Asymmetric band gap shift in electrically addressed blue phase photonic crystal fibers

: In this paper, we present electrooptic experiments on photonic crystal ﬁbers ﬁlled with a liquid crystalline blue phase. These ﬁbers guide light via photonic band gaps (PBGs). The blue phase is isotropic in the ﬁeld-oﬀ state but becomes birefringent under an electric ﬁeld. This leads to a polarization dependent shift of the PBGs. Interestingly, the eﬀect on the PBGs is asymmetrical: while the short wavelength edges of the PBGs shift, the long wavelength edges are almost unaﬀected. By performing band gap and modal analyses via the ﬁnite element simulations, we ﬁnd that the asymmetric shift is the result of the mixed polarization of the involved photonic bands. Finally, we use the band gap shifts to calculate eﬀective Kerr constants of the blue phase.


Introduction
Blue phases (BPs) are chiral liquid crystalline phases which exhibit a three dimensional structure. The lattice constant of this structure is commonly in the range of the wavelengths of visible light. This leads to brilliantly reflecting domains seen in a polarizing microscope [ Fig. 1(a)]. Three different blue phase modifications can appear in the absence of external fields, but only two of them, BP I and II, show a periodic structure [1]. For the BP I and BP II, the three dimensional structure is composed of double twist cylinders [ Fig. 1(b)] stacked in a cubic lattice [1,2]. This cubic super structure makes BPs optically isotropic in the absence of external electric or magnetic fields. Under the influence of an external electric field, the BP becomes birefringent with its optical axis oriented along the direction of the electric field. This is usually a fast process ( 1 ms [3][4][5][6]) compared to nematic liquid crystals (∼ 10 ms). Thus, BPs are of current interest in display research. Another advantage is that BPs do not require surface alignment for a homogeneous orientation. This gives them an advantage over other liquid crystalline phases, which usually require high quality alignment layers. As there is no need for this additional layer, blue phases are excellent candidates for use in confined geometries like photonic crystal fibers (PCFs).
PCFs are optical fibers which confine light by a periodic array of air inclusions [7] in a glass The air inclusions can be easily filled with various liquids by capillary action [8][9][10][11][12]. If the refractive index of the liquid is higher than the index of the surrounding material, light guidance is supported by photonic band gaps (PBGs) of the cladding [13][14][15]. The PBGs reflect the light impinging on the walls and confine it to the core. But there are wavelength regions, where light is not confined because a single PBG only covers a certain wavelength range. Thus, the transmission spectrum of such a fiber exhibits multiple regions of low transmission. These regions are referred to as photonic bands, which exist inside the cladding. The positions of the photonic bands (and photonic band gaps) are determined by the refractive index of the inclusions and the surrounding materials [16]. Thus liquid crystals have been commonly used to control the optical transmission through these fibers. The refractive indices of the LCs can be influenced via temperature change [17,18] or electromagnetic fields [19][20][21].
Choosing blue phases instead of nematic liquid crystals to infiltrate PCFs provides some advantages. There is no need for an alignment, layer which simplifies the experiment. Further, the theoretical description is simplified as BPs are isotropic in the field off-state. Thus, complicated liquid crystal director fields have3 not to be taken into account.
So far, only thermal tuning of blue phase photonic crystal fibers (BPPCFs) has been investigated [22,23] in order to change the transmission. This is generally a slow process. From our point of view, the tuning via electrical fields is a more promising concept. This option has become viable only recently, when blue phases with high sensitivity to electric fields (high Kerr constants) have become available [4,5,24].
In this article, we discuss the effect of electrically induced uniaxiality of the blue phase on the polarization dependent transmission of BPPCFs. We interpret the experimental results with the help of modal analysis and band gap analysis via the finite element method. Finally, we connect the changes in the band gap guidance to the Kerr constant of the blue phase material.

Blue phases
The cubic arrangement of double twist cylinders [ Fig. 1(b)] leads to a macroscopically isotropic liquid crystalline blue phase with the refractive index n BP . An external electric field induces a threshold-less reorientation of molecules; the BP becomes birefringent. The blue phase then resembles a uniaxial medium. Its optical axis is oriented along direction of the external electric field as shown in Fig. 2(a). For small electric field strengths, field-induced biaxiality and saturation can be neglected. If so, the birefringence δn can be modeled by the Kerr effect of an initially isotropic medium [25,26] where K is the Kerr constant, λ the wavelength of the light and E the electric field. The refractive indices parallel (n e ) and perpendicular (n o ) to the optical axis can be expressed by where δn is the field-induced birefringence. For very large electric field strengths, the chiral structure becomes unwound to a nematic state. δn then approaches a saturation value ∆n(T ), which corresponds approximately to the birefringence of the nematic phase that appears in a racemic mixture instead of a BP.

Photonic crystal fibers
The periodic arrangement of air holes in photonic crystal fibers can be filled with a high index material. 'High' means that the refractive index of the material is larger than the refractive index of the background material. Then the light is confined by photonic band gaps. These fibers are therefore also referred to as photonic band gap fibers. However, the cladding does not provide a full band gap over a broad spectral range but is interrupted by photonic bands. Within these photonic bands the light escapes from the core into the cladding and a spectral region of low transmission arises. A closer inspection of these photonic bands reveals that they closely resemble waveguide modes of the individual inclusions [14]. The regions of low transmission can then be interpreted as wavelengths where the core mode is in resonance with the photonic bands. Then, strong coupling occurs and light is distributed over the cladding. Consequently, light is only guided if the core modes and the photonic bands are anti-resonant. This interpretation is called the anti-resonant reflecting optical waveguide (ARROW) model [16,27,28]. According to this model, a strong interaction between the core mode and a photonic band is expected only close to the cutoff of the involved inclusion mode. The cutoff conditions for the inclusion modes are well known [29]. The positions can be approximated by [16,30] where λ m is the m-th expected ARROW resonance wavelength, d is the inclusion diameter, n is the refractive index of the inclusion and n s is the refractive index of the surrounding material. Figure 2(b) shows schematically a transmission spectrum of an ARROW fiber. The positions of the dips in transmission can be predicted and labeled with Eq. (3).

Blue phase photonic crystal fibers
Equation (3) gives a relation between the transmission minima in a BPPCF and the refractive index of the inclusions. If we consider the blue phase to be the high index material, we can calculate the refractive index from the transmission dips.
If an electric field is applied along the fiber cross section, the refractive index of the blue phase changes from n BP toñ BP for a specific direction. From Eq. (3) it is clear that then λ m shifts tõ λ m . The shift in wavelength can be measured. In order to get a relation between the shift in wavelength and the change in refractive index, we useñ BP = n BP + dn and assume that dn 1. The latter condition is reasonable as we work with low electric fields. Then we can approximate the square of the refractive index in Eq. (3) byñ 2 BP ≈ n 2 BP + 2n BP dn. Equation (3) can then be rearranged to This equation relates the shift in wavelength of a band to the corresponding change in refractive index.

Materials and setup
For the experiments, a commercial PCF (LMA10, NKT photonics) made of silica with inclusion diameter d = 2.908 µm, pitch Λ = 6.525 µm [cf.  The transmission of the sample is measured with the setup shown in Fig. 3(a). All measurements are performed upon heating. The BPPCF of 3.3 cm length is fixed between conducting ITO plates [ Fig. 3(b)] to apply an external voltage V . The electric field inside the inclusions E incl can be approximated by where D is the distance between the conduction plates, which is identical with the fiber diameter, s and BP are the relative permittivites of the surrounding medium (silica) and the blue phase, respectively. Details about this equation can be found in the Appendix A.

Measuring the transmission under the influence of an electric field
After filling the BP into the PCF, polarizing microscope images of the fiber were studied upon heating [ Fig. 4]. The transition from the cholesteric phase (a) to BP I (b,c) to BP II (d,e) and then the BP II-isotropic coexistence (f) is observed between 65.5 • C and 67.5 • C. Within the blue phase regions, indeed blue phase platelets are observed. For BP I, also different orientations of the BP crystals can be observed.   Fig. 3(b)]. At 0 V, the spectra show transmission windows which are expected for a photonic band gap fiber. The spectra for xand y-polarization at 0 V confirm that the blue phase is isotropic as both spectra are almost identical.
Under application of a square wave at 1 kHz, the average intensity in the case of x-polarization [ Fig. 5(a)] decreases by 23 % at 600 V. This loss of intensity can be attributed to scattering by the change in refractive index when the electric field switches polarity. The loss increases with increasing voltage, i. e. with increasing induced birefringence. The position of the band gaps, however, is not affected in any way.
In the case of y polarized light [ Fig. 5(b)], a much stronger decrease in intensity with increasing voltage (45 % at 600 V) is observed. The scattering in this case is higher because it relates to the birefringence which is larger for the y-polarization [+2/3 δn, cf. Eq. (2)] than for the x-polarization (−1/3 δn). In contrast to the x-polarized light, the y-polarized light also shows a shift of the band gap positions, the magnitude of which increases with the voltage. To make this effect more visible we focus on a single band gap [gray shaded in Fig. 5(a) and 5(b)].
In Fig. 5(c) and 5(d) the band gap located between 530 nm and 620 nm is plotted for (c) xand (d) y-polarized light. The intensities are normalized to unity for each voltage in both graphs. This procedure reveals that there is almost no change of the band gap shape for the x-polarized case. In contrast, for y-polarized light, there is a significant shift of the left edge of the transmission window by up to 12 nm at 600 V while the right edge remains constant. This effect of an asymmetric shift of the transmission windows will be discussed in the following section.

Explaining the asymmetric shift
From the ARROW model and Eq. (4), a symmetric shift of the band gap is expected. To investigate this behavior, we carried out a modal analysis via the finite element method (FEM). In Fig. 6(a) the fraction of the electric field guided in the core is plotted for three different cases: (i) in the absence of an electric field (blue line) meaning at 0 V where the blue phase is isotropic; (ii) with an applied voltage of 650 V taking into account the correct anisotropy with n x = n z = n o , n y = n e (green line) and (iii) also at 650 V but with an 'artificial' refractive index distribution n x = n y = n e , n z = n o (red line). The last case is an anisotropic waveguide with its optical axis parallel to the fiber axis. The cases (i) and (ii) represent the experiment at 0 V and 650 V, respectively.
The simulation in fact shows a shift of the short wavelength edge of the transmission window from case (i) to (ii). On the long wavelength edge in case (ii), the core power drops to 0.2 at 610 nm which coincides with edge of the transmission window in case (i). In the range from 610 nm to 615 nm the core power stays low indicating weak confinement of the core mode. This is the region we do not see in the experiment due to the high losses. Beyond 615 nm the core mode is not confined anymore. In case (iii), the wavelength region between 610 nm and 615 nm exhibits good confinement of the core mode. Thus, we conclude that the asymmetric shift originates from the anisotropy when the optical axis is perpendicular to the fiber axis. This result is supported by the modal analysis [ Fig. 6(b)] at 610 nm. It reveals that the mode in case (ii) couples to a photonic band (or cladding mode) which is mainly x-polarized. The coupling is weak because the core mode is y-polarized. The photonic band does not move to longer wavelengths under voltage as its polarization is mainly perpendicular to the polarization direction with increasing refractive index. In case (iii), this photonic band is not present because it has shifted to longer wavelengths due to n e = n x = n y .
The above interpretation is supported by the band gap plots in Fig. 6(c) and 6(d). Details on the simulations are given in Appendix B. Fig. 6(c) shows the band gap diagram at 0 V [case (i)]. The effective refractive index of the guided mode is shown in green and has been obtained by modal analysis. It is flanked by two photonic bands located around 537 nm and 612 nm. The interaction of the core mode with the photonic bands leads to a bending of the effective refractive index close to the edges. There are some bands (at 548 nm and 585 nm) within the transmission window which intersect with the core mode but do not interact. These are caused by higher order cladding modes, the high order of which does not allow for significant coupling [16].
At a voltage of 650 V [case (ii), Fig. 6(d)], the bands start to shift and polarization dependencies of the photonic bands start to appear. To distinguish between the polarizations of the bands we introduce a parameter η = |E y | / |E x | , where |E x | and |E y | are the absolute values of electric fields E x and E y averaged for a photonic band. The plot now shows two different kinds of photonic bands. Bands with η > 1 [orange in Fig. 6(d)] are mainly y-polarized and bands with 0.1 < η < 1 (blue) are predominantly x-polarized but still carry some amount of y-polarization.
The region in which the core mode (green line) is guided becomes smaller with increasing voltage. On the short wavelength side, a mainly y-polarized band (η > 1) flanks the core mode at 543 nm. In contrast, the core mode is stopped on the long wavelength edge at 615 nm by a band which is predominantly x-polarized (blue). The left band is shifted significantly due to its strong response to the increase of the refractive index n y . The right edge approximately stays constant because it is mostly x-polarized and interacts with n x . This leads to the asymmetric shift of the transmission window.

Applying the ARROW model to determine the Kerr constant
From the previous discussion it is clear that the induced change in refractive index of the blue phase causes the left edges of the transmission windows to shift. We now want to connect the applied electric field to the shift in wavelength and then to the induced birefringence. Finally, we would like to evaluate if the birefringence is proportional to the electric field squared as described by Eq. (1). Equation (4) gives us a relation between the shift of the photonic band and the corresponding change in refractive index. This equation is based on the ARROW model and we require the band or resonance order m to use Eq. (4). To this end, we plot in Fig. 7 the measured transmission spectrum and the transmission dips predicted by the ARROW model. The results show good agreement between the ARROW model and the experiment. There are some differences towards the shorter wavelengths, though. These differences are due to the inaccuracy of the measured refractive index of the blue phase n BP . To verify this, we plotted the normalized power guided inside the core region. This is in perfect agreement with the ARROW model and in good agreement with the experiment. The numbering of the photonic bands (or resonances) can then be performed: m = 3, . . . , 7 from right to left. Now that we have identified the resonance order m for the corresponding band gaps, we determine the electrically induced wavelength shiftλ m of the short wavelength edges in Fig. 5(b). For each band gap and each voltageλ m are analyzed at five positions of the normalized intensity (I = 0.3, 0.4, . . . , 0.7).
The change in refractive index is then obtained with Eq. (4). If we then assume that the shift in wavelength and therefore the change in refractive index is solely caused by the extraordinary refractive index n e , i. e. dn = n e − n BP , we are able to relate dn to the induced birefingence in Eq. (1) by δn = 3/2dn. From Eq. (1) it is expected that δn is linear with the electric field squared. Therefore, in Fig. 8(a)  The Kerr constants also change with temperature as can be seen in Fig. 8(b). In general Kerr constants seem to decrease, which is commonly observed in blue phases. The band with m = 6 is distinct from the remaining bands by its high Kerr constant of about 8 nm V −2 . This value comes close to the values of the Kerr constant measured in a vertical field switching (VFS) cell (dashed grey line). At all temperatures, the Kerr constants of all photonic bands are lower than the ones measured in the VFS cell. This is again an effect of the bands carrying a mixed polarization. Even though they are mainly influenced by the y-component of the refractive index (n y = n e ), they still weakly interact with the lower index n x = n o . Each photonic band carries different portions of x and y-polarizations and hence shows different Kerr constants.
These results seem to limit the applicability of the ARROW model, which is made for isotropic materials. Nevertheless, there still is a linear relation between the shift of the photonic bands and the electric field squared as it is predicted by Eqs. 1 and 4. The refractive index change we calculated above has to be seen as an effective value and consequently the Kerr constant has to be seen this way. The merit of these values is that with them we are able to predict how the left edge of a specific transmission window shifts for a given electric field.

Conclusion
In this work, we infiltrated a photonic crystal fiber with a liquid crystalline blue phase and electrically induced a birefringence of the blue phase with an optical axis perpendicular to the fiber axis. Consequently, the transmission spectrum of the fiber became strongly polarization dependent. For the polarization of light perpendicular to the optical axis, almost no change in transmission windows was found. However, the transmission windows for light polarized parallel to the optical axis shifted to longer wavelengths. In the latter case, the change was strongly asymmetric. While the long wavelength edge remained constant, the short wavelength edge shifted proportional to the square of the electric field.
By band gap analysis of the photonic cladding, we found that the photonic bands of the short wavelength edge were mainly polarized along the optical axis. The increased refractive index had therefore strong effect on these bands, which led to a red-shift. However, the long wavelength edge was formed by bands mainly polarized perpendicular to the optical axis. They did therefore not shift. Nevertheless, these photonic bands still carried sufficient polarization to stop the core mode.
We then used the shift of the short wavelength edge to determine the change in refractive index. We discovered that the induced shift is indeed proportional to the electric field squared as expected of a Kerr medium. From this, we calculated the Kerr constants, which are lower than the Kerr constant measured in the bulk material. This can be attributed to the mixed polarization of the photonic bands.
Blue phases offer the unique opportunity to study a uniaxial medium within a photonic crystal fiber where the optical axis is distinct from the fiber axis. This provides a new application for the electrooptic effects observed in blue phases.

Appendix A: Electric field inside inclusions
Equation (5) describes the electric field inside the inclusion. This field is different from the electric field within the surrounding material due to the difference in permittivity [31]. The field can be approximated by the limit of a single inclusion which is embedded in the middle of an otherwise perfect plate capacitor. This is possible because the relative permittivities of silica [32] and the optical adhesive are very similar (≈ 3.9). Then, by performing a multipole expansion [31], it can be shown that the external field E ex = V /D and the inclusion field are related by Eq. (5). For the given parameters this results in E incl ≈ 0.25E ex . It is important to note, that within this approximation the field inside the inclusion is homogeneous. However, as there are not one but multiple inclusions close to each other inside the silica surrounding, there will be deviations both inside a specific inclusion and also when comparing different inclusions. The variations can be up to 15 %. The variations are not considered in the calculations of photonic band or the Kerr constant.