We study experimentally the Goos-Hanchen effect observed at the reflection of
a self-collimated beam from the surface of a two-dimensional photonic crystal
and describe a method for controlling the beam reflection through surface
engineering. The microwave photonic crystal, fabricated from alumina rods,
allows control of the output position of a reflected beam undergoing an
internal Goos-Hanchen shift by changing the rod diameter at the reflection
surface. The experimental data is in good agreement with the results of the
finite-difference time-domain numerical calculations.Comment: 3 pages 4 figures, submitted to AP

Kivshar, Yuri

Matthews, Aaron

Applied Physics Letters

English

arXiv

We study experimentally the Goos-Hanchen effect observed at the reflection of
a self-collimated beam from the surface of a two-dimensional photonic crystal
and describe a method for controlling the beam reflection through surface
engineering. The microwave photonic crystal, fabricated from alumina rods,
allows control of the output position of a reflected beam undergoing an
internal Goos-Hanchen shift by changing the rod diameter at the reflection
surface. The experimental data is in good agreement with the results of the
finite-difference time-domain numerical calculations.This work was supported by an award under the Merit
Allocation Scheme on the National Facility of the Australian
Partnership for Advanced Computing and also by the Australian
Research Council through the Center of Excellence Program

The Australian National University

Experimental studies of the internal Goos–Hänchen shift for self-collimatedbeams in two-dimensional microwave photonic crystalsAaron Matthewsa and Yuri KivsharNonlinear Physics Centre and Centre for Ultra-high Bandwidth Devices for Optical Systems (CUDOS),Research School of Physical Sciences and Engineering, Australian National University, Canberra ACT 0200,AustraliaReceived 12 August 2008; accepted 19 August 2008; published online 29 September 2008We study experimentally the Goos–Hänchen effect observed at the reflection of a self-collimatedbeam from the surface of a two-dimensional photonic crystal and describe a method for controllingthe beam reflection through surface engineering. The microwave photonic crystal, fabricated fromalumina rods, allows control of the output position of a reflected beam undergoing an internalGoos–Hänchen shift by changing the rod diameter at the reflection surface. The experimental dataare in good agreement with the results of the finite-difference time-domain numericalcalculations. © 2008 American Institute of Physics. DOI: 10.1063/1.2980033Periodic photonic structures such as photonic crystalsallow the substantial modification of both the dispersion anddiffraction of electromagnetic waves. Many uniquedispersion-modifying phenomena have been predicted andobserved for photonic crystals including negative refraction,superprism effects, and light self-collimation. In particular,electromagnetic waves propagating in homogeneous dielec-tric media diffract, and in order to preserve the size of anarrow beam one needs to trap light by employing differenttypes of waveguiding structures. In contrast, photonic crys-tals allow almost diffractionless propagation of light undercertain conditions, this is known as light self-collimation.When a photonic crystal operates in the self-collimation re-gime, electromagnetic waves propagate without diffraction-induced divergence even in the absence of nonlinear effects.The self-collimation of light beams in photonic crystals hasattracted special attention due to many promising applica-tions in photonic-crystal based structures and circuits see,e.g., Refs. 1–11, and references therein.The self-collimation effect originates from the strong an-isotropic dispersion that makes the equal-frequency contoursultraflat within certain angular and frequency ranges. Sincethe propagation of energy is always normal to the equal-frequency contours, the light only propagates along the di-rection normal to the equal-frequency contours inside thephotonic crystal. Based on this self-collimation effect, somesimple and effective bending and splitting concepts havebeen suggested.9–11 In particular, it was shown that a surfaceof a truncated photonic crystal can operate as a total-internal-reflection mirror for the self-collimated beams, and this ef-fect can be employed to create sharp bends.The surface of a truncated photonic crystal shows manyinteresting properties due to both the surface structure and itsability to support surface states. Surface states in photoniccrystals have only been shown to exist under an appropriatechange in the surface layer, such as a termination though thesurface cell or a change in the surface geometry or materialproperties see, e.g., Refs. 12–14. In addition, the corruga-tion inherent to the surface of truncated photonic crystalsleads to a number of interesting effects including beam shap-ing and enhanced beaming.15–17Recently, we employed finite-difference time-domainFDTD calculations to study numerically the internal Goos–Hänchen GH shift observed at the reflection of a self-collimated beam from the modified surface of a truncatedtwo-dimensional photonic crystal.18 By assuming that thesurface rods can be modified or made of a nonlinear material,we demonstrated numerically a method for tuning the shift ofthe internally reflected beam by engineering the surface ofthe photonic crystal. In this letter, we study the problem ofbeam reflection experimentally for truncated and engineeredsurfaces of a microwave photonic crystal fabricated fromalumina rods and operating at microwave frequencies. Thisrelatively simple experimental setup allows us to demon-strate a proof of principle of our theoretical concept for con-trolling the output position of a reflected beam and tunabilityof the internal GH shift through a change in the rod diameterat the reflection surface. Our experimental data are in goodagreement with the results of the FDTD numerical calcula-tions carried out for the experimental parameters.In our experiment, we use a microwave photonic crystalcreated by a lattice of high purity alumina ceramic rods withthe diameter of r=7.990.02 mm and the refractive indexof n2.4 at the self-collimation frequency. These rods areset in a two-dimensional square lattice with a lattice spacingof a=10.59 mm; the rods are placed in a frame of expandedpolystyrene foam with n1.01. This setup demonstrates theself-collimation effect for the lowest order TE mode at thefrequency of 7.64 GHz equating to a wavelength of =39.24 mm.First, we employ these experimental parameters andstudy numerically the properties of the photonic crystal usingthe plane wave expansion method, implemented as Band-solve, to produce the equal-frequency contours; these dataallow us to determine the specific conditions for realizing thebeam self-collimation effect. Next, the numerical simulationresults are used to determine an optimal structure to ensurethe frequency region of the self-collimation effect are withinthe constraints of our experimental system. In particular, theself-collimation is directed at 45° to the lattice direction andas such the structure is rotated in the experiment.aElectronic mail: afm124@rsphysse.anu.edu.au.APPLIED PHYSICS LETTERS 93, 131901 20080003-6951/2008/9313/131901/3/$23.00 © 2008 American Institute of Physics93, 131901-1Author complimentary copy. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jspNext, we study numerically the reflection of self-collimated beams from the photonic-crystal surface similarto the analysis reported earlier in Ref. 18 for other param-eters. We employ the FDTD method implemented as Full-wave, the propagation of a point source, set adjacent to thesixth surface rod from the short side of the photonic crystal,through our square lattice photonic crystals used in the ex-periments. In our study, the photonic crystal is 26 rods longby 15 rods wide with a cleaved surface at 45° along thelattice direction. The cleaved surface layer is modifiedthrough a change in the diameter of the rods. The sets of rodstested experimentally have respective diameters of2.0050.015 mm, 2.980.01 mm, 4.180.02 mm,4.9150.005 mm, 5.920.01 mm, 7.1750.015 mm, and7.990.02 mm. To measure the GH shift we take an outputplane perpendicular to the input surface set at a distance ofone lattice constant from the output surface. An example ofthe reflected beam experiencing the GH shift, calculated nu-merically, is shown in Fig. 1. For convenience we normalizethe output peak magnitudes to emphasize the GH shift. Theposition of the input plane is translated to the center of thecentral reflection rod on the cleaved surface.Figure 2 shows our experimental setup. The photoniccrystal, consisting of high purity alumina ceramic rods, isheld in place by a frame of expanded polystyrene foam,manufactured using a computer controlled three-dimensionalmilling machine Roland DG MDX-40. The completed pho-tonic crystal is placed in a parallel-plate waveguide see Fig.2 with a separation between the plates of 11 mm with asource antenna of 1.3 mm diameter and 10 mm height in thelower plate, and an detection antenna set flush with the upperplate. A shaped carbon impregnated foam surrounds the mea-surement region to suppress reflections in the measurements.The lower plate Fig. 2, including the input antenna and thephotonic crystal, is mounted on a planar translation stageParker, 803-0936A, two-axis translation stage to allowscanning of the field on the top surface of the photonic crys-tals. A vector network analyzer Rohde and Schwarz modelZVB20 is used to provide both the input point source andreceiving antenna to measure both the amplitude and phaseof the transmission response S12 parameter. This responseis highly correlated to the vertical component of the electricfield, equivalent to a TE mode.Using this experimental setup, we study the internalbeam reflection and measure the GH shift for different sur-face rods, reproducing well the results obtained by numericalFDTD simulations cf. Figs. 3 and 1. In experiment, thelimitations of the antenna resolution lead to a smoothing ofthe double peak observed in the numerical simulations Fig.1 due to the averaging of the field from the surroundingarea. The center position of the experimental peak is used todetermine the GH shift for a given surface diameter. Theresulting peak shifts, measured in lattice constants relative tothe perfect reflection center, are plotted in Fig. 4 for differentvalues of the surface rods. We observe that the output beamposition is sensitive to the diameter of the surface rods, asdiscussed earlier.18 In particular, we note that as the diameterof 7.99 mm in our graph is the same as adding a layer with asurface diameter of 0 mm. As such the shift at a rod diameterFIG. 1. Color online Numerical FDTD simulations of a reflected self-collimated beam experienced a GH shift for the modified surface of5.92 mm rods. Top: input solid green and output dashed blue normalizedfield profiles. Bottom: Electric field generated by a point source and re-flected by the surface.FIG. 2. Color online Experimental setup. Included are photographs of theparallel-plate waveguide mounted on a two-axis translation stage and thefabricated photonic crystal.FIG. 3. Color online Experimental beam propagation in the photonic crys-tal of 7.990.02 mm rods at the self-collimation frequency 7.64 GHzreflected from the surface of 5.920.01 mm rods. The measured GH isapproximately four lattice spacings. Upper plot shows the output field takenone lattice period beyond the surface.131901-2 A. Matthews and Y. Kivshar Appl. Phys. Lett. 93, 131901 2008Author complimentary copy. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jspof zero should be one lattice constant less than the value at7.99 mm and from our results at 2.005 mm it can be seenthat this result is well reproduced.The change in the effective index as a result of thechange in the surface rod diameter provides us with an ana-log to the nonlinearity-induced GH beam shift discussed ear-lier in our theoretical paper.18 While the effective indexchange used here to obtain a significant shift is large, a well-tuned system near a resonance could show a significant shiftfor a far smaller index change.In conclusion, we have studied experimentally theGoos–Hänchen shift experienced by a self-collimated beamreflected from the surface of a truncated two-dimensionalphotonic crystal operating at microwave frequencies. Wehave demonstrated efficient control over the output positionof the beam reflected through 90° by modifying the surfacerods and measuring the internal Goos–Hänchen shift be-tween one and six lattice constants. The experimental resultsare in a good agreement with the FDTD numerical calcula-tions.This work was supported by an award under the MeritAllocation Scheme on the National Facility of the AustralianPartnership for Advanced Computing and also by the Austra-lian Research Council through the Center of Excellence Pro-gram. The authors thank David Powell and Ilya Shadrivovfor useful discussions and help with microwave experiments.1S.-Y. Lin, V. M. Hietala, W. Li, and E. D. Jones, Opt. Lett. 21, 17711996.2H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato,and S. Kawakami, Phys. Rev. B 58, R10096 1998.3H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato,and S. Kawakami, Appl. Phys. Lett. 74, 1212 1999.4M. Notomi, Phys. Rev. B 62, 10696 2000.5C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, Phys. Rev. B65, 201104 2002.6D. N. Chigrin, S. Enoch, C. M. Sotomayor Torres, and G. Tayeb, Opt.Express 11, 1203 2003.7J. Wiltzens and A. Scherer, J. Opt. Soc. Am. A 20, 935 2003.8L. Wu, M. Mazilu, and T. F. Krauss, J. Lightwave Technol. 21, 5612003.9X. Yu and S. Fan, Appl. Phys. Lett. 83, 3251 2003.10S. Shi, A. Sharkawy, C. Chen, D. M. Pustai, and D. W. Prather, Opt. Lett.29, 617 2004.11D. M. Pustai, S. Shi, C. Chen, A. Sharkawy, and D. W. Prather, Opt.Express 12, 1823 2004.12W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M.Rappe, and J. D. Joannopoulos, Opt. Lett. 18, 528 1994.13F. Ramos-Mendieta and P. Halevi, J. Opt. Soc. Am. B 14, 370 1997.14Y. A. Vlasov, N. Moll, and S. J. McNab, Opt. Lett. 29, 2175 2004.15E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, Phys. Rev. B 69,121402R 2004.16P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehr-spohn, U. Gösele, and V. Sandoghdar, Phys. Rev. Lett. 92, 113903 2004.17S. K. Morrison and Yu. S. Kivshar, Appl. Phys. B: Lasers Opt. 81, 3432005.18A. Matthews and Yu. S. Kivshar, Phys. Lett. A 372, 3098 2008.FIG. 4. Color online The experimental GH shift measured in lattice pe-riods for a set of different surface rods. The maximal shift occurs for asmall reduction in the diameter of the surface rod 7.1750.015 mm andthe smallest shift occurs for a large reduction in the surface rod diameter2.0050.015 mm.131901-3 A. Matthews and Y. Kivshar Appl. Phys. Lett. 93, 131901 2008Author complimentary copy. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp

Experimental studies of the internal Goos-Hanchen shift for  self-collimated beams in two-dimensional microwave photonic crystals

Aaron Matthews

Yuri Kivshar

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Experimental studies of the internal Goos–Hänchen shift for self-collimated beams in two-dimensional microwave photonic crystals

Experimental studies of the internal Goos-Hanchen shift for
  self-collimated beams in two-dimensional microwave photonic crystals

Experimental studies of the internal Goos-Hanchen shift for self-collimated beams in two-dimensional microwave photonic crystals

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