fig.8 of ref.[5].

## 5. Comparison of the simulations with the scalar model

When we compare the results of the simulations with those of the predictions made by the scalar model [6], [7], we can conclude that the scalar model predicts lower values of *P*/*P _{layer}* than those that are generated by the finite element simulations (and those that are obtained by the experiments described in ref. [5]). The reason for this discrepancy seems to be, that the contrast between “pit” and “land” visibility is, in the scalar model, caused by destructive interference between waves emanating from the bottom and those emanating from the top of the pits, and is taken to be maximal. However, the simulations show that there is no such interference. Instead the mechanism seems to be the occurrence of induced currents within the top and the sides of the pit, thus generating a dipole field. (Note that this effect is different from plain absorption, for in that case there would be no influence on the field outside of the metal). Also, the assumption of the scalar model that refraction occurs only at the top and the bottom of the pits, does not hold. In addition, singularities in the electric field occur at the top and bottom sides of the pits, thus making the field strength locally more than two times the maximum of the incoming field strength This effect is also not predicted by the scalar model. Thus, it is a matter of lucky coincidence that the scalar diffraction model yields results that are qualitatively correct.

## 6. Summary and conclusions

With the aid of a new finite element program we performed simulations of light diffraction on one- and two-pit structures, similar to those that are used in TwoDOS systems. The pit parameters were varied, and showed that the minimum pit width *b* for acceptable visibility is about 80 nm. Interference with light that is reflected from the bottom of the pit does not play a significant role in pit visibility. Visibility is strongly influenced by the thickness of the metal. Therefore, a pit would be a more effective scatterer when it would be entirely filled with Al, while the land structure would only be filled with a thin layer of Al. In practice, to produce such a structure would probably be quite difficult, however. In the case of two pits, the reflected far field resulting from a spot focussed in between the two pits, differs only slightly from the field resulting from a spot that is focussed on the center of one of the pits.

## A. Scattered far field

We consider the scattered field in the half space *z* > 0 with refractive index *n*. Let **r** = (*x*,*y*,*z*) be a point in this half space and let *r* = (*x*
^{2} +*y*
^{2} +*z*
^{2})^{1/2}. The method of stationary phase applied to the plane wave expansion formula, gives the Fraunhofer approximation for the reflected electric field in points with *r* → ∞:

where λ is the wavelength in vacuum, *k*
_{0} = 2*π*/λ and ℱ is the Fourier transform with respect to (*x*,*y*):

We write: *k _{x}* = 2

*π*ξ and

*k*= 2

_{y}*πη*. In terms of

*x*,

*y*and

*r*we have:

The Fourier transform with respect to *x* and *y* of the magnetic field is for *z* ≥ 0 related to that of the electric field by:

where *μ _{r}* is the relative permeability in the half space,

**k**= (

*k*,

_{x}*k*,

_{y}*k*)

_{z}^{T}, with

The time averaged flow of energy is given by the Poynting vector. In the far field:

$$\phantom{\rule{1.6em}{0ex}}=\frac{{k}_{0}^{2}{n}^{2}}{2{r}^{2}}\frac{{z}^{2}}{{r}^{2}}\mathrm{Re}\left[\mathcal{F}\left[\mathbf{E}\left(.,.,0\right)\right]\left(\frac{\mathit{nx}}{\lambda r},\frac{\mathit{ny}}{\lambda r}\right)\times \mathcal{F}\left[\mathbf{H}\left(.,.,0\right)\right]{\left(\frac{\mathit{nx}}{\lambda r},\frac{\mathit{ny}}{\lambda r}\right)}^{*}\right].$$

The total reflected energy in the half space *z* > 0 is given by the integral of the radial component of (35) over a sphere with large radius. The radial component of the Poynting vector is

The total power reflected into the half space *z* > 0 is then

$$\phantom{\rule{1.6em}{0ex}}=\frac{{k}_{0}^{2}{n}^{2}}{2}{\int}_{0}^{2\pi}{\int}_{0}^{\pi}\mathrm{Re}\left[\mathcal{F}\left[\mathbf{E}\left(.,.,0\right)\right]\left(\frac{\mathit{nx}}{\lambda r},\frac{\mathit{ny}}{\lambda r}\right)\times \mathcal{F}\left[\mathbf{H}\left(.,.,0\right)\right]{\left(\frac{\mathit{nx}}{\lambda r},\frac{\mathit{ny}}{\lambda r}\right)}^{*}\right]\cdot \frac{\mathbf{r}}{r}{\mathrm{cos}}^{2}\phantom{\rule{.2em}{0ex}}\theta \phantom{\rule{.2em}{0ex}}\mathrm{sin}\theta \phantom{\rule{.2em}{0ex}}d\phantom{\rule{.2em}{0ex}}\theta d\varphi $$

where *r*,*θ*,*ϕ* are spherical coordinates:

We can rewrite this integral as an integral over *k _{x}* and

*k*over the circle with radius

_{y}*k*

_{0}

*n*. Eqns. (31), (32) imply

The determinant of the Jacobian matrix is cos *θ* sin *θ*, hence

and therefore *P* can be expressed in terms of an integral over *k _{x}* and

*k*as follows:

_{y}$$\cdot \frac{\mathbf{k}}{{k}_{0}n}\phantom{\rule{.2em}{0ex}}\frac{{k}_{z}}{{k}_{0}n}d{k}_{x}d{k}_{y},$$

where we used that cos(*θ*) = *k _{z}*/

*k*

_{0}

*n*and

By using that *k*
_{0}
*n* = *ω*(*ϵ*
_{0}
*ϵ _{r}*

*μ*

_{0}

*μ*)

_{r}^{1/2}, we can express (35) and (44) in terms of the electric field only:

and

## Acknowledgments

We acknowledge the help of drs. J.J. Rusch for help in the construction of some of the figures, as well as in various other computer problems

## References and links

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