Polarization Rotation and Mode Splitting in Photonic Crystal Line-Defect Waveguides

In a line-defect waveguide of a planer photonic crystal (PhC), we found a new rotational state of polarized light, which exhibits “polarization rotation” on the PhC plane, when a phase mismatch m was added to the air-hole alignment of the waveguide, where mode splitting was simultaneously observed in the dispersion curve. To account for the polarization rotation together with the mode splitting, we propose a two-state model that is constructed from Schrödinger equation obtained from the equation for electromagnetic waves. The proposed two-state model gives an explanation on the relation between the polarization-rotational angle θ and the mismatch m and on its rotational direction (i.e., clockwise or anticlockwise direction) that depends on the mode. Using the two-state model, we also discuss the angular momenta of the polarized light in the PhC waveguide, which are directly related to the Stokes parameters that characterize the polarization rotations.


INTRODUCTION
In a line-defect waveguide of a planer photonic crystal (PhC), the polarization of light can rotate on the two dimensional PhC plane by addition of a phase mismatch to the air-hole alignment of the waveguide, which occurs without non-linear optical interactions (e.g., optical Kerr effects and photorefractive effects). Originally, the presence of light rotation, given as an optical vortex that carries angular momentum, in photonic bandgap media [1][2][3][4] via such non-linear interactions has been found in theoretical and experimental investigations [5][6][7][8][9][10]; This mechanism is attributed to the localized vortex state induced by those non-linear effects in the bandgap media, which is thus sometimes called a gap vortex. This is an analogous concept to a gap soliton in the bandgap media [11,12].
However, without such non-linear effects, we can show the presence of a rotational state of polarized light just by adding the mismatch m (0 < m < a) to the air-hole alignment of a sidewall of the waveguide [13], as illustrated in Figure 1, where a is the air-hole period of the PhC waveguide. Mock et al. also studied the same waveguide [14], but did not report such rotation. They just reported that the light path in the waveguide was "zigzag." An interesting feature of the rotation that we found is that if light propagates from a non-mismatched region to the mismatched region with a gradual mismatch change between them, the light polarization gradually rotates. Here we can define its rotational angle θ for a given m, and as will be shown later, θ is proportional to m (for small m compared with a) via numerical simulations. Moreover, with the addition of m, mode splitting (or level splitting) is observed in the dispersion curve [13,14], which is created from an originally degenerated mode that contains different polarizations. The main cause of this mode splitting is symmetry breaking; When m = 0, structural symmetry in the x and y directions is maintained (despite the presence of a linedefect waveguide at y = z = 0), which causes highly degenerated states in the dispersion curve. But, structural symmetry breaking caused by m = 0 can resolve the degenerated states, creating two split modes, as will be shown in detail in section 2.
We infer from our numerical simulations that the polarization rotation is intimately connected with the mode splitting, because they occur simultaneously, triggered by the addition of m. (Because of the above reasons, the mechanism of the rotational light obtained in our research is completely different from that of a gap vortex with non-linear interactions; the rotation that we found is spatial rotation, not temporal rotation seen for the localized vortex.) In this paper, we will gain a qualitative understanding of the simultaneous polarization rotation and mode splitting from our FIGURE 2 | Band structure of the PhC line-defect waveguide. Here, the air-hole period a is 450 nm, the air-hole radius is 0.29 a, and the waveguide width is 1.2 √ 3 a. Mode-crossing points exist for m/a = 0. As m increases (e.g., m/a = 1/8, 1/4, 3/8, 1/2), each mode-crossing point splits into two modes (the upper and lower modes) and the split-mode spacing (or gap size) becomes large. The light line for the waveguide with upper and lower air-claddings is indicated by the slanted dashed line. two-state model constructed via Schrödinger equation derived from the equation for electromagnetic (EM) waves in the waveguide. In this analysis, we can show that there is a relation between the polarization-rotational angle θ and the mismatch m (via a relation between the mode-splitting spacing ω and the mismatch m), which is an analogous relation to that between an electron-energy-level splittinghω and an applied magnetic field B, orhω ∝ B.
In the next section, we will give some numerical simulation results, and then introduce a theory that connects the mode splitting with the polarization rotation within linear optics.

NUMERICAL RESULTS FOR MODE SPLITTING AND POLARIZATION ROTATION
Finite-difference time-domain computations [13,14] show that when m = 0, the band structure (or the dispersion curve) of a PhC line-defect waveguide has mode-crossing points, as depicted in Figure 2. Each of the mode-crossing points consists of two degenerated modes with different polarizations, which are yielded by band-folding (realized in the reduced zone for the wavenumber k at −π/a ≤ k ≤ π/a). With the addition of non-zero m, the two-fold degeneracy is resolved (see   clearer polarization rotation can be seen in H z , and this will also be shown below.) Figure 5 with the blue (TE + 1 ) and red (TE − 1 ) curves depicts a calculated m-dependence of the split modes, TE + 1 and TE − 1 , that correspond to those in Figure 3B, which are produced from the lowest degenerated TE 1 mode at 190.13 THz with k = 0.2925 × 2π/a. Here, we chiefly focus on the lowest mode (TE 1 ) because its crossing point at 190.13 THz with k = 0.2925 × 2π/a is placed inside the light cone, that is, the light at that point is a reallypropagating mode, not a radiation mode. Also, we observed that a maximum gap size f max = 8.68 THz was obtained at m/a = 0.5 and that the splitting behavior was symmetric for (a) 0 < m/a ≤ 0.5 and (b) 0.5 < m/a < 1. Thus we plotted the figure for Case (a) only, which is sufficient to look at the essence of the behavior. The inset of Figure 5 shows the m-dependence of the normalized gap size, f /f max . Here, we phenomenologically found that it had good linearity, i.e., f /f max = g m/a (g = 2.5615) at m/a 0.3 (but f /f max leveled off near m/a = 0.5). The g = 2.5615 was obtained via the least square fitting with a straight line for the data points at m/a less than or equal to 0.2875. Note that we also observed that the center (say¯ = ( 0 + 1 )/2) of the TE + 1 and TE − 1 modes (say 0 and 1 ) showed little mdependence (almost no m-dependence), as indicated by the black dashed curve in Figure 5 (although 1 − 0 = ω = 2πf showed a large m-dependence, as seen in Figure 5).
We then give clearer field and intensity profiles of the TE − 1 and TE + 1 modes for H z : Figure 6A shows the field profile or the real part of H z in a unit cell at k = π/a of the Brillouin-zone (BZ) end, and Figure 6B shows the intensity profile |H z | 2 at a mode-crossing point, k = 0.2925 × 2π/a. We then observe clearer tilted field and intensity profiles with a tilted angle (or a rotational angle) θ that increases with increasing m. In this case, we found that the rotation of the intensity profiles of TE + 1 (TE − 1 ) in Figure 6B was in the anticlockwise (clockwise) direction; A similar tendency remained in the field rotations of Figure 6A at the BZ end (since we observed in numerical simulations that the mode-splitting spacing became large even at the BZ end as m increased).
The above results were obtained only from the numerical simulations (i.e., with no theoretical explanations via analytical methods). In the next section, we will explain the observed simultaneous polarization rotation and mode splitting by use of an analytical method with a two-state model derived from the equation for EM waves.

ANALYTICAL INTERPRETATION AND DISCUSSION
To derive the two-state model to account for the above phenomena, we start with the following wave equation (obtained from Maxwell's equations) for light propagating in the x direction with polarization E i : where ∇ 2 = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 + ∂ 2 /∂z 2 , n is the refractive index that includes the PhC spatial modulation, and c is the velocity of light.
In the following, we make an approximation that the inputpulse spatial-width W(> λ) is larger than the air-hole period a in the waveguide, where the input wavelength λ also needs to suffices λ > a. This approximation makes it possible to deal with the phenomenon (light rotation) analytically, but cannot well describe it near the PhC band edge because the backward reflection of propagating light is strong due to Bragg reflection with the air-hole arrays. Within the above approximation, an average nature of the rotation of light in a size of order ∼ λ can be obtained.
While keeping the above approximation, we insert E i = u i e i(βx−wt) into Equation (1) and obtain the next Schrödinger equation [15] (see Appendix): where we used the notations:h ≡ λ/2π, t ≡ x, V ≡ n eff − n, β = n eff k = 2πn eff /λ, ∇ 2 2 = ∂ 2 /∂y 2 + ∂ 2 /∂z 2 , and n eff is the effective refractive index in the waveguide. Here we also used the slowlyvarying-envelope approximation, |∂ 2 u i /∂x 2 | ≪ |2β ∂u i /∂x|. In addition, we omitted a small difference in β for waves with different polarizations (and with the same node numbers). In this situation, we can perfectly utilize ideas and descriptions in quantum mechanics to study the phenomena. Hereafter, we will deal with all quantities in units ofh = c = 1, as often used in quantum mechanics. Now we concentrate on the lowest eigenvalue of the righthand side of Equation (2) when m = 0, as described in section 2. In this case, we denote¯ as the lowest eigenvalue, or the lowest frequency at the crossing point (with m = 0). When m is added (i.e., m = 0), since the mode with¯ with degeneracy splits into two modes, we set their eigenvalues to be 0 and 1 , where 0 =¯ − ω/2 and 1 =¯ + ω/2 with a gap size of ω(= 2πf ) between the two split modes, where ω has a large m-dependence but¯ has little m-dependence, as already shown in Figure 5.
For those two split modes, the Hamiltonian H(t) in Equation (3) can be written in the matrix representation as which is known as a two-state model [16]. In Equation (4),1 is a unit matrix andσ 3 is the z-component of the Pauli matrices (σ 1 , σ 2 ,σ 3 ):σ Frontiers in Physics | www.frontiersin.org that suffice and thusσ where [X, Y] = XY − YX, {X, Y} = XY + YX, and i, j, k run from 1 to 3. ǫ ijk is the Levi-Civita symbol, and δ ij is the Kronecker delta.
In what follows, we also use the vector notation of Equation (5), i.e.,σ = (σ 1 ,σ 2 ,σ 3 ). For the Hamiltonian (4), the Schrödinger equation is of the form: = ¯ 1 + ωσ where |φ(t) is a two-component wave function, or a spinor [17,18]. (Its detail is described below.) In Equation (10), using the following transformation we can simplify the Equation (10) as where In Equation (11), deriving the e −i¯ t -term from |φ(t) is accepted, because we observed a sizable field/intensity tilt induced by a change in m or ω (because ω has a large m-dependence, but not¯ because¯ has little m-dependence). Thus clearly, we can say that the tilt does not depend on¯ . Furthermore, when we calculate the expectation value for an operatorÔ, we can rigorously say that no¯ term contributes to the expectation value: In fact, we obtain via the transformation (11).
Further, at the initial state (or at t = 0), the next relation holds: where |ψ is the (time-independent) eigenvector of Equation (13), or a spinor: Here, u 1 and u 2 are polarized electric fields or wave functions that will come into the TE − 1 mode and TE + 1 mode, respectively. Equation (16) is almost the same as the Jones vector constructed with polarized electric fields, which is expressed as a spinor with an appropriate basis [19,20]. There is a slight difference in definition between Equation (16) and the Jones vector defined with plane EM waves, because our EM waves are the waves propagating along the PhC waveguide with different polarization directions; nonetheless, almost the same definition can be used (except for the polarization directions).
The above spinor of a two-state model with Equation (13) can be regarded as that of an electron with spin 1 2 . In this case, as is well-known, the size of ω in Equation (13) is proportional to the strength B of an applied magnetic field, i.e., ω ∝ B (for weak B) [16]. In this situation, we can look upon m as B phenomenologically by help of the "equivalence" between ω ∝ m and ω ∝ B (for small m and B).
Next, to see the time evolution of the system with H (t), we use the following equation for a unitary operator U(t) [16,21].
Equation (17) can be derived from the derivative of the identity which indicates that the left-hand side of Equation (18) is hermitian. If we set this hermitian part as H (t), Equation (17) can be obtained from Equation (18). By inserting Equation (13) into Equation (17) and integrating Equation (17) with the initial condition U(0) =1, we obtain where the time-integration parts have been simply calculated as because H (t) = ωσ 3 /2 has no time-dependence. In this case, the time-evolution |ψ(t) of the spinor |ψ in Equation (16) is of the form: where if we interpret that ω t is an angle θ for the polarization rotation, then Equation (26) gives a double-valuedness to u 2 (or u 1 ) in the θ -rotation. Here, t should be interpreted as the length of the interconnection part between the non-mismatched entrance and the mismatched exit (when we use the units ofh = c = 1); The "t" is a constant when the interconnection length has a fixed value. Even in this situation, θ can vary as m changes because the insertion of f = g f max m/a into ω = 2πf in Equation (26) gives θ = ω t = 2πgf max t m/a = θ 0 m/a, where θ 0 = 2πgf max t is a constant. Inserting ω t = θ into Equation (26), we obtain which provides the relation |ψ(2π) = −|ψ(0) . Furthermore, Equation (27) can explain the aforementioned anticlockwise (clockwise) rotation of the TE + 1 (TE − 1 ) mode, because the explicit expression of Equation (27) with is of the form: Thus, if u 2 (θ ) rotates anticlockwise, then u 1 (θ ) rotates clockwise (and vice versa). Note that in terms of |φ(t) (not |ψ(t) ), we obtain in place of Equation (29): where 0 =¯ t is a constant when the "length" t is a constant, as described above. If m = 0, then θ = 0, but 0 remains as a constant (because¯ in 0 has (almost) no m-dependence); in this case (or m = 0), Equation (30) becomes Equation (31) corresponds to the zero-m fields, as given at the leftmost ones in Figure 6A. We can see that adjusting the parameter "t" in 0 of Equation (31) enables setting various initial polarized states at m = 0. We should examine the field tilt for m = 0 as a difference from those initial field profiles, (u 1 (0) u 2 (0)), and thus what we should look upon as a field tilt is given by and hence we obtain the same result as that in Equation (29).
As for the evolution of the angular momentum operatorσ , we can calculate it asσ Using Equations (21),(33), we obtain where Equations (8), (22) have been used in the calculations. For convenience, we rewrite Equation (34) aŝ where j is a dummy index and runs from 1 to 3. Sinceσ i is defined in the SU(2) space [22], it is not yet related to a vector in our real space, i.e., in the SO(3) space. Next, we show how a vector rotating in SO(3) space is related toσ i in the SU(2) space. To perform this, we start with the following Schrödinger equation: By multiplying ψ(t)|σ i from the left to Equation (37), we obtain where Equation (13) was also used. We then multiplyσ i |ψ(t) from the right to the hermitian conjugate of Equation (37) and obtain By subtracting Equation (39) from Equation (38), we have where [σ i ,σ 3 ] = 2iǫ i3kσk = −2iǫ ij3σj was used. In Equation (41), ψ(t)|σ i |ψ(t) is an expectation value ofσ i at "time" t and is observed as a vector in the SO(3) space. Also, we can show that ψ(t)|σ i |ψ(t) is actually proportional to M i (t) of a general vector M(t) rotating in the SO(3) space, where it suffices the same-form equation as Equation (41): where M(t) is fixed in a moving system with an angular frequency vector ω = ωe 3 and e 3 is a unit vector in the e 3 -direction. Furthermore, in order to check the transformation property of ψ(t)|σ i |ψ(t) , by use of |ψ(t) = U(t)|ψ , we get This means that ψ(t)|σ i |ψ(t) is a vector that transforms via a ij (t), i.e., the rotation in SO(3). In the above, we have assumed ω t = θ since its use in Equations (27),(29). Thus, we consistently use it in the rotation of the angular momentum. We then obtain as the explicit matrix representation of Equation (45), where σ i = ψ|σ i |ψ and σ i θ = ψ(t)|σ i |ψ(t) with ω t = θ (or σ i θ = ψ(θ )|σ i |ψ(θ ) ). The relation of σ i θ with the electric fields is easily obtained from the direct calculations of σ i θ = ψ(θ )|σ i |ψ(θ ) with Equations (5),(28): where S 1 , S 2 , and S 3 are the Stokes parameters that characterize polarization rotations [24,25]. The electric-field intensity relates to the rest (S 0 ) of the Stokes parameters: Note that the rotation of the angular momentum in Equation (46) shows a single-valuedness with respect to θ , which is completely different from the double-valuedness of the wave function |ψ(θ ) ; If we treated an electron of spin 1 2 , |ψ(θ ) would be the wave function of the electron, and the angular-momentum motion would correspond to spin precession [26].
In this paper, we have pointed out the "equivalence" between the PhC line-defect waveguide with an added phase mismatch m and the spin-half electron with an applied magnetic field B. In particular, the EM wave equation in the PhC waveguide could be reduced to the Schrödinger equation; Both systems showed the same two-level splitting and had the split mode or level spacing in proportion to the size of the perturbation, m or B. At the present stage of the theory with some approximations, we cannot prove the "equivalence" mathematically, but we showed it from some supporting evidence. A further development of the theory with much less approximations will be able to explain the equivalence.

SUMMARY
Using the Schrödinger equation derived from the EM wave equation, we have built a two-state model that can explain the observed polarization rotation and mode splitting that occur simultaneously when a phase mismatch m is added to a PhC linedefect waveguide. The theory has given a double-valuedness to the light field (or the wave function) with different polarizations and a single-valuedness to the motion of angular momenta. Using a spinor representation, the former has explained the difference in the rotational direction (i.e., clockwise or anticlockwise direction) of TE − 1 or TE + 1 mode, and the latter has clarified the relation between the angular momenta and the Stokes parameters that define the polarization rotations. Also, the theory has given a relation between the rotational angle θ and the mismatch m for small m (via the numerical result between f and m). In this analysis, the "equivalence" between both systems has been indicated for the light field with two split modes and the electron wave function with two split levels and for the light angular motion and the electron spin precession.

AUTHOR CONTRIBUTIONS
MS proposed the idea to introduce the mismatch. MS, IT, and SS developed theories. MS and KD conducted photonic crystal simulations. IT drafted the main text. All authors contributed to discussions.