Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution. Application to trapezoidal gratings

Gérard Granet; Jean Chandezon; Jean-Pierre Plumey; Karyl Raniriharinosy

doi:10.1364/JOSAA.18.002102

Journal of the Optical Society of America A
Vol. 18,
Issue 9,
pp. 2102-2108
(2001)
•https://doi.org/10.1364/JOSAA.18.002102

Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution. Application to trapezoidal gratings

Gérard Granet, Jean Chandezon, Jean-Pierre Plumey, and Karyl Raniriharinosy

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Gérard Granet, Jean Chandezon, Jean-Pierre Plumey, and Karyl Raniriharinosy, "Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution. Application to trapezoidal gratings," J. Opt. Soc. Am. A 18, 2102-2108 (2001)

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Abstract

We reformulate the coordinate transformation method for profiles represented by parametric equations. Numerically, the eigenvalue problem is solved by expanding both the electromagnetic field and the periodic coefficients of Maxwell’s equations into Fourier series. For trapezoidal gratings, it is shown that the convergence of the method is closely related to the representation of the profile. A proper choice of the representation permits handling profiles that previously had been out of reach owing to their sharp edges. From a practical point of view, we are now able to analyze gratings with two vertical facets by using the coordinate transformation method.

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Parameter	Parametric C Method			FMM
	$d_{1}$
	0.01	0.001	0.00001
TE
$R_{- 3}$	0.0060	0.0060	0.0060	0.0060
$R_{- 2}$	0.0009	0.0007	0.0006	0.0006
$R_{- 1}$	0.0016	0.0015	0.0015	0.0015
$R_{0}$	0.0555	0.0552	0.0552	0.0551
$T_{- 4}$	0.0835	0.0827	0.0824	0.0823
$T_{- 3}$	0.5223	0.5248	0.5245	0.5254
$T_{- 2}$	0.1319	0.1281	0.1280	0.1276
$T_{- 1}$	0.1148	0.1154	0.1158	0.1154
$T_{0}$	0.0045	0.0052	0.0052	0.0053
$T_{1}$	0.0790	0.0805	0.0808	0.0808

TM
$R_{- 3}$	0.0067	0.0063	0.0063	0.0063
$R_{- 2}$	0.0004	0.0003	0.0003	0.0003
$R_{- 1}$	0.0003	0.0003	0.0003	0.0004
$R_{0}$	0.0071	0.0067	0.0067	0.0067
$T_{- 4}$	0.0450	0.0469	0.0471	0.0473
$T_{- 3}$	0.3770	0.3793	0.3790	0.3794
$T_{- 2}$	0.2122	0.2078	0.2081	0.2073
$T_{- 1}$	0.2222	0.2254	0.2261	0.2258
$T_{0}$	0.0841	0.0826	0.0826	0.0825
$T_{1}$	0.0451	0.0442	0.0441	0.0441

Parameter	Parametric C Method				FMM
	$d_{1}$
	0.01	0.001	0.00001
TE
$R_{- 3}$	0.0092	0.0084	0.0084		0.0083
$R_{- 2}$	0.0040	0.0022	0.0021		0.0021
$R_{- 1}$	0.0718	0.0543	0.0525		0.0526
$R_{0}$	0.8870	0.9068	0.9087		0.9087
TM
$d_{1}$	0.01	0.001	0.00001	0.00001^b
$R_{- 3}$	0.5032	0.4624	0.4561	0.4605	0.4600
$R_{- 2}$	0.0760	0.0590	0.0567	0.0583	0.0584
$R_{- 1}$	0.1487	0.1541	0.1543	0.1542	0.1542
$R_{0}$	0.0895	0.1289	0.1358	0.1293	0.1290

Parameter	Parametric C Method			FMM
	$d_{1}$
	0.01	0.001	0.00001
TE
$R_{- 3}$	0.0060	0.0060	0.0060	0.0060
$R_{- 2}$	0.0009	0.0007	0.0006	0.0006
$R_{- 1}$	0.0016	0.0015	0.0015	0.0015
$R_{0}$	0.0555	0.0552	0.0552	0.0551
$T_{- 4}$	0.0835	0.0827	0.0824	0.0823
$T_{- 3}$	0.5223	0.5248	0.5245	0.5254
$T_{- 2}$	0.1319	0.1281	0.1280	0.1276
$T_{- 1}$	0.1148	0.1154	0.1158	0.1154
$T_{0}$	0.0045	0.0052	0.0052	0.0053
$T_{1}$	0.0790	0.0805	0.0808	0.0808

TM
$R_{- 3}$	0.0067	0.0063	0.0063	0.0063
$R_{- 2}$	0.0004	0.0003	0.0003	0.0003
$R_{- 1}$	0.0003	0.0003	0.0003	0.0004
$R_{0}$	0.0071	0.0067	0.0067	0.0067
$T_{- 4}$	0.0450	0.0469	0.0471	0.0473
$T_{- 3}$	0.3770	0.3793	0.3790	0.3794
$T_{- 2}$	0.2122	0.2078	0.2081	0.2073
$T_{- 1}$	0.2222	0.2254	0.2261	0.2258
$T_{0}$	0.0841	0.0826	0.0826	0.0825
$T_{1}$	0.0451	0.0442	0.0441	0.0441

Parameter	Parametric C Method				FMM
	$d_{1}$
	0.01	0.001	0.00001
TE
$R_{- 3}$	0.0092	0.0084	0.0084		0.0083
$R_{- 2}$	0.0040	0.0022	0.0021		0.0021
$R_{- 1}$	0.0718	0.0543	0.0525		0.0526
$R_{0}$	0.8870	0.9068	0.9087		0.9087
TM
$d_{1}$	0.01	0.001	0.00001	0.00001^b
$R_{- 3}$	0.5032	0.4624	0.4561	0.4605	0.4600
$R_{- 2}$	0.0760	0.0590	0.0567	0.0583	0.0584
$R_{- 1}$	0.1487	0.1541	0.1543	0.1542	0.1542
$R_{0}$	0.0895	0.1289	0.1358	0.1293	0.1290

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Equations (43)

Journal of the Optical Society of America A