In this chapter, fundamentals of photonic crystals are mentioned and information on background is given. Macroscopic Maxwell’s equations are discussed to under- stand the phenomenon band-gap crystals and model wave propagation in photonic crystals by Bloch waves and Brillouin zone. Band-gap diagram of photonic crystals are also presented for 1D and 2D photonic crystals. The chapter is concluded with brief information about photonic crystal based devices.
In this thesis, band-gap diagram and wave propagation are studied using finite ele- ment method (FEM). In the next chapter, FEM and the other most common numer- ical methods used in photonic crystal study will be mentioned.
References
[1] M. Johri, Y. A. Ahmed, and T. Bezboruah, “Photonic band gap materials: Technology, applications and challenges,” Current Science, vol. 92, no. 10, pp. 1361–1365, 2007.
[2] P. H. Siegel, T. Loffler, D. Mittleman, K. Mizuno, and X. C. Zhang, “Guest EditorialTerahertz technology: Bridging the Microwave-to-Photonics gap,” IEEE Transactions on Microwave Theory and Techniques, vol. 58, pp. 1901– 1902, July 2010.
[3] M. Tonouchi, “Cutting-edge terahertz technology,” Nat Photon, vol. 1, pp. 97– 105, Feb. 2007.
[4] J. D. Joannopoulos, Photonic crystals: molding the flow of light. Princeton Univ Pr, 2008.
[5] A. R. Parker and H. E. Townley, “Biomimetics of photonic nanostructures,” Nature Nanotechnology, vol. 2, no. 6, pp. 347 – 353, 2007.
[6] J. W. Galusha, L. R. Richey, J. S. Gardner, J. N. Cha, and M. H. Bartl, “Discov- ery of a diamond-based photonic crystal structure in beetle scales,” Physical Review E, vol. 77, p. 050904, May 2008.
[7] J. V. Sanders, “Colour of precious opal,” Nature, vol. 204, pp. 1151–1153, Dec. 1964.
[8] P. Vukusic and J. R. Sambles, “Photonic structures in biology,” Nature, vol. 424, no. 6950, pp. 852 – 855, 2003.
[9] J. P. Vigneron, J. F. Colomer, M. Rassart, A. L. Ingram, and V. Lousse, “Struc- tural origin of the colored reflections from the black-billed magpie feathers,” Physical Review E, vol. 73, no. 2, p. 021914, 2006.
[10] S. John, “Strong localization of photons in certain disordered dielectric super- lattices,” Physical Review Letters, vol. 58, no. 23, pp. 2486 – 2489, 1987. [11] E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and
electronics,” Physical review letters, vol. 58, no. 20, pp. 2059 – 2062, 1987. [12] T. F. Krauss, R. M. Rue, and S. Brand, “Two-dimensional photonic-bandgap
structures operating at near-infrared wavelengths,” Nature, vol. 383, no. 6602, pp. 699 – 702, 1996.
[13] J. O’Brien, W. Kuang, J. R. Cao, M. H. Shih, P. T. Lee, M. Bagheri, A. Mock, and W. K. Marshall, “Photonic crystal microcavity lasers,” Journal of Physics D: Applied Physics, vol. 40, p. 2671, 2007.
[14] P. Russell, “Photonic crystal fibers,” Science, vol. 299, no. 5605, p. 358, 2003. [15] K. Cho, T. Hirai, and T. Ikawa, “Propagating mode in the photonic gap of 1D resonant bragg reflector,” Journal of Luminescence, vol. 100, pp. 283–289, Dec. 2002.
[16] E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: The face-centered-cubic case employing nonspherical atoms,” Physical Re- view Letters, vol. 67, p. 2295, Oct. 1991.
[17] E. Ozbay, G. Tuttle, M. Sigalas, C. M. Soukoulis, and K. M. Ho, “Defect structures in a layer-by-layer photonic band-gap crystal,” Physical Review B, vol. 51, p. 13961, May 1995.
[18] S. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three- dimensional photonic crystal operating at infrared wavelengths,” Nature, vol. 394, no. 6690, pp. 251 – 253, 1998.
[19] K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Physical Review E, vol. 58, no. 3, p. 3896, 1998.
[20] S. G. Johnson and J. D. Joannopoulos, “Introduction to photonic crystals: Blochs theorem, band diagrams, and gaps (but no defects),” Photonic Crys- tal Tutorial, pp. 1 – 16, 2003.
[21] L. Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Phil. Mag, vol. 24, no. 147, p. 145159, 1887.
[22] R. M. Martin, Electronic structure: basic theory and practical methods. Cam- bridge Univ Pr, 2004.
[23] C. Kittel, Introduction to Solid State Physics. New York: John Wiley & Sons, Inc., 6th ed., 1986.
[24] F. Genereux, S. W. Leonard, H. M. van Driel, A. Birner, and U. Gsele, “Large birefringence in two-dimensional silicon photonic crystals,” Physical Review B, vol. 63, p. 161101, Apr. 2001.
[25] D. Cassagne, C. Jouanin, and D. Bertho, “Hexagonal photonic-band-gap structures,” Physical Review B, vol. 53, p. 7134, Mar. 1996.
[26] Z. Li, J. Wang, and B. Gu, “Creation of partial band gaps in anisotropic photonic-band-gap structures,” Physical Review B, vol. 58, no. 7, p. 3721, 1998.
[27] Y. Kalra and R. K. Sinha, “Photonic band gap engineering in 2D photonic crystals,” Pramana, vol. 67, pp. 1155–1164, Dec. 2006.
[29] Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science, vol. 282, pp. 1679 –1682, Nov. 1998.
[30] E. R. Brown, O. B. McMahon, and C. D. Parker, “Photonic-crystal antenna substrates,” Lincoln Laboratory Journal, vol. 11, no. 2, 1998.
[31] K. Inoue, M. Sasada, J. Kawamata, K. Sakoda, and J. W. Haus, “A Two- Dimensional photonic crystal laser,” Japanese Journal of Applied Physics, vol. 38, pp. L157–L159, Feb. 1999.
[32] O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-Dimensional photonic Band-Gap defect mode laser,” Science, vol. 284, pp. 1819 –1821, June 1999.
[33] S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polar- ization mode control of Two-Dimensional photonic crystal laser by unit cell structure design,” Science, vol. 293, no. 5532, pp. 1123 –1125, 2001.
[34] J. Vukovi, M. Lonar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Physical Review E, vol. 65, p. 016608, Dec. 2001.
[35] M. Loncar, T. Yoshie, A. Scherer, P. Gogna, and Y. Qiu, “Low-threshold photonic crystal laser,” Applied Physics Letters, vol. 81, no. 15, p. 2680, 2002. [36] H. Hirayama, T. Hamano, and Y. Aoyagi, “Novel surface emitting laser di- ode using photonic band-gap crystal cavity,” Applied Physics Letters, vol. 69, no. 6, p. 791, 1996.
[37] M. Boroditsky, T. F. Krauss, R. Coccioli, R. Vrijen, R. Bhat, and E. Yablonovitch, “Light extraction from optically pumped light-emitting di- ode by thin-slab photonic crystals,” Applied Physics Letters, vol. 75, no. 8, p. 1036, 1999.
[38] A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joan- nopoulos, “High transmission through sharp bends in photonic crystal wave- guides,” Physical Review Letters, vol. 77, p. 3787, Oct. 1996.
[39] S. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Exper- imental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science, vol. 282, pp. 274 – 276, Oct. 1998.
[40] S. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, “Waveguide branches in photonic crystals,” Journal of the Optical Society of America B, vol. 18, pp. 162–165, Feb. 2001.
[41] S. G. Johnson, C. Manolatou, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Elimination of cross talk in waveguide intersections,” Optics Letters, vol. 23, pp. 1855–1857, Dec. 1998.
[42] S. Boscolo, M. Midrio, and T. F. Krauss, “Y junctions in photonic crystal channel waveguides: high transmission and impedance matching,” Optics Let- ters, vol. 27, pp. 1001–1003, June 2002.
[43] S. Kim, I. Park, and H. Lim, “Proposal for ideal 3-dB splitters-combinersin photonic crystals,” Optics Letters, vol. 30, pp. 257–259, Feb. 2005.
[44] A. Mekis, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Two- dimensional photonic crystal couplers for unidirectional light output,” Optics Letters, vol. 25, pp. 942–944, July 2000.
[45] S. Boscolo, M. Midrio, and C. Someda, “Coupling and decoupling of electro- magnetic waves in parallel 2D photonic crystal waveguides,” IEEE Journal of Quantum Electronics, vol. 38, pp. 47–53, Jan. 2002.
[46] V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature, vol. 431, pp. 1081–1084, Oct. 2004.
[47] K. Sakoda, “Enhanced light amplification due to group-velocity anomaly pe- culiar to two- and three-dimensional photonic crystals,” Optics Express, vol. 4, pp. 167–176, Mar. 1999.
[48] J. Chen, H. Haus, S. Fan, P. Villeneuve, and J. Joannopoulos, “Optical fil- ters from photonic band gap air bridges,” Journal of Lightwave Technology, vol. 14, pp. 2575–2580, Nov. 1996.
[49] D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, “Photonic crystal polarizers and polarizing beam splitters,” Journal of Applied Physics, vol. 93, no. 12, p. 9429, 2003.
[50] S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop filters in photonic crystals,” Opt. Express, vol. 3, no. 1, pp. 4 – 11, 1998. [51] A. Chutinan, M. Mochizuki, M. Imada, and S. Noda, “Surface-emitting chan-
nel drop filters using single defects in two-dimensional photonic crystal slabs,” Applied Physics Letters, vol. 79, no. 17, p. 2690, 2001.
[52] B. Min, J. Kim, and H. Y. Park, “Channel drop filters using resonant tunneling processes in two-dimensional triangular lattice photonic crystal slabs,” Optics Communications, vol. 237, pp. 59–63, July 2004.
[53] J. Knight, J. Arriaga, T. Birks, A. Ortigosa-Blanch, W. Wadsworth, and P. Rus- sell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photonics Tech- nology Letters, vol. 12, pp. 807–809, July 2000.
[54] T. A. Birks, J. C. Knight, and P. S. Russell, “Endlessly single-mode photonic crystal fiber,” Optics Letters, vol. 22, pp. 961–963, July 1997.
[55] H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals: Toward micro- scale lightwave circuits,” Journal of Lightwave Technology, vol. 17, p. 2032, Nov. 1999.
[56] J. Pendry, A. Holden, D. Robbins, and W. Stewart, “Magnetism from conduct- ors and enhanced nonlinear phenomena,” IEEE Transactions on Microwave Theory and Techniques, vol. 47, pp. 2075–2084, Nov. 1999.
[57] J. B. Pendry, “Negative refraction makes a perfect lens,” Physical Review Let- ters, vol. 85, p. 3966, Oct. 2000.
[58] E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Subwavelength resolution in a Two-Dimensional Photonic-Crystal-Based su- perlens,” Physical Review Letters, vol. 91, p. 207401, Nov. 2003.
Chapter 3
MODELLING TOOLS
The problem of wave propagation in a photonic crystal is essentially an electro- magnetic wave propagation problem. The behaviour of electromagnetic field in a photonic crystal is described by macroscopic Maxwell’s equations. To simulate the wave propagation it is required to solve these equations with appropriate boundary conditions. Maxwell’s equations are simply a group of partial differential equations. Except in a few cases, there are no analytical solutions for these equations. There are several numerical methods to solve Maxwell’s equations and model photonic crystal structure. Maxwell’s Equations can be expressed either in time domain or frequency domain form. Therefore, the numerical methods used to model photonic crystals, typically, can be classified as frequency or time domain methods. In time domain methods, time-dependent Maxwell’s equations are solved directly on spa- tial grids and the solution has a time dependency. This is particularly advantageous for obtaining transient response and comparing with experimental results. On the other hand frequency domain methods provide the frequency response of the system assuming the harmonic time dependency, such as calculating eigenstates, studying band structures, or highly dispersive materials. The relation between time and fre- quency can be obtained by Fourier transformation.
In the previous chapter, the basic theory of band-gaps was discussed, Bloch modes were introduced and band-gap diagrams were obtained by means of the plane wave expansion method. The details of the plane wave method and other methods mostly used for analysing photonic crystals deeply in terms of transmission, loss and other aspects for modelling will be discussed.
All numerical methods have advantages and drawbacks; some are efficient for a given problem while the others are only efficient in limited cases. A single method is not enough for analysing all kinds of problems in photonic crystals since not all methods are suitable, flexible or efficient to analyse photonic crystals to all extents. Moreover, accuracy and computation time and sources are always factors one needs to consider.
Many numerical methods have been developed in order to study photonic crystals. Among them, the most used and accepted methods are Plane Wave Method (PWM), the Transfer Matrix Method (TMM), Finite Difference Time Domain (FDTD) method, the Finite Element Method (FEM). These will be discussed briefly.
3.1
Plane Wave Expansion Method
The PWE method is one of the most commonly used methods in photonic crystal study and is primarily used for obtaining band-gap diagrams of photonic crystals. In photonic crystals where the electromagnetic field distribution is periodic as well as the refractive index, the field can be represented as a sum of plane waves. PWE method is based on the Fourier expansion method to solve Maxwell’s equations by super positioning the plane waves. The solution of problem results in an eigenvalue matrix of vectors where the matrix size is theoretically infinite, and it needs to be
truncated by numerical calculations. Ho, Chan and Soukoulis conducted the first theoretical demonstration of complete band-gap in 3D photonic crystals in 1990 (1). In their study, the band-gap of a diamond lattice of air spheres in a solid back- ground was predicted correctly by means of PWE method (2). Soon after this study, it became possible to predict a complete band-gap in three-dimensional photonic crystals correctly before fabrication and experimental characterisation.
Here, it is shown how to derive Maxwell’s equations as a set of plane waves us- ing Fourier expansion. Fourier series are expanded for calculation of eigenvalues for electromagnetic fields using the periodicity and Bloch form with plane waves. As long as the periodicity is defined in reciprocal lattice by reducing the symmetry according to high symmetry points, the band gap structure of any type of crystal lattice can be calculated.
The periodic material with periodicity of lattice vector satisfies,
ε(~r) = ε(~r + ~R) (3.1)
where ~R the lattice vectors. By solving the Maxwell’s equations for magnetic field considering the periodicity of the dielectric constant in the reciprocal space the Bloch theorem can be applied (3):
∇ × 1 ε(~r)∇ × ~H~k(~r) =ω c 2 ~ H~k(~r) (3.2) ~ H~k(~r) = ~hn,~k(~r)ei~k~r (3.3)
where ~hn,~k(~r) is the periodic function with periodicity of the lattice for wave vector ~k and discrete band index, n. Expanding the appropriate Fourier series and expan-
sions into the equation results in infinite matrix-eigenvalue problem and the num- ber of basis function, ei ~G~rneeds to be truncated into N plane waves to numerically solve the problem.
~hn,~k(~r) = X
~ G
~hG~ei ~G~r (3.4)
The magnetic field is expanded into plane waves of wave vector ~k with respect to the reciprocal lattice vector ~G.
~ H~k(~r) = X ~ Gλ hGλ~ eˆλe[i(~k+ ~G)~r] (3.5)
~hGλ~ , the coefficients of the wave vector and can be written as :
~hGλ~ = 2
X
λ=1
hGλ~ ˆeλ (3.6)
where the polarization vectors ˆeλ characterize two independent polarizations λ,
which are perpendicular to ~k + ~G. If the equation of superposition of plane waves for magnetic field is put into the master equation, we obtain
X ~ G0λ0 EGλ, ~~k~ G0λ0~hG~0λ0 = ω c 2 ~hGλ~ (3.7)
The matrix E in the eigenvalue equation is a 2N × 2N Hermitian matrix, where N is the number of plane waves, given by:
The Fourier transform of inverse dielectric constant depends on the difference of the reciprocal lattice vectors only
ε−1( ~G, ~G0) = ε−1( ~G − ~G0) (3.9) The equation finally forms as follows:
X ~ G0 |~k + ~G||~k + ~G0|ε−1( ~G − ~G0) ˆ e2· ˆe02 −ˆe2· ˆe01 −ˆe1· ˆe02 eˆ1· ˆe01 ~h0 1 ~h0 2 = ω c 2 ~h1 ~h2 (3.10) Given that the dielectric constant is periodic, its inverse ε−1( ~G, ~G0) can be expanded in Fourier series as 1 ε(~r) = X ~ G ˆ κ( ~G)ei ~G~r (3.11)
where ˆκ( ~G) are the Fourier coefficients obtained through an integration over the primitive unit cell and Ω is the volume of the unit cell.
ˆ κ( ~G) = 1 Ω Z Ω ε−1(~r)e−i ~G~rd3(~r) (3.12) By using sufficient a number of plane waves, this standard eigenvalue problem can be solved numerically and the photonic band-gap structure is obtained. The size of the coefficient matrix depends on whether electric field or magnetic field is used in the equation. Due to the transversality of the magnetic field (∇ · ~H = 0), it is possible to write wave vector coefficients for only two polarizations resulting in 2N ×2N matrix, instead of the 3N ×3N matrix that would have been if the equation was solved for electric field. When Fourier transform is performed, the selection of the integration region, the unit cell, becomes important. If inversion symmetry
is provided, the Fourier transform matrix will be real. However, the real part of complex Fourier coefficients does not provide the full solution. For instance, if we consider a 2D photonic crystal in cylindrical rods in square lattice configuration, the Fourier coefficients are given by (4):
ˆ κ( ~G) = 1 εrf + 1 εb(1 − f ) | ~G| = 0 1 εr − 1 εb f2J1( ~Grr) ~ Grr | ~G| 6= 0 (3.13)
where, εr, εb are the dielectric constants of rods and the background, respectively.
rris the radius of rods and J1is the first-order Bessel function. The filling fraction,
f is the proportion of rod area to unit cell area and is equal to π rr
a
2
for a square lattice, where a is the lattice period.
When the equation is solved, the photonic band-gap diagram is obtained for two- polarizations: transverse electric, TE and transverse magnetic TM polarization. The plane wave equations can be rewritten for TE and TM polarization by assuming wave propagation along the xy-plane. The unit vector ˆe1 is in the xy-plane and
ˆ
e2 = ˆaz. The polarization vectors are simplify as follows:
ˆ
e1 = [cos θ, sin θ, 0] (3.14)
ˆ
e2 = [0, 0, 1] (3.15)
In a TM wave, H component is along ˆe1and has no components along ˆe1 = ˆaz. In
a TE wave, H component is along ˆe2 = ˆaz and E has no component along ˆaz (5).
TM X ~ G |~k + ~G||~k + ~G0|ε−1( ~G − ~G0)~h1( ~G0) = ω c 2 ~h1( ~G) (3.16)
TE X ~ G |~k + ~G||~k + ~G0|ε−1( ~G − ~G0)~h2( ~G0) = ω c 2 ~h2( ~G) (3.17)
The accuracy of the method relies on the number of plane waves used in the cal- culations. The method is very effective for rather simple cases where the dielectric contrast is not too large and unit cell is less than a wavelength. In order to analyse photonic crystal cavities and waveguides where either point or line defects are in- troduced the structure, the super cell approach can be used (6; 7). Computation time and number of plane wave required to solve the problem accurately will increase depending on the size of super cell. The super cell approach cannot be applied to the cases such as bends and splitters where periodicity is not kept through out the whole structure.
Conditions that require using large amount of plane waves results not only in large dimensions of matrix equations but also in some instabilities related to slow conver- gence due to the Gibb’s phenomena that occurs when Fourier series are applied to a periodic function with discontinuities. The convergence problem has been studied to improve results (8; 9; 10; 11).
PWE method is one of the most popular methods for calculating dispersion dia- grams for photonic crystals and has being used to plot band gap-map atlas. This method is easy to implement and straightforward as well as it gives reliable res- ults in most circumstances. Although PWE method precisely indicates the band gap positions, it is hard to observe how much power is transmitted from the struc- tures, PWE has some limitations and it can only be applied to lossless and dis- persion free materials which means that the method cannot consider losses. PWE method searches for frequencies at a fixed wave vector each time. When the dielec-
tric constant is frequency dependent this method becomes inefficient. Therefore, a band-gap diagram of photonic crystals with a frequency-dependent dielectric function, i.e. metallic photonic crystal, cannot be calculated accurately. How- ever, there are some studies of studying metallic band-gap calculations in which the dielectric function is expressed in different forms to solve the eigenvalue problem (12; 13; 14; 15; 16). Using a plane wave solution it is possible to obtain allowed or forbidden frequencies of a periodic structure, which is especially important to control the confinement of the electromagnetic radiation. The dielectric photonic crystal structures without loss are easy to apply in the PWE method. For lossy and dispersive materials and more complex structures, other methods i.e. transfer matrix method (TMM) is better choice.