Visions form due to the processing of different wavelengths reflected by electromagnetic waves. Objects created by man are mostly colored by pigmentation and the color is consistent at different angles. Objects of nature, however, can be perceived differently. A chameleon, for example, can change its colors to protect itself from predators, but how? An animal with camouflaging abilities is due to the complex material structure, unlike the pigmentation of solid colors. These complex, multilayered Nano-reflectors along with pigmentation are the cause of color change [1]. Such Nano-structures are called photonic crystals [2,3]. This paper explores the fundamentals of photonic crystals, mainly 1-D and 2-D, and future implementation possibilities of these materials.
Photonic crystals are artificial periodic structures where the control of electromagnetic wave dispersion can be achieved. These photonic crystals are made of dielectric materials which perform as electric insulators but which electromagnetic fields can be propagated [2,4]. Lord Rayleigh first discovered 1-D photonic crystals in 1887. He found out that 1-D photonic crystals have narrow band gaps, which prohibits light propagation through the planes [3]. Charles Darwin conducted continued expansion of his research fifty years later. 100 years later, in 1987, Yablonovitch and Sajeev John discovered 2-D and 3-D photonic crystals. In fact, Yablonovitch formed the first 3-D photonic crystal, which he coined as “Yablonovite” structure, four years later in 1991 [2].
First an understanding of wave propagation is necessary. When two reflected waves are in phase, a constructive interference occurs and when two reflected waves are out of phase, a destructive interference occurs. When obtaining a constructive interference, a color is shown, while with a destructive one, no color occurs associated with that wavelength. These interferences are caused by refractive index mediums and are material dependent. According to Bloch’s theorem with respect to the Schrodinger equation, the potential of the wavefunction is periodic and only exist in discrete modes [5]. Waves in photonic crystals are governed by Bloch’s Theorem and thus behave periodically without scattering. This indicates that electrons in periodic potentials behave similarly to light in periodic dielectrics [6]. The time-dependent Schrödinger solutions can be solved to determine the existence of electronic band gaps. Maxwell’s equation, similarly (when applied to a periodic environment), can show that photonic band gaps indeed do exist. Solving Maxwell’s equation with periodic conditions as an eigenvalue problem yields the existence of photonic band gaps, analogously to electronic band gaps by solving Schrödinger’s equation.
Ψ(r,t)=
Ψ(r,t)=Ψ(r) e^(-iωt)
H ̂Ψ(r)=EΨ(r)
(H=(〖-ℏ〗^2 ∇^2)/2m) ̂+V(r)
H(r,t)=H(r)e^(-iωt)
ΘH(r)=ω^2/c^2 H(r)
Θ=∇x 1/(E(r)) ∇x
The equations correspond to the field, eigenvalue problem and the operator. In quantum mechanics, the Hamiltonian operator is used but in electromagnetics, the Hermitian operator is used. E resembles the dielectric constant, Ψ is the wavefunction, m is the mass, w is the frequency and c is the speed of light.
Photonic band gaps can trap certain frequencies of wavelengths and reflect other “forbidden” ones. The forbidden region is where such range of frequencies with a specific wavelength is prohibited to enter the photonic crystal. Creating such material with a band gap can lead to engineering materials that can confine and manage electromagnetic radiation [6]. For example the chameleon discussed earlier, one can see that light isn’t forbidden in all directions such that it can propagate in some. When the chameleon is relaxed, short wavelengths are reflected due to the crystals getting tightly packed, causing the skin to appear green. On the other hand, when the chameleon is excited, longer wavelengths are reflected due to the crystals widening, causing the skin to appear orange [1]. These are incomplete photonic band gaps and light isn’t confined in all directions. According to Yablonovitch, such discoveries are extremely useful in the development of pigmentation. He states that engineering titanium oxide, the white pigment used to make paper white, can lead to band gap structured titanium oxide, leading to more whiteness using less mass of the pigment [2].
To create a photonic gap, several criteria’s need to be met. Let’s consider a 1-D photonic crystal. Multiple papers have concluded of using two dielectric materials with different refractive indices and different dielectric constants, although other uses of different materials do exist [7-10]. As the dielectric constant contrast increases, the band gap gets bigger. The dielectric contrast ratio is at least 4:1 in 3-D lattices so frequencies would overlap [11]. These dielectric materials need to also have a high refractive index, generally higher than 3 [2,7-9,12], although recent development in photonic crystals has shown that refractive indices of over 2 are ideal candidates [13,14]. This is why commonly; the same materials used for semi-conductors are used for photonic crystals (Si, GaAs, etc.). Alternative solution to increasing the refractive index is introducing an annealing step after imprint [15].
When dealing 2-D photonic crystal and wave propagation, both TM (transverse magnetic) and TE (transverse electric) polarizations need to be considered [11]. For each polarization, a photonic band gap is exhibited. Such band gaps, however, are considered incomplete and only exist for a certain polarization mode. A complete 2-D band gaps occur when both these polarizations are present and overlap. The different lattice orientations and corresponding photonic band gaps are shown in Fig.(1) [16].
In Fig. 1, the dielectric lattice structure is also shown. The lattice structure in Fig.1 (a) is created by having an array of high dielectric cylinders in a square lattice. The lattice structure in Fig.1 (b) is created by high dielectric veins connected with square holes in the middle. Having dielectric cylinders with a square lattice causes a band gap in the TM mode, but not the TE mode as shown while having veins with square holes leads to a band Gap in the TE mode but not the TM mode [16,17]. For an optimal design that exhibits a complete band gap, where both TE and TM polarizations overlap, Fig. 2 shows a combination of both lattice structures to achieve a complete band gap. The largest band gap was achieved for a two-dimensional photonic crystal for a square air and GaAs lattice by optimally connecting dielectric rods with veins [18].
The holes are drilled relevant to the order of wavelength into the dielectric lattice, repeatedly and identically. The spacing array needs to be close to the wavelength of light for the electromagnetic waves to be controlled [2]. Other methods of attaining holes have been used such as femto-laser techniques [19].
Two-dimensional photonic crystal methods can be applied to three-dimensional cases to form three-dimensional photonic band gaps.
A technique that has been widely used in photonic crystals is introducing controlled defects, analogous to the introduction of doping in semiconductors. Defects are what make photonic crystals highly applicable in many different fields, such as biomedical, communications, optics, and many others. The ability to confine, control and direct light opens up endless possibilities of applications. A main use of photonic crystals is as waveguides. Planar defects are introduced and waves are guided through the gap where the defect is created. Other current uses are Bragg deflectors, micro cavities, drop in filters, lasers, omnidirectional mirrors and even artificial opals.
Main interests of photonic crystals in developing technologies are photonic crystal fibers. PCF (photonic crystal fibers) are the modern optical fibers. Optical fibers, or sometimes referred to as Bragg fibers, are one-dimensional photonic crystals while PCF’s are two-dimensional. While Bragg fibers utilize total internal reflection for wave propagation and confinement, PCF’s utilize photonic crystal guiding and confinement. By creating a circular-planar defect in the photonic crystal, a waveguide is formed. Bragg reflectors, as shown in Fig. 3 are also known as hollow core fibers. These photonic crystal fibers are currently being researched to use for higher power fiber lasers, high power light transmission, and biochemical sensors [20]. Benefits for such applications include low transmission loss and broad transmission width [21]. Experimentation and testing is being conducted with filling the hollow core with different gases for
In conclusion, photonic crystals are artificial periodic structures that can be used and manipulated to control, confine, and modulate light. Fabrication challenges and the ability to widen the frequency limitation are being researched to further advance the usage of photonic crystals in everyday applications. Applications of photonic crystals include micro cavities, photonic crystal fibers, and using photonic crystals as wave-guides. These applications have been possible by introducing controlled defects into the crystal lattice. Design optimization of photonic crystals is a vast and growing field, with merits to Lord Rayleigh for the discovery of one-dimensional photonic crystals and Eli Yablonovitch for the discovery of two and three-dimensional crystals.
References
1J. Teyssier, S.V. Saenko, D. Van Der Marel, and M.C. Milinkovitch, Nature Communications; London 6, 6368 (2015).
2 E. Yablonovitch, Journal of Modern Optics 41, 173 (1994).
3 Lord Rayleigh, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 24, 145 (1887).
4 S. Soussi, Advances in Applied Mathematics 36, 288 (2006).
5 G. Chen, Nanoscale energy transport and conversion: a parallel treatment of electrons, molecules, phonons, and photons (Oxford University Press, Oxford, 2005).
6 K. Inoue, in Photonic Crystals (Springer, Berlin, Heidelberg, 2004), pp. 1–8.
7 J. Rarity, C. Weissbruch), Microactivities and Photonic Bandgaps: Physics and Applications
Kluwer, Dordrecht, The Netherlands (1996).
8 C. López, L. V. Vµsquez, F. Meseguer, R. Mayoral, M. Oscan Äa, H. Miguez, Superlattices Microstruct. 1997, 22, 399.
9 M. Jacoby, Chem. Eng. News 1998, 23 November, 38.
10Y. Xia, B. Gates, Y. Yin, Y. Lu, Adv. Matter. 12, 693. (2000).
11 K. Ohtaka, in Photonic Crystals (Springer, Berlin, Heidelberg, 2004), pp. 39–63.
12 D.G. Angelakis, P.L. Knight, and E. Paspalakis, Contemporary Physics 45, 303 (2004).
13 T. Kartaloglu, Z. G. Figen, and O. Aytur, J. Opt. Soc. Am. B 20, 343 (2003).
14 B. Andreas, K. Peithmann, K. Buse, and K. Maier, Appl. Phys. Lett. 84, 3813 (2004).
15 G. Calafiore, Q. Fillot, S. Dhuey, S. Sassolini, F. Salvadori, C.A. Mejia, K. Munechika, C. Peroz, S. Cabrini, and C. Piña-Hernandez, Nanotechnology 27, 115303 (2016).
16 R.D. Meade, A.M. Rappe, K.D. Brommer, and J.D. Joannopoulos, J. Opt. Soc. Am. B, JOSAB 10, 328 (1993).
17 W Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Rev. Lett. 68, 2023 (1992).
18 M. Qiu and S. He, J. Opt. Soc. Am. B, JOSAB 17, 1027 (2000).
19 J. Ju, H.F. Xuan, W. Jin, S. Liu, and H.L. Ho, Opt. Lett., OL 35, 3886 (2010).
20 A.M.R. Pinto and M. Lopez-Amo, Journal of Sensors (2012).
21 Y.Y. Wang, N.V. Wheeler, F. Couny, P.J. Roberts, and F. Benabid, Opt. Lett., OL 36, 669 (2011).