Abstract

The “projection operator” method is a variation of the normal-vector method for simulating optical diffraction from biperiodic gratings. A projection operator defined by the normal vector is interpolated over the grating volume rather than interpolating the vector itself. This approach circumvents difficulties associated with sign reversals and discontinuities encountered with the normal-vector method, and it facilitates implementation for general grating geometries. The method is readily extensible to anisotropic and bianisotropic materials. Several numerical examples of the new method are presented, including comparisons to previously published test cases.

© 2014 Optical Society of America

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References

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  1. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
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    [CrossRef]
  3. L. Li, “Mathematical reflections on the Fourier modal method in grating theory,” in Mathematical Modeling in Optical Science, G. Bao, L. Cowsar, and W. Masters, eds. (Society for Industrial and Applied Mathematics, 2001), pp. 111–139.
  4. L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A 5, 345–355 (2003).
    [CrossRef]
  5. M. Onishi, K. Crabtree, and R. A. Chipman, “Formulation of rigorous coupled-wave theory for gratings in bianisotropic media,” J. Opt. Soc. Am. A 28, 1747–1758 (2011).
    [CrossRef]
  6. E. Popov and M. Nevière, “Fast Fourier factorization (FFF) method,” in Light Propagation in Periodic Media, Differential Theory and Design (Dekker, 2003), pp. 77–98.
  7. E. Popov and M. Nevière, “Maxwell equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18, 2886–2894 (2001).
    [CrossRef]
  8. A. David, H. Benisty, and C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B 73, 075107 (2006).
    [CrossRef]
  9. T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, and W. Osten, “Normal vector method for convergence improvement using the RCWA for crossed gratings,” J. Opt. Soc. Am. A 24, 2880–2890 (2007).
    [CrossRef]
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    [CrossRef]
  11. V. Liu and S. Fan, “S4: a free electromagnetic solver for layered periodic structures,” Comput. Phys. Commun. 183, 2233–2244 (2012).
    [CrossRef]
  12. H. Deng and S. Chen, “Preferred NV field for arbitrary crossed-gratings with analytic boundary,” ECS Trans. 52, 859–864 (2013).
  13. R. Antos, “Fourier factorization with complex polarization bases in modeling optics of discontinuous bi-periodic structures,” Opt. Express 17, 7269–7274 (2009).
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  14. R. Antos and M. Veis, “Fourier factorization with complex polarization bases in the plane-wave expansion method applied to two-dimensional photonic crystals,” Opt. Express 18, 27511–27524 (2010).
    [CrossRef]
  15. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]

2013

H. Deng and S. Chen, “Preferred NV field for arbitrary crossed-gratings with analytic boundary,” ECS Trans. 52, 859–864 (2013).

2012

V. Liu and S. Fan, “S4: a free electromagnetic solver for layered periodic structures,” Comput. Phys. Commun. 183, 2233–2244 (2012).
[CrossRef]

2011

2010

2009

2008

2007

2006

A. David, H. Benisty, and C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B 73, 075107 (2006).
[CrossRef]

2003

L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A 5, 345–355 (2003).
[CrossRef]

2001

1996

1995

Antos, R.

Benisty, H.

A. David, H. Benisty, and C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B 73, 075107 (2006).
[CrossRef]

Chen, S.

H. Deng and S. Chen, “Preferred NV field for arbitrary crossed-gratings with analytic boundary,” ECS Trans. 52, 859–864 (2013).

Chipman, R. A.

Crabtree, K.

David, A.

A. David, H. Benisty, and C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B 73, 075107 (2006).
[CrossRef]

Deng, H.

H. Deng and S. Chen, “Preferred NV field for arbitrary crossed-gratings with analytic boundary,” ECS Trans. 52, 859–864 (2013).

Fan, S.

V. Liu and S. Fan, “S4: a free electromagnetic solver for layered periodic structures,” Comput. Phys. Commun. 183, 2233–2244 (2012).
[CrossRef]

Frenner, K.

Gaylord, T. K.

Götz, P.

Grann, E. B.

Kerwien, N.

Li, L.

L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A 5, 345–355 (2003).
[CrossRef]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
[CrossRef]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[CrossRef]

L. Li, “Mathematical reflections on the Fourier modal method in grating theory,” in Mathematical Modeling in Optical Science, G. Bao, L. Cowsar, and W. Masters, eds. (Society for Industrial and Applied Mathematics, 2001), pp. 111–139.

Liu, V.

V. Liu and S. Fan, “S4: a free electromagnetic solver for layered periodic structures,” Comput. Phys. Commun. 183, 2233–2244 (2012).
[CrossRef]

Moharam, M. G.

Nevière, M.

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

E. Popov and M. Nevière, “Fast Fourier factorization (FFF) method,” in Light Propagation in Periodic Media, Differential Theory and Design (Dekker, 2003), pp. 77–98.

Onishi, M.

Osten, W.

Pommet, D. A.

Popov, E.

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

E. Popov and M. Nevière, “Fast Fourier factorization (FFF) method,” in Light Propagation in Periodic Media, Differential Theory and Design (Dekker, 2003), pp. 77–98.

Rafler, S.

Ruoff, J.

Schuster, T.

Veis, M.

Weisbuch, C.

A. David, H. Benisty, and C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B 73, 075107 (2006).
[CrossRef]

Comput. Phys. Commun.

V. Liu and S. Fan, “S4: a free electromagnetic solver for layered periodic structures,” Comput. Phys. Commun. 183, 2233–2244 (2012).
[CrossRef]

ECS Trans.

H. Deng and S. Chen, “Preferred NV field for arbitrary crossed-gratings with analytic boundary,” ECS Trans. 52, 859–864 (2013).

J. Opt. A

L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A 5, 345–355 (2003).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Phys. Rev. B

A. David, H. Benisty, and C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B 73, 075107 (2006).
[CrossRef]

Other

L. Li, “Mathematical reflections on the Fourier modal method in grating theory,” in Mathematical Modeling in Optical Science, G. Bao, L. Cowsar, and W. Masters, eds. (Society for Industrial and Applied Mathematics, 2001), pp. 111–139.

E. Popov and M. Nevière, “Fast Fourier factorization (FFF) method,” in Light Propagation in Periodic Media, Differential Theory and Design (Dekker, 2003), pp. 77–98.

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Figures (6)

Fig. 1.
Fig. 1.

Line grating profile and normal-vector field.

Fig. 2.
Fig. 2.

Biperiodic grating (cylindrical posts) and normal-vector field.

Fig. 3.
Fig. 3.

Convergence plot for the isotropic test case (metal hole array).

Fig. 4.
Fig. 4.

Repeat of Fig. 3 plot comparing the isotropic algorithm [Eq. (40)] with the anisotropic algorithm [Eq. (71)].

Fig. 5.
Fig. 5.

Convergence plot for the anisotropic test case with a rectangular region boundary.

Fig. 6.
Fig. 6.

Convergence plot for the anisotropic test case with an elliptical region boundary.

Equations (102)

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ε[x+d]=ε[x],μ[x+d]=μ[x],
d=e^2d2.
ε[x+d1]=ε[x]=ε[x+d2],μ[x+d1]=μ[x]=μ[x+d2].
d1=e^2d2,1+e^3d3,1,d2=e^2d2,2+e^3d3,2.
a*b=@mnamnbn.
a*=@m,namn.
×E=i2πλB,×H=i2πλD;
D=εE,B=μH.
ε[x]=m1,m2ε˜m1,m2[x1]exp[i2π(m1g1+m2g2)·x].
g1=e^2g2,1+e^3g3,1,g2=e^2g2,2+e^3g3,2;
g1·d1=1,g1·d2=0,g2·d1=0,g2·d2=1.
ε˜m1,m2[x1]=0101(ε[e^1x1+d1t1+d2t2]exp[i2π(m1t1+m2t2)])dt1dt2.
f0,0=e^2f2,0,0+e^3f3,0,0.
E[x]=m1,m2E˜m1,m2[x1]exp[i2πfm1,m2·x],
fm1,m2=f0,0+m1g1+m2g2.
d=(d1,d2)=(00d2,1d2,2d3,1d3,2),
g=(g1,g2)=(00g2,1g2,2g3,1g3,3),
m=(m1,m2).
ε[x]=mε˜m[x1]exp[i2πmg·x],
g·d=(1001),
E[x]=mE˜m[x1]exp[i2πfm·x],
fm·=f0,0·+mg·.
(e^1ddx1+i2πfm)×E˜m=i2πλB˜m,(e^1ddx1+i2πfm)×H˜m=i2πλD˜m.
ND=εNE,
TD=εTE;
N=n^n^·on lateral discontinuities.
N+T=I.
ε1ND=NE.
(2t12+2t22)N[e^1x1+d1t1+d2t2]=0.
Nj,k=Nk,j.
N1,j=0(lamellar).
(TD)j=εkTj,kEk,
ε1(ND)j=kNj,kEk.
F[TD]j,m=nε˜mnk,pT˜j,k,npE˜k,p,
n(ε˜1)mnF[ND]j,n=k,nN˜j,k,mnE˜k,n.
F[TD]j=ε˜*kT˜j,k*E˜k,
ε˜1*F[ND]j=kN˜j,k*E˜k.
F[ND]j=(ε˜1*)1kN˜j,k*E˜k.
D˜j=kε˜j,kE˜k,
ε˜j,k=(ε˜*)δj,k+((ε˜1*)1ε˜*)N˜j,k*.
D˜j,m=k,nε˜j,k,m,nE˜k,n.
ND=Nε(NE+TE),
TD=Tε(NE+TE).
n^·εn^=j,kn^jεj,kn^k=j,kNk,jεj,k=Tr[Nε],
(1Tr[Nε])ND=NεNTr[Nε]E+NεTr[Nε]TE.
TD=TεTr[Nε]ND+(TTεTr[Nε]N)εNE+(TTεTr[Nε]N)εTE.
h=fpg,
F[h]=F[f]*F[pg].
F[h]=F[fp]*F[g](conjecture).
12(f+g++fg)=12(f++f)12(g++g).
(f+f)(g+g)=0.
(f+f)p(g+g)=0.
UND=VE,
TD=PND+QE;
U=1Tr[Nε],
V=UNε,
P=UTε=UεV,
Q=(TPN)ε=ε(I+P)Nε.
U(ND)j=kVj,kEk,
(TD)j=k(Pj,k(ND)k+Qj,kEk).
nU˜mnF[ND]j,n=k,nV˜j,k,mnE˜k,n,
F[TD]j,m=k,n(P˜j,k,mnF[ND]k,n+Q˜j,k,mnE˜k,n).
U˜*=@m,nU˜mn,U=@(j,m),(k,n)δj,k(U˜*)m,n,
V=@(j,m),(k,n)V˜j,k,mn,P=@(j,m),(k,n)P˜j,k,mn,Q=@(j,m),(k,n)Q˜j,k,mn.
U(@(j,m)F[ND]j,m)=V(@(j,m)E˜j,m),
(@(j,m)F[TD]j,m)=P(@(j,m)F[ND]j,m)+Q(@(j,m)E˜j,m).
(@(j,m)F[ND]j,m)=U1V(@(j,m)E˜j,m).
U1=@(j,m),(k,n)δj,k((U˜*)1)m,n.
(@(j,m)F[TD]j,m)=(PU1V+Q)(@(j,m)E˜j,m).
(@(j,m)D˜j,m)=((I+P)U1V+Q)(@(j,m)E˜j,m).
(@(j,m),(k,n)ε˜j,k,m,n)=(I+P)U1V+Q.
ε=(2.00002.50.5i00.5i2.5).
D=εE+ξH,B=μH+ζE,
D˜j,m=kn(ε˜j,k,m,nE˜k,n+ξ˜j,k,m,nH˜k,n),B˜j,m=kn(μ˜j,k,m,nH˜k,n+ζ˜j,k,m,nE˜k,n).
(NDNB)=(NεNξNζNμ)(NE+TENH+TH),
(TDTB)=(TεTξTζTμ)(NE+TENH+TH).
(NDNB)=(NεNNξNNζNNμN)(EH)+=((n^·εn^)N(n^·ξn^)N(n^·ζn^)N(n^·μn^)N)(EH)+on lateral discontinuities.
(NDNB)=(Tr[Nε]NTr[Nξ]NTr[Nζ]NTr[Nμ]N)(EH)+=(Tr[Nε]ITr[Nξ]ITr[Nζ]ITr[Nμ]I)(NENH)+on lateral discontinuities,
(U[E,D]IU[E,B]IU[H,D]IU[H,B]I)(NDNB)=(NENH)+on lateral discontinuities,
(U[E,D]U[E,B]U[H,D]U[H,B])=(Tr[Nε]Tr[Nξ]Tr[Nζ]Tr[Nμ])1.
(U[E,D]IU[E,B]IU[H,D]IU[H,B]I)(NDNB)=(U[E,D]IU[E,B]IU[H,D]IU[H,B]I)(NεNNξNNζNNμN)(EH)+(U[E,D]IU[E,B]IU[H,D]IU[H,B]I)(NεNξNζNμ)(TETH).
(U[E,D]IU[E,B]IU[H,D]IU[H,B]I)(NDNB)=(NENH)+(U[E,D]IU[E,B]IU[H,D]IU[H,B]I)(NεNξNζNμ)(TETH)on lateral discontinuities.
(TDTB)=(TεTξTζTμ)(U[E,D]IU[E,B]IU[H,D]IU[H,B]I)(NDNB)+(TεTξTζTμ)((N00N)(U[E,D]IU[E,B]IU[H,D]IU[H,B]I)(NεNNξNNζNNμN))(EH)+(TεTξTζTμ)((I00I)(U[E,D]IU[E,B]IU[H,D]IU[H,B]I)(NεNξNζNμ))(TETH).
(U[E,D]IU[E,B]IU[H,D]IU[H,B]I)(NDNB)=(V[E,E]V[E,H]V[H,E]V[H,H])(EH),
(TDTB)=(P[D,D]P[D,B]P[B,D]P[B,B])(NDNB)+(Q[D,E]Q[D,H]Q[B,E]Q[B,H])(EH);
(V[E,E]V[E,H]V[H,E]V[H,H])=(U[E,D]IU[E,B]IU[H,D]IU[H,B]I)(NεNξNζNμ),
(P[D,D]P[D,B]P[B,D]P[B,B])=(TεTξTζTμ)(U[E,D]IU[E,B]IU[H,D]IU[H,B]I),
(Q[D,E]Q[D,H]Q[B,E]Q[B,H])=(TεTξTζTμ)((I00I)(U[E,D]IU[E,B]IU[H,D]IU[H,B]I)(NεNξNζNμ)).
(U[E,D]U[E,B]U[H,D]U[H,B])((ND)j(NB)j)=k(Vj,k[E,E]Vj,k[E,H]Vj,k[H,E]Vj,k[H,H])(EkHk),
((TD)j(TB)j)=k((Pj,k[D,D]Pj,k[D,B]Pj,k[B,D]Pj,k[B,B])((ND)k(NB)k)+(Qj,k[D,E]Qj,k[D,H]Qj,k[B,E]Qj,k[B,H])(EkHk)).
n(U˜mn[E,D]U˜mn[E,B]U˜mn[H,D]U˜mn[H,B])(F[ND]j,nF[NB]j,n)=k,n(V˜j,k,mn[E,E]V˜j,k,mn[E,H]V˜j,k,mn[H,E]V˜j,k,mn[H,H])(E˜k,nH˜k,n),
(F[TD]j,mF[TB]j,m)=k,n((P˜j,k,mn[D,D]P˜j,k,mn[D,B]P˜j,k,mn[B,D]P˜j,k,mn[B,B])(F[ND]k,nF[NB]k,n)+(Q˜j,k,mn[D,E]Q˜j,k,mn[D,H]Q˜j,k,mn[B,E]Q˜j,k,mn[B,H])(E˜k,nH˜k,n)).
U˜[]*=@m,nU˜mn[],U[]=@(j,m),(k,n)δj,k(U˜[]*)m,n,
V[]=@(j,m),(k,n)V˜j,k,mn[],P[]=@(j,m),(k,n)P˜j,k,mn[],Q[]=@(j,m),(k,n)Q˜j,k,mn[].
(U[E,D]U[E,B]U[H,D]U[H,B])(@(j,m)F[ND]j,m@(j,m)F[NB]j,m)=(V[E,E]V[E,H]V[H,E]V[H,H])(@(j,m)E˜j,m@(j,m)H˜j,m),
(@(j,m)F[TD]j,m@(j,m)F[TB]j,m)=(P[D,D]P[D,B]P[B,D]P[B,B])(@(j,m)F[ND]j,m@(j,m)F[NB]j,m)+(Q[D,E]Q[D,H]Q[B,E]Q[B,H])(@(j,m)E˜j,m@(j,m)H˜j,m).
(@(j,m)F[ND]j,m@(j,m)F[NB]j,m)=(U[E,D]U[E,B]U[H,D]U[H,B])1(V[E,E]V[E,H]V[H,E]V[H,H])(@(j,m)E˜j,m@(j,m)H˜j,m).
(U[E,D]U[E,B]U[H,D]U[H,B])1=((U˜[E,D]*)I(U˜[E,B]*)I(U˜[H,D]*)I(U˜[H,D]*)I)1=(((U˜*)1)[D,E]I((U˜*)1)[D,H]I((U˜*)1)[B,E]I((U˜*)1)[B,H]I),
(((U˜*)1)[D,E]((U˜*)1)[D,H]((U˜*)1)[B,E]((U˜*)1)[B,H])=((U˜[E,D]*)(U˜[E,B]*)(U˜[H,D]*)(U˜[H,B]*))1.
(@(j,m)F[TD]j,m@(j,m)F[TB]j,m)=((P[D,D]P[D,B]P[B,D]P[B,B])(U[E,D]U[E,B]U[H,D]U[H,B])1(V[E,E]V[E,H]V[H,E]V[H,H])+(Q[D,E]Q[D,H]Q[B,E]Q[B,H]))(@(j,m)E˜j,m@(j,m)H˜j,m).
(@(j,m)D˜j,m@(j,m)B˜j,m)=((I+P[D,D]P[D,B]P[B,D]I+P[B,B])(U[E,D]U[E,B]U[H,D]U[H,B])1(V[E,E]V[E,H]V[H,E]V[H,H])+(Q[D,E]Q[D,H]Q[B,E]Q[B,H]))(@(j,m)E˜j,m@(j,m)H˜j,m).
(@(j,m),(k,n)(ε˜j,k,m,nξ˜j,k,m,nμ˜j,k,m,nζ˜j,k,m,n))=(I+P[D,D]P[D,B]P[B,D]I+P[B,B])(U[E,D]U[E,B]U[H,D]U[H,B])1(V[E,E]V[E,H]V[H,E]V[H,H])+(Q[D,E]Q[D,H]Q[B,E]Q[B,H]).

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