Using symmetries of grating groove profiles to reduce computation cost of the C method

Lifeng Li

Author Affiliations

Lifeng Li^{1}

^{1}State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instruments, Tsinghua University, Beijing 100084, China

This work reduces the computation cost of the C method by taking into account the symmetries of grating grooves. All one-dimensionally periodic, single-interface, surface-relief gratings are classified into five categories according to the symmetries of the planar periodic curves describing the interface. The five categories are reflection symmetry, inversion symmetry, reflection-translation symmetry, complete symmetry (i.e., simultaneous existence of all three aforementioned symmetries), and no symmetry. Reductions of the eigenvalue problem in the C method are first carried out in real space and then in Fourier space by taking advantage of the four types of symmetries. The reflection-translation symmetry can be used without any restriction on the incident angle, but the other symmetries require a Littrow mounting; simultaneous use of the reflection-translation symmetry with any other symmetry further requires an even-order Littrow mounting. The types of eigenfunctions to be solved and boundary conditions to be matched, as well as the time reduction ratios in solving the eigenvalue problem, are given for all possible combinations of groove symmetries and incident configurations. The time reduction ratios range from 1/4 to 1/64.

Lifeng Li J. Opt. Soc. Am. A 32(10) 1744-1753 (2015)

References

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${\phi}_{\mathrm{e}}^{1},{\phi}_{\mathrm{o}}^{1}$, or ${\phi}_{\mathrm{p}},{\phi}_{\mathrm{i}}$

Separable or full

Σ denotes the complete symmetry. NA=not available or not used.
“Even” and “odd” stand for even- and odd-order Littrow mountings (understood in the general sense), and “any” stands for any values of ${\alpha}_{0}$ or γ.
“Full” means boundary conditions are matched as if no symmetry is available or no symmetry is used; “separable” means the full boundary conditions are separable into smaller subsets as explained in Section 5; “even” means only matching of even fields is necessary.

Table 4

Relative Measures of Number of Floating-Point Operations Required in Solving the Eigenvalue Problem in the C Method When Different Types of Groove Symmetries^{
a
} Are Considered

Incident Mounting^{
b
}

${\sigma}_{y}$

i

RT

Σ

Arbitrary

1

1

1/8

1/8

Normal

1/8

1/8

1/8

1/64

Even Littrow

1/4

1/8

1/8

1/32

Odd Littrow

1/4

1/8

1/8

1/4,^{
c
} 1/8^{
d
}

Σ denotes the complete symmetry.
The term Littrow mounting is understood in the general sense.
If the ${\sigma}_{y}$ symmetry is used.
If the i symmetry or RT symmetry is used.

Table 5

Diffraction Efficiencies of Two Gratings with Only Reflection-Translation Symmetry

Diffraction Order

Efficiency

Metallic (TM)

Dielectric (TE)

${R}_{-2}$

0.2598253

0.0149095

${R}_{-1}$

0.6231641

0.0070012

${R}_{0}$

0.0214963

0.0012585

${R}_{+1}$

0.0955127

0.0021955

${T}_{-2}$

—

0.0089153

${T}_{-1}$

—

0.1397674

${T}_{0}$

—

0.6063240

${T}_{+1}$

—

0.1955081

${T}_{+2}$

—

0.0241210

Sum

0.9999984

1.0000005

Tables (5)

Table 1

Symmetries of Fourier Coefficients of Periodic Curves

${\phi}_{\mathrm{e}}^{1},{\phi}_{\mathrm{o}}^{1}$, or ${\phi}_{\mathrm{p}},{\phi}_{\mathrm{i}}$

Separable or full

Σ denotes the complete symmetry. NA=not available or not used.
“Even” and “odd” stand for even- and odd-order Littrow mountings (understood in the general sense), and “any” stands for any values of ${\alpha}_{0}$ or γ.
“Full” means boundary conditions are matched as if no symmetry is available or no symmetry is used; “separable” means the full boundary conditions are separable into smaller subsets as explained in Section 5; “even” means only matching of even fields is necessary.

Table 4

Relative Measures of Number of Floating-Point Operations Required in Solving the Eigenvalue Problem in the C Method When Different Types of Groove Symmetries^{
a
} Are Considered

Incident Mounting^{
b
}

${\sigma}_{y}$

i

RT

Σ

Arbitrary

1

1

1/8

1/8

Normal

1/8

1/8

1/8

1/64

Even Littrow

1/4

1/8

1/8

1/32

Odd Littrow

1/4

1/8

1/8

1/4,^{
c
} 1/8^{
d
}

Σ denotes the complete symmetry.
The term Littrow mounting is understood in the general sense.
If the ${\sigma}_{y}$ symmetry is used.
If the i symmetry or RT symmetry is used.

Table 5

Diffraction Efficiencies of Two Gratings with Only Reflection-Translation Symmetry