A numerically stable method is presented for the analysis of diffraction gratings of arbitrary profile, depth, and in conical mountings. It is based on the classical modal method and uses a stack of lamellar gratpermittivity to approximate an arbitrary profile. A numerical procedure known as the R-matrix propagation aling layers gorithm is used to propagate the modal fields through the layers. This procedure renders the implementation of this new method completely immune to the numerical instability that is associated with the conventional algorithm. Numerical examples including diffraction efficiencies of both dielectric and metallic propagation gratings of depths that range from subwavelength to hundreds of wavelengths are presented. Information about the convergence and the computation time of the method is also included.

S. Campbell, L. C. Botten, R. C. McPhedran, and C. Martijn de Sterke J. Opt. Soc. Am. A 26(4) 938-948 (2009)

References

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Illustration of the Numerical Instability of the T-Matrix Propagation Algorithm^{a}

N

R_{−1}

T_{−1}

e_{∑}

5

0.40296(−2)

0.34246

0.203(−1)

13

0.33724(−2)

0.33892

0.128(−3)

21

0.33710(−2)

0.33888

0.879(−5)

29

0.33405(−2)

0.33865

−0.184(−3)

37

0.20989

0.89279(−5)

0.809(−1)

45

0.72681(+2)

0.11770(−6)

0.171(+3)

Exact

0.33706(−2)

0.33888

−0.122(−14)

R_{−1} and T_{−1} are diffraction efficiencies, and e_{∑} is the sum of the efficiencies minus one. The integers in parentheses are base-ten exponents. The grating is symmetrical. The plane wave is incident normally and TE polarized. Other values of the parameters are ∊^{(+1)} = 1.0, ∊^{(−1)} = (1.65)^{2}, λ = 0.6328d, h_{1} = h_{2} = 0.25d, x_{2,1} − x_{1,1} = 0.6d, and x_{2,2} − x_{1,2} = 0.2d.

Table 2

Comparison of the Numerical Results of the Present Study and the Results of Chuang and Kong16 for a Sinusoidal Dielectric Grating in a Conical Mounting^{a}

Diffraction Order

${I}_{z}^{(h)}=0$

${I}_{z}^{(e)}=0$

Chuang and Kong

Present Paper

Chuang and Kong

Present Paper

R_{−3}

0.1121(−1)

0.11230(−1)

0.2076(−1)

0.20818(−1)

R_{−2}

0.3738(−1)

0.37471(−1)

0.3564(−1)

0.35725(−1)

R_{−1}

0.3875(−1)

0.38568(−1)

0.7006(−2)

0.70720(−2)

R_{0}

0.1032

0.10300

0.1071(−1)

0.10716(−1)

T_{−5}

0.1555(−3)

0.19070(−3)

0.3122(−3)

0.49460(−3)

T_{−4}

0.2934(−4)

0.25231(−4)

0.4914(−3)

0.59074(−3)

T_{−3}

0.7523(−2)

0.74235(−2)

0.1637(−1)

0.16219(−1)

T_{−2}

0.4893(−1)

0.49520(−1)

0.9429(−1)

0.96267(−1)

T_{−1}

0.9950(−1)

0.99081(−1)

0.1676

0.16902

T_{0}

0.7127(−1)

0.71571(−1)

0.4624(−1)

0.46401(−1)

T_{1}

0.5181

0.51845

0.5597

0.55621

T_{2}

0.6381(−1)

0.63408(−1)

0.4108(−1)

0.40467(−1)

Sum

1.0000

0.99993

1.0000

1.00000

The case is taken from Tables 1 and 2 of Ref. 16. The incident wave vector forms a 75° angle with the z axis, and its projection in the x–y plane forms a 60° angle with the y axis. Other values of the parameters are ∊^{(−1)}/∊^{(+1)} = 4.0, d/λ = 2.0, h/λ = 0.6, M = 50, and N = 41.

Table 3

Diffraction Efficiencies of Sinusoidal Dielectric Gratings of Four Groove Depth-to-Period Ratios^{a}

Diffraction Order

h/d = 0.1

h/d = 1.0

h/d = 10 MMM

h/d = 100 MMM

MMM

RFM

MMM

IM

(a) TE Polarization

R_{−2}

0.61092(−3)

0.60804(−3)

0.33921(−2)

0.417(−2)

0.95722(−3)

0.11639(−2)’

R_{−1}

0.96767(−2)

0.96660(−2)

0.76680(−3)

0.945(−3)

0.17554(−3)

0.77516(−3)

R_{0}

0.41656(−1)

0.41686(−1)

0.20552(−2)

0.256(−2)

0.14991(−2)

0.10110(−2)

T_{−3}

0.70696(−5)

0.67940(−5)

0.19936(−1)

0.173(−1)

0.29696(−2)

0.16525(−2)

T_{−2}

0.46550(−4)

0.45917(−4)

0.15202

0.145

0.49744

0.19831

T_{−1}

0.16766(−1)

0.16740(−1)

0.49591

0.496

0.41107

0.79592(−1)

T_{0}

0.87272

0.87282

0.20795

0.211

0.67499(−1)

0.68959

T_{1}

0.58522(−1)

0.58430(−1)

0.11798

0.128

0.18383(−1)

0.27908(−1)

(b) TM Polarization

R_{−2}

0.77607(−3)

0.76830(−3)

0.17302(−2)

0.145(−2)

0.58514(−4)

0.19653(−2)

R_{−1}

0.91238(−2)

0.90746(−2)

0.66721(−3)

0.816(−3)

0.12167(−5)

0.21009(−3)

R_{0}

0.14948(−1)

0.14876(−1)

0.18662(−3)

0.306(−3)

0.10463(−3)

0.24559(−3)

T_{3}

0.78879(−5)

0.52827(−5)

0.12821(−1)

0.122(−1)

0.64171(−3)

0.41496(−2)

T_{−2}

0.59184(−4)

0.59931(−4)

0.21339

0.212

0.18693

0.13827

T_{−1}

0.14850(−1)

0.14826(−1)

0.46158

0.464

0.27573

0.85497(−1)

T_{0}

0.93539

0.93586

0.18362

0.186

0.52608

0.75031

T_{1}

0.24849(−1)

0.24527(−1)

0.12600

0.127

0.10448(−1)

0.19358(−1)

The values of the parameters are ∊^{(+)} = 1.0, ∊^{(−1)} = (0.3 + i7.0)^{2}, θ = 30°, ϕ = 0, d/λ = 1.7, M = 50, and N = 41. MMM stands for the present multilayer modal method, RFM stands for the Rayleigh–Fourier method, and IM stands for the integral method.

Table 4

Diffraction Efficiencies of Sinusoidal Metallic Gratings of Four Groove Depth-to-Period Ratios^{a}

The values of the parameters are ∊^{(+1)} = 1.0, ∊^{(−1)} = (0.3 + i7.0)^{2}, θ = 30°, ϕ = 0, d/λ = 1.7, M = 50, and N = 51. MMM stands for the present multilayer modal method, RFM stands for the Rayleigh–Fourier method, and IM stands for the integral method.
Same parameter values as in (a), except that M = 10 and N = 105.

Tables (4)

Table 1

Illustration of the Numerical Instability of the T-Matrix Propagation Algorithm^{a}

N

R_{−1}

T_{−1}

e_{∑}

5

0.40296(−2)

0.34246

0.203(−1)

13

0.33724(−2)

0.33892

0.128(−3)

21

0.33710(−2)

0.33888

0.879(−5)

29

0.33405(−2)

0.33865

−0.184(−3)

37

0.20989

0.89279(−5)

0.809(−1)

45

0.72681(+2)

0.11770(−6)

0.171(+3)

Exact

0.33706(−2)

0.33888

−0.122(−14)

R_{−1} and T_{−1} are diffraction efficiencies, and e_{∑} is the sum of the efficiencies minus one. The integers in parentheses are base-ten exponents. The grating is symmetrical. The plane wave is incident normally and TE polarized. Other values of the parameters are ∊^{(+1)} = 1.0, ∊^{(−1)} = (1.65)^{2}, λ = 0.6328d, h_{1} = h_{2} = 0.25d, x_{2,1} − x_{1,1} = 0.6d, and x_{2,2} − x_{1,2} = 0.2d.

Table 2

Comparison of the Numerical Results of the Present Study and the Results of Chuang and Kong16 for a Sinusoidal Dielectric Grating in a Conical Mounting^{a}

Diffraction Order

${I}_{z}^{(h)}=0$

${I}_{z}^{(e)}=0$

Chuang and Kong

Present Paper

Chuang and Kong

Present Paper

R_{−3}

0.1121(−1)

0.11230(−1)

0.2076(−1)

0.20818(−1)

R_{−2}

0.3738(−1)

0.37471(−1)

0.3564(−1)

0.35725(−1)

R_{−1}

0.3875(−1)

0.38568(−1)

0.7006(−2)

0.70720(−2)

R_{0}

0.1032

0.10300

0.1071(−1)

0.10716(−1)

T_{−5}

0.1555(−3)

0.19070(−3)

0.3122(−3)

0.49460(−3)

T_{−4}

0.2934(−4)

0.25231(−4)

0.4914(−3)

0.59074(−3)

T_{−3}

0.7523(−2)

0.74235(−2)

0.1637(−1)

0.16219(−1)

T_{−2}

0.4893(−1)

0.49520(−1)

0.9429(−1)

0.96267(−1)

T_{−1}

0.9950(−1)

0.99081(−1)

0.1676

0.16902

T_{0}

0.7127(−1)

0.71571(−1)

0.4624(−1)

0.46401(−1)

T_{1}

0.5181

0.51845

0.5597

0.55621

T_{2}

0.6381(−1)

0.63408(−1)

0.4108(−1)

0.40467(−1)

Sum

1.0000

0.99993

1.0000

1.00000

The case is taken from Tables 1 and 2 of Ref. 16. The incident wave vector forms a 75° angle with the z axis, and its projection in the x–y plane forms a 60° angle with the y axis. Other values of the parameters are ∊^{(−1)}/∊^{(+1)} = 4.0, d/λ = 2.0, h/λ = 0.6, M = 50, and N = 41.

Table 3

Diffraction Efficiencies of Sinusoidal Dielectric Gratings of Four Groove Depth-to-Period Ratios^{a}

Diffraction Order

h/d = 0.1

h/d = 1.0

h/d = 10 MMM

h/d = 100 MMM

MMM

RFM

MMM

IM

(a) TE Polarization

R_{−2}

0.61092(−3)

0.60804(−3)

0.33921(−2)

0.417(−2)

0.95722(−3)

0.11639(−2)’

R_{−1}

0.96767(−2)

0.96660(−2)

0.76680(−3)

0.945(−3)

0.17554(−3)

0.77516(−3)

R_{0}

0.41656(−1)

0.41686(−1)

0.20552(−2)

0.256(−2)

0.14991(−2)

0.10110(−2)

T_{−3}

0.70696(−5)

0.67940(−5)

0.19936(−1)

0.173(−1)

0.29696(−2)

0.16525(−2)

T_{−2}

0.46550(−4)

0.45917(−4)

0.15202

0.145

0.49744

0.19831

T_{−1}

0.16766(−1)

0.16740(−1)

0.49591

0.496

0.41107

0.79592(−1)

T_{0}

0.87272

0.87282

0.20795

0.211

0.67499(−1)

0.68959

T_{1}

0.58522(−1)

0.58430(−1)

0.11798

0.128

0.18383(−1)

0.27908(−1)

(b) TM Polarization

R_{−2}

0.77607(−3)

0.76830(−3)

0.17302(−2)

0.145(−2)

0.58514(−4)

0.19653(−2)

R_{−1}

0.91238(−2)

0.90746(−2)

0.66721(−3)

0.816(−3)

0.12167(−5)

0.21009(−3)

R_{0}

0.14948(−1)

0.14876(−1)

0.18662(−3)

0.306(−3)

0.10463(−3)

0.24559(−3)

T_{3}

0.78879(−5)

0.52827(−5)

0.12821(−1)

0.122(−1)

0.64171(−3)

0.41496(−2)

T_{−2}

0.59184(−4)

0.59931(−4)

0.21339

0.212

0.18693

0.13827

T_{−1}

0.14850(−1)

0.14826(−1)

0.46158

0.464

0.27573

0.85497(−1)

T_{0}

0.93539

0.93586

0.18362

0.186

0.52608

0.75031

T_{1}

0.24849(−1)

0.24527(−1)

0.12600

0.127

0.10448(−1)

0.19358(−1)

The values of the parameters are ∊^{(+)} = 1.0, ∊^{(−1)} = (0.3 + i7.0)^{2}, θ = 30°, ϕ = 0, d/λ = 1.7, M = 50, and N = 41. MMM stands for the present multilayer modal method, RFM stands for the Rayleigh–Fourier method, and IM stands for the integral method.

Table 4

Diffraction Efficiencies of Sinusoidal Metallic Gratings of Four Groove Depth-to-Period Ratios^{a}

The values of the parameters are ∊^{(+1)} = 1.0, ∊^{(−1)} = (0.3 + i7.0)^{2}, θ = 30°, ϕ = 0, d/λ = 1.7, M = 50, and N = 51. MMM stands for the present multilayer modal method, RFM stands for the Rayleigh–Fourier method, and IM stands for the integral method.
Same parameter values as in (a), except that M = 10 and N = 105.