Group-theoretical description of the triangular air-silica photonic crystal-out-of-plane propagation

The method of assigning irreducible representations to modes in three-dimensional photonic structures is applied to the two-dimensional triangular air-silica lattice with out-of-plane wave propagation. In particular prediction of spatial symmetries of the crystal modes is addressed. We show how the photonic bands are affected by different rod radii and out-of-plane components from a group-theoretical point of view. One particular defect mode is analyzed and the structure which is optimal for air-guidance is found. © 2004 Optical Society of America OCIS codes: (060.2310) Fiber optics; (260.2110) Electromagnetic theory. References and links 1. S. G. Johnson and J. Joannopoulus, “Block-iterative frequency domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173 (2001). 2. D. Cassagne, C. Jouanin and D. Bertho, “Hexagonal photonic-band-gap structures,” Phys. Rev. B 53, 7134 (1996). 3. T. Ochiai and K. Sakoda, “Nearly free-photon approximation for two-dimensional photonic crystal slabs,” Phys. Rev. B64, 045108 (2001). 4. M. Plihal and A. Maradudin, “Photonic band structures of two-dimensional systems: The triangular lattice,” Phys. Rev. B44, 8565 (1991). 5. K. Ohtaka and Y. Tanabe, “Photonic Bands Using Vector Spherical Waves. III Group-Theoretical Treatment,” J. Phys. Soc. Japan 65, 2670 (1996) 6. K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001). 7. K. Sakoda, “Group-theoretical classification of eigenmodes in three-dimensional photonic lattices,” Phys. Rev. B 55, 15345 (1997). 8. F. Bassani and G. P. Parravicini, Electronic States and Optical Transitions in Solids (Pergamon Press, Oxford, 1975), 8th ed.. 9. R. Meade, K. Brommer, A. Rappe and J. Joannopoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett.61, 495 (1992). 10. J. Broeng, S. E. Barkou, T. Søndergaard and A. Bjarklev, “Analysis of air-guiding photonic bandgap fibers,” Opt. Lett. 25, 96 (2000). 11. Following the procedure in Ref.[7] we can, for incident waves in certain directions, also use the information about the irreducible representations to identify uncoupled modes which behave as band gaps. 12. M. Løkke, “Triangular photonic band gap crystals implementation in air-guiding optical fibers,” Thesis, University of Aarhus 2003 (unpublished). 13. M. M. Sigalas, R Biswas, K. M. Ho and C. M. Soukoulis, “Theoretical investigation of off-plane propagation of electromagnetic waves in two-dimensional photonic crystals,” Phys. Rev. B 58, 6791 (1998). 14. M. L. Povinelli, Steven G. Johnson, Shanhui Fan and J. D. Joannopoulos, “Emulation of two-dimensional photonic crystal defect modes in a photonic crystal with a three-dimensional photonic band gap,” Phys. Rev. B 64, 075313 (2001).


I. INTRODUCTION
The optical response of photonic band gap crystals is determined by the dispersion relation.By exploring the symmetry of the crystal lattice through a grouptheoretical treatment much information can be retrieved about the dispersion relation, without having to solve Maxwell's equations numerically.In this article we will focus on photonic crystals which can be used as optical fibers to guide light in air.Triangular structures seem to be the most promising as airguiding optical fibers.Therefore, we will consider the two-dimensional photonic crystal composed of silica with circular rods of air arranged in a triangular lattice.We let the periodic lattice plane coincide with the xy-plane and the z-axis defines the uniform direction.The lattice constant is denoted by a in the following.We assume lossless materials with a frequency-independent scalar dielectric function and restrict ourselves to the linear regime.For the use of photonic band gap crystals as optical fibers we have to consider light propagating out-of-plane, i.e. with a nonzero propagation constant in the z-direction, k z = 0.This case however is different from in-plane propagation, because the fields do not split up in two independent polarizations TE and TM.Instead we have to treat transverse vector fields.By the use of group theory we will determine the spatial symmetry of the out-of-plane eigenmodes in the photonic crystal, and try to predict the splitting and position of the bands when a small variation is introduced to the dielectric function.Furthermore, we will analyze one optimal lattice structure for which air-guiding is possible.All the theoretical predictions are supported by numeri- * Electronic address: loekke@phys.au.dk cal calculations.Fully-vectorial eigenmodes of Maxwell's equations with periodic boundary conditions were computed in a plane wave basis, using a freely available software package [1].

II. THEORY
The two-dimensional hexagonal crystal with waves propagating in the periodic lattice plane has been dealt with previously.In [2] the emphasis is on predicting in-plane band gaps for various materials, i.e. different index contrast between the rods and dielectric background.Here, we will instead focus on the twodimensional photonic crystal with modes containing a non-zero k z -component (out-of-plane), see Fig. 1.We examine the band structure for various sizes of the airrods and k z -component, while keeping the material fixed as an air-silica structure.The uniform photonic crystal is considered and we apply the theory to near-uniform structures with small variations in the dielectric function and in the long wavelength limit.

A. Maxwell's equations
In an uniform crystal, with ( r) = , Maxwell's equations reduce to Helmholtz's equation: where X i describes both the electric and magnetic fields.The equations in (1) are scalar equations and together with the restriction of the divergence equations the modes of a uniform or near-uniform crystal can be determined.Using the fact that Bloch's theorem applies in photonic crystals we can rewrite equation ( 1) to a Hermitian eigenvalue problem in Fourier space [3]: with η( G) being the Fourier transform of the inverse of ( r) and G a reciprocal lattice vector.When we consider a triangular lattice with one circular cylindrical rod in the center of the unit cell this Fourier transform can be calculated explicitly [3].η( G) depends on the crystal structure but in the present structure we obtain the same result when k z = 0 as for the in-plane case: , where f denotes the filling fraction and is given by 2 for the triangular crystal structure.r a is the radius of the circular rods and a and b denote the dielectric constants of the rods and the background medium, respectively.For the air-silica crystal considered here the dielectric constants are a = 1 and b = 2.1025.J 1 denotes the Bessel function of the first kind.For a given set of material parameters the Fourier transform depends on the magnitude of the reciprocal lattice vector lying in the periodic lattice plane and the rod radius, η( G) = η(| G |, r a ).Spatial dimensions will be given in units of the lattice constant, a, and vectors in k-space in units of 2π/a.

B. Classification of out-of-plane eigenmodes
The dielectric function is uniform in the z-direction which implies that no dispersion occurs in this direction.The hexagonal photonic crystal therefore belongs to the point group C 6v , and we can assign the modes of the photonic crystal uniquely according to the irreducible representations of C 6v , even when k z = 0.This follows from the fact that Maxwell's equations are invariant with respect to the symmetry operations from C 6v [4].In [5] the irreducible representations were assigned for the two-dimensional hexagonal lattice with the electromagnetic waves propagating in the periodic lattice plane.In this case the modes split into two different polarizations TE and TM and the fields can be treated as scalars.However when k z = 0 we have to consider vector fields.If we consider photonic crystals with magnetic permeability equal to unity both the magnetic field and the electric displacement is purely transverse due to the divergence equations.Furthermore, because the electric displacement and the electric field have the same characters they belong to the same irreducible representations from C 6v [6].This means that we can work out the irreducible representations for transverse vector fields and assign the same sets of irreducible representations to the H-field and E-field, but they do not in general follow the same order of assignment.The procedure of obtaining the compatibility relations is the same as for the in-plane case.Let H ⊆ G be a subgroup of order h H .The number of times the irreducible representation Γ (i) ∈ H, having characters χ (i) (R), appears in the reducible representation Γ, with characters χ(R) is Letting G i denote the group of k-point i we have that we can obtain the compatibility relations using equation (3).The results are summarized in Table I, with the notation following Fig. 1 and [7].We see that Z + 3 , Z − 3 and H + 3 are doubly-degenerated as expected, because these irreducible representations are two-dimensional.
When considering the case with k z = 0 we have to take two independent and orthogonal polarizations into account.We let these two polarizations be given by ê1 and ê2 : When a symmetry operation, R, from the point group acts on a vector field both the field vector and the argument is altered: TABLE I: Compatibility relations for the hexagonal lattice.
TABLE II: Characters for the rotations and mirror reflections.
Using this relation we can determine the character for one point in the extended zone scheme, χ (1) (R).The character is given by the trace of the matrix representation.For counterclockwise rotations of angle θ, around the axis containing the wave vector, R θ , and for mirror reflections, σ, in the plane containing k we obtain We can therefore obtain the one-point characters in Table II.
To determine the character as a whole for the points in the extended zone scheme we have to obtain the number of truly invariant points, N R , under the symmetry operation, R ∈ C 6v .The character as a whole for the reducible representation is then given by χ As an example consider the single Z(0) point in the extended zone scheme having the lowest angular frequency, and the six equivalent Z(1)-points having second lowest angular frequency in the extended zone scheme.For these two points Table III lists the characters obtained.
Carrying out the reduction procedure once again described in equation ( 3) for all the lowest representations we obtain Table IV.From Table IV we can determine the degeneracy of the free-photon bands by the number of irreducible representations and their dimensions.In Table IV we have also given the frequencies of the freephoton hexagonal bands as function of k z .We notice the differences when going from the in-plane to the out-of-plane case [5].First of all we obtain twice as many bands in the band structure as for TE and TM polarizations separately, and the lowest band becomes doubly-degenerated.But new spatial symmetries are also introduced to the out-of-plane band structure.For in- FIG. 2: Free-photon hexagonal bands with kz = 1.4.The dispersion relation is slightly convex due to the non-zero value of kz.
stance the irreducible representation, Γ + 2 , is not present at the high symmetry points for the lowest TE and TM bands.In Fig. 2 we have calculated the free-photon hexagonal bands for k z = 1.4.We obtain a discrete solution for the eigenmodes in the photonic crystal because we deal with a system bounded by the Brillouin zone.We see the agreement with Table IV with respect to the free-photon angular frequencies.When we introduce a small variation to the dielectric constant the angular frequencies are altered and the bands split up.The degeneracies from the free-photon case are lifted and the modes assigned to each irreducible representation have different eigenfrequencies.However, the modes corresponding to the two-dimensional irreducible representations remain doubly-degenerated.Even when the variation of the dielectric constant becomes large the symmetry of the eigenmodes remains unchanged.This is true because the eigenfunctions are formed as a linear combination of unperturbed wave functions according to Bloch's theorem, each possessing the same symmetry.The symmetry information however does not lie in the exponential time factor, due to the expansion in harmonic modes, such that this symmetry is not affected by a variation enlargement in the dielectric function.It is therefore possible to assign the ir- In vacuum the relation ω = c| k| holds.
reducible representations even for modes in hexagonal photonic crystals having large variations in the dielectric structure.This assignment can be carried out by consulting Table I and Table IV.
For the actual assignment we have to know only a few of the irreducible representations, which we can obtain by analyzing the symmetry of the eigenmodes numerically.The rest of the irreducible representations can be assigned by Table I that connect the irreducible representations for adjacent wave vectors, and by using the non-crossing selection rule of bands with the same symmetry.
Once the irreducible representations have been assigned to the band structure we can predict the spatial symmetry of all the lowest crystal modes.This can be done because the character tables of the point groups C 6v , C 2v , C 3v and C 1h describe whether or not the modes are even or odd under the different symmetry operations belonging to the groups.Because a symmetry operation also acts on the vector field, equation (5), this change in spatial symmetry has to be considered as well if one wants to predict the spatial symmetry of the modes.However, if we only look at the z-component of the fields, i.e.E z or H z , we can immediately apply the information about the symmetry from the irreducible representations, because these components are invariant under operations in the xy-plane.In Fig. 3 we have sketched the spatial symmetry of the modes which are classified according to the one-dimensional irreducible representations for the different symmetry points in the Brillouin zone.Γ denotes a general irreducible representation.In addition to Fig. 3 we know that the two-dimensional irreducible representations Z + 3 and Z − 3 are even and odd under the symmetry operation, C 2 , respectively.

C. Prediction of out-of-plane band structures
Ones the irreducible representations are assigned to the crystal modes it is possible to make an estimate of the band frequencies as the structure is changed and for different out-of-plane components, k z .If we consider the three equivalent H-points in the Bril- louin zone the lowest six-fold degeneracy seen in the freephoton case, H(1), will split up when a small variation is introduced to the structure.The degenerate mode will split up in four states, two non-degenerate modes H + 1 and H + 2 and two doubly-degenerate modes 2H + 3 .The symmetrized linear combinations of plane waves that transform like the rows of the irreducible representations can be obtained by the projection operator [7].When k z = 0 we have to take the two independent polarizations into account.This means that the vector field transforms like equation (5).When using the explicit matrix representations of the symmetry operations the symmetrized and normalized eigenfunctions becomes: where k 1 , k 2 and k 3 denote the three equivalent H points in the first Brillouin zone and the polarizations are given by equation ( 4).
In the basis of these symmetrized photon states the eigenvalue equation ( 2) is diagonal because the operator on the left side commutes with the operator, PR , where R ∈ C 6v .Calculating the normalized eigenfrequencies from the symmetrized photon states we can obtain a lower and an upper limit for the frequency bands at the H(1)-point: where η 0 (r a ) = η(0, r a ) and η 1 (r a ) = η( 2 √ 3 , r a ).We see that the frequencies are dependent on the out-of-plane component, k z , and on the structure, r a .Indeed the last factor is responsible for the variation of the crystal structure and in the limit r a → 0 we obtain the frequencies for the uniform crystal.For the Z(0)-point we obtain a single frequency because the degeneracy is not lifted when a variation is introduced Equation (7) shows that when we introduce a variation in the crystal, the lowest frequency is no longer just equal to ω = υ| k| as we saw in Fig. 2, but is suppressed by the factor η 0 (r a ).The boundary frequencies for the Z(1)-point is where η 2 = η(2, r a ) and η 3 = η( 4 √ 3 , r a ).The upper limit is determined by whatever factor is the largest.The reason for the ambiguity is that η 2 and η 3 becomes negative for r a 0.3 making the second factor largest.For the L(1)-point we obtain the frequency limits and the L(2)-point reads with η 0 + η 2 > η 0 − η 2 for small rod radii, r a .
The method above can be used to predict the position of the frequencies and the band split widths in the band structure.However, the approximation is only valid in the near-uniform crystal because of equation (1).That is in photonic crystals where either (i) the dieletric contrast is small (ii) the variation of the structure is small or (iii) in the long-wavelength limit where the waves do not "see" the variation.
As an example of the three limits of approximation we have chosen to look at crystals with (k z , r a ) equal to (0.1, 0.1), (1.4,0.1) and (0.1, 0.44), respectively.For these values we get the frequency boundaries in Table V.
It should be mentioned that the same theory can be used if one wants to analyze photonic band gap crystals with interstitial holes present.The assignment of irreducible representations is exactly the same, but the band split width and position of the bands have to change because the expression for η( G) becomes different.

III. OUT-OF-PLANE BAND STRUCTURES
In Fig. 4 we have plotted the out-of-plane band structure for the triangular air-silica photonic crystal with parameters (k z , r a ) equal to (0.1, 0.1), (1.4,0.1), (0.1, 0.44) and (1.4,0.44), respectively.The predicted positions and band split widths obtained in Table V have been added as error bars.In (a) we see a small split of the modes due to the small variation in the dielectric function.The predicted frequencies are in good agreement with the numerical results as expected in the near-uniform crystal.In (b) the out-of-plane component, k z , has been increased compared to (a).This increase in k z only, induces a larger split between the bands as expected from the theory, because of the k z -dependence.We also notice that the bandwidth becomes smaller when k z is increased.This tendency has been explained in [8] and is due to the entrapment of light with large k z inside high dielectric regions by total internal reflection, resulting in very small overlap.In (c) only the rod radius, r a , has been increased compared to (a).The theory predicts a larger split between the bands when the rod radius increases.However, the predicted split is underestimated and the position tends to be a little too high.The analytical ap- proximation is clearly more sensitive to changes in the rod radius than the out-of-plane component.Still the predicted value at the Z(0)-point is in good agreement with the numerical result because of the long wavelength approximation.In (d) we are neither in the near-uniform limit nor in the long wavelength limit and therefore the approximation does not apply.However, we can assign the irreducible representations using Table I, Table IV and by studying only a few of the eigenmodes numerically.We have made the assignment for the electrical field.The irreducible representations could also have been assigned in (a) -(c) but have been omitted for clarity.
From the irreducible representations we can determine the spatial symmetry of the crystal modes, E z , or ,H z , directly by using Fig. 3. Fig. 5 shows the numerically calculated E z -fields for the crystal, (k z , r a ) = (1.4,0.44).
We see that the spatial distributions of the fields are in accordance with Fig. 3 and the assignment of irreducible representations in Fig. 4 (d) [10].
The band gap appearing in Fig. 4 (d) is just one out of many band gaps that can open up when enlarging the out-of-plane component.It is well known that twodimensional triangular air-silica crystals do not exhibit complete in-plane band gaps, i.e. overlap in frequency of in-plane band gaps for the TE and TM polarizations.However, as shown in [9] several band gaps open up when k z is increased.In Fig. 6 we have calculated the photonic band gap boundaries as function of k z in an air-silica crystal structure with r a = 0.44.The first band gap appears around k z = 1.3 and is located between band number four and band number five.A part of this fundamental band gap exists above the air line, which is necessary to sustain a localized defect mode inside a hollow air-core.The air line is defined by ω = k z c and indicates that the structure is able to reflect electromagnetic waves incident from air.The importance of being able to predict such out-of-plane band gaps are therefore essential in the search for air-guiding photonic crystal fibers.Section IV describes an attempt to find the optimal air-guiding crystal structure in this particular part of the band gap with k z = 1.4.Fig. 6 shows that some of the band gaps also disappear again when k z is further increased.We should notice that no complete band gap exists for all values of k z .We cannot use the simple theory to predict the particular band gap referred to above in the air-silica crystal because it is not covered by the approximation.But since the band gap should appear between band number four and band number five we can conclude from the theory that the cases (0.1, 0.1), (1.4,0.1) and (0.1, 0.44) cannot exhibit this particular band gap.This follows from the fact that band number five has to lie within the error bar at H(1) and that this error bar would have to overlap with that at Z(1) for a band gap to exist.These two error bars however do not overlap in either of the three cases.If we were to consider [11] the triangular air-GaAs crystal instead this necessary condition for band gaps would indeed be fulfilled and in fact this structure exhibits a complete band gap between band number three and band number four for k z ∈ [0; 0.8].
In general, if a band gap exists in-plane we can conclude that this gap will stretch out to non-zero k z , due to continuous dependence on the wave vector.

IV. DEFECT MODES
In this section we will take a closer look at the lowest fundamental band gap above the air line in Fig. 6.In particular at k z = 1.4.In this area it is possible to introduce an air defect to the crystal and obtain a localized defect mode which is localized partly within this defect.In [9] such defects have been discussed.Here, we will introduce a circular air defect in the center and determine the optimal radius of the defect, r d , for which light is maximally confined within this air region.In this way to determine the theoretical parameters for an efficient air-guiding two-dimensional air-silica crystal.The physical energy stored in the electric and magnetic fields are equal for plane wave modes.In Fig. 7 we have calculated the normalized energy fraction of the electromagnetic field confined within the air defect as the radius, r d , is varied.The plot shows a maximum confinement at a rod radius of r d = 1.42,where 92.5% of the field energy is localized within the air region.The corresponding normalized frequency of the defect mode is also shown as function of r d .For the case of r d = 1.42 it is 1.424.We see that the frequencies lie within the band gap boundaries, the horizontal lines, as they should in order for the defect modes to be localized.The frequency increases as r d increases.When r d increases the defect mode "feels" a smaller dielectric constant on average, which implies a scale of the eigenfrequency of the defect mode, ω d ∝ 1/ √ .
For the actual calculation of Fig. 7, a supercell of size 7 × 7 of the unit cell has been used.With this supercell the overlap of adjacent defect modes are negligible, thus the width of the defect band across the unit cell is quite small.In Fig. 8 we have plotted two of the field distributions from Fig. 7 together with the structure of the lattice.The maximally confined field at r d = 1.42 in (a), and for r d = 1.50, where the normalized fraction of energy within the air defect is 76.1% in (b).We see a clear difference in the field intensity in the air core and the defect structure in (a) would be preferable in an air-guiding photonic band gap fiber.The increasing skills in fabrication techniques will make it possible to construct crystals with an accuracy for which it is interesting to optimize defect modes on the length scale required for application to telecommunication.In order to obtain the most efficient photonic band gap fibers, thorough investigations of the lattice structures have to be done.The defect structure considered here is just one out of many possible candidates for air-guiding photonic band gap fibers, and it is indeed likely to be possible to find even more efficient parameters such as different lattices, materials and crystal defects that are more suitable for air-guiding fibers.

V. CONCLUSION
We have classified the out-of-plane modes in the hexagonal structure according to the irreducible representations and confirmed that they agree with the numerical calculations.These spatial symmetries have been used to predict band split widths and positions of the bands in the dispersion relation.Furthermore, one particular defect mode has been analyzed and the optimal structure was found with respect to air-guidance.

FIG. 1 :
FIG. 1: Two-dimensional Brillouin zone for the hexagonal lattice.The irreducible Brillouin zone is bounded by ΓQP in-plane.The highly symmetric lattice points with a nonzero kz-component are denoted by ZLH, respectively (outof-plane).The three equivalent P points, which only differ by a linear combination of reciprocal lattice vectors are also shown.

FIG. 3 :
FIG. 3: The spatial symmetry of the z-component of the modes on each symmetry point in the irreducible Brillouin zone belonging to the different one-dimensional irreducible representations.The black and white colors correspond to opposite signs of the field amplitude.

FIG. 4 :
FIG. 4: The triangular air-silica band structure as kz and ra are varied.The error bars show the predicted positions and widths of the band splits at the high symmetry points calculated in Table V.Notice the different frequency scales.(a) (kz, ra) = (0.1, 0.1) the near-uniform crystal.(b) (kz, ra) = (1.4,0.1), small variation in the crystal structure (c) (kz, ra) = (0.1, 0.44), long wavelength limit.(d) (kz, ra) = (1.4,0.44), no error bars have been added because this case is beyond the approximation.The irreducible representations for the electrical field have been assigned according toTable I and Table IV.A band gap above the air line appears for the normalized frequency in [1.405; 1.434].

FIG. 5 :
FIG. 5: (Color online) The Ez field distribution for the lowest 8 bands at the different symmetry points.The lattice considered has a rod radius of ra = 0.44 and kz = 1.4.The blue and red colors correspond to opposite signs of the field amplitude.

FIG. 6 :
FIG.6:The largest complete band gaps for the lowest 20 bands in the triangular air-silica structure with ra = 0.44 (The white areas in the inset are silica).The dashed line is the air-line[9].

FIG. 7 :
FIG. 7: The fraction of electromagnetic field energy localized inside the air defect as the defect radius, r d , is varied.The dashed line shows the frequency of the corresponding defect mode and the horizontal lines represent the boundaries of the photonic band gap for the triangular air-silica crystal with (kz, ra) = (1.4,0.44).

FIG. 8 :
FIG. 8: (Color online) The field distribution of the Γ + 2 defect mode at two different defect radii.(a) r d = 1.42 with an energy fraction of 92.5% confined in the air defect.(b) r d = 1.50 where the confined energy fraction is 76.1%.

TABLE III :
The characters of the two lowest representations at the Z-point in the extended zone scheme of the hexagonal lattice.

TABLE IV :
The irreducible representations for out-of-plane electromagnetic waves in free space, whose wave vectors (in units of 2π/a) are reduced in the Brillouin zone of the hexagonal lattice.