 Nonlinear photonic crystal

Nonlinear photonic crystals are usually used as quasiphasematching materials. They can be either onedimensional^{[1]} or twodimensional.^{[2]}
Nonlinear Photonic Crystals
Broadly speaking, nonlinear photonic crystals (PC) are periodic structures whose optical response depends on the intensity of the optical field that propagates into the crystal. An immediate consequence is that such structures have new optical properties with improved or new functionalities that cannot be obtained by using their linear counterpart, namely linear pPCs. One such example is optical tunability, that is, optical control of the response of devices based on PC. Although tunability of optical properties of photonic crystals can be achieved, for instance, by applying an electric field to an inverse opal PC infiltrated with nematic liquid crystal, by modulating the PC's index of refraction through the electrooptic effect induced by an external electric field, or through temperatureinduced changes in the PC's index of refraction, highspeed operability desired for certain advanced optical communication systems can be obtained only if intrinsic optical nonlinearities in the PC material are employed. The reason for this is the ultrafast response of certain nonlinear dielelectric materials to optical fields. In contrast to the now very extensive body of research in the properties and devices in linear photonic crystals, research into the theoretical and experimental behavior of these structures under conditions of intense optical fields, e.g. in the nonlinear regime, is still in its formative stages.
In this rapidly growing area of nonlinear photonic crystals, I have been involved in several projects, namely analysis of superprism effect in nonlinear photonic crystals, optically tunable photonic crystal cavities, and optical properties of nonlinear waveguides embedded in linear photonic crystals. The main player in the latter project has been Mayank Bahl (Micky), a graduate student in Prof. Osgood's Optics group.
Short description of projects
 Superprism Effect in Nonlinear Photonic Crystals
A photonic crystal superprism is a newly conceived device, which has a remarkably strong dependence of the direction of propagation of the refracted wave at the incidence facet of a photonic crystal on such input beam parameters as angle of incidence, frequency, etc. In our approach, a pump beam changes the PC's refractive index, through the induced Kerr effect, so that it modulates the photonic band structure (PBS) of the PC, and, consequently, its dispersive properties. A schematic description of these ideas is shown in Fig. 1^{[where?]}.
Fig. 1. Schematic design of a device based on an optically controlled superprism effect in a PC.
Thus, we considered a 2D PC composed of a hexagonal lattice of circular air holes in a background medium made from a nonlinear instantaneous Kerr dielectric with refractive index n = 3.1, a typical value for the modal index of GaAsbased PC slab waveguides, and Kerr coefficient n2 = 3×1016 m2/W. In a complementary geometry, we considered a triangular lattice of rods, surrounded by air, and made from nonlinear instantaneous Kerr dielectric, with the same values for the refractive index and Kerr coefficient. The value chosen for structural parameter of the PC was r/a = 0.33, where r and a are the hole/rod radius and the lattice constant, respectively. The pump beam, with frequency wp, controls the band structure of the PC by changing the refractive index n through the Kerr effect. A signal beam, with frequency ws and wave vector ki, impinges from air onto the PC, at the incidence angle qi. The incidence angle is chosen such that the equifrequency dispersion surface (EDS) around the refracted wave vector, kr, is strongly anisotropic. The angle of refraction of the transmitted signal beam is qr. An example of a region of highly anisotropic dispersion, which corresponds to the 7th band, in the TE polarization, is shown in Fig. 2. Also shown in Fig. 2 is the full band structure of the crystal, as well as the shift in frequency of the signal band, induced by the pump beam.
Fig. 2. PBS of the TE modes for an air hole hexagonal PC at P = 0. Right, band corresponding to the signal frequency ws calculated for P = 0 (green) and P/a = 1 GW/cm2. In the latter case the pump frequency is wp = 0.15 (brown) and wp = 0.21 (red).
The angle of refraction can be determined from an analysis of the equifrequency dispersion curves at ws, calculated for the pumped crystal at P = 0 and P ¹ 0. These curves are shown in Fig. 3. Thus, since the tangential component of ki is conserved upon refraction, the intersection between the vertical line through the tip of ki and the EDS determines the wave vector kr. Then the normal onto the dispersion curve ws = w(k), at k = kr, yields the propagation direction of the refracted beam. If the position of the dispersion curve is modified by tuning the PC’s n through the Kerr effect induced by the pump and if ws = w(k) is highly anisotropic around k = kr, the angle qr varies strongly with P.
A quantitative characterization of the nonlinear superprism effect is given by the dependence of the refraction angle qr of the signal beam on the pump powers P. This dependence, calculated for qi = 62° is shown in Fig. 4. I chose this particular value because for this angle of incidence the equifrequency dispersion curve corresponding to ws = 0.672 is highly anisotropic around k = kr. The results illustrate first that large variations, of tens of degrees, in qr can be induced by optically tuning the PBS of the crystal. Moreover, if the PC is pumped at frequencies close to the PBG edge, dramatically reduced pump powers are required to deflect the transmitted signal beam by a given angle. Two factors contribute to this effect. First, close to the PBG edge, the group velocity of the pump beam is small. Consequently, the intensity of the electric field that corresponds to a given optical power P of the pump increases, so that the induced Kerr effect is stronger. Second, for selffocusing Kerr dielectric materials, the pump beam will induce a positive variation of the refractive index of the crystal, which leads to a further decrease in the group velocity.
Fig. 4. Refraction angle qr versus the pump intensity P/a. The red curve shows qr versus P/a for the case in which only the signal beam is present.
 Alloptical Tunability of Nonlinear Photonic Crystal Cavities
When photonic crystals are made of optically nonlinear materials the optical properties of localized modes supported by such media depend on the mode power. In this connection, I introduced a numerical method that allows one to calculate the field profile and dispersion curves of waveguide modes, as well as the modes and resonant frequencies corresponding to resonant cavities; in both cases these structural defects are embedded in nonlinear PCs. Furthermore, as an example, I applied this numerical method to investigate the optical response of a channel drop filter consisting of a resonant cavity sidecoupled to a waveguide PC whose optical properties can be tuned by means of a pump beam.
Fig. 5. Mode dispersion of the waveguide PC (red line), at P = 0; the blue line corresponds to the resonant frequency of the cavity, also at P = 0 and FDTD simulation of propagation of a low power CW, at ω0, in the waveguide PC sidecoupled to the cavity (top panels). Lower panels, nonlinear mode dispersion and power dependence of the resonant frequency ω0.
 Optical Properties of Nonlinear Waveguides Embedded in Linear Photonic Crystals
A different project I have been involved in addressed the nonlinear optical effects induced by a 1D line defect, made of Kerr material, embedded in a 2D linear photonic crystal (see Fig. 6). Comprehensive ab initio numerical simulations based on the nonlinear FDTD method showed relatively efficient thirdharmonic generation, switching, and optical limiting. More about this subject can be found in this conference talk that Micky gave at OSA 2002 and in the third paper in the following list.
Fig. 6. Schematic of a 1D nonlinear waveguide (red rods are made from Kerr material) embedded in a 2D linear PC (blue rods).
 “Optically tunable superprism effect in nonlinear photonic crystals”, N.  C. Panoiu, M. Bahl, and R. M. Osgood, Jr., Opt. Lett. 28, 2503 (2003).
 “Ultrafast optical tuning of superprism effect in nonlinear photonic crystals”, N.  C. Panoiu, M. Bahl, and R. M. Osgood, Jr., J. Opt. Soc. Am. B 21, 1500 (2004).
 “Alloptical tunability of a nonlinear photonic crystal channel drop filter”, N.  C. Panoiu, M. Bahl, and R. M. Osgood, Jr., Opt. Express 12, 1605 (2004).
 “Nonlinear optical effects in a twodimensional photonic crystal containing onedimensional Kerr defects”, M. Bahl, N.  C. Panoiu, and R. M. Osgood, Jr., Phys. Rev. E 67, 056604 (2003).
References
 ^ J.A. Armstrong, N. Bloembergen, J. Ducuing, P.S. Pershan (1962). "Interaction between light waves in a nonlinear dielectric". Physical Review 127: 1918. Bibcode 1962PhRv..127.1918A. doi:10.1103/PhysRev.127.1918.
 ^ V. Berger (1998). "Nonlinear photonic crystals". Physical Review Letters 81: 4136. Bibcode 1998PhRvL..81.4136B. doi:10.1103/PhysRevLett.81.4136.
http://cumsl.msl.columbia.edu/optics/nicolae/NonlinPhotCryst.html
http://cumsl.msl.columbia.edu/optics/nicolae/NonlinPhotCryst.html
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