1. Introduction

Under certain conditions guided resonances in a 2D photonic crystal slab (PCS) can couple to outside radiation, and give rise to interferences in the reflection or transmission spectra of the PCS. [1,2] The resonant features due to this interference can be either very sharp (linewidth under 1 nm), or very broad (linewidth larger than 150 nm) [3,4], enabling both narrow optical filters or broadband mirrors. Certain guided resonances cannot couple to incident radiation and therefore do not influence the reflection or transmission spectra of a PCS. The presence of such uncoupled modes was first observed experimentally in two-dimensional photonic crystals. [5–7] This absence of coupling was interpreted using group theory arguments applied to photonic crystals, and was associated with a symmetry mismatch with external radiation. [7–10] Afterwards, absence of coupling to certain PCS modes above the light line was also observed experimentally [11], confirmed theoretically using a vector-coupled-mode theory [12], and clarified with group-theoretical symmetry arguments [13] (similar to Refs. [7–10]). In this paper, we will discuss how to excite these uncoupled modes, and how to control the strength of coupling to external radiation.

Controlling the coupling to modes can be useful especially for applications where tailoring the linewidth of a specific spectral line is of interest. An important characteristic of the uncoupled modes is that they are non-degenerate in a square lattice (uncoupled modes in, e.g., a hexagonal lattice are not necessarily non-degenerate). This property can be useful for non-linear applications, including photonic-crystal lasers, where non-degeneracy of a mode can be relevant to the efficient operation of the laser. A good example is Ref. [14] where Srinivasan et al analyze mode symmetries in a surface-emitting photonic-crystal microcavity laser and employ uncoupled non-degenerate modes in the operation of the laser. In principle, these uncoupled modes have infinite vertical Q. Therefore, a way to tune the coupling to these modes can give rise to very sharp resonant features that can be especially useful in sensor applications where the sensitivity is typically dependent upon the width of a spectral line. A method for adjusting the linewidth is also important for filter applications. An interesting filter application is presented in Ref. [15] where Fehrembach and Sentenac employ both degenerate and non-degenerate modes in a square lattice to design efficient polarization independent filters that operate at oblique incidences. The non-degenerate nature of these modes also means that they will couple only to a single polarization (assuming the polarization axes are properly aligned). In applications where coupling to a single polarization is desired, the method usually employed is breaking the 90° rotational symmetry, so that the regular doubly-degenerate modes become separated. The drawback of this strategy is that it is difficult to separate the two modes by a large amount. A non-degenerate mode would not present this challenge. Another useful aspect of exciting uncoupled modes is the ability to couple to select resonances, and not to others, through careful design of the hole shape of the PCS, as explained further in detail.

2. Symmetry concepts

Throughout this paper, we will focus our analysis on square-lattice PCS, although our results can be easily generalized to other 2D lattices, such as the hexagonal lattice (see e.g. Ref. [13]), or even lattices containing point defects. Painter and Srinivasan analyzed the symmetry properties of defect modes in detail in Ref. [16], and then exploited the symmetry properties of these modes to design high-Q cavities [17]. For concreteness we will analyze modes at the Γ point in the Brillouin zone, i.e., we will consider the case where the PCS is illuminated at normal incidence, since it is the most common and practical configuration. We are going to consider what occurs in a square lattice with four mirror symmetries (Fig. 1(a)) and with three of them being broken (Fig. 1(b)).

Figure 1(a) shows the unit cell of a square-lattice PCS, and the four mirror symmetries that are relevant to our analysis, which are the mirror symmetries with respect to the x̑ axis, the y̑ axis, and the two diagonal axes. In addition to these, the PCS also has three rotational symmetries, and one additional mirror symmetry with respect to the z̑ axis. We will denote operators, such as mirror-reflection operators, by $\widehat{\sigma }$, and their respective eigenvalues by σ.

 

Fig. 1. (a) Mirror symmetries in a square-lattice PCS unit cell. (b) A square-lattice PCS with broken mirror symmetry.

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The lateral symmetries of the PCS support six types of eigenmodes classified according to their symmetries. Two of these modes form a doubly-degenerate pair, denoted usually as E ⁽¹⁾ and E ⁽²⁾ (to classify the modes with respect to their symmetries, we will employ the irreducible representation notation in the convention used in Ref. [8]). These two modes are 90° rotated versions of each other in a square lattice. If this 90° rotational symmetry is broken, such as in a rectangular lattice, the degeneracy of these modes is lifted. The other four modes are non-degenerate, and are denoted by A ₁, A ₂, B ₁, and B ₂. The symmetries of these modes is depicted in Fig. 2 (after Ref. [18]), along with the symmetries of plane waves for reference. The symmetries are illustrated with (+) and (-) signs. The signs depict the symmetry of the mode with respect to a mirror-symmetry operation. It means that if a mirror reflection overlaps a (+) and a (+) region, or a (-) and a (-) region, then that mode is symmetric with respect to that symmetry axis. Similarly, if a symmetry operation overlaps a (+) and a (-) region, then that mode is antisymmetric with respect to that symmetry axis. For an example of how the electric field corresponding to these symmetries is distributed in a non-degenerate mode and doubly-degenerate mode, see e.g. Ref. [4]. In addition to the classification of these modes according to their symmetries with respect to the lateral mirror-symmetry axes, we can also classify them with respect to the mirror plane parallel to the slab ($\widehat{\sigma }$_z) as symmetric (σ_z=+1) or antisymmetric (σ_z=-1), which are usually referred to as even and odd modes, respectively.

 

Fig. 2. The six modes of a square-lattice PCS classified with respect to their mirror symmetries, along with the symmetries of polarized plane waves at normal incidence for reference (after Ref. [18]).

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Now we will analyze what implications these symmetries have on the behavior of the modes with respect to the coupling to outside radiation. We know that by reflecting twice through a symmetry axis, we get the same result, i.e., $\widehat{\sigma }$ ²=I, where I is the identity operator. Also, the reflection operator has real eigenvalues σ=+1 or -1, since the eigenmodes can have either odd or even symmetries (i.e., can be symmetric or antisymmetric). These two properties show that the reflection operator is unitary, i.e., $\widehat{\sigma }$†$\widehat{\sigma }$=I. By using this identity, it is straightforward to show that non-degenerate modes are uncoupled. As an example, an A ₁ mode is uncoupled to external radiation because:

where |e_x〉 and |e_y〉 represent plane waves that are x̑-polarized and y̑-polarized, |A ₁〉 represents an A ₁-type non-degenerate mode, and $\widehat{\sigma }$_x and $\widehat{\sigma }$_y represent mirror symmetry reflections with respect to the x̑ axis and y̑ axis, respectively. We can demonstrate the absence of coupling to outside radiation in the same way for the other non-degenerate modes, i.e., A ₂, B ₁, and B ₂.

This example shows that it is the mirror symmetry of the modes that prevents them from coupling to outside radiation. In other words, the situation is a simple symmetry-mismatch problem. Therefore, it is natural to assume that if we perturb the mirror symmetry of a mode, then we should be able to perturb its coupling to outside radiation also. To perturb the mirror symmetry, we can add a small protrusion on one side of a PCS hole, so that a mirror symmetry of the crystal is broken. Instead of circular holes, we now have structures that we will refer to as “keyholes” as depicted in Fig. 1(b). To see some previous work analyzing the effect of different hole shapes on two-dimensional photonic-crystal bands, see Refs. [19,20].

 

Fig. 3. Demonstration of how the perturbed non-degenerate modes A′₁ (a) and B′₂ (b) can couple to plane waves due to their mixed symmetries. The arrows are illustrative electric fields, while the colors depict the symmetries of the modes that are related to the directions of the fields.

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Figure 3 illustrates how an A ₁ -type and a B ₂-type non-degenerate mode is perturbed in a keyhole PCS. We will refer to these perturbed modes through a prime sign, such as A′₁ or B′₂. Note that in Fig. 3, we depict the A ₁, B ₂, A′₁, and B′₂ modes with representative electric fields. Slightly different arrow sizes mean that the mode is not symmetric with respect to that axis. We observe that by breaking the PCS symmetry along the y̑ axis ($\widehat{\sigma }$_y), we also break the symmetry of the modes along that axis, while the symmetry along the orthogonal x̑ axis is not affected. A non-symmetric mode can be separated into a symmetric (s) and an antisymmetric (as) part:

Since these two constituents of the non-symmetric mode will have the mirror symmetry along the y̑ axis, we can analyze through our symmetry arguments how these parts couple to outside radiation. These two parts correspond to a mode with a non-degenerate-mode symmetry, and a mode with a degenerate-mode symmetry. This means that the original non-symmetric mode can be considered as a sum of a non-degenerate mode and a degenerate mode, hence it is able to couple to outside radiation due to its degenerate component. More interesting, we see that for the same type of protrusion (along the y̑ axis) the degenerate parts of the A′₁ and B′₂ modes have different polarizations. Therefore, the protrusion perturbs the A ₁ and B ₂ modes such that they couple to y̑ -polarized and x̑ -polarized plane waves, respectively. Similarly, the other modes A′₂ and B′₁ can be shown to couple to x̑ polarization and y̑ polarization, respectively. This shows the potential for exploiting these non-degenerate modes to control polarization.

It is straightforward to show that just breaking the 90° rotational symmetry, e.g., by introducing an equivalent protrusion on the opposite side of the first protrusion (we will refer to these structure as a “double-keyhole”), or by making the holes elliptical, the perturbed non-degenerate modes do not couple to outside radiation. Let us denote the perturbed modes of these double-keyhole structures with a double-prime, as in A″₁. In this structure, neither the x̑ nor the y̑ symmetry is broken, so we can write $\widehat{\sigma }$_x|A″₁〉=+|A″₁〉, and $\widehat{\sigma }$_y|A″₁〉=+|A″₁〉. This shows that a perturbed non-degenerate mode in a double-keyhole structure does not have a degenerate part. These symmetry properties make the coupling zero when calculated as in Eq. (1) above by using the unitary reflection operators.

3. Results

To verify this concept, we have simulated and measured PCS structures with circular holes, keyholes, and double-keyholes. We have chosen a PCS defined on a free-standing silicon diaphragm with the following parameters: 1000 nm pitch, 450 nm hole diameter, 450 nm slab thickness, and an assumed dielectric constant of 11.9. These parameters were chosen such that we would have guided resonances in our measurement wavelength range of 1300 nm to 1600 nm.

 

Fig. 4. Simulated mode profiles of six different modes of the PCS, showing the symmetries, the power in the displacement field, and the mode number. The power is normalized to the peak power. The color scale is linear as shown in the color bar. The coloring is in the same convention used in Figs. 2 and 3.

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The mode profiles of this PCS were calculated using the MIT Photonic-Bands (MPB) package [21]. After taking the mode profile data from MPB, we used a simple code we developed to evaluate the symmetries of the modes by calculating how the displacement field changes under reflection operations. The code then assigns specific symmetries to the mode profile. A similar approach was employed in Ref. [22] where a two-dimensional square-lattice photonic crystal was investigated using a group theory package of Mathematica to assign symmetries to modes calculated with MPB. The profile of the modes is plotted as the electric field energy density (power in the displacement field) at the center plane, i.e., at the middle of the PCS. The values are normalized to the peak power in the mode.

Figure 4 shows mode profiles of the six different types of modes available in our square-lattice PCS. The coloring was done by our symmetry calculating code, in the same convention as used in Figs. 2 and 3. The E ⁽¹⁾ and E ⁽²⁾ modes are not degenerate (they have different eigenfrequencies) only because of a small numerical birefringence due to the finite grid size used in the simulations. This breaks the 90° rotational symmetry and artificially lifts the degeneracy.

Figure 5 shows how the modes (only A′₁ and B′₂ shown) are perturbed when a protrusion is introduced. The protrusion is a circle of smaller radius placed at the top of the original circular hole. This shape was chosen so that the hole shapes in the simulations were similar to the ones that were fabricated. In the same figure, the symmetric and antisymmetric parts of these modes are shown, verifying our symmetry argument that a perturbed non-degenerate mode will have a degenerate part to it.

 

Fig. 5. Simulated mode profiles of the perturbed modes A′₁ and B′₂, showing the symmetric and antisymmetric parts. The power is normalized to the peak power. The color scale is linear as shown in the color bar. The symmetry coloring is in the same convention used in Fig. 4, while the non-symmetric perturbed modes are shown in a single color (purple).

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To investigate how these modes affect the spectra of a PCS under normal incidence, we simulated these structures using the finite-difference time-domain method. For comparison, we also simulated circular holes, keyholes, and double-keyholes, and plotted their respective spectra in Fig. 6. We see that additional interferences appear for a specific polarization only for the keyhole structures. The fact that these resonant features are present for only one polarization shows that they are due to the non-degenerate modes. We also simulated the modes corresponding to these resonances with MPB, and verified that they exhibit the expected non-degenerate mode symmetries.

 

Fig. 6. Transmission spectra of three PCS with the three different hole shapes, circular (dotted and blue), keyhole (solid and red), and double-keyhole (dashed and green). The spectra were plotted for two different polarizations to see the different non-degenerate resonances in each polarization. The arrows are pointing to non-degenerate resonances of the keyhole PCS.

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Fig. 7. Simulated spectra showing how the linewidth of the non-degenerate resonance changes for different protrusion sizes, which are (a) 1.5%, (b) 2.3%, and (c) 3.0% of the hole size respectively. The arrows are pointing to the non-degenerate resonances of the keyhole PCS.

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We have repeated the simulations for different protrusion sizes, as presented in Fig. 7, to see how it affects the spectra of the PCS, and more specifically how it changes the coupling to the perturbed non-degenerate modes. As the size of the protrusion increases, coupling to the non-degenerate mode also increases. This increase manifests itself in an increase in the linewidth of the resonance. This is expected from the symmetry arguments presented in the first section, since a larger protrusion should perturb the mode more strongly, so that more of its power is in the degenerate part that is able to couple to outside radiation.

The conclusion is that we can control the width of these spectral lines by adjusting the protrusion size. In the next section we will see, however, that controlling the coupling is more complicated than just adjusting the size of the protrusion, and that we have more control over the spectrum than just controlling the linewidth of spectral features due to non-degenerate modes.

To verify the predictions of our symmetry arguments and simulations, we fabricated PCS on free-standing silicon diaphragms that are 100 µm×100 µm. The diaphragms were defined on silicon-on-insulator wafers by etching the backside with a tetramethylammonium hydroxide wet etch, and afterwards removing the underlying oxide. To fabricate the PCS structures, we employed standard lithography techniques. The patterns were defined with an e-beam lithography tool on a polymethylmethacrylate resist layer, and were transferred to the silicon diaphragm by etching through the diaphragm with a dry etch. The size of the PCS itself matched the size of the diaphragm. (For a more detailed description of the fabrication procedure, see Ref. [23].) The fabricated structures with the three different hole shapes are shown in Fig. 8.

 

Fig. 8. Scanning-electron micrographs (SEMs) of the fabricated PCS with three different hole shapes. Colored circles are overlaid on the SEMs for reference. The scale bar on the SEMs and the pitch are 1000 nm, and the hole diameter and slab thickness are 450 nm.

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To test these structures, we used a broadband light source that covers the wavelength range of 1300 nm to 1600 nm. The light was focused to a 1/e² intensity spot diameter of ~100 µm and was launched on the PCS at normal incidence. The light transmitted by the PCS was collected with an optical spectrum analyzer, which provided us with the spectral data shown in Fig. 9. To obtain the spectra, we passed the light first through a linear polarizer, oriented to the x̑ axis, so that we only observed a single polarization. We see that in the keyhole structure at 1394 nm there is an additional, much sharper resonant feature, with a linewidth of 0.5 nm, that is missing in the two other structures.

 

Fig. 9. Measured spectra of the three different PCS with SEMs shown in Fig. 8. This measurement shows only the x̑ polarization. The arrow is pointing to the non-degenerate resonance of the keyhole PCS.

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Figure 10 compares the simulated spectrum of the keyhole structure with the measurement result of Fig. 9. The close match of the simulated non-degenerate resonance with the measured resonance verifies that the observed additional resonance in the keyhole structure is due to a perturbed non-degenerate mode. Furthermore, the MPB simulations predict that the frequency eigenvalue of the perturbed version of the 24^th odd mode with an A ₂ symmetry (shown in Fig. 4) corresponds to a wavelength of 1393 nm, closely matching the center wavelength of the measured resonance at 1394 nm.

 

Fig. 10. Comparison of the measured and simulated spectra of the keyhole PCS.

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4. Discussions

The MPB simulations predict 11 non-degenerate modes in this wavelength range, and five of them are expected to couple to the x̑ polarization. Hence, our symmetry arguments require that we should see five non-degenerate resonances. However, we see only one non-degenerate resonance in the measured spectrum. Before seeking an explanation for the absence of some of the resonances, we should emphasize that throughout the upcoming discussion the terms symmetry and asymmetry will refer to the mirror symmetry of the structure with respect to the y̑ axis. When discussing the results shown in Fig. 7, we were comparing different keyhole PCS structures that had different amounts of asymmetry. Specifically, we were suggesting that a larger protrusion size should induce a larger asymmetry in the PCS structure, and hence lead to larger coupling to the non-degenerate modes. The results presented in Fig. 7 were supporting this simple argument. We will now elaborate on this issue to see if we can relate the coupling strength directly to the protrusion size or position.

 

Fig. 11. A gradual increase in the distance of the protrusion or its size does not necessarily mean that the asymmetry is increased. When we gradually increase the distance of the protrusion from the center through (a), (b), and (c), we observe that although it appears that the asymmetry increases, we end up with a structure that is symmetric. Similarly, when we gradually increase the size of the protrusion through (d), (e), and (f), we end up with a symmetric structure (although the symmetry axis is shifted upwards to the middle of the holes).

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Figure 11 shows two interesting examples of attempts to increase the asymmetry by either moving the protrusion away from the hole, or increasing the protrusion size. We see that as we gradually increase the distance of the protrusion from the PCS hole, or as we gradually increase the size of the protrusion, at some point we end up with a symmetric structure. This example shows that our simple argument that relates the asymmetry to the protrusion size and position is limited to small perturbations. We need a more quantitative way of describing symmetry, so that we can speak about different amounts of symmetry. If we treat symmetry as a continuous measure, then the convolution of the dielectric function ε(x,y) can give us information about the amount of symmetry, which we will denote as S, in the following way:

Here the asterisk operator denotes a convolution. The convolution overlaps the mirror image of the dielectric function with itself at different y̑ positions, so that we can seek the maximum overlap, and normalize to obtain a measure of how close the dielectric function is to being symmetric. We assume that the fill factor is large enough so that there is sufficient overlap between the original hole and its mirror image. With this definition, we can say that for S=1 the PCS is symmetric and there will be no coupling to any non-degenerate mode. For S<1 the PCS is not symmetric, and although we know that for this case non-degenerate modes will couple to outside radiation, we cannot quantify the coupling through this parameter. To quantify the effect of asymmetry, we need to investigate the mode shapes for different perturbations.

 

Fig. 12. Comparison of four modes perturbed by two different protrusions. (a), (b), (c), and (d) correspond to the 21^st odd, the 24^th odd, the 15^th odd, and the 28^th even modes respectively. (a1), (b1), (c1), and (d1) show the unperturbed modes. (a2), (b2), (c2), and (d2) show the modes perturbed with a circular protrusion close to the PCS hole. (a3), (b3), (c3), and (d3) show the modes perturbed with a circular protrusion away from the PCS hole. The amount of normalized power in the respective degenerate part of the perturbed mode is given below each mode profile.

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Figure 12 compares simulations of four distinctive modes (21^st odd, 24^th odd, 15^th odd, and 28^th even) perturbed by two types of protrusions that have the same size but different positions. The amount of normalized power in the respective degenerate part of the perturbed mode is given below each mode profile, for a more quantitative understanding of the effect of the protrusions. In the top row in the figure, the four unperturbed non-degenerate modes are shown for reference. In the middle row, the effect of introducing a small protrusion close to the circular hole is demonstrated. While the first (21^st odd) and third modes (15^th odd) are marginally affected by the protrusion, the second (24^th odd) and the last modes (28^th even) are strongly perturbed. Finally, in the bottom row, when the protrusion is positioned away from the circular hole, now the first (21^st odd) and second modes (24^th odd) are marginally affected, while the third (15^th odd) and last modes (28^th even) are strongly perturbed. This shows that while some modes are immune to the various protrusions (21^st odd), some are very sensitive to all of them (28^th even), and some are perturbed by only one of the protrusions. The perturbation on the modes depends on several factors, such as the mode profile, field distribution, eigenfrequency, and whether the power is concentrated in the region where the protrusion is introduced. For example, it is quite obvious in Fig. 12(b2) that the protrusion has the biggest effect when the energy density is large in the original mode at the site of the intended perturbation. Modes that are immune to protrusions can be employed to make optical devices (such as filters or mirrors) that are more robust to fabrication-related disorders. Sensitive modes, on the other hand, can be employed in various sensing applications that rely on perturbing the structure a certain way.

This example shows that when we increase the geometrical asymmetry (decrease S), the four modes are affected differently. This example also demonstrates that by choosing the proper shape and position of the protrusion, we can select a specific non-degenerate mode to couple to, and also pick out the polarization that it couples to. This gives us good control over the strength of coupling to these modes, and therefore over the transmission and reflection spectra of the PCS.

 

Fig. 13. Comparison of two perturbed non-degenerate modes in the keyhole PCS. These two modes have the strongest degenerate parts, i.e. are perturbed most out of the five non-degenerate modes in our wavelength range and polarization.

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The information that the same protrusion introduced into a PCS can affect distinct modes very differently suggests a probable answer to why we observe only one resonance in the measured spectrum, while there are five non-degenerate modes in our measurement wavelength range. Figure 13 shows two perturbed non-degenerate modes (24^th odd and 16^th even) in the keyhole PCS structure. The two modes are separated into their symmetric and antisymmetric parts. The normalized power in each part is given below the mode profiles. While the power in the degenerate part does not directly provide the quantitative strength of coupling, it still shows how much the mode is perturbed, so that we can assume that the coupling will be larger for a more strongly perturbed mode. The reason that we chose these two particular modes is because they are perturbed the most (the other modes have at least six times less power in their degenerate parts compared to the second strongest perturbed mode). The mode in Fig. 13 that is perturbed the strongest is actually the 24^th odd mode centered at the wavelength 1393 nm, as we saw in our measurements. We see that the degenerate part of this mode carries more than an order of magnitude more power compared to the other mode (16^th even). This shows that due to the mode profile, the 24^th odd mode is affected utmost by the protrusion that breaks the mirror symmetry. This fact is also evident from the strong asymmetry in the mode (Fig. 13). The coupling to the other modes is one to two orders of magnitude weaker, so that the expected sharp resonances are averaged out probably either due to fabrication disorders in the hole size and lattice constant [3], or limitations due to the 1-nm-resolution bandwidth employed in the measurement.

Finally, it is worth mentioning that there are other methods for exciting uncoupled resonances, as an alternative to breaking the mirror symmetry through a protrusion. When a PCS is illuminated at on oblique angle, light will be able to couple to inactive modes. This could be a feasible alternative for making narrow-band filters. Also, it is possible to employ the anisotropies of liquid crystals to excite uncoupled modes, and control their coupling efficiency [24].

5. Conclusions

In conclusion, we have presented simple symmetry arguments, simulations, and measurements to show that modes that are uncoupled to outside radiation in a symmetric PCS structure can be excited by breaking the mirror symmetry, for example by introducing keyholes. We have shown that the coupling to these inactive modes can be controlled by the strength of asymmetry introduced to the crystal, and that it is also possible to choose the modes to couple to through the shape of the asymmetry.