Asymmetric Defect Mode Grating

• The topic defines asymmetric defect mode gratings as periodic structures with a localized defect in a non-equivalent environment, altering resonant spectral properties.
• Analytical methods such as characteristic-matrix and coupled-mode equations reveal how asymmetry affects TE/TM responses, dispersion, and angular behavior.
• Applications include narrowband filters, optical isolators, and polarization converters where deliberate asymmetry tailors mode count, resonance shifts, and coupling efficiency.

Searching arXiv for relevant papers on asymmetric defect modes in gratings and related photonic structures. An asymmetric defect mode grating is a periodic wave structure in which a deliberately introduced defect is embedded in a left–right non-equivalent environment, so that the localized resonant states associated with the defect acquire altered spectral position, parity, angular response, polarization response, or coupling strength relative to the corresponding symmetric configuration. In the canonical one-dimensional case, the grating is a Bragg multilayer or photonic crystal with a defect layer inserted at or near its center; in broader usage, closely related behavior appears in shifted microring gratings, topological-orientation defects, and surface-defect gratings on photonic band-gap crystals, where asymmetry is imposed by geometry, constitutive contrast, or rotational displacement rather than by a single missing period (Aghajamali et al., 2013, Lu et al., 2023).

1. Canonical structures and meanings of asymmetry

The most standard asymmetric defect mode grating is a one-dimensional defective multilayer. In a representative metamaterial photonic crystal, layer $A$ is a negative-index material (NIM), layer $B$ is a positive-index material (PIM) taken as vacuum with $n_B=1$, and layer $C$ is a defect layer. The asymmetric defective sequence is

$(AB)^{N/2}\,C\,(AB)^{N/2},$

whereas the symmetric sequence is

$(AB)^{N/2}\,C\,(BA)^{N/2}.$

Here the asymmetry is purely geometric: the order of layers to the right of the defect is not the mirror image of the order on the left (Aghajamali et al., 2013).

A second archetype replaces the localized defect layer by a finite grating-free interval between two Bragg gratings. In that model, the central layer satisfies $\kappa(x)=0$ for $-L<x<0$, while the right and left semi-infinite gratings have unequal reflectivities. The asymmetric structure is defined by

$\kappa(x)= \begin{cases} 1, & x>0,\[2mm] 0, & -L<x<0,\[2mm] \cos\alpha, & x<-L, \end{cases} \qquad 0\le \alpha < \pi/2,$

so asymmetry is a mismatch of bandgap widths rather than a mirror-ordering change (Mayteevarunyoo et al., 2015).

A third meaning appears in magnetophotonic multilayers with a nonlinear defect. There the transfer sequence is

$$(\mathbf M_1\mathbf M_2)^m\,\mathbf M_d\,(\mathbf M_2\mathbf M_1)^n,$$

and asymmetry means off-center placement of the defect, $B$0, while the symmetric case has $B$1. In that setting the defect-bearing Bragg stack remains periodic away from the defect, but the cavity localization pattern becomes direction dependent because the two Bragg mirrors are unequal (Tuz et al., 2011).

These definitions imply that “asymmetric defect mode grating” is not a single geometry but a family of defect-bearing periodic media. The common element is the coexistence of a periodic stopband-forming background and a localized perturbation whose environment is not mirror equivalent.

2. Electromagnetic and coupled-mode formulations

For stratified electromagnetic gratings, the standard formalism is the characteristic-matrix method. For TE waves in layer $B$2,

$B$3

The refractive index is

$B$4

with positive sign for PIMs and negative sign for NIMs. For TE polarization,

$B$5

while for TM polarization,

$B$6

The total matrix is formed by ordered multiplication of layer matrices, and the transmission coefficient is

$B$7

with transmissivity

$B$8

In the NIM/PIM asymmetric defective structure, the defect modes are not given by a closed-form resonance equation; they are identified numerically as narrow transmission peaks inside the photonic band gap (Aghajamali et al., 2013).

For Bragg gratings with a gapless defect layer, the appropriate reduced model is a pair of coupled-mode equations for forward and backward envelopes $B$9 and $n_B=1$0: $n_B=1$1

$n_B=1$2

In the linear asymmetric system, the defect-mode eigenfrequency satisfies

$n_B=1$3

and the defect mode exists only if

$n_B=1$4

For the fundamental mode, this reduces to

$n_B=1$5

This exact threshold is a concrete statement of how asymmetry can suppress localization unless the defect region is sufficiently wide (Mayteevarunyoo et al., 2015).

3. Linear defect spectra and what asymmetry changes

In one-dimensional lossy NIM/PIM photonic crystals, asymmetry does not necessarily control the number of localized resonances. With a type-I NIM defect layer, the transmission spectrum contains one defect mode in both the asymmetric and symmetric geometries for both TE and TM polarizations. When the defect layer is changed to a type-II NIM, both geometries exhibit three defect modes. In that model, the mode count depends on the constitutive type of the NIM defect layer and is independent of whether the layer sequence is symmetric or asymmetric (Aghajamali et al., 2013).

The same study shows, however, that asymmetry still affects spectral placement and angular persistence. Defect-mode frequencies in the asymmetric structure occur at somewhat higher frequencies than in the symmetric structure. For both type-I and type-II defects, all defect-mode frequencies shift to higher frequency as incidence angle increases; in TE they remain nearly unchanged with angle, while in TM they increase noticeably. In the type-II case, defect modes 1 and 2 disappear above about $n_B=1$6 and $n_B=1$7 in the asymmetric geometry, compared with about $n_B=1$8 and $n_B=1$9 in the symmetric geometry. Asymmetry therefore modifies the detailed dispersion and visibility of defect channels even when it does not alter their number (Aghajamali et al., 2013).

The gapless-layer model reaches a different conclusion because asymmetry there is a reflectivity mismatch. The left and right gratings no longer share the same bandgap, so the common localization window is reduced to

$C$0

This produces a class of semi-delocalized states in the interval

$C$1

localized on the stronger-grating side but delocalized on the weaker side. The asymmetric defect grating therefore supports a competition between attraction to the defect region and repulsion from the reflectivity step, a mechanism absent from the symmetric limit (Mayteevarunyoo et al., 2015).

A common misconception is that structural asymmetry alone always creates additional defect modes. The NIM/PIM multilayer demonstrates that this is not generally true: asymmetry can shift, broaden, or extinguish resonances without changing the defect-mode count, while constitutive dispersion of the defect layer remains the decisive control parameter for the number of in-gap channels (Aghajamali et al., 2013).

4. Nonlinearity, gyrotropy, and direction-sensitive defect localization

When asymmetry is combined with Kerr nonlinearity and magneto-optic gyrotropy, defect-mode gratings acquire a substantially richer response. In a one-dimensional magnetophotonic crystal with a nonlinear defect, the magnetic layers are biased in the Faraday configuration, and the defect-mode resonance splits into a Zeeman-like doublet associated with LCP and RCP eigenwaves. In the symmetric linear case, the split defect resonances occur at approximately

$C$2

with $C$3 (Tuz et al., 2011).

Off-center defect placement, realized by $C$4 or $C$5 instead of $C$6, does not merely shift those resonances. It changes the intracavity localization asymmetrically, so the nonlinear resonance bending differs strongly for the two mirror-related structures. In the asymmetric case, the resonant transmission no longer reaches unity,

$C$7

and the reflection minima remain nonzero,

$C$8

The reflected field also ceases to become circular at resonance and remains elliptical,

$C$9

The co- and cross-polarized reflection coefficients become strongly defect-position dependent: for $(AB)^{N/2}\,C\,(AB)^{N/2},$0, $(AB)^{N/2}\,C\,(AB)^{N/2},$1 and $(AB)^{N/2}\,C\,(AB)^{N/2},$2, while for $(AB)^{N/2}\,C\,(AB)^{N/2},$3 those values are approximately reversed. In this regime, asymmetry is not only spectral; it becomes a mechanism for reversible nonreciprocity and polarization conversion (Tuz et al., 2011).

The asymmetric gapless-layer Bragg model extends the same theme into nonlinear localization. In the weak-interface limit, the effective interaction force on a gap soliton centered at $(AB)^{N/2}\,C\,(AB)^{N/2},$4 is

$(AB)^{N/2}\,C\,(AB)^{N/2},$5

Hence no soliton trapping occurs if

$(AB)^{N/2}\,C\,(AB)^{N/2},$6

This result is the nonlinear analogue of the linear defect-mode existence threshold: asymmetry introduces a repulsive contribution that can overwhelm defect trapping unless the gapless region is sufficiently wide (Mayteevarunyoo et al., 2015).

These models establish that in asymmetric defect mode gratings, nonlinearity does not simply tune an already existing defect resonance. It can change which side of the structure preferentially localizes energy, whether the reflected or transmitted state is circular or elliptical, and whether localization survives at all.

5. Broader realizations beyond the 1D multilayer archetype

The concept broadens significantly outside simple layered Bragg stacks. In photonic-crystal microrings, a shifted sinusoidal grating on the inner boundary creates rotational asymmetry by displacing the center of the grating relative to the center of the outer ring. The ring width becomes

$(AB)^{N/2}\,C\,(AB)^{N/2},$7

where the $(AB)^{N/2}\,C\,(AB)^{N/2},$8 term is the signature of the shift. In the unshifted case, a grating with $(AB)^{N/2}\,C\,(AB)^{N/2},$9 periods selectively splits a single azimuthal mode $(AB)^{N/2}\,C\,(BA)^{N/2}.$0; in the shifted case, multiple adjacent resonances are split. The targeted splitting obeys

$(AB)^{N/2}\,C\,(BA)^{N/2}.$1

while nearby modes follow the approximate relation

$(AB)^{N/2}\,C\,(BA)^{N/2}.$2

This is a defect-like asymmetric grating perturbation in which the asymmetry is azimuthal rather than longitudinal, and the localized states are standing-wave supermodes rather than traditional cavity modes (Lu et al., 2023).

A different realization uses a topological defect in a two-dimensional photonic crystal of elliptical cylinders. There the ellipse orientation is prescribed by

$(AB)^{N/2}\,C\,(BA)^{N/2}.$3

with a principal example $(AB)^{N/2}\,C\,(BA)^{N/2}.$4, $(AB)^{N/2}\,C\,(BA)^{N/2}.$5. Translational symmetry is broken, but long-range positional order is retained, leaving a residual photonic bandgap. The resulting band-edge modes show asymmetric output flux because neighboring crystalline domains rotated by $(AB)^{N/2}\,C\,(BA)^{N/2}.$6 block propagation in one direction while allowing it in another. When a $(AB)^{N/2}\,C\,(BA)^{N/2}.$7 block of cylinders is removed at the center, localized defect states appear whose power flow is dominated by either clockwise or counter-clockwise circulation, and the preferred circulation direction switches when all ellipses are rotated by $(AB)^{N/2}\,C\,(BA)^{N/2}.$8 (Liew et al., 2014).

Surface-defect gratings on three-dimensional photonic band-gap crystals provide another extension. A periodic nanopore defect layer on an inverse-woodpile silicon crystal supports surface defect modes inside the 3D band gap with narrow relative linewidth

$(AB)^{N/2}\,C\,(BA)^{N/2}.$9

Along one in-plane direction the dispersion is negative, corresponding to backward-propagating waves, while along the orthogonal direction it is nearly flat. An analytic model shows that the backward propagation is caused by the surface grating, whereas the 3D band gap supplies vertical confinement. This is not a left–right asymmetric grating in the strict mirror sense, but it is a defect-mode grating whose anisotropic dispersion produces strongly direction-selective behavior (Vreman et al., 10 Feb 2025).

These examples show that the topic includes more than a central defect layer between two Bragg mirrors. The essential ingredients are a periodic wave-scattering background, a defect-induced localized state or split resonance, and an imposed asymmetry that redistributes coupling in frequency, angle, or circulation space.

6. Applications, adjacent architectures, and conceptual boundaries

The most direct application of asymmetric defect mode gratings is narrowband and multichannel filtering. In the NIM/PIM multilayer, a type-I NIM defect yields a single in-gap transmission channel, while a type-II NIM defect yields three; the reported implication is the design of narrowband and multichannel transmission filters (Aghajamali et al., 2013). In magnetophotonic multilayers, the same defect-mode logic supports thin-film optical isolators, tunable polarization-rotating mirrors, and polarization transformation devices because the asymmetric nonlinear cavity can favor co- or cross-polarized reflection depending on defect position (Tuz et al., 2011). In shifted photonic-crystal microrings, asymmetric grating-induced multiple mode splitting is used for multi-frequency engineering in optical parametric oscillation, enabling OPO in a normal-dispersion device that otherwise would not support it (Lu et al., 2023). In surface-defect gratings on 3D band-gap crystals, the proposed application is a device whose output direction depends on emitter frequency (Vreman et al., 10 Feb 2025).

The topic also has important near-neighbors that are not canonical defect-mode gratings. In plasmonic grating–film–grating stacks, asymmetric transmission is generated by localized surface plasmon resonance excitation, tunneling through a thin Ag film, and asymmetric input/output grating coupling. A representative G-F-G structure produces forward transmittivity peaks of about $\kappa(x)=0$0, $\kappa(x)=0$1, and $\kappa(x)=0$2 at $\kappa(x)=0$3, $\kappa(x)=0$4, and $\kappa(x)=0$5 nm, respectively, with isolation ratios all above $\kappa(x)=0$6 dB, but the underlying physics is described in terms of LSPR excitation and tunneling rather than photonic-bandgap defect states (Ma et al., 7 Nov 2025). Likewise, the asymmetric and orthogonal AO-GFG structure supports two localized plasmonic resonances, $\kappa(x)=0$7 nm and $\kappa(x)=0$8 nm, with polarization-dependent and polarization-independent asymmetric transmission channels, again resembling defect-like localized pass states without being formal defect modes in a Bragg-gap sense (Ma et al., 17 Nov 2025).

A second misconception is therefore that any passive asymmetric transmission grating is automatically a defect-mode grating. The plasmonic G-F-G and AO-GFG devices are more accurately described as asymmetric resonant tunneling gratings. Conversely, the absence of gross mirror asymmetry does not exclude defect-mode behavior: topological-orientation defects and surface-defect gratings show that asymmetry can reside in orientational texture or anisotropic dispersion rather than in a simple left–right profile (Liew et al., 2014, Vreman et al., 10 Feb 2025).

Taken together, the literature defines the asymmetric defect mode grating as a design class rather than a single architecture. Its unifying principle is that localized resonant states generated by a defect in a periodic medium are reshaped by asymmetry, and that reshaping can control mode count, parity, linewidth visibility, polarization conversion, directional leakage, circulation handedness, or spectral routing depending on the physical platform.