Abstract

We describe a rigorous model for the scattering of a three-dimensional focused spot by a one-dimensional periodic grating. The incident field is decomposed into a sum of quasi-periodic fields, and the scattering of each of these is computed inside one unit cell of the grating. The model is applied to the simulation of the readout of a DVD disk. The polarization dependence of the reflected near and far fields is studied, and, for a TM-polarized incident spot, plasmons are observed in the reflected far-field intensity.

© 2003 Optical Society of America

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References

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  1. W. Ophey, “Depolarizing grating structures for optical recording,” (Philips Research, Eindhoven, The Netherlands, 1994).
  2. J. B. Judkins, R. W. Ziolkowski, “Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings,” J. Opt. Soc. Am. A 12, 1974–1983 (1995).
    [CrossRef]
  3. J. B. Judkins, C. W. Haggans, R. W. Ziolkowski, “Two-dimensional finite-difference time-domain simulation for rewritable optical disk surface structure design,” Appl. Opt. 35, 2477–2487 (1996).
    [CrossRef] [PubMed]
  4. W.-C. Liu, M. W. Kowarz, “Vector diffraction from subwavelength optical disk structures: two-dimensional modeling of near-field profiles, far-field intensities, and detector signals from a DVD,” Appl. Opt. 38, 3787–3797 (1999).
    [CrossRef]
  5. D. S. Marx, D. Psaltis, “Optical diffraction and focused spots and subwavelength structures,” J. Opt. Soc. Am. A 14, 1268–1278 (1997).
    [CrossRef]
  6. J.-P. Berenger, “A perfectly matched layer for the absorbing of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  7. W.-H. Yeh, L. Li, M. Mansuripur, “Vector diffraction and polarization effects in an optical disk system,” Appl. Opt. 37, 6983–6988 (1998).
    [CrossRef]
  8. J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1985).
    [CrossRef]
  9. L. Li, “Multilayer diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994).
    [CrossRef]
  10. D. S. Marx, D. Psaltis, “Polarization quadrature measurement of subwavelength diffracting structures,” Appl. Opt. 36, 6434–6440 (1997).
    [CrossRef]
  11. R. E. Gerber, L. Li, M. Mansuripur, “Effect of surface plasmon excitations on the irradiance pattern of the return beam in optical disk storage,” Appl. Opt. 34, 4929–4936 (1995).
    [CrossRef] [PubMed]
  12. H. P. Urbach, R. T. M. Merkx, “Finite element simulation of electromagnetic plane wave diffraction at gratings for arbitrary angles of incidence,” in Mathematical and Numerical Aspects of Wave Propagation Phenomena, G. Cohen, L. Halpern, P. Joly, eds. (SIAM, Philadelphia, Pa., 1991), pp. 89–99.
  13. H. P. Urbach, “Convergence of the Galerkin method for two-dimensional electromagnetic problems,” Siam J. Numer. Anal. 28, 697–710 (1991).
    [CrossRef]
  14. G. N. Zhizhin, M. A. Moskalova, E. V. Shomina, V. A. Yakovlev, in Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces, V. M. Agranovich, D. L. Mills, eds. (North-Holland, Amsterdam, 1982), pp. 93.
  15. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, 1988).
  16. 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. A 6, 786–805 (1989).
    [CrossRef]
  17. V. S. Ignatowsky, “Diffraction by an objective lens with arbitrary aperture,” Trans. Opt. Inst. Petrograd I(IV), 1–36 (1921).
  18. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems: II. Structure of the image in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  19. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

1999

1998

1997

1996

1995

1994

L. Li, “Multilayer diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994).
[CrossRef]

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

1991

H. P. Urbach, “Convergence of the Galerkin method for two-dimensional electromagnetic problems,” Siam J. Numer. Anal. 28, 697–710 (1991).
[CrossRef]

1985

J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1985).
[CrossRef]

1959

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems: II. Structure of the image in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

1921

V. S. Ignatowsky, “Diffraction by an objective lens with arbitrary aperture,” Trans. Opt. Inst. Petrograd I(IV), 1–36 (1921).

Berenger, J.-P.

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

Chandezon, J.

J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1985).
[CrossRef]

Gerber, R. E.

Haggans, C. W.

Ignatowsky, V. S.

V. S. Ignatowsky, “Diffraction by an objective lens with arbitrary aperture,” Trans. Opt. Inst. Petrograd I(IV), 1–36 (1921).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Judkins, J. B.

Kowarz, M. W.

Li, L.

Liu, W.-C.

Mansuripur, M.

Marx, D. S.

Maystre, D.

J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1985).
[CrossRef]

Merkx, R. T. M.

H. P. Urbach, R. T. M. Merkx, “Finite element simulation of electromagnetic plane wave diffraction at gratings for arbitrary angles of incidence,” in Mathematical and Numerical Aspects of Wave Propagation Phenomena, G. Cohen, L. Halpern, P. Joly, eds. (SIAM, Philadelphia, Pa., 1991), pp. 89–99.

Moskalova, M. A.

G. N. Zhizhin, M. A. Moskalova, E. V. Shomina, V. A. Yakovlev, in Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces, V. M. Agranovich, D. L. Mills, eds. (North-Holland, Amsterdam, 1982), pp. 93.

Ophey, W.

W. Ophey, “Depolarizing grating structures for optical recording,” (Philips Research, Eindhoven, The Netherlands, 1994).

Psaltis, D.

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, 1988).

Raoult, G.

J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1985).
[CrossRef]

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems: II. Structure of the image in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Shomina, E. V.

G. N. Zhizhin, M. A. Moskalova, E. V. Shomina, V. A. Yakovlev, in Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces, V. M. Agranovich, D. L. Mills, eds. (North-Holland, Amsterdam, 1982), pp. 93.

Urbach, H. P.

H. P. Urbach, “Convergence of the Galerkin method for two-dimensional electromagnetic problems,” Siam J. Numer. Anal. 28, 697–710 (1991).
[CrossRef]

H. P. Urbach, R. T. M. Merkx, “Finite element simulation of electromagnetic plane wave diffraction at gratings for arbitrary angles of incidence,” in Mathematical and Numerical Aspects of Wave Propagation Phenomena, G. Cohen, L. Halpern, P. Joly, eds. (SIAM, Philadelphia, Pa., 1991), pp. 89–99.

Wolf, E.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems: II. Structure of the image in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Yakovlev, V. A.

G. N. Zhizhin, M. A. Moskalova, E. V. Shomina, V. A. Yakovlev, in Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces, V. M. Agranovich, D. L. Mills, eds. (North-Holland, Amsterdam, 1982), pp. 93.

Yeh, W.-H.

Zhizhin, G. N.

G. N. Zhizhin, M. A. Moskalova, E. V. Shomina, V. A. Yakovlev, in Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces, V. M. Agranovich, D. L. Mills, eds. (North-Holland, Amsterdam, 1982), pp. 93.

Ziolkowski, R. W.

Appl. Opt.

J. Comput. Phys.

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

J. Opt.

J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1985).
[CrossRef]

J. Opt. Soc. Am. A

Proc. R. Soc. London Ser. A

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems: II. Structure of the image in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Siam J. Numer. Anal.

H. P. Urbach, “Convergence of the Galerkin method for two-dimensional electromagnetic problems,” Siam J. Numer. Anal. 28, 697–710 (1991).
[CrossRef]

Trans. Opt. Inst. Petrograd

V. S. Ignatowsky, “Diffraction by an objective lens with arbitrary aperture,” Trans. Opt. Inst. Petrograd I(IV), 1–36 (1921).

Other

W. Ophey, “Depolarizing grating structures for optical recording,” (Philips Research, Eindhoven, The Netherlands, 1994).

H. P. Urbach, R. T. M. Merkx, “Finite element simulation of electromagnetic plane wave diffraction at gratings for arbitrary angles of incidence,” in Mathematical and Numerical Aspects of Wave Propagation Phenomena, G. Cohen, L. Halpern, P. Joly, eds. (SIAM, Philadelphia, Pa., 1991), pp. 89–99.

G. N. Zhizhin, M. A. Moskalova, E. V. Shomina, V. A. Yakovlev, in Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces, V. M. Agranovich, D. L. Mills, eds. (North-Holland, Amsterdam, 1982), pp. 93.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, 1988).

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. A 6, 786–805 (1989).
[CrossRef]

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

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Figures (16)

Fig. 1
Fig. 1

One-dimensional periodic grating with the computational cell Ω.

Fig. 2
Fig. 2

Amplitudes of the Ex and the Ey components |Exi| and |Eyi|, respectively, of the incident spot when a TE-polarized plane wave is focused by the objective.

Fig. 3
Fig. 3

Amplitude of the Hy component of the near field for the case of a dominantly TM-polarized incident spot on a DVD disk with groove depth of 50 nm and groove width of 200 nm. Five tracks are shown. The values to the right of the bar give the correspondence between the shades and the amplitude; the correspondence applies not only to this figure but to all contour plots of near-field amplitudes. The values at the left of the bar show the correspondence between the shades and the phases in all contour plots of the phase of near-field components.

Fig. 4
Fig. 4

Amplitude of the y component of the total electric field inside and near the pit for a TE-polarized incident plane wave of which the y component of the electric-field vector has unit amplitude. The behavior of the field in the pit depends strongly on the width of the pit.

Fig. 5
Fig. 5

Phase of the y component of the total electric field inside and near the pit for a TE-polarized incident plane wave of which the y component of the electric-field vector has unit amplitude. The depth of the pit is 800 nm.

Fig. 6
Fig. 6

Amplitude and phase of the y component of the total magnetic field inside and near the pit for a TM-polarized incident plane wave of which the y component of the magnetic field vector has unit amplitude. The pit depth is 800 nm.

Fig. 7
Fig. 7

Values for β1 in the complex plane for various pit widths: (a) TE polarization, (b) TM polarization. For TE-polarization, when the width w of the pit increases from 150 to 220 nm, the real part of β1 increases, while its imaginary part rapidly decreases when w increases. In the case of TM polarization, for increasing width, β1 approaches the real axis.

Fig. 8
Fig. 8

Effective wavelength (left axis) and penetration depth (right axis) inside the pit as a function of pit width. (a) TE polarization, (b) TM polarization.

Fig. 9
Fig. 9

Influence of metal on the wave vector β1 inside an infinite waveguide. For different values of the width w the solution of β1 is shown in the complex plane. (a) TE polarization, (b) TM polarization.

Fig. 10
Fig. 10

Intensity and phase of the reflected Fraunhofer far field for a plane wave falling on a rectangular groove of width 200 nm. The depth varies from 0 to 500 nm. (a) Reflected intensity of the 0th order, (b) phase of the 0th order, (c) reflected intensity of the +1st and -1st orders, (d) phase of the +1st order.

Fig. 11
Fig. 11

Reflected field of plane wave normally incident on grooves with various widths as a function of the depth. (a) TE polarization, (b) TM polarization.

Fig. 12
Fig. 12

Amplitude of the reflected field in the complex plane for a pit width of 200 nm. The Cyclop calculations are printed in white, and results from the extended physical optics model are printed in black. The arrows indicate the directions of increasing depths, from 0 to 500 nm. (a) 0th reflected order, (b) +1st reflected-order reflected field. The phase changes much more slowly with depth for TE than for TM polarization.

Fig. 13
Fig. 13

Sz in the (kx, ky) plane and cross sections with ky=0 for a TE-polarized spot incident on a grating with p=740 nm, w=200 nm. The depth of the pit is varied between 0 and 200 nm.

Fig. 14
Fig. 14

Sz in the (kx, ky) plane and cross sections with ky=0 for a TM-polarized spot incident on a grating with p=740 nm, w=200 nm. The depth of the pit is varied between 0 and 100 nm. Other depths in the next figure.

Fig. 15
Fig. 15

Sz in the (kx, ky) plane and cross sections with ky=0 for a TM-polarized spot incident on a grating with p=740 nm, w=200 nm. The depth of the pit is varied between 125 and 200 nm.

Fig. 16
Fig. 16

Energy flow in the z direction in the reflected far field. The incident spot is TM polarized and is focused on the edge of a pit with a width of half the period of the grating (p=740 nm). The second order does not appear, and the small bands caused by surface plasmons are absent.

Tables (1)

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Table 1 Parameters of the DVD Standard

Equations (78)

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Ei(r, t)=Ai exp[i(k-  r-ωt)],
Hi(r, t)=k×Aiωμ0exp[i(k-  r-ωt)],
kz=(ω20μ0u-kx2-ky2)1/2,
Aik-=0.
×H=-iω0E,
×E=iωμ0H.
˜(x, z)=u,z>0l,z<0.
×H˜=-iω0˜E˜,
×E˜=iωμ0H˜.
U(x+p, y, z)=U(x, y, z)exp(ikxp).
E=E˜+E1,
H=H˜+H1.
×H1+iω0E1=iω0(˜-)E˜,
×E1-iωμ0H1=0.
U(x, y, z)y=ikyU(x, y, z).
ikyHz-Hyz=-iω0Ex,
Hxz-Hzx=-iω0Ey,
Hyx-ikyHx=-iω0Ez,
ikyEz-Eyz=iωμ0Hx,
Exz-Ezx=iωμ0Hy,
Eyx-ikyEx=iωμ0Hz,
ExHz=iω20μ0-ky2ky-ωμ0-ω0kyEyxHyz,
EzHx=iω20μ0-ky2kyωμ0ω0kyEyzHyx.
xAxU+zAzU
+xBzU-zBxU
+(ω20μ0-ky2)AU=0,
A=1ω20μ0-ky2ω000ωμ0,
B=1ω20μ0-ky20-kyky0.
Ω=(x, z);-p2<x<p2,a<z<b,
U(p/2, y, z)=U(-p/2, y, z)exp(ikxp).
Ey(r)=Ai exp(ik-  r)+mAms exp(ikm  r),
Hy(r)=Bi exp(ik-  r)+mBms exp(ikm  r),
km=(kmx, kmy, kmz)=kx+2πmp, ky, [ω20uμ0-(kx+2πm/p)2-ky2]1/2.
Ey(r)=mEˆy(m, z)exp{i[(kx+2mπ/p)x+kyy]},
Hy(r)=mHˆy(m, z)exp{i[(kx+2mπ/p)x+kyy]},
Eˆy(m, z)=1p-p/2p/2Ey(x, y, z)×exp{-i[(kx+2mπ/p)x+kyy]}dx,
Ams exp(ibkm,z)=Eˆy(m, b)-Ai exp(-ibkz)δm0,
Bms exp(ibkm,z)=Hˆy(m, b)-Bi exp(-ibkz)δm0,
Ex=iω20μ0-ky2kyEyx-ωμ0Hyz,
Hx=iω20μ0-ky2ω0Eyz+kyHyx.
Ex(r)=iω20uμ0-ky2(ikxkyAi+iωμ0kzBi)×exp(ik-r)+iω20uμ0-ky2×m(ikm,xkyAms-iωμ0km,zBms)×exp(ikmr),
Hx(r)=iω20uμ0-ky2(-iω0ukzAi+ikxkyBi)×exp(ik-r)+iω20uμ0-ky2×m(iω0ukm,zAms+ikm,xkyBms)×exp(ikmr).
Ex(r)=iω20μ0-ky2×m-ωμ0Hˆy(m, z)z+ikm,xkyEˆy(m, z)×exp[i(km,xx+kyy)].
Hx(r)=iω20μ0-ky2×mω0Eˆy(m, z)z+ikm,xkyHˆy(m, z)×exp[i(km,xx+kyy)].
A()Uˆ(m, b)z-ikm,zA(u)Uˆ(m, b)-ikm,x[B()-B(u)]Uˆ(m, b)=-2ikm,zA(u)AiBiexp(-ikm,zb)δm0,
Uˆ(m, b)=Eˆy(m, b)Hˆy(m, b).
A()Uˆ(m, a)z+ikm,zA(l)Uˆ(m, a)-ikm,x[B()-B(l)]Uˆ(m, a)=0,
km,x=kx+2mπ/p,
km,z=[ω20lμ0-(kx+2πm/p)2-ky2]1/2.
Eyi(r)=mAmi exp(ikm-r),
Hyi(r)=mBmi exp(ikm-r),
km-={kx+2πm/p,ky,-[ω20uμ0-(kx+2πm/p)2+ky2]1/2},
F(Ei)(kx, ky, z)=--Ei(x, y, z)×exp[-i(kx+kyy)]dxdy.
Ei(x, y, zi)
=14π2--F(Ei)(kx, ky, zi)×exp[i(kxx+kyy)]dkxdky=14π2m=---π/p+2πm/pπ/p+2πm/pF(Ei)(kx, ky, zi)×exp[i(kxx+kyy)]dkxdky=14π2m=---π/pπ/pF(Ei)(kx+2πm/p, ky, zi)×exp{i[(kx+2πm/p)x+kyy)]}dkxdky=--π/pπ/pEkxi(x, ky, zi)exp(ikyy)dkxdky,
Ekxi(x, ky, z)exp(ikyy)=14π2m=-F(Ei)(kx+2πm/p, ky, z)×exp{i[(kx+2πm/p)x+kyy]}
Ekxi(x+p, ky, z)=Ekxi(x, ky, z)exp(ikxp)
2=2+i2,
2<-1,
U(x, y, z)=A1 exp[i(kxx-κ1z)],z>0A2 exp[i(kxx+κ2z)],z<0,
kx2+κj2=ω20μ0jforj=1, 2.
H=(0, Hy, 0)T,E=iω0j(-Hy/z, 0, Hy/x)T,
A2=A1,
kx2=k2121+2,
kxkn1=21+21/2,
kxkn1=21+21/22-22.
kx>kn1.
[(k˜x+2πm/p)2+k˜y2]1/2=Re kxpl
β1TM=kn1,
β1TE=(k2n12-π2/w2)1/2,
p2<λn1<p.
U(x)=r,-p/2x<-w/2r exp(2iβ1d),-w/2x<ω/2r,w/2x<p/2,
Uˆ0=rp-wp+wpexp(2iβ1d),
Uˆ±1=-rπsin(πw/p)[1-exp(2iβ1d)].
Sz=12Re[F(Ex)F(Hy)*-F(Ey)F(Hx)*],
Re kxpl=1.63k.
kxk+mλp2+kyk2=(1.63)2,
Imsin2mπwp|1-exp(2iβ1d)|2,

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