3.3 Photonic bandgap (PBG) fi ber- based
3.3.2 Understanding colors of photonic band- gap fi bers
In our laboratory, we fabricate solid and hollow core PBG Bragg fi bers using layer- by-layer deposition of polymer fi lms, as well as co- rolling of commercial and custom- made polymer fi lms around the core mandrel (Gao et al. , 2006;
Dupuis et al. , 2007). Photos of a typical solid core fi ber preform and a resultant fi ber are presented in Fig. 3.4(a) . For fabrication of Bragg fi bers, we mainly use two material combinations (the high – n h and low – n l refractive index layers), which are polystyrene (PS)/poly(methyl methylacrylate) (PMMA) and polycarbonate (PC)/poly(vinylene difl oride) (PVDF), featuring the refractive index contrasts of n h / n l = 1.6/1.48 and n h / n l = 1.58/1.4, respectively. To describe guided states in such fi bers, we typically start with fi nding the bandgaps of a Bragg refl ector. In Fig. 3.4(b) , we present a typical band diagram (frequency versus the propagation constant, which is defi ned by the ratio of the amplitude at
the source of the wave to the amplitude at a certain distance) of the guided modes of an infi nite planar periodic refl ector fabricated from PMMA/PS and having layers of equal thicknesses d h = d l = d = 430 nm.
Gray regions in the band diagram describe states delocalized over the whole periodic refl ector. Such states are effi ciently irradiated out of the fi ber on the imperfections at the fi ber/air interface. Thus, when launching white light into the fi ber (Plate I(a)), states delocalized over the whole fi ber cross section are typically irradiated after the fi rst few cm of propagation. Clear regions in Fig. 3.4(b) defi ne regions of phase space where no delocalized states exist inside the periodic refl ector; these are the refl ector bandgaps. Bragg refl ector can therefore, confi ne light in the fi ber core if the frequency and angle of incidence (propagation constant) of guided light falls into the refl ector bandgap. As the core size of a Bragg fi ber is very large (compared to the wavelength of operation), light propagation inside the fi ber core can be envisioned as a sequence of consecutive bounces of rays traveling at shallow angles with respect to the core/refl ector interface.
The effective refractive index of such rays is close, while somewhat smaller than that of a core material. Dispersion relation of the Gaussian- like fundamental core guided mode (shown in Fig. 3.4(b) as a solid red curve), therefore, appears inside the Bragg refl ector bandgap, and is positioned somewhat above the light
3.4 (a) Solid core plastic Bragg fi ber preform and a resultant fi ber.
(b) Band diagram of the modes of a solid PMMA core Bragg fi ber with a PMMA/PS refl ector. Colors of the emitted and refl ected light from the Bragg fi bers are determined by the positions of the fi ber refl ector band gaps. Direction Z corresponds to the central axis of the fi ber and η is the incident angle on the core- cladding interface.
line of the core material. Dispersion relations of the higher- order, higher- loss core modes (not shown in Fig. 3.4(b) ) are positioned further above the light line of the core material, while propagation of such modes within the fi ber core is characterized by steeper incidence angles onto the core/refl ector interface. The color guided by the fi ber core is, therefore, defi ned by the spectral region corresponding to the intersection of the core material light line with the refl ector bandgap. Spectral position of a refl ector bandgap (guided color) can be varied at will by changing the thicknesses of the refl ector layers, with thicker layers resulting in bandgaps positioned at longer wavelengths. Practically, layer thicknesses are varied by drawing the same preform to fi bers of various diameters.
It is important to note that, although the bandgap position is determined solely by the geometry of a refl ector, the color of guided light is rather determined by the intersection of the light line of a fi ber core material with the refl ector bandgap (solid line in Fig. 3.4(b) ). From basic theory of the low refractive index- contract fi bers (Skorobogatiy, 2005), it follows that the center wavelength λ c of the refl ector bandgap is given by:
[3.1]
where n c is the core refractive index, n h,l are the refractive indices of the high and low refractive index layers in the fi ber Bragg refl ector, while d h,l are the corresponding layer thicknesses. As follows from Fig. 3.4(b) and Eq. 3.1, we can actively change the color of the guided light by either changing the thickness of the refl ector layers, or by changing the value of the core refractive index. The former can be implemented by stretching the fi ber. The latter can be implemented by fi lling the hollow core of a PBG Bragg fi ber with a material whose refractive index can be changed by varying certain environmental parameters such as temperature or electric fi eld. One class of such materials is liquid crystals that have already been successfully applied to tune bandgap positions in various PBG systems (Busch and John, 1999; Fudouzi and Xia, 2003; Larsen et al. , 2003).
In addition, even with no light traveling inside a fi ber, while only under the ambient (external) illumination, the PBG Bragg fi bers still appear colored (Plate I(b)). Typically, the fi ber color in the far fi eld is determined by the refl ection properties of the fi ber Bragg refl ector under the normal light incidence ( β = 0).
From Fig. 3.4(b) , it is clear that for low refractive index- contrast all- polymer Bragg fi bers, the bandgap of a refl ector at normal incidence is, generally, located at a different spectral position than the refl ector bandgap that supports the core guided mode. Therefore, the fi ber color under ambient illumination is different from the fi ber color due to irradiation of the core guided light. This provides the interesting opportunity to adjust the overall fi ber color by controlling the relative intensities of the ambient and propagating light. Finally, we note that when
operating within higher- order bandgaps of the fi ber Bragg refl ector, the color of guided light can be both of higher or lower frequency than the color of the fi ber under ambient illumination.