a2619_1.pdf JThD121.pdf
Design of Gradient Index (GRIN) Lens using Photonic Non-Crystals a
Paul Stellmana, Kehan Tianb, and George Barbastathisa Dept. of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Ave., 3-466, Cambridge, MA 02139 b IBM Semiconductor Research and Development Center, Hopewell Junction, NY 12533 Author e-mail address:
[email protected] Abstract: We use analytical and numerical techniques to design a cylindrical lens with a gradient index of refraction. In our device, we design the desired index distribution by using a photonic crystal with slowly-varying lattice parameters. © 2007 Optical Society of America OCIS Codes:
(080.0080) Geometrical Optics; (220.0220) Optical design and fabrication
1. Introduction The well-known dispersion properties of photonic crystals [1] are utilized in a new class of material in which the crystal lattice properties (e.g. lattice spacing) vary slowly throughout the device. As an example, Figure 1 illustrates a photonic non-crystal GRIN lens in which the horizontal lattice spacing between cylindrical dielectric rods increases gradually with distance from the optical axis. The non-uniformity of the material is a degree-of-freedom which can be engineered to manipulate light in novel ways. Mathematical techniques such as finite-difference time-domain (FDTD) simulations and Hamiltonian optics methods are employed to optimize system performance subject to manufacturing parameters.
r
z
Figure 1. FDTD simulation of electric field (out-of-plane) propagating through photonic non-crystal GRIN lens. 2. Numerical Simulation of GRIN lens Gradient-index lenses are very useful in optical systems due to their exceptional focusing capabilities. The index distribution for a radial-GRIN device is given by the quadratic expression
n(r ) = nmax −
r2 . As 2 fd
an initial design approach, we use an approximate effective index distribution in which the refractive index is proportional to the lattice spacing. The FDTD simulation results shown in Figure 1 demonstrate that the Gaussian source at the left end of the material is expanded into a plane wave and then refocused at the second focal plane within the material.
a2619_1.pdf JThD121.pdf
3. Analytical Design of Index Distribution Hamiltonian methods are convenient for calculating ray trajectories in both bulk media [2] and slowlyvarying optical materials [3, 4]. After defining the Hamiltonian, H, for the particular medium, the ray trajectory is determined by solving the set of coupled, ordinary differential equations
dr ∂H = dσ ∂k and dk ∂H , =− dσ ∂x
(1)
where r, k, and denote the position vector, wavevector, and an arbitrary parameter describing the trajectory, respectively. For the case of bulk media, the Hamiltonian is given by
(
)
H (r , z ) = k x2 + k y2 + k z2 − n(r , z ) k 02 . We have simulated the ray trajectories for the standard GRIN 2
lens with a quadratic index distribution. Figure 2a illustrates the focusing of both on- and off-axis rays with the lens, but aberrations are still evident since the rays do not intersect at a point. To correct the aberrations, we assume that the index distribution is a higher order polynomial, and the coefficients of the polynomial are optimized such that the rays intersect at a point. Figure 2b demonstrates the perfect focusing capability of the optimized index distribution.
a)
b)
Figure 2. a) Ray trajectories for standard GRIN lens and b) trajectories for optimized index distribution 4. Discussion While the device in Figure 2b does focus light free of aberrations, it does not assist in designing the material using non-crystals. Therefore, we are also using Hamilton’s equations (1) to optimize the lattice spacings for the GRIN device in Figure 1. In this case, the Hamiltonian is given by the local Bloch dispersion relation, and the manufacturing parameters, i.e. the lattice parameters, are optimized. 5. References [1] J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton University Press, 1995. [2] H.A. Buchdahl, Introduction to Hamiltonian Optics, Dover Publications, 1970. [3] P. St. J. Russell and T.A. Birks, “Hamiltonian Optics of Nonuniform Photonic Crystals”, J. of Lightwave Tech. 17 no. 11 (1999). [4] Y. Jiao, S. Fan, and D.A.B. Miller, “Designing for beam propagation in periodic and nonperiodic photonic nanostructures: Extended Hamiltonian method”, PR E 70 036612 (2004).
