Principle of photonic crystal fibers

Principle of photonic crystal fibers Jan Sporik 1, Miloslav Filka 1, Vladimír Tejkal 1, Pavel Reichert 1 1 Fakulta elektrotechniky a komunikačních technologií VUT v Brně Email: {xspori1, filka, xtejka, xreich1}@feec.vutbr.cz Abstract in this paper, principle of PCF (Photonic Crystal Fiber) and its application is described. Simulations of one dimension photonic crystal are made. Possibilities of conventional fibers are shown and the comparison with PCF is done. Various types of commercially available PCF are listed. The Photonic Crystal Gaps are calculated for various one dimension periodic structures. Plane wave expansion method is used. The solutions are found by solving the eigenvalue problem for the intensity of Electric field. The influence of Photonic Crystal Gap on the layer width and the parameter gap-midgap ratio is discussed. The scalar solution for one dimension photonic crystal is found. stresses to the fiber, microscopic fluctuations in density, and imperfect splicing techniques. PCF is now finding applications in fiber-optic communications, nonlinear devices, fiber lasers, amplifiers, high-power transmission, highly sensitive gas sensors, and other areas. The principle of light guiding is different for various types of PCFs. With of analogy of the theory of quantum mechanics, macroscopic media with a different dielectric constant and a periodic dielectric function are used in the photonic crystal. The periodicity of different crystals is shown in Fig.1. 1 Introduction Structured optical fibers are also called microstructured optical fibers and sometimes photonic crystal fiber in case the arrays of holes are periodic. PCF (Photonic Crystal Fiber) is a type of optical fiber using the properties of photonic crystals. Its advantages against a conventional optical fiber are possibility to control optical properties and confinement characteristics of material [1]. The conventional optical fibers simple guide light and they have started revolution in telecommunication with CH. K. Kao s and G. Hockham s proposition in 1966 [2]. The principle of total internal reflection has been used for guiding of light in the fiber. Nowadays, we almost have reached the maximum of its the best properties, which are limited by the optical properties of their solid cores (attenuation.2 db/km, zero dispersion shifted to the minimum loss window at 155 nm on used wavelength with fiber specified in ITU-T G.653 [2], etc). The maximum transmission distance reached by modern fiber is not limited by the absorption of material but also by dispersions. They cause spreading of optical pulses during their travel along the fiber. Dispersion is caused by a variety of factors for optical fibers [2]. Chromatic dispersion primarily limits performance of single-mode fiber. Polarization mode dispersion occurs because imperfections or distortions in a fiber can alter the propagation velocities for the two polarizations of spreading mode. Dispersion limits the bandwidth of the fiber because the spreading optical pulse limits the rate that pulses can follow one another on the fiber and still be distinguishable at the receiver. Fiber attenuation is caused by a combination of material absorption, connection losses, and scattering (Rayleigh and Mie). Modern fiber has the attenuation about.2 db/km. The original Kao s optical fibers had attenuation about 1 db/km. Other forms of attenuation are caused by physical Fig. 1. 1-D (a), 2-D (b), and 3-D (c) photonic crystals for light spreading in z-direction Low-loss dielectric periodic material with sufficiently different dielectric constants in crystals can control flow of the light. Crystals with photonic band gaps can be design. We can design such structure which can prevent the light from propagation in certain directions with specified range of wavelengths. Another principle used in PCFs comes from multilayer dielectric mirror. It consists of layers of material with different dielectric constant. The light with wavelength according to the band gap of this material can be completely reflected. If the periodicity of dielectrics is only in one direction, we can call this material as one dimensional photonic crystal. This prin- 12

ciple is used for Brag fiber. If the periodicity of dielectrics is in two directions, we call crystals as a two dimensional photonic crystals. Almost all PCF have periodicity in two directions (e.g. Index Guiding Photonic Crystal Fiber, Hollow Core Photonic Crystal Fiber). Also, planar waveguides have primarily periodicity in two dimensions (directions). Three dimensions photonics crystals exist too, if the periodicity is in three directions. Single-Mode Double Clad Fibers With Large Mode Area They are also called Air-clad fibers. This type of fiber is similar to LMA-PCF, but it uses double clad and it has active doped core. 2 Types of Photonic Crystal Fibers A few types of PCFs and a value of refractive index inside are shown in Fig.2. 1-D Photonic Crystal Fiber Bragg Fiber The core may have a much lower refractive index than the cladding. It uses Bragg PBG (Photonic Band Gap) mechanism for reach the omnidirectional-mirror, more in [1, 3, 4, and 5]. Current air-core Bragg fibers are based on a combination of polymer and soft glass [6]. 2-D Photonic Crystal Fiber Index Guiding Photonic Crystal Fiber (IG-PCF) The holey cladding of the fiber forms effective index of material. The effective index is smaller than refractive index of solid core (silica) due to the holey are filed by material with smaller refractive index (air). Like in conventional fibers, the principle of the total reflection is used [1]. Hollow Core Photonic Crystal Fibers (HC-PCF) It is also called holey fiber. It enables the guidance of the light in the hollow core with lower attenuation than in the solid silica core. The core can be filled by air or gas. Commercially available IG PCFs [7]: Endlessly Single Mode PCF The principle is based on the fact than fundamental mode may not escape the core region of solid core PCF due to it does not fit into the gaps between the air holes. Higher order modes may escape the core. Large Mode Area PCF (LMA-PCF) The very large mode area enables high power levels without material damage and a nonlinear effect. Highly Nonlinear Photonic Crystal Fiber (HN-PCF) Small core size (less than 1 µm) and the large contrast of index core-cladding enable to create fibers with extremely small effective areas and high nonlinear coefficients. Polarization Maintaining PCF (PM-PCF) This type of fiber allows the polarization maintaining. Fig. 2. Refractive index for HC-PCF (up), IG-PCF (middle), and Brag fiber (down) 3 Advantages of using PCF There are many advantages against a conventional optical fiber. The biggest ones are possibility to control optical properties and confinement characteristics of material. Allow for guidance through hollow fibers (air holes). Smaller attenuation than with fiber with solid core. PCFs with larger cores may carry more power than conventional fibers. Larger contrast available for effective-index guidance. 13

Attenuation effects not worse than for conventional fibers. Control over dispersion: size of air holes may be tuned to shift point of zero dispersion into visible range of the light. 4 Disadvantages of using PCF Short manufacture length and high price are the main disadvantages of using PCF as transmission media for telecommunications as conventional fibers. Next problem is with coupling and possibility to connect them with other waveguides and devices. 5 Simulation of 1-D photonic crystal The material is homogeneous in the xy-plane, and it is periodic in the z-direction. The Plane Wave Expansion method (PWE) is used to solve the Maxwell s equations by formulating an eigenvalue problem out of the equation, and for study of the periodic structure properties. Transfer matrix method can be used for properties like the reflectance R and the transmittance T. The periodic structure is shown in Fig. 3. Dielectric constant ε(z) is function of variable z and can be described as a plane wave: Solution for 1D wave equation for periodic structures can be written (Bloch s theorem): (2), (3) where U k (z) is periodic function with the same period as structure, k is wave number. From the wave equation (4) are obtained Fouriercoefficients for E-field: where Equation (5) can be written in matrix form:, (4) (5) is vector of Fourier-coefficients obtained as solution (5), is diagonal matrix, is Toeplitz matrix, and is eigenvalue. This eigenvalue problem can be solved using MATLAB. Photonic Band Gaps (PBGs) for He-Ne laser grating are shown in Fig. 4. There is normalized value of frequency on the y-axis and normalized value of wave number k on the x-axis. Values of dielectric coefficients and their diameters are written bellow the figure. Fig. 3. 1-D periodic structure in z-direction with a period Λ and a width of layer a. (1) Coefficients for structure from Fig.2. are equal: Fig. 4. Photonic band diagrams for HeNe, ε 1 =2.32 2, ε 2 =1.38 2, d 1 =633 nm/4/2.32, d 2 =633 nm/4/1.38, a=d 1, Λ=d 1 +d 2 14

VOL. 2, NO. 2, JUNE 211 The dependence of photonic band gap on dielectric constants is shown in Fig. 5. a=1, Λ=2, each layer equal.5λ. In this case, it doesn t matter if dielectric coefficients are switched ε 1 to ε 2 and ε 2 to ε 1. The values of wave vector k are normalized by G (1) and they are on the x-axis. There is normalized frequency on the y-axis. There is shown the gap closing due to the decreasing ratio (contrast) between dielectric constants. The grey rectangle represents the size of the photonic band gap..3.3 The size of photonic band gap can be described by its frequency width Δω. Because of the results of Maxwell Equations are scalable, the using of a gap-midgap ratio is more useful for description of the gap than frequency width Δω. The dependence of the gap size is inversely dependent to the scale of the crystal. The gap-midgap ratio is independent of scale of the crystal and is defined in [1] as the gap frequency width/middle frequency of the gap (Δω/ω m ). The dependence of the gap-midgap ratio is shown in Fig.7. The gap-midgap ratio is genrally expressed as a percentage. The 2%-gap coresponds to the mid-midgap ratio.2. -.5.5 -.5.5 a- b-.3.3 Δω/ωm Δω/ωm -.5.5 -.5 c- d- Fig. 5. Photonic band diagrams for one-axis propagation for different multilayer films. Λ=2, a=1 for all films. a: ε 1 =1 represents air, ε 2 = 12 represents high dielectric material (GaAlAs), b: ε 1 =6, ε 2 = 12, c: ε 1 =13 represents GaAs, ε 2 = 12 represents GaAlAs, d: ε 1 = ε 2 = 12 represents homogenous material in all three directions. [1] The dependence of the gap size on the ratio between the width of layer (a) and the period Λ is plotted in Fig. 6..5 Fig. 7. Gap-midgap ratio of band gap for changing ratio a/λ. Left y-axis is given for structure with ε 1 =1, ε 2 =12 and right y- axis is for structure consists of ε 1 =13, ε 2 =12. Ration of width a of layer with dielectric constant ε 1 and period Λ is given on x-axis. 6 Conclusions This work was supported in part by MSM 2163513 research program. S1,12 S13,12 7 Conclusions Fig. 6. Size of band gap S for gram for changing ratio a/ Λ. Left y-axis (S 1,12 ) is given for structure with ε 1 =1, ε 2 =12 and right y-axis (S 13,12 ) is for structure consist of ε 1 =13, ε 2 =12. Ration of thickness a of layer with dielectric constant ε 1 and period Λ is given on x-axis. Photonic crystal fibers have been studied for more than 2 years and there are still found the possibilities to improve its properties. The basic comparison between conventional fiber and PCF has been done. On the assumption that minimization of PCF fabrication price they can offer much better possibilities than conventional fibers. Nowadays, they are not used in communication instead of convention fiber as media. They are used in fabrication of gas lasers, sensing, and high power pulse transmission. The principle of PBG for one dimensional photonic crystal was shown in simulations in Matlab. The scalar solution has been found for one dimension photonic crystal. The PBG can show which range of frequencies is refracted, but it does not have to mean that this 15

structure can be used as mirrors for waveguiding of the light. The full vectorial solution of the problem must be done to make decision about it. References [1] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light. Princeton University Press, second ed., February 28. [2] M. Filka, Optoelectronics for telecommunications and information. Texas: Inc. Publishers, 29. [3] P.Yeh, A. Yariv, and E. Marom, "Theory of Bragg fiber," J. Opt. Soc. Am. 68, 1196-121 (1978). [4] Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, Guiding optical light in air using an all-dielectric structure, J.Lightwave Technol. 17, 239-241 (1999). [5] G. Vienne, Y. Xu, C. Jakobsen, H. J. Deyerl, T. P. Hansen, B. H. Larsen, J. B. Jensen, T. Sørensen, M. Terrel, Y. Huang, R. K. Lee, N. A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, A. Yariv, First demonstration of air-silica Bragg fiber, Proc. OFC 24 (24). [6] S. D. Hart, G. R. Maskaly, B. Temelkuran, P. H. Prideaux, J. D. Joannopoulos, and Y. Fink, External reflection from omnidirectional dielectric mirror fibers, Science 296, 51-513 (22). [7] http://www.nktphotonics.com 16