Abstract

The lamellar grating problem is reformulated through the concept of adaptive spatial resolution. We introduce a new coordinate system such that spatial resolution is increased around the discontinuities of the permittivity function. We derive a new eigenproblem that we solve by using Fourier expansions for both the field and the coefficients of Maxwell’s equations. We provide numerical evidence that highly improved convergence rates can be obtained.

© 1999 Optical Society of America

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References

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  1. J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
    [CrossRef]
  2. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  3. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  4. G. Granet, J. Chandezon, “The method of curvilinear coordinates applied to the problem of scattering from surface-relief gratings defined by parametric equations: application to scattering from a cycloidal grating,” Pure Appl. Opt. 6, 727–740 (1997).
    [CrossRef]
  5. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
    [CrossRef]
  6. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar grating,” Opt. Acta 28, 1087–1102 (1981).
    [CrossRef]
  7. L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
    [CrossRef]

1997 (1)

G. Granet, J. Chandezon, “The method of curvilinear coordinates applied to the problem of scattering from surface-relief gratings defined by parametric equations: application to scattering from a cycloidal grating,” Pure Appl. Opt. 6, 727–740 (1997).
[CrossRef]

1996 (1)

1993 (1)

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

1982 (2)

1981 (2)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Chandezon, J.

G. Granet, J. Chandezon, “The method of curvilinear coordinates applied to the problem of scattering from surface-relief gratings defined by parametric equations: application to scattering from a cycloidal grating,” Pure Appl. Opt. 6, 727–740 (1997).
[CrossRef]

J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
[CrossRef]

Cornet, G.

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Dupuis, M. T.

Gaylord, T. K.

Granet, G.

G. Granet, J. Chandezon, “The method of curvilinear coordinates applied to the problem of scattering from surface-relief gratings defined by parametric equations: application to scattering from a cycloidal grating,” Pure Appl. Opt. 6, 727–740 (1997).
[CrossRef]

Li, L.

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, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

Maystre, D.

McPhedran, R. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Moharam, M. G.

J. Mod. Opt. (1)

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Acta (2)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Pure Appl. Opt. (1)

G. Granet, J. Chandezon, “The method of curvilinear coordinates applied to the problem of scattering from surface-relief gratings defined by parametric equations: application to scattering from a cycloidal grating,” Pure Appl. Opt. 6, 727–740 (1997).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Geometry of the diffraction problem.

Fig. 2
Fig. 2

Change of variable x=x(u) for two values of the η parameter, f=0.5.

Fig. 3
Fig. 3

Convergence of the second-smallest real eigenvalue for TE polarization. The exact eigenvalue is found to be 3.35101975722312. The grating and incidence parameters are f=0.4, θ=29, d=1, λ=1, ν21=1, ν22=5.

Fig. 4
Fig. 4

Convergence of the second-smallest real eigenvalue for TM polarization. The exact eigenvalue is found to be 2.81329903403930. The grating and incidence parameters are the same as in Fig. 3.

Fig. 5
Fig. 5

Convergence of the zeroth-order diffraction efficiencies for TE polarization. The grating and incidence parameters are: f=0.5, h=1, θ=30, d=1, λ=1, ν21=1, ν22=0.22-i*6.71, ν1=1, ν3=ν22, η=0.99.

Fig. 6
Fig. 6

Convergence of the zeroth-order diffraction efficiencies for TM polarization. The grating and incidence parameters are the same as in Fig. 5.

Tables (2)

Tables Icon

Table 1 Zero and -1-Order Efficiencies and Computation Time for Various Truncation Orders and Two Values of the η Parameter, for the Grating of Figs. 5 and 6

Tables Icon

Table 2 Efficiencies for Various Truncation Orders and Three Values of the η Parameter for a Grating That Supports Five Diffraction Ordersa

Equations (45)

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ν2(x)=ν21if0<x<fdν22iffd<x<d,
kx=k sin θ,ky=-k cos θ,
k=2πλ=ωμ00,
ψi(x, y)=q[aqi exp(-ikrqiy)+bqi×exp(ikrqiy)]ϕqi(x),
Li(x)ϕqi(x)=rqi2ϕqi(x),
ϕqi(x+d)=exp(-ikx d)ϕqi(x).
Li(x)
=Lei=2x2+k2νi2(x)forTEpolarizationLmi=x 1νi2(x) x+k2forTMpolarization.
ϕq1(x)=ϕq3(x)=ϕq(x)=exp(-ikaqx),
aq=sin θ+q λd,
rqi=νi2-aq2,i[1, 3].
δmq=1ifm=q0ifmq.
Rq=rq1r01 |aq1|2
Tq=rq3r01 |bq3|2forTEpolarizationν1ν32 rq3r01 |bq3|2forTMpolarization
ϕq2(x)=mϕmq2 exp(-ikamx),
x=u-η f2π sin2πufdif0<u<fdu+η 1-f2π sin2π(d-u)(1-f )diffd<u<d,
curl A=1h(u) h(u)exeyezuyzh(u)AxAyAz,
h(u)=xu.
Ezy=-jωμ0Hx,
1h(u) Ezu=-jωμ0Hy,
1h(u) Hyu-[h(u)Hx]y=jω0ν2(u)Ez,
Hzy=jω0ν2(u)Ex,
1h(u) Hzu=jω0ν2(u)Ey,
1h(u) Eyu-[h(u)Ex]y=-jωμ0Hz,
y h(u) Ezy+u 1h(u) Ezu
+ω2μ00ν2(u)h(u)Ez=0
y h(u)ν2(u) Hzy+u 1ν2(u)h(u) Hzu
+ω2μ00h(u)Hz=0
a2+β2=1.
a(u)=ν2(u)h(u),
b(u)=h(u)ν2(u).
Le(u)Ez=-2Ezy2
Le(u)=1h(u) k2a(u)+u 1h(u) u
Lm(u)Hz=-2Hzy2
Lm(u)=1b(u) k2h(u)+u 1a(u) u
[Le]=[hmn]-1[k2[amn]-[a][hmn]-1[a]]
[Lm]=[bmn]-1[k2[hmn]-[a][amn]-1[a]],
hmn=hm-n=1d 0dh(u)exp-i2π(m-n) uddu.
Ez=mq[aeqEmq exp(-ikreqy)+beqEmq exp(ikreqy)]exp(-ikamu),
Hz=mq[ahqHmq exp(-ikrhqy)+bhqHmq exp(ikrhqy)]exp(-ikamu),
ZHx=mqreq[aeqEmq exp(-ikreqy)-beqEmq exp(ikreqy)]exp(-ikamu),
Ex=mqrhq[ahqExmq exp(-ikrhqy)-bhqExmq exp(ikrhqy)]exp(-ikamu),
Z=μ0/0.
[Exmq]=1ν2mq[Hmq],
1ν2mq=1d 0d1ν2(u)exp-i2π(m-q) uddu.

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