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 dPoly/4 (where λPoly 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

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References

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  1. X. Wei, A. J. H. Wachters and H. P. Urbach: "Finite element model for three-dimensional optical storage problem," to be published.
  2. J. M. Brok and H. P. Urbach, "Rigorous model of the scattering of a focussed spot by a grating, and its application in optical recording," J. Opt. Soc. Am. A 20256-272 (2003).
    [CrossRef]
  3. see e.g. P. G. Ciarlet, "The Finite Element Method for Elliptic Problems," SIAM 2002.
  4. X. Wei, H. P. Urbach, A. Wachters and Y. Aksenov: "3D Rigorous simulation of mask induced polarization," to be published.
  5. W. M. J. Coene, D. M. Bruls, A. H. J. Immink, A. M. van der Lee, A. P. Hekstra, J. Riani, S. van Beneden, M. Ciacci, J. W. M. Bergmans, M. Furuki, "Two-dimensional Optical Storage," IEEE Proceedings of the International Conference on Acoustics, Speed and Signal Processing 5, 749-752 (2005).
  6. W. M. J. Coene, "Nonlinear Signal-Processing Model for Scalar Diffraction in Optical Recording," Appl. Opt. 42, 6525-6535 (2003)
    [CrossRef] [PubMed]
  7. L. Fagoonee, W. M. J. Coene, A. Moinian, and B. Honary, "Nonlinear signal-processing model for signal generation in multilevel two-dimensional optical storage," Opt. Lett.,  29, 385-387 (2004)
    [CrossRef] [PubMed]
  8. J. D. Jackson, Classical Electrodynamics (Wiley, 1975)mj
  9. J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys 114, 185-200 (1994).
    [CrossRef]
  10. http://www.sara.nl/userinfo/reservoir/sepran/index.html
  11. J. Nedelec, "Mixed finite elements in R3," Numer. Math. 35, 315-341 (1980).
    [CrossRef]
  12. Y. Saad, " Iterative methods for sparse linear systems," SIAM, 2nd edition, 2003.
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  14. DIFFRACT is a product of MM Research Inc. Tucson Ariz. Its theoretical basis has been described in, e.g. M.Mansuripur, "Certain computational aspects of vector diffraction problems," J. Opt. Soc. Am. 6, 786-805 (1989).
    [CrossRef]
  15. J. van Bladel, Singular Electromagnetic Fields and Sources (Oxford 1991).

2004 (1)

2003 (2)

1994 (1)

J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys 114, 185-200 (1994).
[CrossRef]

1989 (1)

DIFFRACT is a product of MM Research Inc. Tucson Ariz. Its theoretical basis has been described in, e.g. M.Mansuripur, "Certain computational aspects of vector diffraction problems," J. Opt. Soc. Am. 6, 786-805 (1989).
[CrossRef]

1980 (1)

J. Nedelec, "Mixed finite elements in R3," Numer. Math. 35, 315-341 (1980).
[CrossRef]

Aksenov, Y.

X. Wei, H. P. Urbach, A. Wachters and Y. Aksenov: "3D Rigorous simulation of mask induced polarization," to be published.

Berenger, J. P.

J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys 114, 185-200 (1994).
[CrossRef]

Brok, J. M.

Coene, W. M. J.

Fagoonee, L.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Honary, B.

Moinian, A.

Urbach, H. P.

J. M. Brok and H. P. Urbach, "Rigorous model of the scattering of a focussed spot by a grating, and its application in optical recording," J. Opt. Soc. Am. A 20256-272 (2003).
[CrossRef]

X. Wei, A. J. H. Wachters and H. P. Urbach: "Finite element model for three-dimensional optical storage problem," to be published.

X. Wei, H. P. Urbach, A. Wachters and Y. Aksenov: "3D Rigorous simulation of mask induced polarization," to be published.

Wachters, A.

X. Wei, H. P. Urbach, A. Wachters and Y. Aksenov: "3D Rigorous simulation of mask induced polarization," to be published.

Wachters, A. J. H.

X. Wei, A. J. H. Wachters and H. P. Urbach: "Finite element model for three-dimensional optical storage problem," to be published.

Wei, X.

X. Wei, A. J. H. Wachters and H. P. Urbach: "Finite element model for three-dimensional optical storage problem," to be published.

X. Wei, H. P. Urbach, A. Wachters and Y. Aksenov: "3D Rigorous simulation of mask induced polarization," to be published.

Appl. Opt. (1)

Finite element model for three-dimensional optical storage problem (1)

X. Wei, A. J. H. Wachters and H. P. Urbach: "Finite element model for three-dimensional optical storage problem," to be published.

J. Comput. Phys (1)

J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys 114, 185-200 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

DIFFRACT is a product of MM Research Inc. Tucson Ariz. Its theoretical basis has been described in, e.g. M.Mansuripur, "Certain computational aspects of vector diffraction problems," J. Opt. Soc. Am. 6, 786-805 (1989).
[CrossRef]

J. Opt. Soc. Am. A (1)

Numer. Math. (1)

J. Nedelec, "Mixed finite elements in R3," Numer. Math. 35, 315-341 (1980).
[CrossRef]

Opt. Lett. (1)

Other (8)

J. D. Jackson, Classical Electrodynamics (Wiley, 1975)mj

Y. Saad, " Iterative methods for sparse linear systems," SIAM, 2nd edition, 2003.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

http://www.sara.nl/userinfo/reservoir/sepran/index.html

J. van Bladel, Singular Electromagnetic Fields and Sources (Oxford 1991).

see e.g. P. G. Ciarlet, "The Finite Element Method for Elliptic Problems," SIAM 2002.

X. Wei, H. P. Urbach, A. Wachters and Y. Aksenov: "3D Rigorous simulation of mask induced polarization," to be published.

W. M. J. Coene, D. M. Bruls, A. H. J. Immink, A. M. van der Lee, A. P. Hekstra, J. Riani, S. van Beneden, M. Ciacci, J. W. M. Bergmans, M. Furuki, "Two-dimensional Optical Storage," IEEE Proceedings of the International Conference on Acoustics, Speed and Signal Processing 5, 749-752 (2005).

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Figures (18)

Fig. 1.
Fig. 1.

Absolute value of Ex in the focal plane, for an x-polarized Airy spot

Fig. 2.
Fig. 2.

Absolute value of Ey in the focal plane, for an x-polarized Airy spot

Fig. 3.
Fig. 3.

Amplitude of the x-component of the electric near-field Ex in the xz-plane for a layered structure of Poly-Al-Poly, with a thickness of the Al layer of 15 nm. The computational domain Ω consists of the region {(x,z)∣(-70 < x < 70) ∧ (60 ≤ z ≤ 110)}. The surrounding region is the PML. The Al-layer lies at 80 ≤ z ≤ 95. The Airy spot is focussed at the top of the layer (z = 95). The incident Airy spot is predominantly x-polarized.

Fig. 4.
Fig. 4.

Absolute value of the flux of the reflected Fraunhofer field, r 2S∣ for a layered structure of Poly-Al-Poly, with a thickness of the Al layer of 15 nm. k 0 = 2π/λ is the wave vector in vacuum, and kx and ky are the x- and y-coordinates of the wave vector, respectively. The incident Airy spot is predominantly x-polarized.

Fig. 5.
Fig. 5.

Side view of the “realistic” pit structure.

Fig. 6.
Fig. 6.

Top view of pit structure, used in the simulations. The xz-plane goes through the line indicating the width b of the pit. The point (x,y) = (0,0) lies at the center of the pit.

Fig. 7.
Fig. 7.

Amplitude of the x-component of the electric field Ex in the xz-plane in the case of the “realistic” pit structure of Fig.5. The incident Airy spot is predominantly x-polarized.

Fig. 8.
Fig. 8.

Absolute value of the flux r 2S∣ of the reflected Fraunhofer field in the case of the “realistic” pit structure of Fig.5. k 0 = 2π/λ is the wave vector in vacuum, and kx and ky are the x- and y-coordinates of the wave vector, respectively. The incident Airy spot is predominantly x-polarized.

Fig. 9.
Fig. 9.

Side view of the “idealized” pit structure, that was used to study trends in the behavior of the Fraunhofer field.

Fig. 10.
Fig. 10.

Amplitude of the x-component of the electric field Ex in the xz-plane in the case of the “straight” pit structure of Fig.9 with b = 80nm, d = 80 nm and t = 15 nm. The incident Airy spot is predominantly x-polarized.

Fig. 11.
Fig. 11.

Absolute value of the flux r 2S∣ of the reflected Fraunhofer field in the case of the “straight” pit structure of Fig.9 with b = 80 nm, d = 80 nm and t = 15 nm. k 0 = 2π/λ is the wave vector in vacuum, and kx and ky are the x- and y-coordinates of the wave vector, respectively. The incident Airy spot is predominantly x-polarized.

Fig. 12.
Fig. 12.

Amplitude of the x-component of the electric field Ex in the xz-plane in the case of the “straight” pit structure of Fig.9 with b = 80 nm, d = 80 nm and t = 5 nm. The incident Airy spot is predominantly x-polarized.

Fig. 13.
Fig. 13.

Absolute value of the flux of the reflected Fraunhofer field, r 2S∣ for a pit structure, with a thickness of the Al layer of 10nm, a depth d = 80 nm, and a width b = 120nm. k 0 = 2π/λ is the wave vector in vacuum, and kx and ky are the x- and y-coordinates of the wave vector, respectively. The incident Airy spot is predominantly x-polarized.

Fig. 14.
Fig. 14.

Amplitude of the x-component of the electric field Ex in the xz-plane in the case of the “straight” pit structure of Fig.9 with b = 80 nm, d = 60 nm and t = 10 nm. The incident Airy spot is predominantly x-polarized.

Fig. 15.
Fig. 15.

Amplitude of the x-component of the electric field Ex in the xz-plane in the case of the “straight” pit structure of Fig.9 with b = 80 nm, d = 120 nm and t = 10 nm. The incident Airy spot is predominantly x-polarized.

Fig. 16.
Fig. 16.

Amplitude of the x-component of the electric field Ex in the xz-plane in the case of a two-pit structure where the spot is focussed just above the right pit at (x,y,z) = (0,0,90). The incident Airy spot is predominantly x-polarized.

Fig. 17.
Fig. 17.

Absolute value of the flux r 2S∣ of the reflected Fraunhofer field in the case of the two-pit pit structure. k 0 = 2π/λ is the wave vector in vacuum, and kx and ky are the x-and y-coordinates of the wave vector, respectively. The incident Airy spot is predominantly x-polarized.

Fig. 18.
Fig. 18.

Amplitude of the x-component of the total electric field Ex in the focal plane (i.e. z = 90 nm) in the case of a two-pit structure where the Airy spot is focussed just above the right pit at (x,y) = (0,0). The incident Airy spot is predominantly x-polarized.

Tables (7)

Tables Icon

Table 1. Refractive Indices and dielectric constants of the materials used in the simulations.

Tables Icon

Table 2. Total reflected power P in the far field for an incident “x-polarized Airy spot” and absolute and relative squares of the absolute value of the reflection coefficient as a function of the thickness t of the Al layer for a plane wave at perpendicular incidence. The last row gives the values in case that the layer consists of a perfect conductor. In this case the reflection is 100%.

Tables Icon

Table 3. Total reflected power P in the far field for a layered structure with width t=15 nm, the “realistic” pit of Fig.5 (oblique walls), and a “straight” pit with d = 80 nm, t = 15 nm, and b = 80 nm (see Fig.9). The incoming spot is the “x-polarized Airy spot”.

Tables Icon

Table 4. Total reflected power P in the far field for a straight pit structure with width b = 80 nm, depth d = 80 nm and variable thickness t of the Al layer. The third column gives the ratio of P of the pit, and that of a flat layer with the same thickness t (see Table 2). The incoming spot is the “x-polarized Airy spot”.

Tables Icon

Table 5. Total reflected power P in the far field for a straight pit structure with depth d = 80 nm, thickness of the Al layer t = 10 nm, and variable width b. The third column gives the ratio of P of the pit, and that of the layer with the same thickness t = 10 nm, i.e. Player = 2.186∙10-15 W (see Table 2). The incoming spot is the ”x-polarized Airy spot”.

Tables Icon

Table 6. Total reflected power P in the far field for a straight pit structure with width b = 80 nm, thickness of the Al layer t = 10 nm, and variable depth d. The third column gives the ratio of P of the pit, and that of the layer with the same thickness t = 10 nm, i.e. Player = 2.186∙10-15 W (see Table 2). The incoming spot is the “x-polarized Airy spot”.

Tables Icon

Table 7. Total reflected power P in the far field for a single- or two-pit structure with width b = 80 nm, thickness of the Al layer t = 10 nm, and depth d = 80 nm, in the case of a “large” computational box: -220 ≤ x ≤ 70. The fifth column gives the ratio of P of the pit, and that of the layer with the same thickness t = 10 nm, i.e. Player = 2.186 ∙ 10-15 W (see Table 2). The incoming spot is either the “x-polarized Airy spot” or the “y-polarized Airy spot”.

Equations (50)

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λ = 405 nm .
NA = 0.85 .
D FWHM = λ 2 NA = 238 nm .
s = 138 nm
d = λ 4 n Ploy = 405 4 1.619 62.5 nm ,
b min = λ 2 n Poly 125 nm
× E = i ω μ 0 μ r ¯ ¯ H .
× H = i ε 0 ε r ¯ ¯ E + J .
( ε 0 ε r ¯ ¯ E ) = ρ .
H = 0 .
ω 2 ε 0 μ 0 ε r ¯ ¯ E ( μ r 1 ¯ ¯ × E ) = i ω μ 0 J .
ω 2 ε 0 μ 0 μ r ¯ ¯ H ( ε r 1 ¯ ¯ × H ) = × ( ε r 1 ¯ ¯ J ) .
E s = E E 0
RA = f NA 1 ( NA ) 2
E x max = 0.803 V m .
S ( r ) = 1 2 Re [ E ( r ) × H * ( r ) ] ,
P = 0 2 π 0 π S r ( r ) r 2 sin θ dθd ϕ .
k x 2 + k y 2 ( NA ) k 0
E r = RE i ; E i = TE i .
R = r [ 1 exp ( 2 ik n 2 t ) ) ] 1 r 2 exp ( 2 ik n 2 t ) ,
T = ( 1 r ) ( 1 + r ) exp ( 2 ik n 2 t ) 1 r 2 exp ( 2 ik n 2 t ) ,
r = n 1 n 2 n 1 + n 2 .
S r = R 2 S i .
I r = R 2 I r .
x = k x f k 0 ; y = k y f k 0
δ = λ 2 πIm ( n Al ) = 405 2 π · 4.95 = 13 nm .
λ Poly = λ n Poly = 405 1.619 250 nm .
b min = λ 2 n Poly 125 nm ,
E ( r ) i k 0 n r z r e i k 0 nr [ E . . 0 ] ( nx λr , ny λr ) ,
( f ) ξ η = e 2 π i ( ξx + ηy ) f ( x y ) dxdy .
k x = 2 πξ = 2 π nx λ r = k 0 n x r ,
k y = 2 πη = 2 π ny λ r = k 0 n y r ,
[ H ( . , . , z ) ] k x 2 π k y 2 π = k ω μ 0 μ r × [ E ( . , . , z ) ] k x 2 π k y 2 π ,
k z = ( k 0 n 2 k x 2 k y 2 ) 1 2 .
S ( r ) = 1 2 Re [ E ( r ) × H ( r ) * ]
= k 0 2 n 2 2 r 2 z 2 r 2 Re [ [ E . . 0 ] nx λ r ny λ r × [ H . . 0 ] nx λ r ny λ r * ] .
S r = S r r = S x x r + S y y r + S z z r .
P = 0 2 π 0 π 2 S r ( r ) r 2 sin θ d θ d ϕ .
= k 0 2 n 2 2 0 2 π 0 π Re [ [ E . . 0 ] nx λ r ny λ r × [ H . . 0 ] nx λ r ny λ r * ] r r cos 2 θ sin θ d θ d ϕ
x = r cos ϕ sin θ ,
y = r sin ϕ sin θ ,
z = r cos θ .
k x k 0 n = cos ϕ sin θ ,
k y k 0 n = sin ϕ sin θ .
cos θ sin θ d θ d ϕ = d ( k x k 0 n ) d ( k y k 0 n ) ,
P = 1 2 ( k x 2 + k y 2 ) 1 2 < k 0 n Re [ [ E . . 0 ] k x 2 π k y 2 π × [ H . . 0 ] k x 2 π k y 2 π * ]
k k 0 n k z k 0 n d k x d k y ,
r r = k k 0 n .
S ( r ) = k 0 2 n 2 2 r 2 z 2 r 2 ( ε 0 ε r μ 0 μ r ) 1 2 [ E . . 0 ] nx λ r ny λ r 2 r r ,
P = 1 2 ( ε 0 ε r μ 0 μ r ) 1 2 ( k x 2 + k y 2 ) 1 2 < k 0 n [ E . . 0 ] k x 2 π k y 2 π 2 k z k 0 n d k x d k y .

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