The Rayleigh method is still occasionally used in grating simulations because it features an extremely simple solution procedure, high computational accuracy, and fast computation. However, for deep-groove gratings (depth-to-period ratio h/d>0.3) or sharp-corner gratings, the Rayleigh method yields unreliable results. For rectangular gratings, the Fourier modal method is complete in both physical and mathematical senses and has achieved notable success; it offers strong computational stability, high accuracy, fast computation, and imposes no restriction on the grating depth-to-period ratio. However, when handling non-rectangular gratings, the Fourier modal method needs to approximate them as multiple layers of rectangular gratings, which not only reduces accuracy but also greatly increases calculation time. The Chandezon method (C-method) performs well for non-rectangular gratings, providing high accuracy and a faster computation speed than the Fourier modal method. However, it involves a complex mathematical procedure with coordinate transformations and covariant inverse tensors, which discourages many researchers.
This paper analyzes and elucidates the root causes of problems in traditional modal methods from the perspective of physical essence and mathematical treatment mechanisms. It presents detailed physical analysis and mathematical derivation for the Huygens modal method, which retains the advantages of traditional modal methods while overcoming their drawbacks, and conducts a comparative study with traditional approaches.
First, this paper analyzes the root cause of the increase in the condition number of the Rayleigh-method modal-field matrix from a mathematical perspective, and reveals that the condition number rises sharply as the groove-depth duty ratio increases. It then identifies the fundamental reason why the Rayleigh method is only applicable to gratings with a depth-to-period ratio h/d<0.3, and proposes that addressing this issue requires a scientific justification of the “Rayleigh hypothesis—that the diffracted field at any grating interface is a superposition of plane waves”. The fundamental modal field in the Huygens method is not a plane wave but a nonuniform wave, which is essentially different from the Rayleigh fundamental modal field based on a globally equal-amplitude plane-wave assumption. At the diffraction interface, the Rayleigh fundamental modal field contains this nonuniform-wave component, whereas the Huygens fundamental modal field does not. When the grating grooves are very deep, the relevant factor varies over a wide numerical range, causing the Rayleigh-method modal-field matrix to become highly ill-conditioned; in contrast, because the Huygens method does not include this factor, its matrix exhibits a much lower degree of ill-conditioning. However, when the grating grooves are very shallow (h/d<0.3), the influence of this factor becomes prominent, which reduces the reliability of the Rayleigh-method results.
The basis functions used in the Huygens method satisfy orthonormality on the interface, whereas those in the Rayleigh method do not meet physical requirements—namely, the interface electromagnetic field is difficult to fit accurately by superposing plane waves (because the plane wave fronts do not match the interface shape). Moreover, from a mathematical viewpoint, they do not possess orthogonality and completeness. Therefore, as the grating depth-to-period ratio increases, the Rayleigh-method basis functions fail to provide an effective fit to the electromagnetic field, and the core hypothesis becomes invalid accordingly. In contrast, the basis functions in the Huygens method satisfy physical requirements and exhibit orthonormality and completeness on the interface. The resulting matrix for the fundamental modal field is weakly ill-conditioned, which directly overcomes the limitations of the Rayleigh hypothesis. In a certain sense, the Rayleigh fundamental modal field can be regarded as a special case of the Huygens fundamental modal field.
The basis function assumption of the Huygens method is consistent with that of the C-method. Although the C-method has shown strong practical effectiveness, it suffers from a complex mathematical derivation process. The C-method transforms the grating boundary surface into a plane via coordinate transformation and takes the “plane wave” in the new coordinate system as the solution of the electromagnetic field. After transforming back to the original coordinates, the nonuniform-wave assumption is obtained; it is then substituted into Maxwell’s equations in the curvilinear coordinate system to derive the true eigenvalue equation. This derivation process is far more complex than that of the Huygens method. The eigenvalue equation of the Huygens fundamental modal field is simpler in form, the associated matrix is less ill-conditioned, and the eigenvectors satisfy orthonormality, which simplifies the computational procedure and improves computation speed.
Finally, this paper performs numerical simulations to compare the reliability of the results obtained by the Rayleigh method, the Huygens method, and the C-method, and provides a general form of the Huygens modal method applicable to arbitrary diffraction gratings.
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