1. Introduction

The state of polarization of the electromagnetic field is one of its most fundamental properties, quantifying the vectorial character of the optical response of systems ranging from the simple to the complex. In general, it is a spatially-dependent quantity, represented by an ellipse extracted at every point by mapping the modulation of the field vector over a temporal period. The polarization state is circumscribed by three distinct types of singularities: pure circular polarization (indeterminate ellipse rotation angle); pure linear polarization (indeterminate polarization handedness); and disclinations (indeterminate polarization direction with a concomitant vanishing field). Polarization singularities in transverse electromagnetic fields were first discussed in depth by Hajnal [1]. A generalization to random vector fields was performed by Berry and Dennis [2]. However, it appears that no formal treatment of polarization singularities in optical systems of high symmetry has yet been reported. We demonstrate that symmetry transformation properties allow us to map the general states of local polarization. Furthermore, we reveal how polarization singularities are expressed, noting that a system’s symmetry will constrain local states of polarization to subsets of the system’s polarization response. A linear two-dimensional photonic crystal is used to illustrate this since it has complex symmetry properties, and tractable analytic solutions in the vanishing dielectric contrast limit. Because the results for any irreducible representation are inherent to system symmetry, they are invariant to dielectric contrast. To the best of our knowledge, ours is the first report quantifying the predictive power of group theory in describing the local states of polarization in optical systems with discrete symmetry. Previous reports have noted polarization singularities in such periodic systems as optical lattices [3], and polarization gradients in photonic crystals [4], but without formal reference to symmetry.

We introduce below the formal language of group theory in the context of the Bloch mode of a two-dimensional hexagonal photonic crystal, extracting local polarization states and singularities. Classification of the optical Bloch mode by an irreducible representation of its point group is a well-established technique [8], which may be used to quantify global transformation properties. We have uniquely extended the treatment to deal with site symmetries and the local optical field response, as encoded in the polarization states. By revealing how the local polarization state transforms under the action of the elements of the symmetry group, through the set of all crystallographic orbits of local field vectors, we express the fundamental domain from which the entire mode can be constructed. Furthermore, we demonstrate that particular polarization singularities must appear at those locations, known as special Wyckoff positions, associated with sites of higher symmetry.

2. Group theory and the symmetry transformation properties of Bloch modes

A photonic crystal is a system where there is periodic spatial symmetry in the dielectric permittivity defined by ε(r) = ε(r + R), where R is the sum of integer multiples of elementary lattice vectors. We consider a two-dimensional photonic crystal comprising a hexagonal array of cylinders, as shown in Fig. 1(a), as an illustration of a system with high symmetry and symmorphic space group representation. The principal axes of symmetry lie at points where the site-symmetry groups are isomorphic to the space group’s point group. It suffices thus to consider the system’s point group, and subgroups thereof.

2.1 Point groups and irreducible representations

The point group of a photonic crystal in two dimensions is determined by the group of symmetry operators, R, such that Rε(r) = ε(r) (i.e., the set of operators that leave the dielectric profile invariant). In two dimensions, the symmetry operators are as follows: the identity operator, E; the rotational operator, C_n, which denotes an n-fold (2π/n) rotation; and the mirror reflection operator, σ_ν, which indicates mirror reflection with respect to the vertical plane ν. The origin of the unit cell is placed at the point of highest symmetry (i.e., at the principal axis of rotation, where the mirror planes intersect), and the symmetry transformation operations are typically performed about that point. Figure 1(b) reveals that the point group of the hexagonal lattice is C _6ν = {E,2C ₆,2C ₃,C ₂,3σ_x,3σ_y} (the coefficients indicate the number of operators that belong to each conjugacy class). In Schoenflies notation, the point group is given by C_nν, where n is the largest rotational symmetry and ν indicates the presence of mirror symmetry. The electromagnetic eigenmodes of the Maxwell equations are found by employing Bloch’s theorem for specific wavevectors k [8], hence the governing differential operators, ℒ[ε(r),k], express not only the real-space symmetries of the lattice, but also the reciprocal space symmetries of the modal propagation directions. For simplicity, we restrict ourselves to the transverse electric case, whence the eigenfunction of the operator is the magnetic field, H _k,n (r), whose polarization is always parallel to the z-axis. The corresponding electric field, E _k,n (r), may be determined directly from the curl of the magnetic field, and its polarization must thus lie within the x-y plane.

 

Fig. 1. Two-dimensional hexagonal lattice photonic crystal; (a) composed of a periodic array of dielectric cylinders, which have periodic symmetry in the x-y plane and infinite translational symmetry in the z direction; (b) top view of the dielectric cylinders and the symmetry operators R which leave the dielectric profile invariant; (c) Brillouin zone and equivalent K points. [8]

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The point group associated with a particular wavevector is identified by a subset of symmetry operators that leave the differential operator invariant: Rℒ[ε(r),k]R ^-1 = ℒ[ε(r),k]. Clearly, this implies that the operators commute, [R, ℒ[ε(r),k] = 0, and in consequence have simultaneous eigenfunctions, H _k,n(r). Because R k = k + G (where G is a reciprocal lattice vector) must be satisfied, H _k,n (r) must be associated with a particular subgroup of C _6ν for each k. For example, at the K point in the Brillouin zone, Fig. 1(c), the eigenfunction is associated with the C _3ν = {E,2C ₃,3σ_y} point group. Such simultaneous eigenfunctions are the partner functions, ψ ^(Γ) _i, of irreducible representations, Γ, of the point group. Each irreducible representation is described by a set of square non-singular matrices, D ^(Γ)(R), for all R in the group as

where the number, l, of partner functions is determined by the degeneracy. A unique set of characters, χ ^(Γ) _R, arising from the trace of each matrix, as

more simply quantifies the transformation of the partner function. For simplicity, we have restricted ourselves to nondegenerate modes, l = 1, whence χ ^(Γ) _R = D ^(Γ)(R). Thus the fundamental transformations of all partner functions (e.g., the Bloch modes) may be obtained directly from known character tables. The character table for C _3ν is shown in Table 1: the rows are the irreducible representations (here denoted A ₁, A ₂ and E), and the columns the symmetry operations [9].

Representing the symmetry operators by orthogonal matrices, R↔, the operation of R on a vector function F(r) alters both the position vector and the field vector, as

If F ^(Γ)(r) is a partner function to one of the irreducible representations of the point group, then application of Eqs. (1) and (2) to Eq. (3) provides a symmetry transformation relation,

Equation (4) holds for any H _k,n (r) and E _k,n (r), with their respective irreducible representations, Γ, fixed by the choice of [ε(r),k].

2.2 Wyckoff positions: crystallographic orbits and site symmetry

The space described by the permittivity profile, ε(r), may be subdivided into sets of symmetry-equivalent points known as crystallographic orbits. Each crystallographic orbit is obtained by applying all the operators of the space group G to a specific point, r = p, within the unit cell. For the symmetry operations g_i ⊂ G, where g_i ε(p) = ε(p), a site symmetry group G _p (G _p ⊂ G) is formed for which p is invariant; G _p is isomorphic, either to the point group or, to one of its subgroups. Note that every point in a crystallographic orbit has site symmetry conjugate to the remaining points (i.e., G _p = g_j G _p′ g ^-1 _j, where g_j ∈ G, g_j ∉ G _p and g_j ∉ G _p′). The orbits can be classified by their Wyckoff positions [12], those with the same site symmetry having the same Wyckoff position. A general Wyckoff position has a site symmetry group containing merely the identity operator E. Special Wyckoff positions have site symmetry groups with more than one element.

Table 1. The C _3ν Character Table

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For each space group G, there is a fundamental domain Ω that, when acted upon by all the elements of the space group, covers all space. The primitive unit cell is the fundamental domain for the subgroup of lattice translational operators, but for the full space group G the fundamental domain is smaller. The number of points in a crystallographic orbit that appear in the unit cell is equal to t = n_Cnν/n _Gp, where n_Cnν is the number of elements of the point group C_nν, and n _Gp is the number of elements in the site symmetry group [12].

2.3 Field expression in the vanishing contrast limit

For the fundamental mode (n = 1) at the K point, k = x̂(4π/3a), the magnetic field is associated with the A ₁ irreducible representation and the electric field is associated with the A ₂ irreducible representation [8]. We focus on the electric field, because it has nontrivial polarization states, and exploit the vanishing contrast limit, ε(r) → ε, whence [7,11]

We note that this is purely for ease of illustration and local polarization state determination.

3. Expressing polarization: local variation and the existence of singularities

3.1 The local polarization state

To quantify the local state of polarization, we note that the electric field is complex, whence

with the components not necessarily orthogonal. Defining a(r) and b(r) such that [P(r) + i Q(r)] = [a(r) + i b(r)]e ^iτ(r), judicious choice of τ(r) as

will ensure a(r)⊥b(r) [5]. These vectors are the semi-major and semi-minor axes, respectively, of polarization ellipses uniquely defined at every field location. Note that τ (r) will be undefined for certain singular states, and that P ² (r) = P(r) · P(r).

We have generated the polarization ellipses for the fundamental mode of a two-dimensional hexagonal photonic crystal at the K point, as described by Eq. (5) in the vanishing contrast limit. The result is shown in Fig. 2(a) by the blue ellipses, which were generated for an equally-spaced array of points by mapping the tip of the electric field vectors over one temporal period, where a common major axis length has been imposed for clarity. The large grey circles are the putative locations of the hexagonally-arrayed cylinders. Figure 2(b) reveals the actual magnitude of the electric field, for comparison. The direction of polarization is determined from P(r)×Q(r), which is positive for the left-hand (counterclockwise) polarization and negative for the right-hand (clockwise) polarization, as shown in Fig. 2(c). Further discussion follows below.

 

Fig. 2. Polarization response for the A ₂ irreducible representation of C _3ν. (a) calculated local polarization states: polarization ellipses (blue), dielectric columns (closed grey circles), linear polarization (black lines), circular polarization (red circles) disclinations (green circles); the primitive unit cell is delineated by the dashed red lines and the fundamental domain is shown in gold; (b) field magnitude, ∣E(r)∣ (arbitrary units); (c) the ẑ component of P(r)×Q(r) (arbitrary units; note that all other vector components are zero); (d) unit cell of the hexagonal lattice (dashed line), fundamental domain (orange region), reflection planes (lines), centers of C ₃ rotational symmetry (red triangles), and the orbit of a general Wyckoff position (black dots).

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3.2 Singularities in the local state of polarization

The local state of polarization may be undefined with respect to its mathematical descriptors at distinct locations, r ₀. These points are known as polarization singularities and three types have been noted: C points, L lines and disclinations [6]. We now illustrate their existence explicitly from the modal response of our optical system, deferring discussion of system symmetries to the following section.

C points are individual points in space (inclusively, lines at higher dimension) where the local state of polarization is circular, as given by a(r ₀) = b (r ₀), whence τ (r ₀) is undefined. From Eq. (7), P(r ₀)-Q(r ₀) = P(r ₀)² -Q(r ₀)² = 0 in consequence. In Fig. 2(a), the C points of the electric field are marked as red circles. These locations correspond, in Fig. 2(c), to extrema in the ẑ component of P (r) × Q (r).

L lines are continuous arrays of points in space (inclusively, planes at higher dimension) where the local state of polarization is linear. On an L line, the direction of the vector normal to the ellipse is undefined, whence P(r ₀)×Q(r ₀) must vanish. Such may occur when either P(r ₀) or Q(r ₀) vanish, or are (anti)parallel. In Fig. 2(a), the L lines are denoted by the straight line segments in black. These lines correspond, in Fig. 2(c), to the demarcation between regions of right-handed (negative) and left-handed (positive) polarization.

Disclinations are singularities in the traditional sense of the term: locations where the field magnitude vanishes and a phase term becomes not just undefined but discontinuous. Such arise because P(r ₀) = Q(r ₀) = 0, and Eq. (7) shows τ(r ₀) is consequently undefined. From Fig. 2(b), we note that the field magnitudes vanish at the column centers. These locations correspond, in Fig. 2(a), to the intersections of L lines, which are shown by the green circles. Note the discontinuity in the rotation angle at these positions.

4. Site symmetry and the polarization state

The symmetry operations of the space group G establish the mapping of a local electric field vector, and hence its polarization state, to all other symmetry-equivalent locations in the system. To derive quantitative insight into the nature of the local polarization response, we must express the full symmetry representation of the space group, both the translation subgroup and the point group. We do this with reference to the crystallographic orbit, as regards locations with particular site symmetries classified by their Wyckoff positions. The latter are shown, via self-consistency arguments, to yield the various polarization singularities.

4.1 Site symmetries and Wyckoff positions

Quantifying the symmetry hierarchy may be done with reference to the primitive unit cell, shown by the dashed line in Fig. 2(d). The cell is comprised of two equilateral subcells, delineated by solid lines showing the mirror reflection planes. All positions on such planes are symmetry-equivalent locations, and their Wyckoff positions are thus equal, the site symmetry being C _1h = {E,σ_ν}. The intersections of the mirror planes are the locations of principal axes, their site symmetries being C _3ν = {E,2C ₃,3σ_y} due to simultaneous existence of three-fold rotational symmetries. The centers of the equilateral sub-cells do not possess any mirror symmetries compatible with the translation subgroup, and hence their site symmetries are merely C ₃ = {E, 2C ₃}. All three-fold rotational symmetries are noted by the red triangles in Fig. 2(d). As will be demonstrated more rigorously below, it is apparent that the trisect of the equilateral sub-cell, noted in orange, tessellates all of space: the operators of the point group map this region across the entire unit cell, while the translation operators replicate it across the lattice. Any position strictly within this fundamental domain has only the identity site symmetry, E ={E}, and its Wyckoff position is thus general. By contrast, the other three site symmetries are the various special Wyckoff positions, located on the boundary of the fundamental domain. In what follows, we relate the type of Wyckoff position to the specific local polarization state.

4.2 Local transformation and the crystallographic orbit

The operation of R on the field, given by Eq. (3), is a vector field description with a complementary local relation, given by

that expresses E(r ₁) and E(r ₂) as symmetry-equivalent vectors at distinct locations r ₁ and r ₂. The set of all r generated by Equation (8), from all symmetry operations including those of the full translation subgroup of the space group G, is the Bloch mode’s crystallographic orbit for the r ₁ generation point.

The transformation of the mode at the generation point, R↔E(R↔^-1 r ₁), drives a corresponding transformation in the local state of polarization, R↔P(R↔^-1 r ₁)×R↔Q(R↔r ₁) [see Eqs. (3) and (6)]. From Eq. (8), we require that members of the crystallographic orbit have states of local polarization related by R↔P(R↔^-1 r ₁)×R↔Q(R ^-1↔r ₁) = P(r ₂)×Q(r ₂). Since the cross product of the transformation of any two vectors, F and G, is R↔F×R↔G = (detR↔)F× G [10], we find

Since R does not change the length of a vector, and det(R↔) = ±1, Eqs. (8) and (9) assert that field magnitudes ∣E(r ₁∣) = ∣R E(r ₁)∣ = ∣E(r ₂)∣ and polarization response magnitudes (i.e., ellipticity), ∣P(r ₁)×Q(r ₁)] = ∣R(P(r ₁)×Q(r ₁)) = ∣P(r ₂)×Q(r ₂)∣, are the same for all members of the crystallographic orbit. The polarization direction, however, depends directly on the nature of the symmetry operation: rotations will preserve the handedness (det(R↔) = 1), while reflections will reverse it (det(R↔) = -1).

The polarization response of the field, Eq. (5), is shown in Fig. 2(a) for the extended hexagonal lattice to more clearly reveal the symmetry. The parallelogram inscribed by the dashed red lines gives the primitive unit cell, and the optical response clearly possesses its full translational invariance. Inspection reveals that the polarization response of the field at an arbitrary site within the fundamental domain (here shown in gold) does indeed map onto the six symmetry-equivalent positions within the unit cell, in accordance with t = n _C3ν/n_E =6/1 = 6. This is confirmed explicitly in Fig. 2(d), where we have selected a generation point r ₁ near the apex of the fundamental domain. The three elements of the crystallographic orbit within this equilateral subcell, deriving from {E,2C ₃}, preserve ellipticity and polarization handedness (e.g., P(r ₂)×Q(r ₂) = P(r ₁)×Q(r ₁)) (c.f. Fig. 2(c)) while sequentially advancing the rotation angle by 2π/3. Likewise, it has three elements in the other equilateral subcell, deriving from the subgroup of reflection operators {3σ_y}, where both orientation and ellipticity of the polarization ellipses are mirror-symmetric, Fig. 2(c) demonstrating their opposite polarization directions (e.g., P(r ₂)×Q(r ₂) = -P(r ₁)×Q(r ₁)). Although we have not explicitly indicated the polarization handedness in Fig. 2(a), it does indeed transform as expected. While we have shown the transformation between members of the crystallographic orbit for the general Wyckoff positions, symmetry does not constrain the local polarization state at such sites to any particular response.

4.3 Polarization singularities and site symmetry

At special Wyckoff positions within the field, we may determine whether a higher site symmetry constrains the local state of polarization to a particular response. A local symmetry transformation relation, analogous to the vector field relationship of Eq. (4), exists for special Wyckoff positions r ₁ as

with the character of the symmetry transformation χ ^(r₁) _R determined by the particular site symmetry group at r ₁. This local character must match the global symmetry transformation character χ ^(Γ) _R of the Bloch mode to within a phase factor determined by the nature of the displacement of r ₁ from the principal axis, as discussed in [11]. To reveal the singular nature of the polarization at a special Wyckoff position, we express the local field, via Eq. (6), as

components D(r ₁), B(r ₁) and C(r ₁) are purely real, while Ω(r ₁) is a complex phase factor. For any r ₁, substitution of Eq. (11) into Eq. (10) will yield a self-consistency condition contingent on the local symmetry.

For the Wyckoff positions associated with site symmetry groups C _1h, elements of the σ_y E(σ ^-1 _y r ₁) = E(r ₂), were found in Sec. 4.2 to P(r ₂) × Q(r ₂) = -P(r ₁) × Q (r ₁). Hence, for σ_y E(σ ^-1 _y r ₁) = E(r ₁), we require P(r ₁)×Q(r ₁) = 0, whence the sense of rotation is undefined, and thus mirror planes must constitute L lines. It now remains to determine the linear polarization rotation angle, vis-à-vis the respective mirror planes. Let us investigate the self-consistency condition, employing a in Eq. (10) for illustration. The argument proceeds identically for σ′_y and σ″_y. Suppressing r ₁, for convenience, we obtain

No simple constraint on the character exists from Eq. (12) itself. However, our choice must be consistent with the character of the A ₂ irreducible representation of the E(r) Bloch mode, since the mirror plane intersects the principal axis, which has the symmetry of the point group. Choosing χ ^(r₁) _σy = χ ^(A₂) _σy =-1, Eq. (12) then requires D = B = 0, with C indeterminate, whence E(r ₁) = Ω(r ₁)(0, C(r ₁)). The polarization direction is thus orthogonal to the mirror plane. Similar expressions may be found for the other two mirror planes, with all polarization directions mutually conforming to the mirror symmetry requirements noted in Sec. 4.2. This is in accord with the observed singular states determined by direct calculation of Eq. (5), where Fig. 2(a) reveals the L lines and Fig. 2(c) their polarization directions (from P × Q = 0).

We now determine if there are any locations where Eq. (10) is valid for rotation operations. However, because polarization handedness is maintained upon rotation, no specific insight is gained from the general local transformation properties, established in Sec. 4.2 for rotation operators. Employing C_n in Eq. (10) yields a general self-consistency condition for any site containing a rotational symmetry:

For n = 3, we find two distinct types of singularities, dependent on the site symmetry group to which the C ₃ operator belongs.

The Wyckoff positions associated with the site symmetry group C ₃ ={E, 2C ₃}, which is a cyclic group, identify points of pure three-fold rotational symmetry in the hexagonal lattice. They are located in the unit cell at (a/2,a/√3)) and (a, a/√3), as shown in Fig. 2(d), and are part of the same crystallographic orbit. Note that we expect two elements of the orbit to appear in the unit cell, since t = n _C3ν/n _C3 = 6/3 = 2. From Eq. (13), the self-consistency condition for these two local states of polarization can be determined as

Taking the ratio of the two vector component equations, to eliminate χ ^(r₁) _C3, cross-multiplying and equating the consequent real and imaginary parts yields B ² = C ² + D ² and BD = 0, respectively. The only possible solution to this constraint system, for non-zero fields, is D = 0, B = ±C or B=0, D = ±iC. For all four possibilities we obtain two unique solutions to Eq. (14): left-hand circularly polarized light, E(r ₁) = C(r ₁)Ω(r ₁)(i,1,0), and right-hand circularly polarized light, E(r ₁) = C(r ₁)Ω(r ₁)(-i, 1,0). With these fields, Eq. (14) yields their respective characters as χ ^(LCP) _C3 = (-1 + i√3)/2 and χ ^(RCP) _C3 = (-1-i√3)/ 2. Such characters are in agreement with those of the degenerate E irreducible representations of the C ₃ point group. More particularly, these are in agreement with the characters determined for the Bloch mode itself, for C ₃ symmetry operators referenced to the points (a/2, a/2√3)) and (a,a/√3], rather than to the principal axis at (0,0) [11]. Figure 2(a) reveals that C points (circular polarization states), denoted by the red open circles, do indeed exist at all points in the crystallographic orbit where (a/2,a/(2√3)) is the generating point. Moreover, the observed polarization handedness, as noted in Fig. 2(c), conforms to that anticipated by the symmetry arguments (i.e., left-handed P × Q positive, right-handed P × Q negative).

The special Wyckoff positions of highest symmetry are found at the intersections of the mirror planes, where they possess both C ₃ and C _1h symmetries, i.e., the point group C _3ν = {E,2C ₃,3σ_y} these positions are the points of the 10) will express local symmetry transformation relations if simultaneously the constraints of Eq. (12) and Eq. (14) are satisfied. Since the electric field is confined to the x-y plane these locations cannot simultaneously be both C points and L points, hence the fields must locally vanish. Moreover, since each site lies at the intersection of L lines, whose polarization directions are orthogonal to their lines, the local state of polarization must be discontinuous. A vanishing field with indeterminate polarization state is the definitive signature of a disclination.

5. Conclusions

Using the fundamental TE mode of a simple two-dimensional hexagonal photonic crystal at the K point, we have shown how insight into the nature of the local polarization state may be extracted by the application of group theory to the system’s site symmetries. The crystallographic orbit was determined as the set of symmetry-equivalent points in the electric field, the local site symmetries identified, and the fundamental domain identified. Local symmetry transformation relations, corresponding to the general and special Wyckoff positions associated with their site symmetries, were found. From the former, the general transformation properties of the local polarization state were established, while from the latter self-consistency conditions were developed and nature of the polarization response deduced. It was thus that the three types of polarization singularities, and their locations, were predicted. Comparison with the local states of polarization derived from the Bloch mode, calculated in the vanishing contrast limit, confirmed the nature and existence of C points, L lines and disclinations in this linear photonic crystal. Since increasing the dielectric contrast of the photonic crystal does not change the system’s group properties, these observations hold for any contrast — indeed, they also hold for the two-dimensional photonic crystal slab geometry [13]. Efforts are underway to experimentally validate these findings in slab structures using near-field optical polarization microscopy. Such arrays of singular polarization states in photonic crystal slabs may have applications in novel optical sensor technologies, or in the laser-cooling of appropriately-confined molecules [4]. We close by noting that the approach we have described may be generally applied to any optical waveguiding system of sufficiently high symmetry, permitting fundamental design insight into the expression of desired optical response. .