Abstract

A new formulation of the Fourier modal method (FMM) that applies the correct rules of Fourier factorization for crossed surface-relief gratings is presented. The new formulation adopts a general nonrectangular Cartesian coordinate system, which gives the FMM greater generality and in some cases the ability to save computer memory and computation time. By numerical examples, the new FMM is shown to converge much faster than the old FMM. In particular, the FMM is used to produce well-converged numerical results for metallic crossed gratings. In addition, two matrix truncation schemes, the parallelogramic truncation and a new circular truncation, are considered. Numerical experiments show that the former is superior.

© 1997 Optical Society of America

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  1. P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
    [CrossRef]
  2. D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
    [CrossRef]
  3. S. T. Han, Y.-L. Tsao, R. M. Walser, M. F. Becker, “Electromagnetic scattering of two-dimensional surface-relief dielectricgratings,” Appl. Opt. 31, 2343–2352 (1992).
    [CrossRef] [PubMed]
  4. G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
    [CrossRef]
  5. R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
    [CrossRef]
  6. G. Granet, “Diffraction par des surfaces bipériodiques: résolutionen coordonnées non-orthogonales,” Pure Appl. Opt. 4, 777–793 (1995).
    [CrossRef]
  7. J. B. Harris, T. W. Preist, J. R. Sambles, R. N. Thorpe, R. A. Watts, “Optical response of bigratings,” J. Opt. Soc. Am. A 13, 2041–2049 (1996).
    [CrossRef]
  8. D. C. Dobson, J. A. Cox, “An integral equation method for biperiodic diffraction structures,” in International Conference on the Application and Theory of Periodic Structures , J. M. Lerner, W. R. McKinney, eds., Proc. SPIE1545, 106–113 (1991).
    [CrossRef]
  9. R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
    [CrossRef]
  10. E. Noponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elementswith three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
    [CrossRef]
  11. J.-J. Greffet, C. Baylard, P. Versaevel, “Diffraction of electromagnetic waves by crossed gratings: a seriessolution,” Opt. Lett. 17, 1740–1742 (1992).
    [CrossRef] [PubMed]
  12. O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation ofboundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993).
    [CrossRef]
  13. O. P. Bruno, F. Reitich, “Calculation of electromagnetic scattering via boundary variations andanalytic continuation,” Appl. Computat. Electromagn. Soc. J. 11, 17–31 (1996).
  14. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  15. P. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  16. G. Granet, B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellargratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
    [CrossRef]
  17. See, for example, R. C. Wrede, Introduction to Vector and Tensor Analysis (Dover, New York, 1972), Chap. 1.
  18. We ignore the electromagnetic edge effect at the vertices of the zigzag contour. In the far field, this artificially introduced edge effect should be negligible.
  19. At the time of the writing, I have not mathematically proven the validity or invalidity of the hypothesis. However, the numerical examples given in Section 7 seem to support this hypothesis. The sum clearly corresponds to the Fourier coefficient of ∊E2 with respect to x1, albeit in a complicated way. The latter, as indicated in Eq. (26b), is continuous with respect to x2.
  20. L. Li, “Formulation and comparison of two recursive matrix algorithms for modelinglayered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  21. S. Zohar, “Toeplitz matrix inversion: the algorithm of W. F. Trench,” J. Assoc. Comput. Mach. 16, 592–601 (1969).
    [CrossRef]
  22. L. Li, C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffractiongratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993).
    [CrossRef]

1996 (6)

1995 (1)

G. Granet, “Diffraction par des surfaces bipériodiques: résolutionen coordonnées non-orthogonales,” Pure Appl. Opt. 4, 777–793 (1995).
[CrossRef]

1994 (1)

1993 (3)

1992 (2)

1982 (1)

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

1979 (1)

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

1978 (2)

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
[CrossRef]

1969 (1)

S. Zohar, “Toeplitz matrix inversion: the algorithm of W. F. Trench,” J. Assoc. Comput. Mach. 16, 592–601 (1969).
[CrossRef]

Baylard, C.

Becker, M. F.

Bräuer, R.

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Bruno, O. P.

O. P. Bruno, F. Reitich, “Calculation of electromagnetic scattering via boundary variations andanalytic continuation,” Appl. Computat. Electromagn. Soc. J. 11, 17–31 (1996).

O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation ofboundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993).
[CrossRef]

Bryngdahl, O.

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Cox, J. A.

D. C. Dobson, J. A. Cox, “An integral equation method for biperiodic diffraction structures,” in International Conference on the Application and Theory of Periodic Structures , J. M. Lerner, W. R. McKinney, eds., Proc. SPIE1545, 106–113 (1991).
[CrossRef]

Derrick, G. H.

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Dobson, D. C.

D. C. Dobson, J. A. Cox, “An integral equation method for biperiodic diffraction structures,” in International Conference on the Application and Theory of Periodic Structures , J. M. Lerner, W. R. McKinney, eds., Proc. SPIE1545, 106–113 (1991).
[CrossRef]

Granet, G.

G. Granet, B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellargratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
[CrossRef]

G. Granet, “Diffraction par des surfaces bipériodiques: résolutionen coordonnées non-orthogonales,” Pure Appl. Opt. 4, 777–793 (1995).
[CrossRef]

Greffet, J.-J.

Guizal, B.

Haggans, C. W.

Han, S. T.

Harris, J. B.

Lalanne, P.

Li, L.

Maystre, D.

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
[CrossRef]

McPhedran, R. C.

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Morris, G. M.

Nevière, M.

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
[CrossRef]

Noponen, E.

Preist, T. W.

Reitich, F.

O. P. Bruno, F. Reitich, “Calculation of electromagnetic scattering via boundary variations andanalytic continuation,” Appl. Computat. Electromagn. Soc. J. 11, 17–31 (1996).

O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation ofboundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993).
[CrossRef]

Sambles, J. R.

Thorpe, R. N.

Tsao, Y.-L.

Turunen, J.

Versaevel, P.

Vincent, P.

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

Walser, R. M.

Watts, R. A.

Wrede, R. C.

See, for example, R. C. Wrede, Introduction to Vector and Tensor Analysis (Dover, New York, 1972), Chap. 1.

Zohar, S.

S. Zohar, “Toeplitz matrix inversion: the algorithm of W. F. Trench,” J. Assoc. Comput. Mach. 16, 592–601 (1969).
[CrossRef]

Appl. Computat. Electromagn. Soc. J. (1)

O. P. Bruno, F. Reitich, “Calculation of electromagnetic scattering via boundary variations andanalytic continuation,” Appl. Computat. Electromagn. Soc. J. 11, 17–31 (1996).

Appl. Opt. (1)

Appl. Phys. (1)

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

J. Assoc. Comput. Mach. (1)

S. Zohar, “Toeplitz matrix inversion: the algorithm of W. F. Trench,” J. Assoc. Comput. Mach. 16, 592–601 (1969).
[CrossRef]

J. Opt. (Paris) (2)

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
[CrossRef]

J. Opt. Soc. Am. A (8)

Opt. Commun. (2)

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

Opt. Lett. (1)

Pure Appl. Opt. (1)

G. Granet, “Diffraction par des surfaces bipériodiques: résolutionen coordonnées non-orthogonales,” Pure Appl. Opt. 4, 777–793 (1995).
[CrossRef]

Other (4)

D. C. Dobson, J. A. Cox, “An integral equation method for biperiodic diffraction structures,” in International Conference on the Application and Theory of Periodic Structures , J. M. Lerner, W. R. McKinney, eds., Proc. SPIE1545, 106–113 (1991).
[CrossRef]

See, for example, R. C. Wrede, Introduction to Vector and Tensor Analysis (Dover, New York, 1972), Chap. 1.

We ignore the electromagnetic edge effect at the vertices of the zigzag contour. In the far field, this artificially introduced edge effect should be negligible.

At the time of the writing, I have not mathematically proven the validity or invalidity of the hypothesis. However, the numerical examples given in Section 7 seem to support this hypothesis. The sum clearly corresponds to the Fourier coefficient of ∊E2 with respect to x1, albeit in a complicated way. The latter, as indicated in Eq. (26b), is continuous with respect to x2.

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Figures (10)

Fig. 1
Fig. 1

Crossed grating illuminated by a plane wave. A rectangular Cartesian coordinate system is attached to the grating so that its x axis is along one of the periodic directions.

Fig. 2
Fig. 2

Top view of a crossed grating and a nonrectangular Cartesian coordinate system in the grating plane. The x1 and x2 axes are parallel to two periodic directions of the grating. The x1 axis is parallel to the x axis, and the x2 axis forms an angle ζ with the y axis.

Fig. 3
Fig. 3

Section of the reciprocal lattice. The diffraction orders are located at crossings of the grid lines, which are parallel to the contravariant basis vectors. The solid circle indicates the disk of propagating orders. The dashed circle and parallelogram show two truncation schemes.

Fig. 4
Fig. 4

Discretization of an arbitrary (in this case, a circular) grating contour Γ by a zigzag contour Γ that consists of line segments of the grid lines parallel to the covariant basis vectors.

Fig. 5
Fig. 5

Top view of the checkerboard grating that is described in Example 1. Areas labeled with letters A, B, and C are three possible unit cells.

Fig. 6
Fig. 6

Convergence of the (0, -1)st transmitted order of the checkerboard grating. The letters in the legend have the following denotations: O for old FMM, N for new FMM, P for parallelogramic truncation, C for circular truncation, A for using unit cell A, and B for using unit cell B.

Fig. 7
Fig. 7

Convergence of the (-1, -2)nd transmitted order of the circular pillar grating in Example 2. The legends read the same way as in Fig. 6.

Fig. 8
Fig. 8

Top view of the metallic grating that is considered in Example 3.

Fig. 9
Fig. 9

Convergence of the (0, -1)th transmitted order of the metallic grating in Example 3. The legends read the same way as in Figs. 6 and 7.

Fig. 10
Fig. 10

The same as Fig. 9, but for the (0, 0)th reflected order.

Tables (5)

Tables Icon

Table 1 Diffraction Efficiencies (%) of the Transmitted Orders of the Checkerboard Grating in Example 1

Tables Icon

Table 2 Diffraction Efficiencies (%) of the Circular Pillar Grating in Example 2: Reflected Orders

Tables Icon

Table 3 Diffraction Efficiencies (%) of the Circular Pillar Grating in Example 2: Transmitted Orders

Tables Icon

Table 4 Diffraction Efficiencies (%) of the Metallic Grating in Example 3: Reflected Orders

Tables Icon

Table 5 Diffraction Efficiencies (%) of the Metallic Grating in Example 3: Transmitted Orders

Equations (61)

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x1=x-y tan ζ,x2=y sec ζ,x3=z.
b1=xˆ,b2=xˆ sin ζ+yˆ cos ζ,b3=zˆ,
b1=xˆ-yˆ tan ζ,b2=yˆ sec ζ,b3=zˆ,
mn=1d1d2 0d20d1(x1, x2)×exp[-i(mK1x1+nK2x2)]dx1dx2,
mn,jl=m-j,n-l.
mn=1d1 0d1(x1, x2)exp[-i(m-n)K1x1]dx1,
mn=1d2 0d2(x1, x2)exp[-i(m-n)K2x2]dx2.
mn,jl={1/-1}mjnl=1d2 0d2{1/-1}mj(x2)×exp[-i(n-l)K2x2]dx2,
mn,jl={1/-1}nlmj=1d1 0d1{1/-1}nl(x1)×exp[-i(m-j)K1x1]dx1.
k=kσbσ=α0b1+β0b2-γ00(+1)b3,
α0=k(+1) sin θ cos φ,
β0=k(+1) sin θ sin(φ+ζ),
γ00(+1)=k(+1) cos θ,
Eσ(r)=Iσ exp[i(α0x1+β0x2-γ00(+1)x3)]+m,nRσmn×exp[i(αmx1+βnx2+γmn(+1)x3)],
(x3>h),
Eσ(r)=m,nTσmn exp[i(αmx1+βnx2-γmn(-1)x3)]
(x3<0),
αm=α0+mK1,βn=β0+nK2,
sec2ζ (αm2+βn2-2αmβn sinζ)+γmn(a)2=k(a)2
(a=±1).
Re[γmn(a)]+Im[γmn(a)]>0(a=±1).
kmn(±1)=αmb1+βnb2±γmn(±1)b3.
U(a)={(m, n)|Im[γmn(a)]=0, m, n}.
πk(a)2(K1K2/cosζ)=π(d1d2 cosζ)λ2/((a)μ),
1γ00(+1) [(k(+1)2-β02)|I1|2+(k(+1)2-α02)|I2|2
+(α0β0-k(+1)2 sin ζ)(I1I2¯+I2I1¯)]=1,
ηmn(a)=1γmn(a) [(k(a)2-βn2)|E1mn(a)|2+(k(a)2-αm2)|E2mn(a)|2+(αmβn-k(a)2 sin ζ)×(E1mn(a)E2mn(a)¯+E2mn(a)E1mn(a)¯)],
Ψ(x1, x2, x3)=m,nΨmn(x3)exp(iαmx1+iβnx2),
2E3-3E2=ik0μ sec ζ (H1-sin ζ H2),
3E1-1E3=ik0μ sec ζ (H2-sin ζ H1),
1E2-2E1=ik0μ cos ζ H3,
2H3-3H2=-ik0 sec ζ (E1-sin ζ E2),
3H1-1H3=-ik0 sec ζ (E2-sin ζ E1),
1H2-2H1=-ik0 cos ζ E3.
E1(x02+0)=E1(x02-0),
E2(x01+0)=E2(x01-0),
(x01+0)E1(x01+0)=(x01-0)E1(x01-0),
(x02+0)E2(x02+0)=(x02-0)E2(x02-0),
cosζ k0i 3E1mn=+k02μ(H2mn-H1mn sinζ)-αmj,lmn,jl-1(αjH2jl-βlH1jl),
cosζ k0i 3E2mn=-k02μ(H1mn-H2mn sinζ)-βnj,lmn,jl-1(αjH2jl-βlH1jl).
E1=sec2ζ (E1-E2 sinζ)=E1-E2 sinζ.
2H3m-3H2mik0 cos ζ+j1mj-1E1j=sinζ j1mj-1Ej2.
j,l1mn,jl2H3jl-3H2jlik0 cos ζ+s,pjl,spE1sp
=sin ζ Emn2.
cos ζ μk0i 3H2mn
=βn(αmE2mn-βnE1mn)-μk02 sin ζ j,l1mn,jl-1E2jl
+μk02j,lcos2ζ mn,jl+sin2ζ 1mn,jl-1E1jl.
cos ζ μk0i 3H1mn
=αm(αmE2mn-βnE1mn)+μk02 sinζ j,l1mn,jl-1E1jl
-μk02j,lcos2ζ mn,jl+sin2ζ 1mn,jl-1E2jl.
cosζ k0i 3E1E2=FH1H2,
cosζ k0i 3H1H2=GE1E2,
F=α-1β-μk02 sinζμk02-α-1αβ-1β-μk02μk02 sinζ-β-1α,
G=μk02 sinζ 1-1-αβα2-μk02cos2ζ +sin2ζ 1-1μk02cos2ζ +sin2 ζ 1-1-β2αβ-μk02 sinζ 1-1.
(FG-μk02 cos2ζ γ2)E1E2=0.
H1H2=secζμk0γ GE1E2.
Re[γ]+Im[γ]>0.
Eσ(r)=m,n,q[uq exp(iγqx3)+dq exp(-iγqx3)]×exp[i(αmx1+βnx2)]Eσmnq,
Hσ(r)=m,n,q[uq exp(iγqx3)-dq exp(-iγqx3)]×exp[i(αmx1+βnx2)]Hσmnq,
E1mnE2mnH1mnH2mn=E1mnqE1mnqE2mnqE2mnqH1mnq-H1mnqH2mnq-H2mnq×exp(iγqx3)00exp(-iγqx3)uqdq.
Φ=Wϕud.

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