PHOTONIC CRYSTALS

NEGATIVE REFRACTION AND FOCUSING USING 2D DIELECTRIC PHOTONIC CRYSTALS

_Initially, the studies regarding photonic crystals were focused on utilizing their band gap, and defect modes realized within the band   gap. It turned out _that photonic crystals can provide much more than that for controlling the propagation of light. These periodically (or   quasi-periodically) modulated _dielectric or metallic structures provide bands for the photons in close analogy to the bands of electrons   in a semiconductor material, by which they _acquire the name “crystal”. By tuning the geometrical and material parameters of the   crystal, the band structure can be sculptured to yield virtually any _type of dispersion for the propagation of electromagnetic   waveswhich reveals novel and unusual optical phenomena. Perhaps the most exciting _phenomenon among these is the negative   refraction of electromagnetic waves at the interface between a positive index medium (e.g. air) and the PC. _This was demonstrated   first by Kosaka et al. in 1998.

Mechanism:

_Negative refraction may occur when the incident field couples to a band with convex equal frequency contours (EFCs) in k-space,   where the _conservation of the surface parallel component of the wavevector, k, combined with the “negative” curvature of the band   causes the incident beam _bend negatively. In this case, neither the group velocity nor the effective index is negative and the PC is   essentially a positive index medium, exhibiting _negative refraction.  In another mechanism, the group velocity and the phase velocity   derived from the band dispersion are antiparallel for all the values _of k, leading to n_eff < 0 for the PC. Both mechanisms are confirmed   by recent experimental observations.

  The photonic crystal is a hexagonal lattice of alumina rods in air. Lattice period is a = 4.79 mm. The rods have dielectric constant ε =   9.61, diameter 2r _= 3.15 mm, and length l = 15 cm. Figure 1 shows the transverse electric (TE) polarized band structure in the first   Brillouin zone. Throughout this paper, _the transverse direction is taken in the plane of the 2D photonic crystal. The 5^th band shaded in   the figure extends from  (f = 40.65 GHz) to  (f = 46.27 GHz), where is the scaled frequency.

Fig. 1 The TE polarized band structure of the PC. 5^th band (shaded) gives negative refraction.

_ In Fig. 2, the band surface in the full Brilloin zone along with the equal-frequency contours (EFCs) are shown. The EFCs of the band   are shrinking with _increasing frequency, contrary to the EFCs in air (n = 1) which are given by the dispersion . As a result, the   effective refractive index ,becomes negative due to sign of the antiparallel group velocity  and   the phase velocity . _Here c is the speed of light in vacuum, and  is the unit wavevector in PC.

Fig. 2 The surface (transparent shell) of the 5^th band in the full Brillouin zone. Some equal frequency contours are projected to the Brillouin zone plane.

_The phase measurements are performed with an HP 8510C network analyzer. The phase of the transmitted signal (S₁₂ in the   S-_parameter convention), is measured between [-π, +π],  as a function of frequency. This raw data is then “unwrapped” by adding _2π   at the +/-π jumps, to obtain the phase spectra. Since the absolute phase is meaningless, the phase is measured with respect to _a   calibration. We first perform the calibration in air, by removing the photonic crystal between the antennas, and then measure the   relative phase shift induced by structure, including the cavity.

Negative Refraction Measurements:

  The refraction spectra are measured by a setup consisting of an HP 8510C network analyzer, a microwave horn antenna as the   transmitter and a waveguide antenna as the receiver. The PC has 7 layers along the incidence (ΓM) direction and 31 layers along the   lateral direction. The horn antenna is on the negative side of the PC with respect to its central axis. The spatial intensity distribution   along the PC-air interface is scanned by Δx ~ 1.27 mm steps, while the frequency is swept from 38.5 GHz to 43.5 GHz in 400 steps,   averaged over 256 measurements at each frequency. Figure 3a displays the transmission spectra as a function of frequency and   lateral position for three different incidence angles of θ_i = 15^o, 30^o, and 45^o. The transmitted beam clearly appears on the negative   side.

     Fig. 3: (a) Measured negative refraction spectra of the 5^th band along the PC-air interface for incidence angles θ=15^o, 30^o, and   45^o. Measured (b) and simulated (c) intensity profiles at f = 41.7 GHz for the respective incidence angles. Solid curves indicate   Gaussian fits.

  To investigate the beam profiles, the measured and simulated spatial cross section at f = 41.7 GHz ()are plotted in Fig.3b   and Fig. 3c, respectively. The incident field has a Gaussian beam profile centered at x = 0 (not shown on figures). Figure 4 depicts a   simulation for the incidence.

     Fig. 4: Simulated negative refraction of a plane wave at f = 41.7 GHz incident at θ=30^o to the PC interface (mark 1). Zero order   (mark 2) and higher order (mark 3) reflections occur. The refracted (mark 4) the transmitted (mark 5) components appear to propagate as single beams.

  Evidently, higher order reflection (mark 3) is present, nevertheless, the refracted beam, appears to be a single component, which is   also suggested by the single transmitted component on the other side of the PC. We therefore assume that most of the propagating   power is coupled to the zero-order diffracted wave, and employ Snell's law for this geometry by , where  is   the  angle of incidence and  is the angle of refraction inside the photonic crystal. For θ_i = 15^o, 30^o, and 45^o we obtain n_eff = -0.52,   -0.66, and -0.86 from the experiment, respectively. The simulation results for the same incidence angles give n_eff = -0.66, -0.72, and   -0.80, respectively.

Focusing of an omnidirectional source by photonic crystal slab lens:

  The presence of negative refraction for large range of incidence angles brings the possibility that the slab structure may act like a   lens for an omni-directional source. For the present PC, we first performed FDTD simulations for a TE polarized point source at f =   42.07 GHz located at a distance d_src = 2λ away from air-PC interface. Figure 5.b shows the resulting spatial intensity distribution in   the image plane, normalized by the maximum intensity value. The PC-air interface is located at z = 0. Note that the focusing occurs   away from the PC-air interface, with peak intensity of ~ -5 dB, observed at z ≈ 8λ.

Fig. 5: (a) Lateral intensity profiles measured at six different positions along the propagation direction: z/λ = 1.78, 3.56, 5.34, 7.12, 8.90, and 10.68. (b) Simulated 2D intensity in the image plane. z = 0 corresponds to PC interface.

  In the experiment, a waveguide aperture is used as the source. The intensity distribution in the image plane is measured by a   monopole antenna. For d_src = 2λ, first the propagation direction, z, is scanned for locating the maximum intensity, and then lateral   cross sections of intensity at several z around the _peak position are measured. In Fig. 4.a the focusing of the beam both in lateral and   longitudinal directions is evident. The maximum intensity (normalized _to unity) is observed at d_focus/λ ≈ 8. The measured and   simulated focusing profiles are quite similar.

_When the source is shifted along the longitudinal direction, the focus pattern is expected to move accordingly. The FDTD simulated   2D map of the _magnetic field, H_y(x,z), plotted in Fig.6 shows this behaviour clearly: when the source is moved from d_src = 2λ to d_src =   4λ away, the focus pattern on _the other side shifts towards the PC-air interface.

Fig. 6: Simulated 2D field distribution H_y(x,z) for d_src = 2λ (top) and d_src = 4λ (bottom).

_The measured lateral profiles of the electric field intensity at focus points for various d_src are plotted in Fig. 7. From the determined   focal positions we _have found that (d_src + d_focus) remains roughly constant. The figure also displays the intensity profile in the absence   of the PC for d_src = 2λ case (dotted _line). At the focal distance, the free space propagation is almost a flat line with no features   indicating the source location. This shows the drastic _enhancement by focusing.

Fig. 7: The measured lateral intensity profiles at respective focal distances for different source distances. The intensity axis is normalized by the d_src = 2.0λ profile peak. The dashed line denotes the intensity profile in the absence of PC for d_src = 2λ.

Related Group Publications

  1- K. Guven, K. Aydin, K. B. Alici, C. M. Soukoulis, and E. Ozbay, Phys. Rev. B 70, 205125 (2005).

    Negative refraction and subwavelength focusing using photonic crystals

    Metallodielectric photonic crystals with negative refraction and focusing

    Highly directional radiation using sources embedded in photonic crystals

    Coupling and phase analysis of cavity structures in photonic structures

    Enhanced transmission and beaming via photonic crystal surface modes

Metamaterials   -   Photonic Crystals  -  GaN/AlGaN Devices  -  Other Semiconductor Devices and Fabrication Techniques

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