An optical diode (OD) and its application based on the photonic band-gap (PBG) property of chiral photonic crystals are presented. The OD is constructed from wave-plates and cholesteric liquid crystals with two different pitches. The component is polarization sensitive and exhibits passive anisotropic transmission analogous to the electronic diode. The non-reciprocal transmission of linearly polarized light at the different PBG regions is discussed. A new scheme of a color-separating system with the OD is proposed. The OD behavior constitutes the operational mechanism for a color splitter.
©2006 Optical Society of America
Much research has been devoted to periodic dielectric structures known as photonic crystals (PCs) during the last years. Because of their optical analogies with semiconductors and their potential to advance photonic technology, PCs are a field with an enormous set of possible applications such as waveguides, holey fibers, multiplexers, and filters . Among many photonic band-gap (PBG) materials, cholesteric liquid crystals (CLCs) with periodic helical structures are very attractive as self-organized chiral PCs. A circularly polarized light with the same handedness as the CLC helix propagating along a helical axis is forbidden in a certain frequency range, and light of opposite chirality is unaffected by the structure . In the planar texture of CLCs, the axes of molecular director rotate from plane to plane to form a helical structure with pitch P. For sufficiently thick films, the reflectance of the specific polarized light with normal incidence is nearly 100% within a band centered at λo = n̄P. The bandwidth is Δλ~≈λo Δn/n̄, where Δn=n ∥-n ⊥ is the birefringence defined by the indices of refraction parallel and perpendicular to the nematic director and n̄ =[(+)/2]1/2 is the average. The selective reflection property of the CLCs has been used for lasers, switches, sensors, and display applications [3–6].
In many electric devices, a diode is an essential component that transmits current in the forward direction but prevents it in the backward direction. The possibility of achieving anisotropic transmission through a passive optical element is useful in the fields of optical isolation and all-optical processing. We can define a device called an optical diode (OD) if it permits light in one direction and blocks light in the reverse direction. Various ODs based on PC structures have been proposed and demonstrated [7–10]. Furthermore, the tunability in OD operation has been designed to show electro-controlled non-reciprocal transmission [11, 12]. The diode behavior depends on the properties of complex asymmetric structures, different nonlinear materials, left handed media and polarization conversions. In this paper, we apply the general idea to achieving diode action by making the structure asymmetric only. We discuss a spatially nonreciprocal device based on a multilayer structure produced by conventional wave-plates and two CLC films with different pitches. From the dispersion relation of the structure, the non-reciprocal transmission of linearly polarized light at the PBG regions shows the diode performance. We also propose a new model of color separating based on the PBG OD. The mechanism is capable of splitting light into red (R), green (G) and blue (B) components. An optical projection system using the OD can offer the advantage of lower system weight than the systems using the prism designs .
2. Structure and analysis of the optical diode
The structure of the OD is represented in Fig. 1(a). There are two CLC films (CLC1 and CLC2) with the same helical handedness but different pitches (P1 <P2 ), and they are joined by a half-wave plate. All of the films are sandwiched by two quarter-wave plates which are used as right circular polarizers for an s-polarized incident light (perpendicular to the incident plane) from directions I and II. For the two quarter-wave plates, the optical axis (OA) of the left plate is oriented at angle ϕ=45° relative to the x-axis, and the other is at ϕ=-45°. Since cholesterics are locally nematic and fluid, the texture must be solidified to stabilize the PBG properties. Polymer-stabilized CLCs have been developed for the purpose . Otherwise, an alternative choice for structurally chiral materials can be exemplified by sculptured thin films, which also exhibit the phenomenon of PBG for normally incident light of only one circular polarization state .
The CLC films are assumed to have a right helix, and the dispersion curves for right-handed circularly polarized (RCP) waves in CLC1 and CLC2 are shown in Fig. 1(b). Blue and green areas are the PBGs of CLC1 and CLC2, respectively. qi =2π/Pi is the wavenumber corresponding to the pitch, where i =1,2. CLC films are used as the key elements of the OD. A phase retard π/2 will convert linearly polarized light to RCP in case of the plate’s OA oriented by 45 ° to the direction of input light polarization. The half-wave plate affects the nonreciprocal transmission property of the OD. This retarder layer changes the circular polarization states of light between right and left handedness. As illustrated in Fig. 1(a), the s-polarized incident waves in direction I become RCP after the left quarter-wave plate. CLC1 can reflect the RCP waves (blue arrow) which satisfy the selective reflection condition and transmit the other waves which are unaffected by PBG1. The reflected waves are RCP and return to their original polarization state after the left quarter-wave plate. The transmitted waves become left-handed circular polarization (LCP) after the half-wave phase retarder and still are unaffected by PBG2. Finally, their polarization state turns into p-polarization (parallel to the incident plane) after the right quarter-wave plate. In direction II, the specific input waves (blue arrow) changing to the RCP state after the right quarter-wave plate can pass through CLC2 if PBG1 and PBG2 do not overlap. After the half-wave plate, the polarization state becomes LCP. The specific waves are unaffected by CLC1, and then they change to the p-polarization state after penetrating the left quarter-wave plate. Due to the PBG1 position depending on the propagation direction, the blue arrow shows non-reciprocal transmission of an OD. For the same case of s-polarized incidence, the green arrow concerned with PBG2 also has the similar OD behavior but propagates in the reverse direction. The dispersion relation for forward and backward propagation of the s-polarized input waves in the OD structure can be characterized as Fig. 1(c). If the incident light is p-polarized, it should be noted that the operation mode of circular polarizers referring to the OA must be altered. Hence, there is no difference to the OD performance between s- and p-polarized incident waves.
The simulation of transmission and reflection for the OD is obtained with the finite element method (FEM)  based on Berreman’s 4×4 transfer matrix . Along the helical z-axis, the equation of light propagation with angular frequency ω is given by
where Ψ(z) is the field vector expressed as Ψ(z) = (Ex ,Hy ,Ey ,-Hx )T, and Δ(z) is a derivative matrix. ε ∥= and ε ⊥= are the dielectric constants. Defining ε = (ε ∥+ε ⊥)/2 and δ = (ε ∥-ε ⊥)/2, the wave equations for the electric field E = (Ex ,Ey ) can be derived from Eq. (1) by eliminating the magnetic field H:
Here s=1 is the right-handed screw sense and s=-1 is the left-handed one. For the incident light with wavelength λ at an oblique angle θ relative to the z-axis, the wavenumber is k0 =2π/λ and its component in the x-axis direction is kx = k0 sinθ. Galerkin’s method is used to Eq. (2) to obtain the matrix equations of the FEM. Besides this numerical method to solve Eq. (1), specific analytical methods are available, and simple solutions are known for the particular case of light propagating along the helical axis [13, 16–18].
3. Results and discussions
To evaluate the performance of the OD, the following physical parameters are used. For the typical nematic liquid crystals in the visible region, refractive indices are within 1.4 < n ⊥ < 1.6 and 1.4 < n ∥ < 1.9 , and the birefringence is in the range of 0.1 ~ 0.2 . The refractive indices of the birefringent planes in the two CLC films are both taken to be n ∥ = 1.7 and n ⊥ = 1.5 . The thickness of CLC1 and CLC2 layers is 12P1 and 12P2, respectively. n̄P 1cosθ = 460 nm and n̄P 2cosθ = 550nm are the central reflection wavelengths of the two CLC films. In order to reduce the Fabry Perot effect , the extraordinary and ordinary refractive indices of the wave-plates are all chosen to be ne = n ∥ and no =n ⊥. The exact retardation for the wave-plates occurs at λ=550 nm, and the thickness of the plates is assumed to be d1 =d3 =λ/(4Δn) and d2 =λ/(2Δn), which can simplify the calculation. To prevent the Fresnel reflection from bringing the fluctuation in transmittance and reflectance curves , it is assumed that the OD sample is nearly index matched to its surroundings with ns =[( +)/2]1/2 . Non-absorbing chiral nematic media and wave-plates are assumed, and the surface roughness at the interfaces is neglected.
For normal incidence in direction I, the device response to s-polarized light is shown in Fig. 2(a) and Fig. 2(b). From the total reflectance and transmittance curves, the stop band (PBG1) can be observed between 430 and 490 nm. In order to examine the polarization of the reflected/transmitted light, the total reflectance/transmittance curve is decomposed into p- and s-polarization components. This makes it possible to see the way in which the reflectance/transmittance is related to the polarization state of the incident light. In direction II, the response to s-polarized light is shown in Fig. 2(c) and Fig. 2(d), and the stop band (PBG2) is located between 513 and 586 nm. Figure 2 reveals that s-polarized light incident in directions I and II suffers different PBGs through the OD, corresponding to Fig. 1(c).
The unequal efficiencies for different polarization output states with respect to the input states result in coupling loss, the polarization dependent loss. For the light within PBG1, the coupling from the s-polarization input to the p-polarization reflection output is insignificant as shown in Fig. 2(a). The coupling loss is less than 8% in magnitude. The reflected light affected by PBG1 nearly keeps the same polarization state as the incoming. The similar results can also be observed in Fig. 2(c), but the reflection band is PBG2. The coupling loss from the s-polarization input to the p-polarization reflection output is less than 3%, maximized within PBG2. In the range of the wavelength of visible light, the dominant polarization component of the transmitted light, except the wavelengths within the PBGs or under 425nm, is the p-polarization as shown in Fig. 2(b) and Fig. 2(d). For different wavelengths, the optical waves will experience distinct phase retardations after passing through the wave-plates. The plates can function exactly only for a particular wavelength. Consequently, it will be incomplete for other wavelengths to convert any given polarization state into any other by the retarders. The s-polarization output in transmittance will be obvious to the wavelengths apart from the specific wavelength (550 nm) as illustrated in Fig. 2(b) and Fig. 2(d). To lower the coupling loss from the s-polarization input to the s-polarization transmission output, broadband wave-plates should be used to increase the bandwidth that can cover the visible region.
4. Color-separating system
Most professional projectors use the three-light valve approach that requires splitting the white light into three colors . A conceptual sketch of the color-separating design shown in Fig. 3 is intended for the potential application of a full-color projector. The system consists of a mirror, a half-wave plate, and the OD. The OD can serve as a color splitter. Figure 3 shows the color-separating mechanism. For an s-polarized light incident at angle θ in the forward direction, the B component of light spectrum is separated first. The reflected light directed on the other side of the surface normal still possesses the s-polarization state. The transmitted light becomes the p-polarization state after the OD. Next, its polarization state turns into s-polarized after the half-wave plate. Then, the transmitted light is fed back by the mirror in the reverse direction, and thus the G and R components of light spectrum can be segregated.
Figure 4 shows the reflection and transmission spectra for s-polarized light beams incident at angle θ=10° in directions I and II. The other values of the parameters used in the case are the same as the former ones. Comparing with Fig. 2, the regions of the two band-gaps are approximately the same as the results for normal incidence. For wavelengths beyond 590nm, the noticeable difference between normal and oblique incidence is the increase of the coupling loss in the transmission spectra as shown in Fig. 4(b) and Fig. 4(d). The residual phase retardation resulting from a multilayer structure with LCs is obvious when light is directed at an oblique angle . The angular dependence of the coupling loss is due to the fact that both the phase retardations and optical path in most LCs are functions of incident angles. Therefore, the coupling between p- and s-polarized states will increase with the incident angle. To neutralize the angular dependence, a phase compensation film can be used to cancel the residual phase retardation [6, 22], and thus can reduce the loss. Besides, the fringes, caused by the interference from the chiral layers, in transmittance curves reveal dips of the intensity at some wavelengths. Because the human eye contains only three kinds of color receptors, many different color spectra will excite the receptors at exactly the same level. Hence, the eye cannot tell them apart, and the dips in transmittance will not affect the feeling of the colors obviously. However, to reduce the fringes, creating an imperfect planar texture in a CLC can be considered .
It has been analyzed in this work that the new combination of wave-plates and CLC films with different pitches can lead to unidirectional light propagation. The proposed OD is sensitive to the polarization of the incident light field, and its construction is not complicated. Based on the FEM, the non-reciprocal transmission of s-polarized light is simulated to reveal the diode performance at the PBG regions. The calculated reflection/transmission spectra are confirmed to show the diode effect with a particular directionality. It is also demonstrated that the optical component could be useful in the implementation of a color-separating system in optical projectors. The separating mechanism without conventional color filters is a first proposal of the integration of common optical elements and the OD. An OD-based device maybe offers simplicity and compactness for laboratory and industrial use in optics.
The authors would like to thank the National Science Council (NSC) of Taiwan for financial support under contract No. NSC 95-2221-E-006-040-MY2.
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