## Abstract

The Fourier modal method (FMM) has advanced greatly by using adaptive coordinates and adaptive spatial resolution. The convergence characteristics were shown to be improved significantly, a construction principle for suitable meshes was demonstrated and a guideline for the optimal choice of the coordinate transformation parameters was found. However, the construction guidelines published so far rely on a certain restriction that is overcome with the formulation presented in this paper. Moreover, a modularization principle is formulated that significantly eases the construction of coordinate transformations in unit cells with reappearing shapes and complex sub-structures.

© 2014 Optical Society of America

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### Equations (20)

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(1)
$${\overline{x}}^{1}={\overline{x}}^{1}({x}^{1},{x}^{2}),$$
(2)
$${\overline{x}}^{2}={\overline{x}}^{2}({x}^{1},{x}^{2}),$$
(3)
$${\overline{x}}^{3}={x}^{3}.$$
(4)
$${\xi}^{\rho \sigma \tau}{\partial}_{\sigma}{E}_{\tau}=i{k}_{0}\sqrt{g}{\mu}^{\rho \sigma}{H}_{\sigma},$$
(5)
$${\xi}^{\rho \sigma \tau}{\partial}_{\sigma}{H}_{\tau}=-i{k}_{0}\sqrt{g}{\epsilon}^{\rho \sigma}{E}_{\sigma}.$$
(6)
$${g}^{\rho \sigma}=\frac{\partial {x}^{\rho}}{\partial {\overline{x}}^{\tau}}\frac{\partial {x}^{\sigma}}{\partial {\overline{x}}^{\tau}},$$
(7)
$$\sqrt{g}{\epsilon}^{\rho \sigma}=\sqrt{g}\frac{\partial {x}^{\rho}}{\partial {\overline{x}}^{\tau}}\frac{\partial {x}^{\sigma}}{\partial {\overline{x}}^{\chi}}{\overline{\epsilon}}^{\tau \chi}.$$
(8)
$$\mathit{LT}(c,\overline{c},d,\overline{d},x)=\frac{\overline{d}-\overline{c}}{d-c}x+\overline{c}-c\frac{\overline{d}-\overline{c}}{d-c}$$
(9)
$${\overline{x}}^{1}({x}^{1},{x}^{2})=\mathit{LT}(0,0,{P}_{0,{x}^{1}},{H}_{1}({x}^{2}),{x}^{1}),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}({x}^{1},{x}^{2})\in \u2460$$
(10)
$$\text{with}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{H}_{1}({x}^{2})=\mathit{LT}\left(0,{P}_{0,{x}^{1}},{P}_{0,{x}^{2}},{P}_{\phi ,{x}^{1}},{x}^{2}\right),$$
(11)
$${\overline{x}}^{2}({x}^{1},{x}^{2})=\mathit{LT}\left(0,0,{P}_{0,{x}^{2}},{H}_{2}({x}^{1}),{x}^{2}\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}({x}^{1},{x}^{2})\in \u2460$$
(12)
$$\text{with}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{H}_{2}({x}^{1})=\mathit{LT}\left(0,{P}_{0,{x}^{2}},{P}_{0,{x}^{1}},{P}_{\phi ,{x}^{2}},{x}^{1}\right).$$
(13)
$${H}_{1}({x}^{2})=h\left({x}_{h}^{2}\right)=h\left(\frac{{P}_{\phi ,{x}^{2}}}{{P}_{0,{x}^{2}}}\cdot {x}^{2}\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}({x}^{1},{x}^{2})\in \u2460,$$
(14)
$${H}_{2}({x}^{1})=g\left({x}_{g}^{1}\right)=g\left(\frac{{P}_{\phi ,{x}^{1}}}{{P}_{0,{x}^{1}}}\cdot {x}^{1}\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}({x}^{1},{x}^{2})\in \u2460,$$
(15)
$${\overline{x}}^{1}({x}^{1},{x}^{2})=\mathit{LT}\left({P}_{1,{x}^{1}},e({x}_{e}^{2}),{S}_{1,{x}^{1}},g({x}_{g}^{2}),{x}^{1}\right),$$
(16)
$${\overline{x}}^{2}({x}^{1},{x}^{2})=\mathit{LT}\left({P}_{1,{x}^{2}},f({x}_{f}^{1}),{Q}_{1,{x}^{2}},h({x}_{h}^{1}),{x}^{2}\right).$$
(17)
$$h\left({x}_{h}^{1}\right)=h\left(\mathit{LT}\left({Q}_{1,{x}^{1}},{Q}_{2,{x}^{1}},{R}_{1,{x}^{1}},{R}_{2,{x}^{1}},{x}^{1}\right)\right),$$
(18)
$$f\left({x}_{f}^{1}\right)=f\left(\mathit{LT}\left({P}_{1,{x}^{1}},{P}_{2,{x}^{1}},{S}_{1,{x}^{1}},{S}_{2,{x}^{1}},{x}^{1}\right)\right),$$
(19)
$$e\left({x}_{e}^{2}\right)=e\left(\mathit{LT}\left({P}_{1,{x}^{2}},{P}_{2,{x}^{2}},{Q}_{1,{x}^{2}},{Q}_{2,{x}^{2}},{x}^{2}\right)\right),$$
(20)
$$g\left({x}_{g}^{2}\right)=g\left(\mathit{LT}\left({S}_{1,{x}^{2}},{S}_{2,{x}^{2}},{R}_{1,{x}^{2}},{R}_{2,{x}^{2}},{x}^{2}\right)\right).$$