Photonic Band-Gap Materials

SOLID STATE PHYSICS. VOL. 49

Photonic Band-Gap Materials P. M. Hur Department o f Physics, The Chinese Uniuersity of Hong Kong, Shatin, New Territories, Hong Kong

NEILF. JOHNSON Department of Physics, Clarendon Laboratory, Uniuersity of Oxford, Oxford OX1 3PU, England, United Kingdom

I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Experimental Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Experimental Systems and Results . . . . . . . . . . . . . . . . . . . . . . . . . 3. Applications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Photonic Band Theory: Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 5. Theoretical Calculations and Results . . . . . . . . . . . . . . . . . . . . . . . . IV. Conclusions and Future Direction . . . . . . . . . . . . . . . . . . . . . . . . . . .

151 151 157 157 165 170 170 186 199

1. Introduction

1. SURVEY

The properties of wave propagation in a periodic system have always been of great interest to physicists. A well-studied example in solid state physics is that of an electron in a periodic potential in either one dimension (lD), two dimensions (2D), or three dimensions (3D).'-3 Such a periodic potential arises in naturally occurring crystalline solids (3D) and in artificially made heterostructures (1D and 2D). The eigenstates of such an electron are known as Bloch states. As a result of Bragg scattering at the Brillouin zone boundary, gaps of disallowed energies appear in the energy spectrum

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N. W. Ashcroft and N.D. Mermin, Solid State Physics, Saunders College, Philadelphia (1976). 'J. Callaway, Quantum Theory of the Solid State, Chaps. 4 and 5, Academic Press, New York (1974). 'J. Callaway, in Solid State Physics (F. Seitz and D. Turnbull, eds.), Vol. 7, p. 100, Academic, New York (1958).

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Copyright 0 1995 by Academic Press, Inc. All rights of reproduction in any form reserved.

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of the electron.' The continuous energy-wavevector dispersion relation characteristic of free space is therefore modified; this energy spectrum is referred to as the electronic band structure of the system. The quantitative details of a given electronic band structure depend on the specific form of the periodicity being considered. As a result, different crystalline solids have different electronic band structures.', 2 * 4 An analogous situation can arise for electromagnetic waves propagating through a medium with a periodic variation in the dielectric constant. These systems, which are generally known as photonic band-gap (PBG) materials or photonic crystals, can be designed with a periodic variation in the dielectric constant in either one, two, or all three spatial A typical structure consists of a regular array of objects (e.g., spheres) of one dielectric material embedded in a second material with a different dielectric constant. Due to the periodic structure of the array, there may be ranges of frequencies at which no allowed modes exist for electromagnetic wave propagation in a given direction. These ranges of frequencies are termed photonic band gaps in analogy with the electronic band gaps. The dispersion relation (i.e., the frequency-wavevector relation) for electromagnetic wave propagation in PBG materials is referred to as the photonic band structure. The photonic band structure of a given PBG material depends on the crystal structure, the lattice constant, the shape of the embedded dielectric object, the dielectric constants of the constituent materials, and the filling fraction, which is the percentage of total crystal volume occupied by any one of the materials. The existence of full photonic band gaps in PBG materials has important potential applications in the area of quantum optics and laser technology.' The idea of combining two different materials in order to create a new structure with novel properties has been repeatedly employed in solid state physics in recent years. In semiconductor superlattices? for example, layers of semiconductors are put together to create an additional periodicity in one direction. This procedure is known as band-gap engineering since the resulting structure can have a band gap that is quite different from those of the constituent semiconductors. Multilayered systems of magnetic and nonmagnetic materials have been shown to exhibit enhanced 4E. 0. Kane, in Narrow Gap Semiconductors (W. Zawadski, ed.), p. 19, Lecture Notes in Physics, Vol. 33, Springer-Verlag, New York (1981). %ee, for example, the articles published in the special issue of J . Opt. SOC. A m . B. 10, No. 2 (1993). bE. Yablonovitch, J . Phys.: Condens. Matter, 5, 2443 (1993). 'P. St. J-Russell, Phys. World, p. 37, Aug. 1992. 'P. R. Villeneuve and M. Piche, Prog. Quant. Electr. 18, 153 (1994). 9D. L. Smith and C. Mailhiot, Reu. Mod. Phys. 62, 173 (1990).

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magnetoresistance, the so-called "giant magnetoresistance" phenomenon." Giant magnetoresistance has also been observed in granular magnetic composites in which small, magnetic particles are randomly dispersed in a nonmagnetic host medium. The latter systems are examples of the larger class of materials collectively known as macroscopically inhomogeneous media, the physics of which has recently been reviewed by Bergman and Stroud." The underlying interest in such systems concerns the effective response (e.g., electrical, thermal, dielectric, optical, or mechanical) when two or more kinds of materials with different macroscopic properties are put together in either an ordered or disordered way. In this respect PBG materials are another example of such composite systems, with the relevant property being the effective optical response. The idea of a system with a photonic band gap is not new. There is a history of work on 1D waveguides and on periodic 1D structures made from slabs of dielectric materials, yielding photonic gaps for electromagnetic waves propagating in the direction normal to the slabs.12 The PBG materials are therefore the logical extension to higher dimensions of the well-known 1D waveguides and attenuators already in use. Surprisingly however, the possibility of fabricating such a PBG system in higher and John.I4 The dimensions was only raised in 1987 by Yablon~vitch'~ important advantage of such higher dimensional PBG materials over their ID counterparts is the possibility of a full photonic band gap throughout the entire Brillouin zone, thereby preventing propagation of electromagnetic waves in all directions as opposed to just along one particular direction. In the five years that followed this initial suggestion, many groups became attracted to PBG material research as a result of advances in fabrication techniques and the increasing awareness of the potential importance of PBG 'OR. E. Camley and R. L. Stamps, J. Phys.: Condens. Matter 5, 3727 (1993). I'D. J. Bergman and D. Stroud, in Solid State Physics (H. Ehrenreich and D. Turnbull, eds.), Vol. 46, p. 147, Academic Press, New York (1992). 12 P. Yeh, Optical Waues in Layered Media, John Wiley & Sons, New York (1991). "E. Yablonovitch, Phys. Reu. Lett. 58, 2059 (1987). I4 S. John, Phys. Reu. Lett. 58, 2486 (1987). "S. L. McCall, P. M. Platzman, R. Dialichaouch, D. Smith, and S. Schultz, Phys. Reu. Lett. 67,2017 (1991). 16 E. N. Economou and A. Zdetsis, Phys. Reu. B 40, 1334 (1989). I'E. Yablonovitch and T. J. Gmitter, Phys. Reu. Lett. 63, 1950 (1989). in E. Yablonovitch and T. J. Gmitter, J. Opt. SOC. Am. 7, 1792 (1990). 19 K. M. Leung and Y. F. Liu, Phys. Reu. B 41, 10,188 (1990). 'OK. M. Leung and Y. F. Liu, Phys. Reu. Lett. 65,2646 (1990). 21K.M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).

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One of the principal motivations driving the PBG field has been to design a PBG material (2D or 3D) that has a full photonic band gap in the optical frequency range and yet which is both lossless and easy to fabricate. Despite numerous attempts, the successful achievement of this goal has yet to be reported. Several of the proposed technological applications for PBG materials rely on a complete photonic band gap throughout the entire Brillouin zone. Nevertheless, other applications have been suggested that are less dependent on such a full gap. Therefore, there is merit in considering the general class of PBG materials, whether or not they actually show a full photonic band gap for all directions of propagation. As is customary in this field, therefore, we will use the general term PBG material irrespective of whether or not a full photonic band gap actually exists throughout the Brillouin zone in the particular sample under consideration. We shall also adopt the terminology photonic band structure even though the band structure arises from a strictly classical treatment of the problem, that is, from solving Maxwell's equations. The aim of this article is twofold. First, we discuss recent experimental and theoretical progress in understanding the properties of PBG materials. Second, we attempt to provide a rigorous formalism describing electromagnetic properties in PBG materials, akin to that currently available in the literature for electronic properties in crystalline solids.', 40 The complica*'R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Reu. B 44, 13,772 (1991); 10,961 (1991). 23J. Pendry and A. M. MacKinnon, Phys. Reu. Lett. 69, 2772 (1992). 24 K. M. Ha, C. T. Chan, and C. M. Soukoulis, Phys. Reu. Lett. 66, 393 (1990). "S. Satpathy, Z. Zhang, and M. R. Salehpour, Phys. Reu. Lett. 64, 1239 (1990); 65, 2478(E). 26 Z. Zhang and S. Satpathy, Phys. Reu. Lett. 65, 2650 (1990). "M. Plihal, A. Shambrook, A. A. Maradudin, and P. Sheng, Opt. Commun. 80, 199 (1991). 28M. Plihal and A. A. Maradudin, Phys. Reu. E 44, 8565 (1991). 29 N. Stefanou, V. Karathanos, and A. Modinos, J . Phys.: Condens. Matter 4, 7389 (1992). ' O S . Datta, C. T. Chan, K. M. Ho, and C. M. Soukoulis, Phys. Reu. E 46 10,650 (1992). "E. Yablonovitch, T. J. Gmitter, R. D. Meade, K. D. Bromrner, A. M. Rappe, and J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991). 32 R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Appl. Phys. Lett. 61, 495 (1992). 33 E. Yablonovitch, T. J. Gmitter, and K. M. b u n g , Phys. Reu. Lett. 67, 2295 (1991). 34H.S. Soziier, J. W. Haus, and R. Inguva, Phys. Reu. E 45, 13,962 (1992). 3s C. T. Chan, K. M. Ho, and C. M. Soukoulis, Europhys. Lett. 16, 563 (1991). 36 G. Henderson, T. K. Gaylord, and E. N. Glytsis, Phys. Rev. B 45, 8404 (1992). 37P. R. Villeneuve and M. Piche, J . Opt. SOC. Am. A 8, 1296 (1991). 38 G. X. Qian, Phys. Reu. B 44, 11,482 (1991). 39P. R. Villeneuve and M. Piche, Phys. Reu. E 46, 4969 (1992); 4973 (1992). 40 E. 1. Blount, in Solid State Physics (F. Seitz and D. Turnbull, eds.), Val. 13, p. 305, Academic, New York (1962).

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tion for PBG crystals compared to electronic crystals is that Maxwell’s equations describing the propagation of electromagnetic waves are vectorial in nature, while the Schroedinger equation describing electronic waves is scalar. It is this feature that can give rise to nontrivial generalizations of the well-known properties for electrons in solids. The sections in this article on experimental fabrication techniques (Section 11.2), potential applications (Section 11.31, and theoretical calculations of photonic band structures (Section 111.5) are intended for general readership. The formalism section (Section 111.4) is more advanced, but is designed to be self-contained and can therefore be read independently. Because PBG materials are generally artificially made structures, there are an infinite number of possible PBG materials depending on the crystal structure, the dimensionality, the lattice parameter, the filling fraction, and the dielectric constants of the constituent materials. The search for the PBG material with the largest photonic band gap is therefore far from easy, given the range of possible parameters. Initially it was thought that an fcc (face-centered-cubic) structure should have a large photonic band gap, because of its nearly spherical Brillouin zone.41 This provoked many theoretical and experimental investigations of PBG materials with an fcc crystalline form. Section 11.2 examines the various sample constructions and experimental arrangements that have been employed so far in the search for a photonic gap in 2D and 3D PBG materials. We also review the experimental results to date. The technical aspects will only be discussed in sufficient detail so as to highlight the number of new experimental techniques that have had to be developed in the PBG field. In this way, we emphasize the extent to which investigation of the photonic band gap differs from that of the electronic band gap. For detailed technical discussions of the experimental apparatus, we refer the reader to the original sources. The proposed applications for PBG materials have been both numerous and diverse. They are all, however, based on the idea of the modification of the free-space photon dispersion relation from w = ck, where k is the wavevector, to the more general form o = w ( k ) . These applications employ either a perfectly periodic PBG material (crystal) or consider a PBG material containing “defects.” In perfect PBG crystals, the introduction of gaps in the photon dispersion prevents propagation of photons at those frequencies, thereby allowing applications of PBG materials in integrated optics as filters and polarizers. The frequency gap also provides a means of tailoring the optoelectronic properties of a semiconductor device placed within the PBG material. In particular, the electron-hole recombination

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E. Yablonovitch, J . Opr. Soc. Am. B 10, 283 (1993).

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process, which is a major form of loss of efficiency in many devices, can be suppressed if the energies of any photons that would be emitted in free space fall in the gap of the PBG material.41Introducing a “defect” into the periodic PBG material can introduce a finite number of localized defect modes, just as in a doped semiconductor system. Such discrete levels lying in the gap imply that such “doped” PBG materials could provide useful single-mode cavity systems operating in the technologically important optical frequency range. Section 11.3 discusses the various applications that have been suggested for PBG materials. Again, technical details are not given in detail because our goal is to emphasize the general mode and diversity of the proposed applications. A major distinction between the properties of electronic and photonic band structures, as mentioned earlier, results from the vector nature of electromagnetic waves as opposed to the scalar nature of electronic waves. There are additional fundamental differences. Charge and the Pauli exclusion principle do not enter into the discussion of PBG materials as photons are chargeless, spin-one entities. Also photon-photon interactions are negligible, in contrast to the important role of electron-electron interactions in solids. A description of photonic band structures therefore has an advantage over that of electronic band structures in that a singleparticle description is exact; that is, solving Maxwell’s equations for a single electromagnetic wave can yield exact results. This is in sharp contrast to the electronic case for which many-electron interactions can qualitatively affect the properties predicted by the single-electron Shroedinger equation. This allows the presentation in Section 111.4 of an exact formalism describing photonic band structures in PBG crystals. The various representations are discussed for the photonic crystal. The corresponding equations are given both in the absence and presence of defects. Similarities and differences between the photonic and electronic problems are emphasized. In practice, photonic band structures have been calculated theoretically for various PBG crystals in 2D and 3D using extensions of traditional, numerical electronic band structure techniques. In Section 111.5, we discuss the various theoretical approaches to date. In addition we demonstrate the extent to which predictions from different theoretical approaches are consistent with the experimental results discussed in Section 11.2. Given that there are an infinite number of possible PBG crystals, it is neither practical nor instructive to provide a complete catalog of all the theoretical band structures that have been reported. We choose rather to focus on a few representative results that have been widely cited in the literature and that can be compared directly to experiment. In so doing, we will attempt to highlight trends that have emerged with respect to the

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photonic properties as the PBG material parameters are varied. The most popular calculational technique employed to date has been a numerical plane-wave expansion (see, for example, Ref. 20). Accurate calculations of photonic band structures must account for the vector nature of the electromagnetic waves. Such vector calculations have so far been numerically intensive and have necessitated powerful computers. Given the wide choice of PBG material parameters, it would be useful for the design and fabrication of PBG materials to have some approximate analytic expressions for the photonic band gap as a function of filling factor, dielectric contrast, lattice parameter, and lattice structure. This is particularly important given the rather unrelated structures in which full photon band gaps have been observed. Because the main frequency region of interest lies in the vicinity of a photonic band gap, it is reasonable to search for alternative theoretical schemes that allow more insight into the nature of the dispersion relation in the vicinity of band maxima and minima. Such a scheme is well known in conventional electronic theory and is the so-called k .p It is shown in Section 111.5 that a photonic k * p approach can be successfully used to calculate the dispersion relation near the photonic band gap. Another method that has proved useful for understanding electronic properties in solids is the tight-binding method.’ Section 111.5 also illustrates that a tight-binding method can be employed successfully in PBG materials. We conclude in Section IV with a discussion of possible future directions for research into PBG materials. II. Experimental Overview

2. EXPERIMENTAL SYSTEMS AND RESULTS

In principle, the construction of a photonic band-gap material is straightfoward. One needs to introduce a periodic variation in the dielectric constant of a material in one, two, or three spatial dimensions, thereby producing a lD, 2D, or 3D periodic dielectric material. In practice, there are many technical complexities and subleties in the fabrication and measurement of PBG materials; these have become apparent through experimental trial and error.41 One-dimensional PBG materials operating in the microwave regime significantly predate the “invention” or 2D or 3D PBG material^.^' Such 1D systems could be built with relatively low precision because of the relatively long microwave wavelengths. In addition, since the periodicity only had to be introduced along one direction, samples could be made fairly easily using slabs of dielectric materials. The challenge of fabricating

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a PBG material with a full photonic band gap at optical frequencies introduces two significant complications. First, the periodicity must be introduced in two or three dimensions because it is only in 2D or 3D PBG materials that full photonic band gaps can exist. Second, the requirement of a photonic band gap in the optical frequency spectrum requires precision machining of dielectric periodicity on the optical length scale. Indeed the optical regime remains the hardest, but arguably the most useful, frontier in the fabrication of PBG materials with a full band gap. So far, the successful fabrication of a PBG sample with a full band gap in the visible spectrum has yet to be reported because of these difficulties at such small length scales. For example, PBG materials fabricated by hole-drilling, as discussed later, must have holes that are uniform on this length scale in order to obtain a true gap in a perfect crystal. To date, investigations of PBG material properties in both 2D and 3D have therefore been mostly limited to the microwave regime, where precision in machining is less important. The search for a full photonic band gap has prompted interest in the fabrication and investigation of both 2D and 3D PBG structures. Typically investigations in the microwave regime have been carried out on 2D and 3D samples about 20 periods long and a few centimeters in length.5 In general, 2D microwave samples can be made by arranging long dielectric rods in a regular lattice or by cutting holes into a dielectric slab; 3D samples are fabricated either by arranging dielectric objects such as spheres in a regular lattice or by creating voids in a dielectric b10ck.~The advantage of 2D structures over 3D is that they are generally easier to fabricate. An important distinction can be made between the nature of experiment needed to determine the photonic band gap in a given sample of a PBG material, and that needed to determine the electronic band gap in conventional solids. In semiconducting and insulating samples, the electronic band gap can be measured reasonably easily using photon absorption. This is because the band gap separates valence bands full of electrons from conduction bands that are essentially empty. A photon will then be absorbed if it has sufficient energy to take an electron from one of the full valence bands up to the empty conduction band.2 The absorption of photons for energies below the band gap is therefore essentially zero. As the band gap energy is reached, the absorption rises sharply, particularly in low-dimensional systems. The band gap is then obtained from this threshold photon energy. The photonic band gap in PBG materials cannot be measured in the same way because, unlike the electronic system, the band gap does not separate a full valence band of carriers from an empty conduction band. The photonic band gap is simply a window of frequencies in which propagation through the crystal cannot occur. Some ingenious

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methods have therefore had to be developed to determine the possible presence of a photonic band gap throughout the Brillouin zone. We now look more closely at the various techniques that have been employed to fabricate a given PBG material sample and measure its photonic band structure in the microwave frequency range. Such microwave experiments have been carried out by a number of groups; the different setups are all based on determining the phase and amplitude of the field transmitted through a PBG material sample for various orientations. The refractive index ratio between the two PBG material constituents in typical microwave experiments has usually been chosen to be similar to that between a common semiconductor such as GaAs and air ( - 3.6:l). In this way, any dispersion relations obtained in the microwave regime can then simply be scaled down in size in order to gain insight into the corresponding photonic band structure in the optical regime.41 One experimental setup for investigating photonic band structures in a given PBG material in the microwave regime involves an anechoic chamber, as used by Yablonovitch and Gmitter.” The arrangement consists of a long chamber with walls that absorb microwaves, which provide essentially plane waves incident on the PBG material sample. This pump-probe setup ” allowed investigation of electromagnetic wave propagation through PBG samples over the frequency range 1 to 20 GHz. To determine the photonic gap, the frequency range was swept with the radiation incident along a given crystal direction, and the signal transmission was measured. To obtain the corresponding wavevector k in the PBG material, Yablonovitch and Gmitter” exploited the fact that the k component parallel to the surface was conserved and that the frequencies at the onset of the gap correspond to wavevectors at the Brilloin zone edge. By rotating the crystal and using both polarizations of incident radiation separately, they were able to map out parts of the photonic band structure. A drawback of this method is that they could only investigate the band structure in the vicinity of the gap; in addition, they could not investigate the bands along the T-X and T-L directions because these directions do not lie in the surface of the Brillouin zone. Using this setup, Yablonovitch and Gmitter ”, built and investigated 3D fcc structures consisting of 8000 dielectric spheres of 6-mm radius. The spheres were made of Al,O, and were supported in the lattice by a thermal-compression-molded dielectric 3, and the foam refractive index foam. The sphere refractive index was 1. A true photonic band gap was not observed, however, at any was filling fraction. They then tried replacing the dielectric spheres with spherical airholes in a dielectric background. This was achieved by drilling holes in opposite faces of dielectric slabs with a spherical drill bit and then stacking these

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slabs on top of each other so that the hemispheres faced each other; this produced the desired fcc array of spherical voids inside a dielectric block. However, no photonic band gap was observed. They then tried increasing the size of the airholes to such an extent that the voids were closer than closed-packed; the voids therefore overlapped and allowed continuous holes to pass through the structure. Initially, they thought they had observed a full photonic band gap centered at 15 GHz and forbidden for both p01arizations.I~ However, it turned out that their finite-size PBG sample, which was only 12 unit cells wide, had not shown up band crossings in the dispersion relation near the W and U symmetry points. These crossings later emerged from theoretical calculations and are discussed further in Section 111.5. The finite size of the sample had the effect of limiting the wavevector resolution and restricting the dynamic range in transmission. The major experimental breakthrough yielding the first observation of a photonic band gap in the microwave regime came when Yablonovitch and co-workers found that these symmetry-induced degeneracies in the fcc structure could be lifted by building the PBG material from nonspherical atoms.33 Using a mask with a triangular array of holes, they drilled three sets of holes at various oblique angles to obtain an fcc structure of nonspherical, elongated, air-filled atom^.^^,^' The resulting structure was a fully 3D fcc crystal. For a dielectric refractive index of 3.6 they observed a full gap corresponding to 19% of the mid-gap frequency in a structure made from 22% dielectric. They also measured the mid-gap attenuation to be 10 dB per unit cell; this indicates that the 3D PBG crystal need not be many layers thick in order to expel the photon field at that frequency. The significant feature of the hole-drilling procedure of Yablonovitch et is that the holes did not need to be individually drilled. Furthermore, similar structures were also grown by reactive ion e t ~ h i n g , ~which ’ left oval holes in the material. It was found33 that the forbidden gap width increased slightly in such structures. This opened the possibility that the nonspherical “atoms,” which were observed to lift the degeneracies and open a complete photonic gap, might be fabricated in the optical frequency regime using reactive ion etching. Meade et used a similar experimental pump-probe setup to investigate 2D PBG materials. Their samples consisted of air columns drilled into a dielectric material of refractive index 3.6, forming a triangular lattice array. They found a gap for both polarizations between 13 and 16 GHz for a sample of air columns with both diameter and lattice constant of almost 1 cm. These two groups ~ o l l a b o r a t e don ~ ~the investigation of defect modes in a 3D PBG sample with a single defect. Their sample consisted of the fcc structure with nonspherical airholes in a dielectric material with a refrac-

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tive index of 3.6. The hole diameter and lattice constant were about 0.5 and 1 cm, respectively. Defects were introduced into the PBG crystal in one of two ways: by adding one dielectric sphere (donor) in a hole of the lattice or by breaking one of the connections between the airholes (acceptor). Transmission through the defect modes was achieved by replacing the plane wave (single k) incident on the sample with a spherical point source. In this way, waves were incident on the sample with a range of possible k component's along the normal to the surface. This allowed generation of a wide range of possible defect modes at a given frequency. The results are shown in Fig. 1. For the sample with a broken connection [Fig. l(b)l the appearance of a single mode in the forbidden gap is quite striking when compared to the strong attenuation in the perfect crystal at this frequency [Fig. l(a)]. Because of its relatively equal frequency separation from the two band edges, the defect mode in Fig. l(b) is labeled as a deep acceptor.

1

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,

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11

12

13

14

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Frequency in GHz

16

17

FIG.1. Experimental results for attenuation of microwave radiation passing through (a) a perfect 3D PBG fcc crystal of thickness 8 to 10 atomic layers. Crystal consists of air atoms cut from a background dielectric; the forbidden gap is indicated. (b) The same PBG crystal with a single acceptor defect made by cutting a slice through the dielectric material in one unit cell; the acceptor yields modes near the mid-gap. (c) The same PBG crystal with a single donor defect consisting of a dielectric sphere; the donor yields modes near the band edge. [Taken from Fig. 3 of E. Yablonovitch, T. J. Gmitter, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Reu. Lett. 67, 3380 (1991).]

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For the sample with an extra dielectric sphere in the middle of an airhole, multiple modes were observed; these modes appeared close to the upper frequency edge of the gap since the defect represented a relatively weak perturbation to the system. These modes were hence labeled as shallow donor modes. A theoretical discussion of PBG defects and the analogies with the more familiar case of doped semiconductors is given in Section III.4.c. A second technique for measuring PBG photonic dispersion is the waveguide method used by McCall et ul.1s,42to look at 2D photonic band structures, both with and without single defects. For frequencies in the range 6 to 20 GHz, they used a parallel-plate metallic waveguide with plate separation of 1 cm and plate dimensions of 46 X 51 cm. At each end of the chamber, which contained the PBG sample, 8- to 12-GHz waveguide fittings allowed injection and detection of microwaves. The waveguide system contained a single TEM mode and was essentially 2D since the electric field for such frequencies was nearly constant along the transverse direction. Their PBG samples were arranged in square or triangular lattices consisting of 1-cm-long low-loss dielectric rods. The rods (about 900) were placed in the form of a finite lattice in a precision-drilled foam template. The rods were positioned in such a way that the electric field was polarized parallel to the rods (TM: transverse magnetic mode) and the radiation propagated in the plane perpendicular to the rods. They were able to sweep the microwave frequency and make measurements of the power transmitted through the sample. Removing a rod introduced a defect (vacancy) into the lattice. Their 2D result.^'^,^^ are similar to the 3D data of Meade et d3*They managed to map out the position-dependent fields in the resulting standing-wave structure (i.e., defect modes) by coupling them to a noninvasive probe through a lattice of small holes in the upper plate. They found experimentally large attenuation consistent with the presence of a photonic band gap in a various directions. They also found localized defect modes with a decay length of the order of a lattice constant.42 A third technique of measuring PBG material properties is the so-called COMITS (coherent microwave transient spectroscopy) technique emThis technique involves the generation of ployed by Robertson et ~

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D. R. Smith, R. Dalichaouch. N, Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, 1. Opf.SOC.Am. B 10, 314 (1993); D. R. Smith, S. Schultz, S. L. McCall, and P. M. Platzmann, J . Mod. Opt. 41, 395 (1994). 43 W. M. Robertson, G. Arjavalingan, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Reu. Lett. 68, 2023 (1992). 44 W. M. Robertson, G. Aqavalingan, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, J . Opt. SOC. Am. B 10, 322 (1993).

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picosecond electromagnetic transients via optoelectronically pulsed antennas. The technique permits investigation of the phase of the transmitted signal and hence mapping out of the photonic dispersion relations. Measurements can be made over a broad frequency range (15 to 140 GHz) and the technique is sensitive to polarization. It has been used by Robertson to investigate both bulk and surface state dispersion in PBG samples. The setup has a coplanar stripline antenna transmitter and receiver placed either side of a sample in air. A dc bias is used to generate a short optical pulse ( 2 ps), which triggers a short current pulse in the transmitter; this current pulse travels down the stripline and then radiates into the air. The radiation pulse is focused toward the sample using a hemispherical lens. The transmitted pulse is then picked up by the transmitter on the other side of the sample. The Fourier transform of the time-dependent signal at the receiver gives the transmitted amplitude as a function of frequency. The ratio of the Fourier transformed signals both with and without the sample gives the phase information. Robertson et a1.43,44investigated two 2D PBG materials made, respectively, of long, 0.37-mm radius, ceramic rods and long, 0.75-mm radius cylindrical holes in a 3.6 refractive index background dielectric. Samples with holes were investigated for both square and triangular lattices. Figure 2 shows an example of their results.43 Although the finite time window of only 200 ps limited the spectral resolution of the signal, some evidence of gaps could still be seen. The agreement between these experimental results and theory is discussed in Section 111.5. Robertson et al.4s also investigated nonradiative surface modes, exponentially decaying away from the surface, at the boundary surface of terminated PBG materials. Their setup used a phase-matching prism to couple microwave radiation to the surface mode; this enabled coupling of the radiation to the surface modes, which, by themselves, could not radiate into the crystal because they lay in the band gap, nor could they radiate into air because of the requirement of energy and wavevector conservation. For a 2D PBG sample of dielectric rods terminated by hemispherical rods in a square array, Robertson et al. found surface modes; however, no such modes were observed for samples of arrays terminated by full cylindrical rods. Finally we mention a recently introduced technique for fabricating 3D PBG materials due to Ozbay ef al.46 They employed the ordered stacking of micromachined (110) silicon wafers in order to build a periodic struc-

-

45 W. M. Robertson, G. Arjavalingan, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Opt. Lett. 18, 528 (1993). 4h E. Ozbay, E. Michel, G. Tuttle, R. Biswas, M. Sigalas, and K. M. Ho, Appl. Phys. Lett. 64, 2059 (1994).

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P. M. HUI AND NEIL F. JOHNSON

0.0

4.2

8.4

12.6

WAVEVECTOR ( c m l )

16.8

FIG. 2. Comparison of experimental and theoretical (plane-wave expansion) photonic band structure for a 2D PBG crystal consisting of a square lattice of alumina-ceramic cylindrical rods with e = 8.9. Cylinder radius is 0 . 2 ~where a is the crystal lattice constant. Propagation of electromagnetic waves is along the (10) direction. Electric-field polarization is parallel to the rod axis in (a) and perpendicular in (b). Solid dots are experimental COMITS (coherent microwave transient spectrum) results; dashed lines are theoretical (plane-wave expansion) results. [Taken from Fig. 3 of W. M. Robertson, G. Arjavalingan, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Reu. Lett. 68, 2023 (1992).]

ture that had been predicted t h e ~ r e t i c a l l yto ~ ~form a full 3D band gap. They demonstrated a full band gap in this structure centered at 100 GHz with a window of about 40 GHz. Their fabrication technique employed the anisotropic etching of silicon by aqueous potassium hydroxide, which etches the (110) planes of silicon very rapidly, leaving the (111) planes relatively untouched. Hence using (110koriented silicon, they could etch arrays of parallel rods into wafers, and then stack these to make the desired structure. The silicon wafers were each 3 in. in diameter and 390 p m thick. High-resistivity wafers were chosen to minimize absorption losses in the silicon. These authors suggested46 that such a technique could also be used to construct PBG materials in the optical frequency regime. 41

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, Solid. State Commun. 89, 413 (1994).

PHOTONIC BAND-GAP MATERIALS

165

We note that a combination of electron-beam lithography and reactive ion-beam etching has recently been used to fabricate a 2D periodic structure in GaAs/GaAlAs with features on the 50-nm scale.48

3. APPLICATIONS Many applications have been proposed for photonic band structure materials, based on the modification of the free-space photon dispersion relation. The two main applications that have most caught the interest of workers in the field are the suppression of the electron-hole recombination process in semiconductor devices, thereby removing a major form of loss of efficiency, and the possibility of constructing single-mode cavities via localized photon modes in PBG materials with defects. In this section we look at these two applications in more detail, before discussing briefly other suggested uses for PBG materials that have appeared in the literature. a. Suppression of Electron-Hole Recombination One of the most interesting applications of a full photonic gap in PBG materials is to alter the radiative recombination rate of electrons and holes in a semiconductor. The importance of this application lies in the fact that such processes often lead to energy loss, noise, and lowering of the characteristic operating speed in semiconductor devices. In heterojunction bipolar transistors, for example, there are certain regions of the transistor current -voltage characteristic where the gain is determined by such electron-hole recombination processes. In solar cells, recombination limits the maximum output voltage. Yablonovitch has suggested an application in the field of laser diodes4’ If a correlated flow of electrons were to be used to drive a laser diode, there would be a constant flux of output photons; this phenomenon is known as photon-number-state squeezing and yields a low bit-error rate for optical communication. In practice, laser diode efficiency is limited by random spontaneous emission in all directions. Employing a PBG material with a full band gap eliminates unwanted electromagnetic modes that would otherwise increase the bit-error rate; the photon squeezing effect would therefore be enhanced. The simplest way of seeing the effect of a PBG material on such electron-hole recombination is from Fermi’s golden The rate of downward transition between filled and empty levels, such as in an atom, is 4xP. L. Gourley, J. R. Wendt, G. A. Vawter, T. M. Brennan, and B. E. Hammons, Appl. Phys. Leti. 64, 687 (1994). 4Y E. A. Hinds, Advances in Atomic, Molecular and Optical Physics, Vol. 28, p. 237, Academic, New York (1991).

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P. M. HUI AND NEIL F. JOHNSON

proportional to I(flVli)l%(E);here ( f l V l i ) is the matrix element of the interaction V coupling the atom to the field, between initial and final The interaction V in the Coulomb gauge is states i and f of the sy~tern.~’ essentially just A p where A is the vector potential and p is the momentum operator; p ( E ) is the density of final states per unit energy. Altering the dispersion relation from its free-space form ( w = c k ) affects the density of photon states per unit frequency and hence p ( E ) . In free space there are modes at every frequency (i.e. the spectrum is continuous), and hence the probability of finding a system such as an atom or semiconductor in a given excited state (exciton) will decay exponentially with time. If the only modes present are evanescent due to the presence of a PBG, then the atom becomes “dressed” by a photonic cloud, and the resulting electronic properties of the atom differ from those in free space. The only decay channels now available to the electron-hole pair (exciton) are nonradiative, such as phonon emission or higher order photon processes. b. Single-Mode Cavities In atomic physics there has been a recent surge of interest in what is called cavity quantum electrodynamics?’ The basic idea is that the boundary conditions imposed by a finite cavity can modify the photon spectrum in such a way that only discrete mode frequencies are available, just like a particle-in-a-box for an electron. The radiation properties of an atom in a cavity depend on the availability of such modes at the frequency of recombination In practice, cavities with no losses have been difficult to build. In particular, metal cavities, which work well at microwave frequencies, become too lossy at optical frequencies. In contrast, PBG materials with full band gaps have an advantage; since they are made of low-loss dielectrics, they can in principle work well at optical frequencies. Optical devices such as lasers often require a single well-defined photon mode to operate. Such defect modes can be introduced into a PBG material by creating a defect in an otherwise perfect PBG crystal. According to the specific defect design, a defect mode can be pulled out of the continuum to form a localized state in the gap; the electromagnetic fields for this defect mode are localized around the defect and decay exponentially with increasing distance from the defect. The decay distance can be chosen to be of the order of the lattice spacing by placing the defect mode near the center of the gap. The dependence of the defect mode frequency on the characteristics of the defect is further discussed in Sections III.4.c and 111.5.

.’”

‘“P. W. Milonni and P. L. Knight, Opt. Commun. 9, 119 (1973).

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167

The defects in PBGs can be thought of as higher dimensional versions of a quarter-wavelength slab at the center of a Fabry-Perot etalon. The only losses in such defect modes lying in a true photonic gap are due to absorption of the dielectric material itself. The Q factor measures the extent of such losses and the coupling of the mode itself to the surroundings. It is given by the number of optical cycles needed for the radiation field energy in the cavity to decay by a factor e-'. The Q factor of the defect modes in doped PBGs materials can be made very large by choosing low-loss dielectrics. At the same time, the frequency range can, in principle, be made large enough by finding a combination of dielectric materials yielding a sufficiently large photonic band gap. It has been suggested that reliable and thresholdless light-emitting diodes (LEDs) could be made from such high-Q single-mode PBG microcavitie~.~' Just as with conventional LEDs, a voltage can be applied to a PBG material. Typical current commercial laser diodes can have several hundred longitudinal modes within the spectral bandwidth of the gain medium, yielding a low quantum efficiency of much less than 1% (the quantum efficiency is given by the ratio of output to input powers). Single-mode operation in such a situation results from mode competition and gain saturation. Although Fabry-Perot microresonators have the advantage that there is only one high-Q longitudinal mode, the open-sided cavity couples the atoms to free-space modes; the resulting quantum efficiency is still less than 1%. The potential advantages of high-Q PBG single-mode LEDs are discussed more fully in Refs. 41 and 6. c. Other Applications There have been many other suggestions for devices based on PBG materials. The possible applications, and hence the level of fabrication precision required, depend very much on the frequency range and hence operating wavelength of the required device. Millimeter detectors can therefore be built fairly cheaply because of their relatively macroscopic periodicity (millimeters). On the contrary, optical devices will need fabrication techniques that are accurate on the scale of microns. As alluded to earlier, PBG materials can provide a new electromagnetic environment for atomic and mesoscopic physics. Mossberg and Lewenstein5' have shown that a PBG material's full band gap can drastically alter the properties of single-atom resonance fluorescence. For example, when the radiative emission on one or both of the strong-field resonance fluorescence sidebands is suppressed using a photonic band gap, the atom 51

T. W. Mossberg and M. Lewenstein, J . Opt. Soc. Am. B 10, 340 (1993).

168

P. M. HUI AND NEIL F. JOHNSON

behaves like a perfect classical dipole and is the source of coherent monochromatic radiation that is insensitive to fluctuations in the drivingfield intensity. Photon antibunching, a fundamental aspect of atomic fluorescence in free space, does not occur in the PBG frequency range. Kurizki et al.52 considered the effect on the physics of an atom of a localized mode arising from a defect in a PBG material; they show the existence of new quantum-electrodynamic effects in resonant field-atom interaction, owing to the spatially modulated standing-wave character of the field. In particular, they use a two-level atom interacting with the quantized field of these defects to show the existence of oscillatory patterns of the atomic population inversion, fluorescence spectra, and nonclassical field states. Dowling and BowdenS3 considered theoretically the behavior of two similar dipoles, with almost the same frequency, that radiate in frequency near a photonic band edge such that one of the dipoles was in the gap, while the other was in the band. They showed that although the dipole in the gap could not radiate directly, its properties could be probed via beats in the output power of the dipole lying in the band. De Martini et af.54have investigated both theoretically and experimentally spontaneous and stimulated emission in the thresholdless microlaser, which is based on a PBG material environment. Chu and Ho55 studied the spontaneous emission from electron-hole pairs in cylindrical dielectric waveguides and the spontaneous emission factor of microcavity ring lasers. They conclude that microlasers based on strongly guided single-mode dielectric waveguides are promising devices for achieving high . ~ experimen~ efficiencies and low lasing thresholds. Erdogan et ~ 1 studied tally and theoretically the enhancement and inhibition of radiation in ~~ theocylindrically symmetric, periodic structures. Bullock et u I . showed retically the possibility of using 2D PBG materials as 2D Bragg reflector mirrors for semiconductor laser-mode control. Dowling et have noted that the zero photon group velocity at the band edge of a 1D PBG corresponds to a very long optical path length; in the presence of an active 52G.Kurizki, B. Sherman, and A. Kadyshevitch, J . Opt. Soc. Am. B 10, 346 (1993); B. Sherman, G. Kurizki, and A. Kadyshevitch, Phys. Reu. Lett. 69, 1927 (1992); A. G. Kofman, G . Kurizki, and B. Sherman, J . Mod. Opt. 41, 353 (1994). 53J. P. Dowling and C. M. Bowden, J . Opt. SOC. Am. B 10, 353 (1993). 54 F. De Martini, M. Marrocco, P. Mataloni, and D. Murra, J . Opt. SOC. Am B 10, 360 (1993). 55D. Y.Chu and S. T. Ho, J . Opt. SOC.Am. B 10, 381 (1993). 56T. Erdogan, K. G. Sullivan, and D. G. Hall, J . Opt. SOC. Am. B 10, 391 (1993). 57 D. L. Bullock, C. C. Shih, and R. S. Margulies, J . Opt. SOC.Am. B 10, 399 (1993). 58 J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, J . Appl. Phys. 75, 1896 (1994). See also J. P. Dowling and C. M. Bowden, J . Mod. Opt. 41, 345 (1994); M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, J . Appl. Phys. 76, 2023 (1994).

PHOTONIC BAND-GAP MATERIALS

169

medium this increase in path length could yield a large gain, suggesting possible application in vertical-cavity surface-emitting lasers. Control of signal propagation in millimeter and microwave integrated circuits is a further application of PBG materials. The photonic gap is chosen so as to reduce the power lost by emission into the dielectric investisubstrate, and hence enhance the emission into air. Brown et gated the radiation properties of a planar bow-tie-shaped antenna based on a PBG material. They concluded that highly efficient planar antennas could be made on PBG material regions fabricated in semiconductor substrates such as GaAs. A related application is that of shielding in Y junctions in integrated optical circuits, in order to reduce leakage of radiation. Delay lines have also been suggested. The possibility of constructing highly anisotropic 3D photonic band structures has provided the motivation for investigations of highly directional radiation for imaging. Nonlinear effects in dielectric materials could be enhanced by PBG materials. By suppressing one-photon events, higher order multiphoton effects, ordinarily swamped by the one-photon effects, might be observable. In particular, two-photon lasers could be built operating at a frequency w / 2 , where w lies in the photonic band gap. These lasers have been predicted to have properties as diverse as frequency tuning, emission of squeezed states, and short-pulse generation.' Disordered PBG materials can also modify the atomic radiative dynamics. In particular, a small amount of disorder can induce localized modes in the gap; the experimental results discussed in Section 11.2 for a single defect represent the extreme limit of such localization. The analogous behavior for electrons is that of Anderson localization. For electrons, however, the presence of electron-phonon scattering and electron-electron interactions hinder experimental observation of the effect. For photons in a disordered PBG, the optical version should be observable, as first pointed out and developed subsequently by J ~ h n . ' ~ This , ~ ~ is? particularly ~' likely because photon-photon interaction and photon-phonon effects are negligible, and careful engineering should be able to control the exact disorder introduced into the PBG. Experimental work has included studies at optical frequencies in 3D random arrangements of scatterers. In a 2D disordered array of rods, localized states have been observed.62As disorder is introduced into a perfect PBG crystal, photonic band gaps will be replaced by pseudo-gaps with a small, finite density of states. Like their E. R. Brown, C. D. Parker, and E. Yablonovitch, J . Opt. Soc. Am. B 10, 404 (1993). S. John, Physics Today, p. 32 (May 1991); Physica B 175, 87 (1991). hl S. John and J. Wang, Phys. Rev. Letr. 64, 2418 (1990); Phys. Rev. B 43, 12,772 (1991). h2 R. Dalichaouch, J. P. Armstrong, S. Schultz, P. M. Platzman, and S. L. McCall, Nature 354, 53 (1991). 59

60

170

P. M. HUI AND NEIL F. JOHNSON

perfect crystal counterparts, disordered PBG materials can also therefore reduce the rate of spontaneous emission in semiconductors at optical frequencies. Genack and Garcia63 have suggested a new class of compact filters, transducers, and switches based on the characteristics of localized waves in nonabsorbing media. These characteristics include narrow resonances, giant transmission fluctuations, strong spatial correlation, and extreme sensitivity to variations in the dielectric function of the medium. More recently, the localization of light in 1D Fibonacci dielectric multilayers has been in~estigated.~~

111. Theoretical Background

4. PHOTONIC BANDTHEORY: FORMALISM In this section we provide a vector formalism for photonic properties in PBG materials that parallels the well-known scalar formalism for electronic properties in s0lids.2,~'In particular, we develop a vector representation theory employing various basis sets whose scalar analogs have proved useful in electronic systems. To date, most theoretical calculations of photonic properties in PBG materials have employed only a plane-wave basis (i.e., plane-wave representation). However, given the success of other representations in electronic systems,' it is worth investigating their photonic, vector analogs with the hope that these will yield additional insight into PBG properties. In particular we develop the crystal momentum representation (CMR), the crystal coordinate representation (CCR), and the effective mass representation (EMR). For a perfect PBG crystal, these representations are shown to lead to the vector analog of the well-known k p theory for electronic systems; the electronic k * p theory has proved to be an invaluable tool for describing the dispersion relation of electrons near an electronic band gap in semiconductor^.^ In addition, a tight-binding description of photonic band structures is obtained. For a PBG crystal plus defect, the representations lead to a vector analog of the electronic effectiveness-mass equation for the case when the defect represents a slowly varying perturbation. When the defect represents a well-localized perturbation, a generalization of the electronic Koster-Slater impurity model2 is obtained. 63

A. 2. Genack and N. Garcia. J . Opt. SOC. Am. B 10, 408 (1993).

W. Gellerman, M. Kohmoto, B. Sutherland, and P. C. Taylor, Phys. Reu. Lett. 72, 633 (1994). 64

171

PHOTONIC BAND-GAP MATERIALS

a. Wave Equations For electromagnetic waves propagating in a medium characterized by a spatially dependent dielectric constant E(r) and magnetic permeability p(r), the electric and magnetic fields satisfy the wave equations. 1 V x -V p(r)

X

1 d2 E(r, t ) + - ,.E(r)E(r, c 2 dt

t)

=

0

(4.1)

and 1 V x -V E (r)

1 d2 c2 d P

x H(r, t ) + - - (r)H(r, t ) = 0.

(4.2)

We are considering, for simplicity, systems where E and p are scalar functions of position. The extension to anisotropic systems, with a tensorial dielectric constant and magnetic permeability, is straightforward. Consider a monochromatic wave of frequency w with e-'"' time dependence; the wave equations become 1 V x -V p(r)

x E(r)

-

w2

7E(r)E(r) C

=

0

(4.3)

=

0.

(4.4)

and V

X

1 -V E (r)

X

w2

H(r) - yCp ( r ) H ( r )

These equations are of identical form and one can be obtained from the other by interchanging the roles of (E, E ) and (H, p). Solving either the E equation [Eq. (4.3)l or the H equation [Eq. (4.411 exactly will lead to the same physical results. The E equation and the H equation are similar in form to the envelopefunction equation for electrons in semiconductor heterostructures within the effective mass approximation.2 In the H equation, E(r) plays the role of a spatially dependent effective mass, u$(r)/c2 plays the role of kinetic energy, and hence p(r) is similar to a spatially dependent potential. In the E equation, p(r) plays the role of effective mass, w2E(r)/c2 plays the role of kinetic energy, and hence E(r) is now analogous to a spatially dependent potential. Further details on the analogy between the photonic and elecThere are, tronic problems have been discussed by Henderson et however, fundamental differences. Photonic problems deal with vector

172

P. M. HUI AND NEIL F. JOHNSON

fields. Charge and the Pauli exclusion principle do not enter into the problem because photons are chargeless, spin-one entities. In addition, the photon-photon interaction is not important, in contrast to the situation in solids where Coulomb interactions can be significant. In PBG materials, the dielectric constant is periodic and obeys

where R is any lattice vector. The PBG materials considered to date have been nonmagnetic, hence p = 1 everywhere. The E and H equations now become V

X

V

X

w2

E(r) - x E ( r ) E ( r ) C

=

0

(4.6)

=

0.

(4.7)

and 1 V x -V 4r)

X

w2

H(r) - x H ( r ) C

Equations (4.6) and (4.7) form the starting point for many of the reported photonic band structure calculations in PBG materials. We note that it is also possible" to start with an equivalent equation, similar to Eqs. (4.6) and (4.71, for the displacement field D(r). As is discussed in Section 111.5, some of the earlier 3D PBG calculations used a scalar wave equation of the form.

The scalar wave equation has the advantage of being computationally easier to solve. However, it can lead to qualitatively incorrect results for 3D photonic band structures. In contrast, for lower spatial dimensions, a scalar wave equation can be exact. In particular, for 2D PBG crystals, the vector wave equations for the TE (transverse electric) and TM (transverse magnetic) polarizations can be exactly reduced to equivalent scalar wave eq~ations.''-~~ In this section, we will keep the formalism as general as possible by dealing directly with the exact vector wave equations. 65

T. K. Gaylord, G. N. Henderson, and E. N. Glytsis, J . Opt. SOC. Am. B 10, 333 (1993). N. F. Johnson, P. M. Hui, and K. H. Luk, Solid Stare Commun. 90, 229 (1994). 67P.M. Hui, W. M. Lee, and N. F. Johnson, Solid State Commun. 91, 65 (1994). 66

PHOTONIC BAND-GAP MATERIALS

173

b. Representation Theory Bloch Functions The eigenfunctions of Eqs. (4.6) and (4.7) satisfy Floquet's therorem and are Bloch functions of the form

(4.10) where n is a band index and k lies within the first Brillouin zone (BZ). The functions u;(k, r) and uy(k, r) are periodic in r with the same periodicity as &), and C! is the volume of the crystal. The corresponding eigenvalues are oik,yielding the band structure or the dispersion relation. The Bloch functions form the basis set for the crystal momentum representation (CMR). The orthogonality relations are

and

where the integral is over the whole volume a. Note the presence of the weight function E(r) in the orthogonality relation for the eigenfunctions of the E equation. The orthogonality relation for the H Bloch functions should similarly be weighted by ,u(r), which has been taken to be unity throughout the system. Plane- Wave Expansion Most photonic band structure calculations are based on the plane-wave expansion (PWE) method. Since u(k,r) is periodic, it can be expanded in terms of plane waves with wave vectors G being reciprocal lattice vectors. Thus, E(r) and H(r) can be expanded as

(4.13) Hk(r) = ~ H k ( G ) e ' ( k + G ) ' r . G

(4.14)

174

P. M. HUI AND NEIL F. JOHNSON

The periodic dielectric constant E(r) can be expanded in a similar fashion as (4.15) The Fourier coefficients E(G) can be chosen to be real and expressed as (4.16) where the integration is over the volume u, of a unit cell. In some systems, such as isolated spheres of one dielectric in another, E ( G )can be calculated analytically; in other systems it has to be calculated numerically.' Substituting Eqs. (4.13) and (4.14) into the original E and H wave equations yields matrix equations for the Fourier coefficients E,(G) and H

and v(G - G')(k

+ G ) x H,(G')

w2

=

0, (4.18)

where v(G) are the Fourier coefficients of the inverse of the dielectric constant, that is, of the function v(r) = l/E(r). For each k, Eqs. (4.17) and (4.18) each represents an infinite-dimensional matrix equation. Diagonalizing the matrix gives infinitely many w,(k), each of which is labeled by a band index n. Repeating the calculation for different wave vectors gives the band structure. In principle, solving either Eq. (4.17) or (4.18) gives identical results. However, the infinte-dimensional matrix problem must be truncated in practice. After truncation, it is not generally the case that the E and H equations give identical results, even when the same finite number of plane waves is used. Equation (4.18) is a standard eigenvalue problem and is often used for photonic band calculations. Two methods have been used to obtain the Fourier coefficients v(G - G I . They can be obtained by Fourier transforming the function l / E ( r ) directly or by inverting the matrix E(G - G') obtained by Fourier transforming the function 4r). It has been found that the latter approach leads to a faster conver-

PHOTONIC BAND-GAP MATERIALS

175

gence of the eigenvalues than direct transformation of l/E(r), and that the eigenvalues obtained are close to those calculated from the E equation when the same finite number or plane waves is used.8 Implementation of PWE methods in practice is discussed further in Sec. 111.5. Equations (4.17) and (4.18) are the general matrix equations for PBG materials. The size of the H equation matrix, for instance, can be reduced by noting that V . H = 0. Equation (4.14) implies that H,(G) can be regarded therefore as having only two components, each of which is orthogonal to k + G. Thus, H,(G) can be expressed as

where Zk, (G) is a unit vector and {k + G, ;,,(GI, zk2(G))forms a triad. Similar reduction in the size of the E equation matrix results from noting that V . V X E = 0. These equations may be further reduced to simpler forms in some special cases, for example, in 2D systems in which an ordered array of parallel rods of one dielectric is embedded in a second dielectric. In this case, the size of the matrix can be reduced by decoupling the fields into two orthogonal polarizations. Of particular interest in such a 2D PBG material is the TM polarization where the magnetic field is transverse to the rods (and E is parallel to the rods) and the TE polarization where the electric field is transverse to the rods (and H is parallel to the rods). These two situations are discussed further in Sec. 111.5. Wannier Functions Whereas the Bloch functions give a k-space description, a real space description can be achieved by defining Wannier functions. Suppose that the Bloch functions Hnk(r) of the H equation are known; then the Wannier functions can be defined in terms of the set of Bloch functions within a band n as (4.20)

where the sum is over k s within the first BZ and N is the number of unit cells. Because there are N allowed k values in the first BZ, Eq. (4.20) simply defines N Wannier functions associated with the nth band by forming suitable linear combinations of N Bloch functions within the band. These functions are localized around the lattice point labeled by R.

176

P. M. HUI AND NEIL F. JOHNSON

The orthogonality relation is

where the integration is over the crystal volume R. The Wannier functions form a complete set for all n and R. They form the basis set for the crystal coordinate representation (CCR), which is useful in describing the effects of impurities. For ordered PBG materials, the Wannier functions can be used to formulate a tight-binding description of the band structure. Similar constructions can be carried out for the Bloch functions of the E equation, leading to Wannier functions E,(r - R), which are localized around the site R. The orthogonality relation for E,(r - R) also carries the weight function 4 r ) as in Eq. (4.11). Kohn-Luttinger Functions and k .p formalism The Kohn-Luttinger functions@ xnk(r)of the H equation can be defined in an analogous way to those in the electronic problems as

(4.22) where k, is some fked wavevector within the BZ. The Kohn-Luttinger functions satisfy the orthogonality relation (4.23) Kohn-Luttinger functions can be similarly constructed for the E equation. To set up a k p formulation for photonic bands, we can start with either the H equation or the E equation. We will work with the H equation. We expand H(r) in Eq. (4.7) in terms of Kohn-Luttinger functions

-

(4.24) where the sum is, in principle, over all bands. The expression of eigenfunctions in terms of the Kohn-Luttinger functions is usually referred to as the efectiue mass representation (EMR). A set of equations for the expansion coefficients anj(k) can be obtained by substituting Eq. (4.24) into the wave 6M

J. M. Luttinger and W. Kohn, Phys. Rev. 97, 869 (1955).

177

PHOTONIC BAND-GAP MATERIALS

equation [Eq. (4.711 and using the orthogonality relation Eq. (4.23). The resulting equation is

[2( 1

I

wj)[) - W i k ) 6 / j - p/,(k)]a.,(k) =

(4.25)

O,

where

Here s = k - ko is the deviation from the wavevector k o . Two types of terms are involved: The first few terms are linear in s = Isl; the last two terms are quadratic in s. The s . p term is analogous to the k p term in the electronic problem in that p(rj, involves integrals over a unit cell of volume u, of two u's and a differential operator,

-

Owing to the vector nature of the problem, p(,j, is more complicated than the momentum matrix element for the electronic problem, which is proportional to /@Vuj dr, where the u's are scalar functions in the electronic case. The simplifying feature is that the whole integral can be treated as a parameter as in the electronic problem, and can either be obtained by fitting to band structures calculated using numerical methods, o,' by fitting to available experimental data. The a@element of the matrix Q(rj,is given by

where [u], is the a component of u. In Eq. (4.261, the matrix and

6

q(/j) = TrG(/j)*

GT is the transpose of (4.29)

178

P. M. HUI AND NEIL F. JOHNSON

The k * p formalism is exact up to this point. Solving for the photonic band structure mik in Eq. (4.25) amounts to finding the eigenvalues of a matrix H with elements Hlj given by

In principle, the band indices 1 and j run over all bands and the matrix is of infinite dimension. In practice, the matrix will be truncated to sizes that can be handled either analytically or numerically without performing heavy computations. As is shown in Section 111.5, analytically solvable two- or three-band models are often sufficient to describe the dispersion of the bands accurately if one is only interested in the dispersion around k, . It is an advantage of the k . p method that, after truncation, only a few parameters are needed to characterize the band structure in the region of interest around the photonic band gap. The dispersion relation of a particular band n around k, can be obtained using perturbation theory. In this case, k = k, and thus s is small. Keeping terms to second order in s, we have

(4.31)

where

Motivated by the form of the dispersion relation in a uniform medium w 2 = c 2 k 2 /(€PI, we define 1

[GF].,

=

1 a2(m,2k) 2c2 aka ak, '

(4.33)

a quantity related to the reciprocal of the refractive index. For nonmagnetic materials, we have p = 1 everywhere, and Eq. (4.33) defines a reciprocal effective dielectric tensor for the material. Using Eq. (4.31) for

179

PHOTONIC BAND-GAP MATERIALS

w : ~ ,we have

which is analogous to the f-sum rule for the reciprocal effective mass tensor in the electronic problem.* A similar k . p approach can be formulate starting with the E equation. Results can be easily obtained from those of the H equation by noting the identical forms of the original wave equations [Eqs. (4.3) and (4.411. We note that this photon k * p formulation may be used for studying “dielectric superlattices” consisting of alternating layers of different periodic dielectric crystals and “dielectric heterostructures” in a way that is analogous to the k * p envelope-function approach employed in treating semiconductor h e t e r o ~ t r u c t u r e-. ~”~ c. Defects As in electronic problems, the introduction of defects and impurities in PBG materials leads to defect states (modes) in the gap. These states behave as donor or acceptor states depending on the characteristics of the defect. As mentioned in Section 11.3, such defect modes are useful because many optical devices require some well-defined photon modes to operate. Numerical calculations of such modes are further discussed in Section 111.5. The disorder of a dielectric structure can be described by a positiondependent dielectric function E(r). This function can be written E(r)

=

E(r)[l

+ V(r)l,

(4.35)

where 4 r ) is periodic in real space and E(r)V(r) describes the deviation from the periodic d r ) at any point r. The detailed mechanisms of the disorder is not important here; it could be positional disorder, a missing or additional dielectric object, shape or size irregularities, or a periodic structure consisting of two or more kinds of dielectric embedded in a host (i.e., a PBG alloy). Substituting Eq. (4.35) into the H equation yields V

X

1

-V E(r)

X H(r)

+ V X U(r>V X H(r)

w2

-

TH(r) C

=

0, (4.36)

N. F. Johnson, H. Ehrenreich, P. M. Hui, and P. M. Young, Phys. Reu. E 41,3655 (1990). N. F. Johnson, P. M. Hui, and H. Ehrenreich, Phys. Rev. Lett. 61, 1993 (1988). ” N. F. Johnson, Ph.D. Thesis, Harvard University (1989). 69

711

180

P. M. HUI AND NEIL F. JOHNSON

where (4.37) Without the second term, this is the wave equation for an ordered PBG material from which the Bloch functions, Wannier functions, and Kohn-Luttinger functions can be constructed (see Section III.4.b). A general method to solve the problem is to expand H(r) in terms of some complete set of basis functions, such as Bloch functions, Kohn-Luttinger functions, or Wannier functions leading to the CMR, EMR, and CCR representations, respectively. Crystal Momentum Representation of Bloch functions as

We expand H(r) in Eq. (4.36) in terms

(4.38)

where A , , =,,(nklH> are the expansion coefficients and thus the wave function in the CMR. Substituting into Eq. (4.36) leads to

(4.39) which is the CMR equation for a defect in a PBG material. Effective Mass Representation Luttinger functions as

We expand H(r) in terms of the Kohn-

(4.40) where the C,, are the expansion coefficients and hence the wave function in the EMR. Substituting Eq. (4.40) into Eq. (4.36) gives the EMR

181

PHOTONIC BAND-GAP MATERIALS

equation for a defect in a PBG material:

where Pnn,is given by Eq. (4.26). In the ordered case, the last term in Eq. (4.41) vanishes and the equation reduces to the k . p formalism of Sec. 111. 4.b. Cystal Coordinate Representation We expand H(r) in terms of Wannier functions as

(4.42) where BnR=,(nRIH) are the expansion coefficients and hence the wave functions in the CCR. Substituting in Eq. (4.36) gives the CCR equation for a defect in a PBG material:

+ c [/H:(r n' R'

-

R) . V

X

U(r)V

X

1

H,,(r - R') dr BntR,= 0, (4.43)

where we have defined the function (4.44) Without the disorder term (U = 0), this CCR formulation in Eq. (4.43) amounts to a tight-binding approach based on the Wannier functions centered around each lattice point. The photonic dispersion relation can hence be regarded as resulting from overlaps of neighboring Wannier functions. In principle, the hopping integrals can be calculated from first

182

P. M. HUI AND NEIL F. JOHNSON

principles. An empirical approach would be to fit the dispersion relation obtained by numerical methods, such as PWE, to the tight-binding form

where a i , y i , and Si are the tight-binding parameters for the ith band. The parameter a describes the position of the band, and y and 6 are the nearest neighbor (nn) and next nearest neighbor (nnn) overlap integrals, respectively. Empirical formulas for these parameters as a function of the system variables would provide a valuable tool for gaining insight into photonic band structures over a wide range of PBG materials, without going through extensive computations. Such an approach is similar to the empirical tight-binding method developed in solid state physi~s.’~In Sec. 111.5, we will show that such an approach can yield highly accurate photonic band structures. For the case of a slowly varying defect perturbation, the CCR equation [Eq. (4.43)] can be used to formulate an envelope-function approach that is analogous to the effective mass approximation treatment of shallow impurities in a semiconductor. Consider Eq. (4.43); for a slowly varying perturbation U (r) (i.e., shallow impurity), Bloch states in the vicinity of the extremum of band n’ will become mixed just as in the electronic problem. Expanding the dispersion relation around the extremum at k , in terms of the x, y, and z components of the wavevector k yields

where A,,, are general coefficients. Using standard summation relations for plane waves2 it follows from Eq. (4.44) that

, where w;,(k’ -+ -iVRr + k,) denotes the dispersion relation o ; , ~from Eq. (4.46) with k’ replaced by -iVR, + k , . The vector operator VR, corresponds to differentiation with respect to the components of R . The 12

W. A. Harrison, Electronic Structure and the Properties of Solids, W. H. Freeman and Company, San Francisco (1980).

183

PHOTONIC BAND-GAP MATERIALS

CCR equation can now be rewritten as

+

nR

[

/H:,(r - R’) * V

X

U(r)V

X

I

H,(r - R) dr B,,

=

0, (4.48)

and is still exact. The integral in the last term can be rewritten as73 M,,,z(R’,R) = / [ V x H,,(r - R’)]* U(r)[V x H,(r - R)] dr. (4.49)

We now assume that U(r) varies slowly over a unit cell and over the spatial extent of the Wannier function. Since V X H,(r - R) is expected to be localized around the site R, following the behavior of H,(r - R), the overlap between the curl of Wannier functions centered around different sites will be small. The perturbation U(r) can be taken to be constant over the region of space for which the integral is appreciable. Assuming that the coupling between different bands is small, the approximation M,,,(R’,R) = 8,,t8RR,AntU(R) can be made, where d,,is an integral defined as in Eq. (4.49) but with U(r) set to unity, n = n’, and R‘ = R. The one-band equation for a slowly varying perturbation is therefore given by

[

$wa(k’

4

-iVRt

+ k,) + U(R’)d,, -

B,,,,

=

0.

(4.50)

The analogy with the effective mass equation for semiconductors becomes clear when we replace R’ by the continuous variable r and we rename the “envelope function” B,,,, as Fnr(r).2,71 The effects of the periodic 4 ) are embedded in the dispersion relation w : ~ ,which appears in the first term. As a simple example, consider the low-frequency limit where the photonic dispersion relation of the ordered system around k, = 0 can be written as

73

K. M. Leung, J . Opt. SOC.Am. B 10,303 (1993).

184

P. M. HUI AND NEIL F. JOHNSON

where l/Eeff is the inverse of the effective dielectric constant. In this case, Eq. (4.50) becomes (4.52) Note that for systems without cubic isotropy the effective dielectric function Eefl will be a tensor, analogous to the effective mass tensor for electrons. Next consider the technologically interesting case where the dispersion relation of the ordered system has a finite-frequency photonic band gap at k, = 0, and the dispersion relation in the vicinity of the gap can be written as w,(k) = o,(O)+&k2.

(4.53)

Squaring w,(k) and retaining terms up to order k 2 , Eq. (4.50) now becomes

We note that this envelope-function formulation can also be obtained starting from the equivalent CMR and EMR equations, respectively. A convenient way to study the impurity problem is by means of Green’s functions. A Green’s function can be defined for the nth band c2

GAo)(R’- R, o) = - C N

&k.(R‘-R) W’

(4.55)

-

The CCR equation can then be rewritten in terms of Green’s functions as

X

[IHz,(r - R’) V *

X

U(r)V

X

H,(r

-

1

R) dr BnR = 0, (4.56)

which is still exact. The Green’s function method will be particularly useful when coupled with the empirical tight-binding approach because G(O) can then usually be expressed in terms of complete elliptical integrals if only 74G.F. Koster and J. C. Slater, Phys. Reu. 95, 1167 (1954).

PHOTONIC BAND-GAP MATERIALS

185

nearest neighbor couplings are important, or obtained numerically using the tight-binding form of wnk when next nearest neighbor couplings have to be taken into account. This method amounts to taking the second term in Eq. (4.36) as the perturbation term and then constructing the total Green’s function G of the perturbed problem using the unperturbed Green’s function G‘”’. The poles of G provide information regarding the energy of the defect modes. As an illustration, we assume that the photonic bands can be treated independently. Equation (4.56) now takes the simpler form

with M given by Eq. (4.49). Consider a single impurity problem in which a region of one of the dielectrics E , in the unit cell, say, at the origin, is replaced by a third dielectric with dielectric constant e C .In a 3D (2D) system with spheres (cylinders) of one dielectric positioned within another, this can be visualized as being the replacement of the sphere (cylinder) centered at the origin by another dielectric sphere (cylinder) of the same = U within the volume of the size. In this case, U(r) = - & / ( E , E , ) replaced region, and is zero elsewhere; here 6~ = E , - E , is the difference in the dielectric constants of the impurity and the original material. Since the Wannier functions are localized around lattice sites, the integral in Eq. (4.57) can be approximated as M,,,,(R’,R ) = ~ 5 S R ~ ,U ~ jn ” ,where j,,is an integral defined in the same way as An but with the integration only carried out over the region of the replaced material. Note that 2, has implicit w dependence through Maxwell’s equations. The defect modes can then be obtained by solving the equation (4.58)

and identifying roots having frequencies outside the continuum of the band. This is reminiscent of the Koster-Slater treatment for a localized impurity in the electronic Finally we note two interesting extensions of the defect problem in PBG materials. The first concerns a single defect (impurity) with nonlinear optical properties situated in a PBG crystal. It has recently been found for the electronic case75.76that nonlinear impurities can lead to results concerning the existence of a bound state that are qualitatively different from ”M. 1. Molina and G. P. Tsironis, Phys. Reu. B 47, 15,330 (1993), Phys. Reu. Lett. 73, 363 (1994). “K. M. Ng, Y. Y. Yiu, and P. M. Hui, Solid State Comrnun. 95, 801 (1995).

186

P. M. HUI AND NEIL F. JOHNSON

the usual linear problem. Similarly in PBG materials, the presence of a nonlinear defect could play an important role in determining whether or not a truly localized photon mode exists in the photonic band gap. Such a defect may, for example, consist of a sphere made of a frequency-dependent dielectric. This would provide an additional degree of freedom to tune the position of localized modes. The above Green’s function formulation can be generalized to study such nonlinear impurities. The second situation concerns a PBG crystal containing many impurities. This problem becomes one that is similar to the alloy problem in solid state physics.2.77 Standard techniques, such as the average t-matrix approximation and the coherent potential a p p r o ~ i m a t i o ncan ~ ~ be applied to study this problem using the tight-binding picture78 or k . p formalism79 as a starting point.

5. THEORETICAL CALCULATIONS AND RESULTS We now turn to a discussion of the various theoretical approaches that have been employed in practice to calculate photonic properties in PBG materials. These methods are basically generalizations of techniques originally developed for electronic calculations. We also discuss the extent to which the theoretical predictions are in agreement with the results of experiments discussed in Section 11.2. As mentioned in Section 1.1, the fact that there are an infinite number of possible PBG crystals means that we cannot provide a complete catalog of all possible photonic band structures. Instead we choose to focus on a few representative theoretical calculations and compare them with the corresponding experimental results, while emphasizing trends that have emerged concerning photonic properties as a function of PBG material parameters. Many theoretical papers have attempted to solve either the exact vector wave equation (Section III.4.a) or a simplified scalar wave equation. These calculations are essentially numerical and computationally intensive. For both vector and scalar equations, the most prevalent approach to obtaining solutions has been based on a plane-wave expansion of the position-dependent fields as discussed in Section III.4.b. This approach will in principle provide exact solutions of both the vector and scalar wave equations if an infinite set of plane waves is considered; the resulting matrix to be diagonalized is infinite dimensional [Eqs. (4.17) and (4.1811. In practice, a 77 H. Ehrenreich and L. M. Schwartz, in Solid Stute Physics (H. Ehrenreich, F. Seitz, and D. Turnbull, eds.), Vol. 31, p. 149, Academic, New York (1976). 78 E. N. Economou, Green’s Functions in Quantum Physics, Springer Series in Solid Stute Sciences, Vol. 7, Springer-Verlag, Berlin (1979). 79 E. Siggia, Phys. Reu. B 10, 5147 (1974).

PHOTONIC BAND-GAP MATERIALS

187

finite set of plane waves must be employed. Given a large enough computer and sufficient computing time, such a calculation would indeed yield numerical answers which converge to the exact result. For the case of the scalar wave equation, the similarity to the scalar Schroedinger equation meant that plane-wave algorithms and numerical techniques developed in standard electronic band theory calculations could be used directly. For the vector wave equation, the numerical routines had to be modified to account for the vector nature of the fields. In addition, attention had to be paid to the errors introduced as a result of the plane-wave truncation procedure. The size and origin of these truncation errors, and hence the resulting accuracy of the calculations, depend on the specific truncation procedure employed. Such errors can be more significant than in the electronic case because of possible discontinuities in, for example, the electric field at a boundary between two dielectrics; a large number of plane waves may be needed for an accurate representation of the field and convergence may be slow. Similar difficulties arise in describing the step-like discontinuity in the dielectric constant at the boundary. This general difficulty of representing a discontinuous function by a finite series of plane waves is a manifestation of Gibb's phenomenon in Fourier transform theory. Gaussian functions and high-order supergaussian functions that provide good convergence have been The results of the numerical calculation after truncation may also depend on whether one is solving for B(r), D(r), E(r), or H(r). A detailed review of PBG photonic band structure properties based on a plane-wave analysis is given by Villeneuve and Piche.8 Initially plane-wave expansions were used to solve the simpler scalar wave equation for 3D structures, as opposed to the full vector equations. The hope was that the results would be qualitatively, and perhaps quantitatively, similar to the results of a full vector theory. Examples of such a numerical calculation include Refs. 16, 19, 25, and 80. It soon became clear, however, that the scalar wave approximation" was not reliable for 3D PBG crystals." In particular, the details of the photonic band gap predicted by the scalar wave theory in 3D were often quantitatively different from those observed experimentally, for example, for the fcc structure. In fact, the scalar wave calculations predicted that many structures should be capable of exhibiting full photonic gaps in 3D,30whereas experimentally the opposite was found to happen. xu

S. John and R. Rangarajan, Phys. Reu. B 38, 10101 (1989). Born and E. Wolf, Principles of Opfics,Pergamon, Oxford (1965). 82 As noted in Section III.4.a. the vector wave eqtiation for TE and T M polarizations in a 2D PBG can be reduced t o scalar form. " M.

P. M. HUI AND NEIL F. JOHNSON

188

Leung and Liu2' and Zhang and Satpathy26 carried out a numerical plane-wave solution for the full vector wave equations in a 3D PBG crystal. It was thought that an fcc structure, with its nearly spherical Brillouin zone, would be the most promising candidate for a full photonic band gap. They therefore chose to examine theoretically a 3D fcc structure of spherical airholes in a background dielectric. These calculations showed degeneracies that were symmetry induced and therefore prevented a photonic band gap. Ho et aL2' turned to the 3D diamond structure, which broke the near-spherical symmetry. They found that a photonic band gap could form in this structure. In particular, for diamond structures comprising either airholes in a dielectric (refractive index 3.6) or dielectric spheres in air, they predicted full photonic band gaps of around 20% of the mid-gap frequency. Their results are summarized in Figs. 3 and 4. Ho et al. chose to consider the wave equation in H(r) rather than E(r) or D(r). The motivation was the fact that H(r) is continuous at an interface between media with different dielectric constants, whereas E(r) and D(r) are both discontinuous. Despite the discontinuity in the first derivative of H(r), the H(r) expansion is therefore expected to have a faster convergence. They then expanded in a basis of just transverse plane waves, rather than expanding in a complete basis, solving for the normal modes, and then projecting the longitudinal solutions. Subsequently, it was realized that any choice of nonspherical atoms should lift these symmetry-induced degeneracies mentioned earlier, yield0.7 0.6

0.5

z

0.4

W

3

a

LL

0.3

0.2 0.11

0.0

*

1 1 U

L

/J/ r

1 1 X

W

K

Wavevector

FIG. 3. Theoretical photonic band structure, calculated numerically using a plane-wave expansion, for a 3D diamond structure of dielectric spheres of refractive index 3.6 in an air background. A filling fraction of 34% for the dielectric implies the spheres are just touching each other. A full photonic band gap appears between the second and third bands. Frequency is in units of c/a where a is the lattice constant. [Taken from Fig. 2 of K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Reu. Lerr. 65, 3152 (19901.1

189

PHOTONIC BAND-GAP MATERIALS

-

-o- air spheres

0 0.25

0.0

0.4

0.2

--c--

0

dielectric spheres

1

1.0

0.8

0.6

filling ratio dielectric spheres

air spheres

2

3

. ....c--*

4

5

6

-...

0’

7

refractive index ratio FIG. 4. Ratio of full photonic band gap width to the mid-gap frequency in a 3D PBG diamond structure, shown as a function of (a) the filling fraction and (b) refractive index ratio. Results were obtained numerically using a plane-wave expansion. The two cases of airholes in a dielectric and dielectric spheres in air are shown. The dashed line in (b) corresponds to air spheres occupying 81% of total volume; the solid line corresponds to dielectric spheres with a filling fraction of 34%. These filling fractions are chosen so as to maximize the gap. As in Fig. 3, the dielectric refractive index is 3.6 and the gap appears between the second and third bands. [Taken from Fig. 3 of K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).]

ing a full 3D photonic band gap. Yablonovith et ~ 1 then . fabricated ~ ~ such a structure with nonspherical airholes (see Section 11.2) and found experimentally a full 3D photonic band gap in agreement with Ho et d ’ s vector ’ predicted, based on plane-wave calculations, theory. Chan et ~ 1 . ~ have that the diamond structure with a refractive index contrast as low as 1.87

190

P. M. HUI AND NEIL F. JOHNSON

to 1 will still open up a full photonic gap. In addition, the simple cubic structure was found to produce a gap.”*R3This finding is interesting since such simple-cubic systems are likely to be easier to fabricate in 3D than those with complex unit cells. Soziier and Hauss3 reported theoretical results for both spheres and scaffold structures in 3D. As mentioned earlier, it had been thought that a spherical Brillouin zone would favor a large photonic band gap. Soziier and Haus showed, however, that the bcc (body-centered-cubic) structure, whose Brillouin zone is rounder than simple cubic, had surprisingly small gaps. By solving the vector equations in 2D using the plane-wave expansion, it was shown theoretically 32.39,R4 that the 2D hcp (hexagonal-close-packed) lattice of long parallel rods should yield a full photonic gap in the perpendicular plane. Also structures such as arrays of air rods with square cross section in a background dielectric have been predicted to give 2D photonic band gaps.39 Maradudin and McGurnR5have studied the effect of finite rod lengths on 2D photonic properties. Using the plane-wave numerical routines of Meade et al., Robertson et u1.43v44obtained good agreement between the theoretical dispersion relation and the experimental results of the COMITS measurements for 2D square and ’triangular lattices of dielectric rods. An example of this agreement between theory and experiment is shown in Fig. 2. In addition, both theory and experiment found that transmission via certain modes was forbidden by symmetry. ~ undertook a study of two complementary 2D PBG Meade et ~ 1 . ’ then structures in order to try to understand what features determine whether a given PBG material will have a large or small photonic band gap. The advantage in carrying out this theoretical investigation in a 2D, as opposed to 3D, PBG material is that the eigenmodes of the 2D system are either TM with electric field parallel to the rods, or TE with the electric field perpendicular to the rods. The polarizations in a 3D system are much harder to visualize. The separation of these modes in 2D allows independent investigation of the factors underlying the presence of a gap for each polarization. Their numerical results for the 2D dispersion relations are shown in Fig. 5. Note that the sample considered in Fig. 5(a) corresponds to the COMITS sample, which was featured in Fig. 2. Meade et ~ 1 . ’ ~ concluded that it is concentrations of dielectric material that are important for developing a gap for the TM polarization, while connectivity within the 83H. S. Soziier and J. W. Haus, J . Opt. SOC. Am. B 10, 296 (1993). 84 J. M. Gerard, A. Izrael, J . Y. Marzin, R. Padjen, and F. R. Ladan, Solid State Electron. 37, 1341 (1994). n5 A. A. Maradudin and A. R. McGurn, J . Opt. SOC. A m . B 10, 307 (1993). 86 R. D. Meade, A. M. Rappe, K. D. Brornrner, and J. D. JWdnnOpOUlOS, J . Opt. SOC.Am. B 10, 328 (1993).

PHOTONIC BAND-GAP MATERIALS

191

0.8 0.7

0.6 0.s 0.4

0.3 0.2 0.1

0

0.6

0.5 0.4

0.3 0.2

0.1

0

Fic;. 5. Theoretical photonic band structure cdkulated numerically using a plane-wave expansion, for two 2D photonic (PBG) crystals. In (a) the PBG crystal is the same as that in Fig. 2. In (b) the PBG consists of a square array of square holes (side length 0 . 8 4 ~ ~in) a dielectric with E = 8.9. Cross sections of the crystals are shown in the insets. Solid lines represent TM modes [fields along ( E , , H,, H,,)]; dashed lines represent TE modes [fields along ( H , , E x , E , )I. Brillouin zones are shown as insets. [Taken from Fig. 1 of R. D. Meade, A. M. Rappe, K. D. Brommer, and J . D. Joannopoulos, J . Opt. Soc. Am. B 10, 328 (1993).]

192

P. M. HUI AND NEIL F. JOHNSON

plane is important for band gaps in TE polarization. At low dielectric contrast (e.g., 13 for GaAs) the structure that optimizes both of these, and therefore produces a gap for both polarizations, is a triangular lattice of air columns; at a higher index contrast the optimal structure is a square lattice of air columns. More recently, Ho et have predicted, using the numerical plane-wave expansion, a novel 3D PBG material with a full photonic band gap. The 3D PBG crystal is made by stacking layers of dielectric rods that repeat every four layers. The rods can be circular, elliptical or rectangular in shape. A sample with this crystal structure was then built by Ozbay et u I . , ~ ~ as discussed in Section 11.2. The sample showed a mid-gap attenuation in good agreement with the theoretical values. Chan et al." have predicted, using the same plane-wave expansion as Ho et af.,a new class of structures with rhombohedra1 symmetry that possess large photonic gaps. They point out that most of the 3D structures known to date to have full gaps fall into this broad class of so-called A7 structures." Chan et note that the local connectivity, that is, the number of rods per joint, of the structure is very important in determining the existence of a full gap; this finding is ' Chan et al. consistent with the earlier conclusions of Meade et ~ 1 . ~Indeed find that structures composed of a high-dielectric component forming a percolating network with the lowest coordination number seem to favor a full gap. Their analysis suggests that diamond, with its near-spherical Brillouin zone and low connectivity, should still be the preferred structure for PBG materials. Theoretical calculations of photonic dispersion relations have also been carried out recently for PBG materials where one of the constituent materials is either or superconducting." In addition the plane-wave expansion method has been use to calculate the effects of anisotropy in a photonic crystal with spheres made from an anisotropic dielectric?' Other anisotropic PBG materials studied have included exotic cholestric blue phases.y3 Enhanced dispersion forces in ordered colloidal suspensions have been shown theoretically to alter the phase diagram for hard-sphere transitions in such system^.'^

87

C. T. Chan, S. Datta, K. M. Ho, and C. M. Soukoulis, Phys. Rev. B 50, 1988 (1994). "J. Slater, Quantum Theory of Molecules and Solids, vol. 2, McGraw-Hill, New York (1965); L. M. Falicov and S. Colin, Phys. Rev. 137, A871 (1965). 89 A. R. McGurn and A. A. Maradudin, Phys. Rev. B 48, 17,576 (1993). 90 R. M. Hornreich, S. Shtrikman, and C. Sommers, Phys. Rev. B 49, 10,914 (1994). "M. W. Lee, P. M. Hui, and D. Stroud, Phys. Rev. B 51, 8634 (1995). y21. H. H. Zabel and D. Stroud, Phys. Reu. B 48, 5004 (1993). 93R.M. Horneigh and S. Shtrikman, Phys. Rev. E 47, 2067 (1993). 94 C. L. Adler and N. M. Lawandy, Phys. Rev. Lett. 66,2617 (1991).

PHOTONIC BAND-GAP MATERIALS

193

For PBG materials with defects, ideally one would like to calculate the properties of a single defect in an otherwise perfect (infinite) PBG material. Such an approach is computationally prohibitive using a plane-wave expansion. Instead the plane-wave expansion has been used to describe defects placed in a finite unit cell of PBG material using the supercell m e t h ~ d . ~The ~ , ~supercell ' method places a single defect in a single cell, which is then repeated. Meade et aLZ2used 8-atom conventional cells of they decided to deal with the H(r) field diamond. Following Ho et because of its continuous nature at dielectric boundaries. They expanded the H(r) field in a basis of plane waves up to a finite frequency; they included about 130 plane waves per polarization per primitive unit cell. Based on a test case calculation for a 1D system of periodic dielectric slabs for which an exact solution was available, they estimated the frequency error of their numerical routine to be within 5%. However, they found that the convergence in frequencies was rather slow compared to electronic structure calculations; they deduced that this slow convergence was related to the discontinuity in the first derivative of H(r) at the dielectric boundaries. They sampled frequencies at 48 k points in the irreducible Brillouin zone of the 8-atom unit cell in order to calculate the density of states of the PBG energy spectrum and then coarse-grained the frequencies. The coarse graining of the Brillouin zone led to some difficulty in reproducing the typical density of states at low frequencies. Meade et a1.22 chose to study defects in the diamond structure studied by Ho et al." with air spheres in a background dielectric, because of the prediction of a large photonic band gap. They considered two types of impurities22: dielectric spheres in air, located in the bond-center site, and air spheres in the dielectric, located at the hexagonal site. The air sphere acted like a repulsive potential in that it attempted to repel the field lines from it; the air sphere therefore pushes a state out of the valence band. The dielectric sphere attracted field lines and hence acted as a negative potential. It therefore pulls a state down from the conduction band. Subsequently, Meade et managed to improve on the accuracy of these theoretical calculations by drastically reducing the computational time required in their plane-wave calculations; the convergence rate of the calculation was subsequently increased. This improvement was achieved by exploiting properties of particular matrix operations involved in the planewave numerical routines. Typically the improved method, which was still supercell-based, employed 750,000 plane waves for each eigenmode; it was estimatedy5 that the resulting frequencies converged to better than 0.5%. R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and 0. L. Alerhand, Phys. Reu. B 48,8434 (1993). 95

194

P. M. HUI AND NEIL F. JOHNSON

2 u. 0.450

’ 1

’

2

’ ’ ‘ 3 4 5 Defect Volume

’ 6

’ 7

-

8

FIG.6. Comparison of experimental and theoretical (plane-wave expansion) defect mode frequencies for the 3D fcc photonic crystal considered in Fig. 1. The defect volume is normalized by A/n’ where n is the refractive index and A is the mid-gap vacuum wavelength. Experimental values (obtained from sample considered in Fig. 1) are shown as solid circles. “Old Theory” from E. Yablonovitch, T. J. Gmitter, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Reu. Lett. 67, 3380 (1991), is shown as a dotted line. “New Theory” with improved plane-wave convergence is from R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and 0. L. Alerhand, Phys. Reu. B 48, 8434 (1993). Solid lines are nondegenerate modes while dotted-dashed lines are doubly degenerate. Modes on the left result from the air impurity (acceptor); modes on the right result from the dielectric defect (donor). [Taken from Fig. 4 of R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and 0. L. Alcrhand, Phys. R ~ L B, . 48, 8434 (1993).]

Figure 6 compares the original calculational method of Meade et a1.22 (labeled “Old Theory”) with the improved scheme” (labeled “New Theory”) for defect modes in the 3D fcc structure of Yablonovitch et aL3’; this structure was discussed in Section 11.2 and is featured in Fig. 1. Also shown in Fig. 6 are the experimental results3’ for this same 3D structure. The agreement between theory and experiment is very good for the air defect (acceptor), but less so for the dielectric sphere defect (donor). In particular, only one donor mode was identified experimentally while three were found theoretically. Meade et aLY5noted that the other two modes had not been investigated experimentally due to the difficulty in performing the experiment. The reason proposed by Meade et al. for the fact that the calculated frequencies were higher than the experimental ones (see Fig. 6) was that the dielectric defect had to be positioned manually in the crystal and held in place; the air defect, on the other hand, could be accurately and directly placed in the PBG crystal by machining. A couple of alternative approaches have been suggested to the planewave expansion method. One of these is by Pendry and MacKinnon.23 Their approach closely reflects the nature of experimental PBG samples, which all consist of a finite number of lattice periods. Pendry and MacKinnon employ a finite-element method for calculating the transmission

PHOTONIC BAND-GAP MATERIALS

195

coefficient through finite numbers of PBG slabs. Their method, while still numerical, avoids having to store large matrices; it also has some other significant features. First, conventional plane-wave calculations calculate w at a given real k-value; unfortunately, this technique proves difficult for complex k, which are of interest for localized modes. Pendry’s method is a photon equivalent of the on-shell scattering method employed in lowenergy electron diffraction theory. The method allows calculation for both real and complex k(w). Second, for systems with a single defect, we mentioned earlier that the plane-wave calculations can converge slowly, and can therefore be extremely time consuming. The on-shell method however expands the field over a surface as opposed to a volume. This calculation is far more efficient. Third, the method of Pendry allows a straightforward application to metallic systems at microwave frequencies. In this limit, E(r) can be complex with large imaginary values, hampering conventional plane-wave methods. The finite-element method divides space into a set of small cells with coupling between adjacent cells. The stability of the method depends crucially on the choice of lattice and coupling constants. There is also a subtle problem related to the finite-element procedure. The longitudinal solutions of Maxwell’s equations can be regarded as zero frequency, dispersionless modes; finite-element models in which the longitudinal modes are only approximately zero will therefore confuse such modes with the low-frequency transverse modes of interest. Hence Pendry and MacKinnon2’ placed a criterion on their model such that it produces a set of zero-frequency, dispersionless modes, which converge to the longitudinal modes in the limit of small unit cell size. The computer time saved by their method can be great. In particular for a dielectric structure containing L x L x L cells, the dimensions of the transfer matrix are only 4L2 instead of the 2 L3 expected otherwise; this leads to a speed increase of a factor of 100 when L = Figure 7 compares the calculations of Pendry and M a ~ K i n n o nto~ ~the COMITS measurements of Robertson et al. (see Fig. 2) for a 2D array of dielectric cylinders. They divided the unit cell of the system into a 10 X 10 X 1 mesh. For each cell an average was taken over the dielectric constant within that cell. The transfer matrix was then found by multiplying together the matrices for each of the 10 slices. The transmission coefficient was calculated by stacking together slabs of one cell thickness using the multiple scattering formula familiar in the theory of low-energy d i f f r a ~ t i o n .The ~ ~ agreement in Fig. 7 is very good, and lies within the experimental error of 5 GHz. Sigalas ef aZ.96used this transfer96 M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, and K. M. Ho, Phys. Reu. B 48, 14,121 (1993); M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, Phys. Reu. B 49,

11,080 (1994).

196

P. M. HUI AND NEIL F. JOHNSON (a)

E

perpendicular to rods theory

-experimeni

tc+f+

.“2

C

3

.-C

‘0

.-9

iw

E

B

b

(b)

100 GHz E parallel to rods

100

t c c theory ~ exuerimenl

-

Ecd

0.0

1.o

FIG.7. Lefi: Comparison of experimental and theoretical photonic band structures for the 2D PBG crystal of dielectric cylinders described in Fig. 2. Solid lines are theoretical [finite-element method of J. Pendry and A. M. MacKinnon, Phys. Reu. Lett. 69, 2772 (1992)l; solid dots are experimental (COMITS data from Fig. 2). Right: Power transmitted through a seven-row array of dielectric cylinders. Dotted curve shows instrument response in absence of cylinders. In each case, E polarization is perpendicular to rods in (a), and parallel to rods in (b). [Taken from Fig. 1 of J. Pendry and A. M. MacKinnon, Phys. Rev. Lett. 69, 2772 (1992).]

matrix technique to investigate 2D ordered and disordered PBG materials, also obtaining good agreement with experiment. A second theoretical approach, which is not directly plane-wave based, is that of L e ~ n gwho ~ ~ presented a calculation of defect modes in PBG crystals using a Green’s function technique within a basis of vector Wannier functions (see Section III.4.b). In addition, Leung and Q u and ~ M ~ o~ r ~ zhave ~ ~ considered a photonic multiple-scattering theory analogous to the well-known KKR (Korringa-Kohn-Rostoker) theory for electrons. This method was found 97K. M. h u n g and Y.Qui, Phys. Reu. B 48, 7767 (1993). 9nA.Moroz, Phys. Reu. B 51, 2068 (1995).

197

PHOTONIC BAND-GAP MATERIALS

to give very good agreement with the plane-wave expansion method for a 2D PBG crystal.y7 These numerical solutions provide a well-defined prescription for obtaining electromagnetic properties in PBG materials. However, it would also be invaluable in aiding in the design of PBG materials to have some approximate analytic expressions for the photonic band gap as a function of the PBG material parameters. Since the main frequency region of interest lies in the vicinity of a gap, we might look for alternative theoretical schemes that allow more insight into the nature of the dispersion relation near photonic band maxima and minima. A well-known scheme in conventional electronic theory is the k * p method." Exploiting the analogy with electronic systems, we have developed''? y9. loo this k p approach to calculate the photonic dispersion relation near the photonic band gap in both the scalar wave approximation and the full vector theory (see Section III.4.b for the general vector formalism). As a demonstration of the ease of implementation and accuracy of the k . p method described in Section III.4.b, we now compare the k p results with the numerical plane-wave expansion method discussed earlier. We consider the 2D PBG sample employed in Figs. 2, %a), and 7 consisting of a square array of dielectric cylinders. Both k and k, have only x and y components, and s in Eq. (4.26) takes on the form ( s x , sy ,0) (see Section III.4.b). Consider the TM modes in which the u for the electric field takes on the form u = [O, 0, u ( x , y ) ] . The zz element is the only pcssible nonvanizhing_matrix element of the matrix Thus, both the s Q * s and the (Q + Q T ) k, terms in Eq. (4.26) vanish. In Eq. (4.27) the vector p(!,, reduces to the form ( p , , p y , 0) and is given by p(,,) = (2i/n,)/uT Vu, dr. The dotted lines in Fig. 8 show the plane-wave band structure for low-lying TM bands near high-symmetry points in the Brillouin zone. [These are the same TM modes as shown in Fig. 5(a).l The eigenstates (i.e., the u functions) at the high-symmetry points ( X and I') obtained from the plane-wave method were then used to calculate the matrix elements PI, in Eq. (4.30). The band structures at X ( l 3 can easily be described within our k . p formalism using a two- (three)-band model. Equation (4.30) now has the form of a 2 X 2 (3 X 3) matrix at X ( T ) and can be analytically diagonalized. The resulting k p band structures are shown by solid lines in Fig. 8, out to 10% of the separation along the line between the high symmetry points. The agreement is remarkably good. In Fig. 8(a) the slight upward curvature of the plane-wave bands from X -+ M can now be explained within the k p formalism by the 673

-

-

6.

-

Y'J

N. F. Johnson and P. M. Hui, Phys. Reu. B 48, 10,118 (1993). N. F. Johnson and P. M. Hui, J . Phys.: Condens. Mutter 5, L355 (1993).

loll

198

P. M. HUI AND NEIL F. JOHNSON

*

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.

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.

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(0,O) FIG. 8. Theoretical TM band structure calculated using k . p theory (solid lines) and plane-wave expansion (dotted lines) for 2D PBG crystal of dielectric cylinders; sample is same as that considered in Figs. 2, Xa), and 7. Analytic diagnalization of truncated k . p matrix is described in text. The X point is (n/a,O), the M point is ( n / a , n / a ) , and r is (0,O). Frequency w is in units of c / a , where c is the speed of light, and wavevector k is in units of l / a . (a) Lowest two bands near the X point. (b) Bands 2, 3, and 4 near the r point.

PHOTONIC BAND-GAP MATERIALS

199

presence or the q(,,,,, terms. The small differences between the results of the two methods are consistent with the neglect of the higher bands in the k . p matrix. The higher bands, if included, would have the effect of “repelling” (i.e., pushing down) those bands shown in Fig. 8, yielding even better agreement. Using similar reasoning, the T E modes can also be found. These are shown in Fig. 9, where they are again compared to the plane-wave method band structure [i.e., to the TE modes in Fig. %a)]. Again the agreement is very good. We emphasize that, for the purpose of comparison, we calculated the k . p input parameters using the numerical plane-wave method eigenstates in order to demonstrate how accurate even a few-band k p model can be. The usefulness of the k . p method in practice is that these parameters can be obtained by fitting either to published band structures calculated using numerical techniques or to experimental data. Finally we note the application to photonic problems of another wellknown approach from electronic band structure theory, the tight-binding method or ETBM (empirical tight binding m e t h ~ d ) . ’ ~ This method has been applied successfully to a variety of semiconductor systems to describe their electronic band structures. We have developed a similar scheme for photonic band structures“” (also see Section 111.4.~).Within this approach, we have demonstrated that a good fit to the 2D photonic band structure can be achieved using just a few tight-binding parameters. Figure 10 shows the lowest five TM-mode bands for the same 2D PBG sample as was considered in Figs. 2, %a), and 7. The solid lines are tight-binding bands, whereas the dotted lines are plane-wave method bands. The parameters used in Fig. 10 are obtained simply by fitting to the plane-wave method band structure at the symmetry points r, X , and M in the BZ. Improved values can be obtained by more careful fits to more points within the BZ. For the lowest band, it is sufficient to include only the nearest neighbor term in order to obtain a good fit. For the higher bands, a reasonable fit can be obtained by taking into account the next nearest neighbor term. An ETBM theory can then be developed by studying the dependence of the tight-binding parameters on the system parameters.’”’ IV. Conclusions and Future Direction

We have presented a review of recent experimental and theoretical results concerning the properties of photonic band-gap materials. Both 2D and 3D PBG systems have been discussed, with and without defects. Most “” W. M. Lee, M. Phil. Thesis, The Chinese University of Hong Kong (1994); W. M. Lee, P. M . Hui, and N. F. Johnson (unpublished).

200

P. M. HUI AND NEIL F. JOHNSON oa/2nc

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0.875

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r

t

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-

+

M

(a) oa/21tc 0.725 1

0.575

1

0.525 A

(C)

FIG. 9. Theoretical TE band structure calculated using k . p theory (solid lines) and plane-wave expansion (dotted lines) for 2D PBG crystal of dielectric cylinders; sample is same as that considered in Figs. 2, 5(a), 7, and 8. Analytic diagonalization of truncated k . p matrix is described in text. The X point is (r/a, O), the M point is ( r / a , r / a ) , and r is (0,O). Frequency w is in units of 2nc/a, where c is the speed of light, and wavevector k is in units of r / a . (a) Lowest two bands near the X point. (b) Bands 2, 3, 4, and 5 near the r point. (c) Lowest four bands near M point.

20 1

PHOTONIC BAND-GAP MATERIALS

0.9

0.0

0.7 0.6

0.5

0.3

0.2 0.1

0

r

X

M

r

FIG.10. Theoretical TM band structure calculated using a next nearest-neighbor empirical tight-binding scheme (ETBM-solid lines) and plane-wave expansion (dotted lines) for 2D PBG crystal of dielectric cylinders; sample is same as that considered in Figs. 2, 5(a), 7, 8, and 9. The X point is ( T / u , O ) , the M point is (?r/a, ?r/a), and r is (0,O).Frequency w is in units of 2?rc/u, where c is the speed of light.

reported theoretical work has relied on numerical solution of the vector wave equation using a plane-wave expansion. The agreement between recent numerical plane-wave results and available experimental data is generally very good in both 2D and 3D systems. We have, in addition, provided a self-contained discussion of the representation theory for PBG materials. Formal results have been obtained that represent the vector-wave analogs of the well-known expressions for electronic systems. In particular,

202

P. M. HUI AND NEIL F. JOHNSON

we have demonstrated that theoretical approaches such as k p theory and the tight-binding method can easily be implemented in PBG materials. These two approaches, which have already provided valuable insight into electronic properties in solids, yield results in PBG materials that are in quantitative agreement with the numerical plane-wave results. The outstanding experimental challenge in the fabrication of PBG materials is to make a 2D or 3D crystal with a full photonic band gap throughout the Brillouin zone in the optical frequency range. From the technological viewpoint, such a structure should have the additional property that it is easy to fabricate. On the more fundamental side, the ability to modify atomic and semiconductor radiation properties via a modification of the radiation channels opens many possibilities for investigating electron-photon effects in such systems. PBG materials may therefore prove very useful in the field of cavity quantum electrodynamics. We have also raised the possibility of introducing nonlinear defects into PBG crystals. The numerical plane-wave routines for calculating PBG properties have become very sophisticated in the last few years. However, one of the main theoretical challenges that has eluded researchers is to provide a simple, preferably analytic, recipe for the approximate value of the photonic band gap as a function of the PBG parameters. Given that the number of possible PBG crystals is infinite, such a result would be invaluable to experimentalists searching for the PBG crystal with the optimum gap at a given frequency. We believe that photonic k p theory, which focuses on the frequency region around the photonic band gap, and the empirical tight-binding method could prove useful in this search. Our discussion throughout this article has focused on “bulk” PBG crystals. Given the analogies that exist between photonic and electronic band structures, one can raise the possibility of following the semiconductor (e.g., GaAs-GaAlAs) heterostructure field by making PBG heterostructures. For example, PBG (quantum) wells, wires, dots, and superlattices are all possible, given the existence of two PBG materials with full photonic band gaps G.e, the analogs of GaAs and GaAlAs, respectively). In the same way as the envelope-function formalism based on bulk k . p theory provides the most useful description for the electronic the formalism of Section 111.4 can similarly be generalized to these new PBG systems. Just as for semiconductor heterostructures,69- 7 1 the extra flexibility in layer widths, etc., implies that such PBG heterostructures should exhibit a wide range of photonic band-gap behavior.

PHOTONIC BAND-GAP MATERIALS

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ACKNOWLEDGMENTS This work was supported in part by grants from the British Council under the UK-HK Joint Research Scheme 1993-94 and 1994-95. PMH acknowledges additional support from the Chinese University of Hong Kong under a Strategic Research Program and a Direct Grant for Research 1994-95, and useful discussions with S. Y . Liu during the course of this work. NFJ acknowledges additional support from the Nuffield Foundation, and the hospitality of thc Universidad de Los Andes (Colombia) where part of this work was prepared. W e thank R. D. Meade for useful discussions and W. M. Lee and K. H. Luk for fruitful collaborations.