Abstract

The calculation of local absorption is very important in the context of lithography and in photo-detector design. We present a rigorous method for calculating the local absorption in periodic structures. The computation depends primarily on the calculation of the electric field inside the structure. Since the standard definitions produce unsatisfactory results, we use a modified version of a method published by Lalanne [J. Modern Opt. 45, 1357 (1998)]. The results for these field definitions agree very accurately with results obtained by the law of conservation of energy. We present some examples which are typical for the application scenarios of lithography and detectors.

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References

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    [CrossRef]
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    [CrossRef]
  5. P. Lalanne and M. P. Jurek, “Computation of the near field pattern with coupled wave method for transverse magnetic polarisation,” J. Mod. Opt. 45(7), 1357–1374 (1998).
    [CrossRef]
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    [CrossRef]
  9. E. Silberstein, P. Lalanne, J.-P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. 18(11), 2865–2875 (2001).
    [CrossRef]

2009 (1)

2007 (1)

2001 (2)

E. Silberstein, P. Lalanne, J.-P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. 18(11), 2865–2875 (2001).
[CrossRef]

E. Popov and M. Neviere, “Maxwell equations in Fourier space: fast converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18(11), 2886–2894 (2001).
[CrossRef]

2000 (1)

1998 (1)

P. Lalanne and M. P. Jurek, “Computation of the near field pattern with coupled wave method for transverse magnetic polarisation,” J. Mod. Opt. 45(7), 1357–1374 (1998).
[CrossRef]

1996 (2)

1995 (1)

Brenner, K.-H.

Cao, Q.

E. Silberstein, P. Lalanne, J.-P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. 18(11), 2865–2875 (2001).
[CrossRef]

Fertig, M.

Gaylord, T. K.

Grann, E. B.

Hugonin, J.-P.

E. Silberstein, P. Lalanne, J.-P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. 18(11), 2865–2875 (2001).
[CrossRef]

Jurek, M. P.

P. Lalanne and M. P. Jurek, “Computation of the near field pattern with coupled wave method for transverse magnetic polarisation,” J. Mod. Opt. 45(7), 1357–1374 (1998).
[CrossRef]

Kerwien, N.

Lalanne, P.

E. Silberstein, P. Lalanne, J.-P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. 18(11), 2865–2875 (2001).
[CrossRef]

P. Lalanne and E. Silberstein, “Fourier-modal methods applied to waveguide computational problems,” Opt. Lett. 25(15), 1092–1094 (2000).
[CrossRef]

P. Lalanne and M. P. Jurek, “Computation of the near field pattern with coupled wave method for transverse magnetic polarisation,” J. Mod. Opt. 45(7), 1357–1374 (1998).
[CrossRef]

Lalanne, Ph.

Li, L.

Mahrt, R. F.

Moharam, M. G.

Moll, N.

Morf, T.

Morris, G. M.

Neviere, M.

Osten, W.

Pfluger, T.

Pommet, D. A.

Popov, E.

Rafler, S.

Ruoff, J.

Schuster, T.

Silberstein, E.

E. Silberstein, P. Lalanne, J.-P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. 18(11), 2865–2875 (2001).
[CrossRef]

P. Lalanne and E. Silberstein, “Fourier-modal methods applied to waveguide computational problems,” Opt. Lett. 25(15), 1092–1094 (2000).
[CrossRef]

Stoferle, T.

Trauter, B.

Weiss, J.

J. Lightwave Technol. (1)

J. Mod. Opt. (1)

P. Lalanne and M. P. Jurek, “Computation of the near field pattern with coupled wave method for transverse magnetic polarisation,” J. Mod. Opt. 45(7), 1357–1374 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

E. Silberstein, P. Lalanne, J.-P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. 18(11), 2865–2875 (2001).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Lett. (1)

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Figures (6)

Fig. 1
Fig. 1

Grating geometry

Fig. 2
Fig. 2

Local absorption for the TE-case. Left with true spatial ε-distribution, right with reconstructed ε-distribution.

Fig. 3
Fig. 3

Local absorption for the TM-case. Left with exact spatialε-distribution, right with reconstructed ε-distribution, using the standard field definitions.

Fig. 4
Fig. 4

Local absorption for the TM-case. Left with exact spatialε-distribution, right with reconstructed ε-distribution, using the modified field definitions from Eq. (20). The results agree much better with the global absorption value of 1.389%.

Fig. 6
Fig. 6

Local absorption in a stacked grating structure for different wavelength. Left. exposure at 425 nm, Right: exposure at 675 nm wavelength.

Fig. 5
Fig. 5

Comparison of global and integrated absorption for different number of modes. The blue curve was calculated with the modified field definition while the green curve was calculated with the standard field definition.

Equations (19)

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S=12Re(E×H*)
rotE=μμ0Η˙rotH=j+εε0E˙
div(S)+μμ04t|H|2+εrε04t|E|2=12Re(Ej*)Im(ε)ε02ω|E|2.
SidA+StdA+SrdA=Im(ε)ε02ω|E|2dV
PiPtPr=Pa
Pa=ε0ω2Im(ε(r))|E(r)|2dV
Pa=k02Z0Im(ε(r))|E(r)|2dV
E(r)=E0E1(r)
Pi=SidA=12Z0εiμi|E0|2Asi,z
PaPi=k02ki,z1AIm(ε(r))|E1(r)|2dV
PaPi=δzNXk02ki,zj=0NX1k=0NZ1Im(ε(xj,zk))|E1(xj,zk)|2
Ag=1m(Tm+Rm)
Ey(x,z)=mSm(z)exp(ikx,mx)
a(xj,zk)=k0ni,zδzNXIm(ε(xj,zk))|Ey(xj,zk)|2
ε(x)=m=MMε˜mexp(2πimxP)
Ex(x,z)=mSm(z)exp(ikx,mx)
Ez(x,z)=mum(z)exp(ikx,mx)
Ex(x,z)=1ε(x)mdm(z)exp(ikx,mx)
Ex(x,z)=mem(z)exp(ikx,mx)

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