Control of light's polarization state is crucial in many applications that require precise manipulation of electromagnetic waves. Polarizers are used to reduce glare, to send and receive radar and satellite communications, and to identify left- or right-handedness in many biological molecules, among others. Almost every application that requires polarized light uses a waveplate to control the polarization state, and so waveplates are valuable in areas as varied as laser optics, astronomy, and biomedicine. A typical waveplate uses a birefringent crystal, such as quartz or calcite, to alter the phase difference between the two linearly polarized components of a light wave in air.1
Light moving through a birefringent crystal is a combination of two different components called eigenstates. Each eigenstate has a distinct phase speed, attenuation rate, and other electromagnetic characteristics, all of which depend on the direction of propagation. Because the atomic binding forces within a birefringent crystal vary in different crystallographic directions, one eigenstate will have a lower phase speed than the other. As light emerges from the exit side of the crystal into air, the phase relationship between its two linearly polarized components is changed relative to the phase relationship on the entry side. Specifically, one of the two components suffers a phase retardation, or delay, with respect to the other. In most crystals, the phase retardation increases with wavelength (color), and the change in the intrinsic optical properties with the wavelength is small enough to be ignored. However, for many applications including CD and DVD storage and displays, the waveplate needs to control the polarization state without dependence on wavelength in the intensity of emerging light. Researchers have struggled to identify a material that compensates for these differences in phase retardation over the entire visible regime.
In 2009, Roberts and co-workers2 demonstrated the presence of achromatic waveplates in the eyes of mantis shrimps. Consisting of many eyelets each, the compound eyes in this species of stomatopod crustaceans can distinguish between left- and right-circularly polarized light (similar to the technology used to produce some 3D movies). The animal's extraordinary vision is possible because an array of tiny hairlike folds, or microvilli, within a light-sensitive cell of each eyelet functions as a quarter waveplate. (A quarter waveplate creates a quarter-wavelength phase shift.) The membrane of every microvillus in the light-sensitive cell is made of a birefringent material. In addition, the array of parallel microvilli displays a kind of birefringence called form birefringence, because the diameter of each microvillus is much smaller than the light's wavelength. Material birefringence and form birefringence combine in the light-sensitive cell to yield achromatic phase retardation—a wavelength-independent phase shift—over the visible regime.
With the biologically inspired understanding that two birefringences need to be combined for achromatic phase retardation, we conceived a periodically multilayered structure (PMS) as an achromatic waveplate. The waveplate is based on two different thin films. Each thin film is an array of parallel nanorods that exhibits form birefringence. The PMS thus exploits two different form birefringences. Two identical thin films labeled A sandwich a thin film labeled B to form the unit cell of the PMS: see Figure 1. These thin films can be deposited using the well-known oblique angle deposition (OAD) method3 and its variants, as we have demonstrated experimentally.4
Figure 1. Schematic showing two unit cells of a periodically multilayered structure.
Suppose that both thin films, A and B, are deposited using a variant of the OAD method called the serial bideposition method,5 which yields an array of upright nanorods. For light that is incident normally (i.e., with its wave vector parallel to the z axis in Figure 1) on the PMS, the electric field of one linearly polarized component may be oriented along the x axis, whereas the electric field of the other linearly polarized component will then be oriented along the y axis. As each thin film is birefringent, the effective refractive index (a measure of the phase speed) that it presents to one linearly polarized component is different from that presented to the other linearly polarized component. Furthermore, the two effective refractive indexes of A are different from their counterparts for B, because the two types of thin films were deposited under different conditions and/or could be made of different materials.
For each linearly polarized component of normally incident light, the symmetric unit cell ABA is equivalent to a homogeneous dielectric layer with a specific refractive index and phase thickness (a measure of phase change). Both of these equivalent parameters depend on the orientation of each component's electric field, the wavelength, the effective refractive index and the thickness of A, and the effective refractive index and the thickness of B. Therefore, the phase retardation of the unit cell, which is the difference between the two phase thicknesses, is also a function of all these parameters. A uniform phase retardation over the visible regime can be derived by carefully selecting the thicknesses of A and B. We make that selection using a merit function that is defined as the rate of change of the unit cell's phase retardation with respect to wavelength.
Figure 2. Calculated phase-retardation spectrum of a 25-cell PMS.
As an example, we selected the effective refractive indexes of A to be 1.369 and 1.494 for the two incident linear polarization states, and those of B as 1.610 and 1.547. We selected these values from a database of experimental results on nanorod arrays of tantalum oxide. We also targeted the merit function—averaged for 301 wavelengths within the 400–700nm visible regime—to be less than 0.07°/nm, with no single value exceeding 0.15°/nm. Then, we varied each film's thickness over ranges that are easily possible for fabrication, and found that the thickness of A can range from 46 to 61nm and the thickness of B can range from 138 to 153nm. For a PMS containing 25 unit cells, and with A's thickness equal to 53 nm and B's thickness equal to 149nm, we calculated that the phase retardation is 86.459°± 4.814° for light with wavelength between 425 and 675nm: see Figure 2. The phase retardation's deviation from its mean value is less than 6% for almost the entire visible regime.
In summary, current waveplates made of crystals and thin films tend to exhibit achromatic phase retardation over a narrow range of wavelengths. We have devised a periodically multilayered structure with a symmetric unit cell of two alternating types of nanorod arrays that can function as an achromatic waveplate for the entire visible regime. The interplay between two different form birefringences makes this achromatic performance possible. Tight control over deviations in phase retardation requires precise control of the thickness and birefringence of each thin film in the unit cell. Our methodology can be applied to design a waveplate for any wavelength range, and is feasible for various achromatic waveplates in display technologies, communications systems, and optical pick-up systems. Our next step will be to reduce the total number of thin films in the structure to make fabrication easier.
Yi-Jun Jen and Meng-Jie Lin thank the National Science Council of the Republic of China, Taiwan, for financially supporting this research under Contracts NSC 99-2221-E-027-043-MY3 and NSC 99-2120-M-002-012. Akhlesh Lakhtakia thanks the Binder Endowment at Penn State for partial support of his research activities.
Yi-Jun Jen, Meng-Jie Lin
National Taipei University of Technology
Pennsylvania State University
University Park, PA
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2. N. W. Roberts, T.-H. Chiou, N. J. Marshall, T. W. Cronin, A biological quarter-wave retarder with excellent achromaticity in the visible wavelength region, Nat. Photon. 3, pp. 641-644, 2009.
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4. Y.-J. Jen, A. Lakhtakia, C.-W. Yu, C.-F. Lin, M.-J. Lin, S.-H. Wang, J.-R. Lai, Biologically inspired achromatic waveplates for visible light, Nat. Commun. 2, Article number 363, 2011.
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