K. Edee, J.-P. Plumey, A. Moreau, and B. Guizal, "Matched coordinates in the framework of polynomial modal methods for complex metasurface modeling," J. Opt. Soc. Am. A 35, 608-615 (2018)
The polynomial modal method (PMM) is one of the most powerful methods for modeling diffraction from lamellar gratings. In the present work, we show that applying it to the so-called matched coordinates leads to important improvement of convergence for crossed lamellar gratings with patterns that are not parallel to the coordinates’ axes. After giving the new formulation of the PMM under matched coordinates in the general framework of biperiodic structures, we provide numerical examples to demonstrate the effectiveness of the proposed approach.
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Effective Index of the Mode () of Circular Dielectric Waveguide with Radius Computed with the PMM Applied to Matched Coordinates with Respect to the Number dim of Polynomial Basis Functions and for Four Different Approximation Orders of the Profile a
dim
4
144
1.1949645
1.1955070
1.1955090
1.1955090
5
225
1.2011819
1.2001330
1.2001357
1.2001358
6
324
1.2002889
1.2003632
1.2003635
1.2003637
7
441
1.2003660
1.2005009
1.2005015
1.2005016
8
576
1.2003735
1.2005010
1.2005019
1.2005019
9
729
1.2003733
1.2005017
1.2005022
1.2005022
10
900
1.2003733
1.2005016
1.2005022
1.2005022
11
1089
1.2003734
1.2005016
1.2005022
1.2005022
Numerical parameters: , , .
Table 2.
Convergence of the Effective Index of the First Two Largest Modes of a Circular Dielectric Waveguide Computed with the PMM Applied to Matched Coordinates with Respect to the Number of the Two-Dimensional Polynomial Basis Functionsa
dim
4
144
1.372012057720
1.308743659129
5
225
1.372076419419
1.313974643438
6
324
1.372089102247
1.314071009093
7
441
1.372089940695
1.314120858592
8
576
1.372090020159
1.314121136568
9
729
1.372090024747
1.314121265267
10
900
1.372090024979
1.314121265785
11
1089
1.372090025026
1.314121265889
12
1296
1.372090025039
1.314121265885
13
1521
1.372090025041
1.314121265892
14
1764
1.372090025040
1.314121265888
Numerical parameters: ; permittivity of the core of the waveguide ; radius ; wavelength ; .
Table 3.
Transmission and Reflection of a Periodic Array of Metallic Nanodisks at Frequency 368 Thz Computed with FMM with Respect to the Truncation Order ()
dim
Transmission
Reflection
4
81
0.255660074523854
0.245518064253931
8
289
0.146768341085604
0.497396017893836
12
625
0.285873144072741
0.218280901442797
16
1089
0.048071941060222
0.617203887903258
20
1681
0.054020330149697
0.650230757913798
24
2401
0.038495466032819
0.645637399604440
Table 4.
Transmission and Reflection of a Periodic Array of Metallic Nanodisks at Frequency 368 Thz Computed with PMM with Respect to the Number of Polynomial Basis Functions
dim
Transmission
Reflection
2
36
0.011001443839646
0.809107113492976
4
144
0.002937102920039
0.894137551402787
6
324
0.003008150208291
0.896787110050819
8
576
0.003026035657107
0.898786758315475
10
900
0.003013688919740
0.899890163916452
12
1296
0.003005338003497
0.900473743443338
14
1764
0.003000626927471
0.900780098923112
16
2304
0.002999846668958
0.900969507356113
Table 5.
Transmission and Reflection of a Periodic Array of Metallic Nanodisks at Frequency 368 Thz Computed with PMM with Respect to the Number of Polynomial Basis Functions
dim
Transmission
Reflection
2
16
0.260723595560106
0.616438528379894
4
100
0.003520530551876
0.885297267544124
6
256
0.003227754760435
0.893745335595428
8
484
0.003055227193656
0.898145097977379
10
784
0.003029559523430
0.899619086848221
12
1156
0.003012874666758
0.900330418521852
14
1600
0.003005435571883
0.900720011231017
16
2116
0.002996901603091
0.900990430238347
Table 6.
Transmission and Reflection of a Periodic Array of Metallic Nanodisks at Frequency 368 Thz Computed with PMM with Respect to the Number of Polynomial Basis Functions
dim
Transmission
Reflection
4
64
0.036123728117308
0.829194436107274
6
196
0.003602920213573
0.887843962967683
8
400
0.003147754652346
0.896887406555596
10
676
0.003054608072025
0.899198859259387
12
1024
0.003023853670667
0.900140229063836
14
1444
0.003010006942582
0.900622772910305
16
1936
0.003004582112426
0.900871972468303
Table 7.
Convergence of the Transmission and Reflection of a Periodic Array of Metallic Nanodisks ( at Wavelength 1530 nm) on Substrate () with Respect to the Number of Polynomial Basis Functions
dim
Transmission
Reflection
4
144
0.006324074919277
0.884736855857434
6
324
0.004734758593923
0.891144139644723
8
576
0.004441152901665
0.893548344101546
10
900
0.004331230399484
0.894679935515439
12
1296
0.004275286003664
0.895293989000282
14
1764
0.004246806558619
0.895616792755514
16
2304
0.004224435131269
0.895871824222813
18
2916
0.004205996990137
0.896084684513263
Tables (7)
Table 1.
Effective Index of the Mode () of Circular Dielectric Waveguide with Radius Computed with the PMM Applied to Matched Coordinates with Respect to the Number dim of Polynomial Basis Functions and for Four Different Approximation Orders of the Profile a
dim
4
144
1.1949645
1.1955070
1.1955090
1.1955090
5
225
1.2011819
1.2001330
1.2001357
1.2001358
6
324
1.2002889
1.2003632
1.2003635
1.2003637
7
441
1.2003660
1.2005009
1.2005015
1.2005016
8
576
1.2003735
1.2005010
1.2005019
1.2005019
9
729
1.2003733
1.2005017
1.2005022
1.2005022
10
900
1.2003733
1.2005016
1.2005022
1.2005022
11
1089
1.2003734
1.2005016
1.2005022
1.2005022
Numerical parameters: , , .
Table 2.
Convergence of the Effective Index of the First Two Largest Modes of a Circular Dielectric Waveguide Computed with the PMM Applied to Matched Coordinates with Respect to the Number of the Two-Dimensional Polynomial Basis Functionsa
dim
4
144
1.372012057720
1.308743659129
5
225
1.372076419419
1.313974643438
6
324
1.372089102247
1.314071009093
7
441
1.372089940695
1.314120858592
8
576
1.372090020159
1.314121136568
9
729
1.372090024747
1.314121265267
10
900
1.372090024979
1.314121265785
11
1089
1.372090025026
1.314121265889
12
1296
1.372090025039
1.314121265885
13
1521
1.372090025041
1.314121265892
14
1764
1.372090025040
1.314121265888
Numerical parameters: ; permittivity of the core of the waveguide ; radius ; wavelength ; .
Table 3.
Transmission and Reflection of a Periodic Array of Metallic Nanodisks at Frequency 368 Thz Computed with FMM with Respect to the Truncation Order ()
dim
Transmission
Reflection
4
81
0.255660074523854
0.245518064253931
8
289
0.146768341085604
0.497396017893836
12
625
0.285873144072741
0.218280901442797
16
1089
0.048071941060222
0.617203887903258
20
1681
0.054020330149697
0.650230757913798
24
2401
0.038495466032819
0.645637399604440
Table 4.
Transmission and Reflection of a Periodic Array of Metallic Nanodisks at Frequency 368 Thz Computed with PMM with Respect to the Number of Polynomial Basis Functions
dim
Transmission
Reflection
2
36
0.011001443839646
0.809107113492976
4
144
0.002937102920039
0.894137551402787
6
324
0.003008150208291
0.896787110050819
8
576
0.003026035657107
0.898786758315475
10
900
0.003013688919740
0.899890163916452
12
1296
0.003005338003497
0.900473743443338
14
1764
0.003000626927471
0.900780098923112
16
2304
0.002999846668958
0.900969507356113
Table 5.
Transmission and Reflection of a Periodic Array of Metallic Nanodisks at Frequency 368 Thz Computed with PMM with Respect to the Number of Polynomial Basis Functions
dim
Transmission
Reflection
2
16
0.260723595560106
0.616438528379894
4
100
0.003520530551876
0.885297267544124
6
256
0.003227754760435
0.893745335595428
8
484
0.003055227193656
0.898145097977379
10
784
0.003029559523430
0.899619086848221
12
1156
0.003012874666758
0.900330418521852
14
1600
0.003005435571883
0.900720011231017
16
2116
0.002996901603091
0.900990430238347
Table 6.
Transmission and Reflection of a Periodic Array of Metallic Nanodisks at Frequency 368 Thz Computed with PMM with Respect to the Number of Polynomial Basis Functions
dim
Transmission
Reflection
4
64
0.036123728117308
0.829194436107274
6
196
0.003602920213573
0.887843962967683
8
400
0.003147754652346
0.896887406555596
10
676
0.003054608072025
0.899198859259387
12
1024
0.003023853670667
0.900140229063836
14
1444
0.003010006942582
0.900622772910305
16
1936
0.003004582112426
0.900871972468303
Table 7.
Convergence of the Transmission and Reflection of a Periodic Array of Metallic Nanodisks ( at Wavelength 1530 nm) on Substrate () with Respect to the Number of Polynomial Basis Functions