Rigorous 3D Calculation of Effects of Pit Structure in TwoDOS Systems

J. A. C. Veerman; A. J. H. Wachters; A. M. van der Lee; H. P. Urbach

doi:10.1364/OE.15.002075

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
$λ = 405 nm .$
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,
$NA = 0.85 .$
• 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 maximum
$D_{FWHM} = \frac{λ}{2 NA} = 238 nm .$
Likewise 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]
$s = 138 nm$
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
$d = \frac{λ}{4 n_{Ploy}} = \frac{405}{4 \cdot 1.619} \approx 62.5 nm,$
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_Poly = 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],
$b_{min} = \frac{λ}{2 n_{Poly}} \approx 125 nm$
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:

\nabla \times E = i ω μ_{0} \underline{\underline{μ_{r}}} H .

\nabla \times H = - i ε_{0} \underline{\underline{ε_{r}}} E + J .

\nabla \cdot (ε_{0} \underline{\underline{ε_{r}}} E) = ρ .

\nabla \cdot H = 0 .

Here ϵ_r͇ and μ_r͇ are the relative permittivity tensor, and relative permeability tensor, respectively. (We will take μ_r͇ = 1 in our simulations). J and ρ are the known external current and charge density, respectively. From equations (7) and (8) the vector Helmholtz equations can be derived:

ω^{2} ε_{0} μ_{0} \underline{\underline{ε_{r}}} E - \nabla (\underline{\underline{μ_{r}^{- 1}}} \nabla \times E) = - i ω μ_{0} J .

ω^{2} ε_{0} μ_{0} \underline{\underline{μ_{r}}} H - \nabla (\underline{\underline{ε_{r}^{- 1}}} \nabla \times H) = - \nabla \times (\underline{\underline{ε_{r}^{- 1}}} J) .

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} = E - E_{0}

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 _s can be restricted to Ω and the PML. Once E _s 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 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 10⁵. 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

RA = \frac{f \cdot NA}{\sqrt{1 - {(NA)}^{2}}}

and the maximum absolute value of E_x (i.e. the absolute value at (x,y) = (0,0) given by:

{∣ E_{x} ∣}_{max} = 0.803 \frac{V}{m} .

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:

S (r) = \frac{1}{2} Re [E (r) \times H^{*} (r)],

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 r̂ 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:

P = \int_{0}^{2 π} \int_{0}^{π} S_{r} (r) r^{2} \sin θ dθd ϕ .

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

\sqrt{k_{x}^{2} + k_{y}^{2}} \leq (NA) k_{0}

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_y) = (512,512) points, with a width of 51.46 wavelengths. A plot of ∣E_x∣ and ∣E_y∣ for the case of an 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_elements = 30 for the Poly, and N_elements = 120 for the Al in the case of a layered structure or a 1-pit structure, and N_elements = 12 for the Poly, and N_elements = 80 for the Al, in the case of a two-pit structure. The number of elements per wavelength in the PML was N_elements = 5 in all cases, and the complex stretching parameter [1] was chosen to be ζ= 15+15i. This resulted in a matrix with 1.5∙10⁵ to 3.5∙10⁵ 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.

Fig. 1. Absolute value of E_x in the focal plane, for an x-polarized Airy spot

Thickness t (nm)	P (10^-15 W)	$\frac{P}{P_{perfect}}$	∣R ²(t)∣
5	1.064	0.27	0.2552
10	2.186	0.55	0.5419
15	2.795	0.71	0.7005
25	3.255	0.82	0.8204
∞	3.397	0.86	0.8580
perfect	3.956	1.00	1.00

number of pits	spot polarization	(x,y) of spot focus	P(10^-15 W)	$\frac{P}{P_{layer}}$
1	x	(0,0)	1.858	0.84
2	x	(0,0)	1.735	0.79
2	y	(0,0)	1.820	0.83
2	x	(-69,0)	1.638	0.77
2	y	(-69,0)	1.779	0.81

Material	n	ϵ_r = n ²
Poly	1.619	2.621
Al	0.65+4.95i	-24.08+6.435i
Perfect	∞	∞

Material	n	ϵ_r = n ²
Poly	1.619	2.621
Al	0.65+4.95i	-24.08+6.435i
Perfect	∞	∞

Abstract

1. Introduction

2. TwoDOS characteristics

3. Measurements performed so far; scalar theory

4. Computer simulations of TwoDOS pit structure

4.1. Finite element program

4.2. Macroscopic set-up

4.3. Simulations

4.3.1. Parameters used

4.3.2. Layered structure

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

4.3.4. Single straight pit: varying the pit ptructure

4.3.5. Two pits

Convergence of “large-box” calculations

Results of calculations

5. Comparison of the simulations with the scalar model

6. Summary and conclusions

A. Scattered far field

Acknowledgments

References and links

Cited By

Figures (18)

Tables (7)

Equations (50)

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