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Optics in Curved Space week ending PRL 105, 143901 (2010) PHYSICAL REVIEW LETTERS 1 OCTOBER 2010 Optics in Curved Space Vincent H. Schultheiss,1,2 Sascha Batz,1,2 Alexander Szameit,3 Felix Dreisow,4 Stefan Nolte,4 Andreas Tu¨nnermann,4 Stefano Longhi,5 and Ulf Peschel2,* 1Max Planck Institute for the Science of Light, 91058 Erlangen, Germany 2Institute for Optics, Information and Photonics, University Erlangen-Nuremberg, 91058 Erlangen, Germany 3Physics Department and Solid State Institute, Technion, 32000 Haifa, Israel 4Institute of Applied Physics, Friedrich-Schiller-University Jena, 07743 Jena, Germany 5Dipartimento di Fisica, Politecnico di Milano, 20133 Milano, Italy (Received 13 April 2010; published 27 September 2010) We experimentally study the impact of intrinsic and extrinsic curvature of space on the evolution of light. We show that the topology of a surface matters for radii of curvature comparable with the wavelength, whereas for macroscopically curved surfaces only intrinsic curvature is relevant. On a surface with constant positive Gaussian curvature we observe periodic refocusing, self-imaging, and diffractionless propagation. In contrast, light spreads exponentially on surfaces with constant negative Gaussian curvature. For the first time we realized two beam interference in negatively curved space. DOI: 10.1103/PhysRevLett.105.143901 PACS numbers: 42.25.Fx, 04.40.Àb, 42.25.Hz, 42.82.Et Introduction.—Regions of considerably curved space- P on the surface two tangent circles with maximal and time remain inaccessible for a direct experimental inves- minimal radii R1 and R2 can be found. The Gaussian ¼ tigation, but have been subject to extensive theoretical curvature K at P is then defined as K 12, where research. The interplay between space-time curvature and 1;2 ¼ 1=R1;2 are the principal curvatures. For tangent electromagnetic wave dynamics is claimed to provoke circles facing opposite sides of the surface, K is negative. ¼ð þ Þ intriguing effects like the generation of Hawking radiation The extrinsic curvature H is given by H 1 2 =2. [1] or the Unruh effect [2]. The extreme conditions neces- Proper lengths on curved surfaces are determined by means sary to witness these phenomena are not found in the of the two-dimensional metric tensor ðgijÞ according to 2 i j vicinity of Earth. Consequently, since the work by Unruh ds ¼ gijdx dx , where i; j ¼ x; z are coordinates parame- [3] several approaches have promoted analog models of trizing the surface of interest. The Gaussian curvature K curved space-time, which share particular features and depends on neither the actual embedding nor on the topol- allow for an investigation of resulting phenomena in the ogy of the surface, but on the metric (gij) alone—and laboratory. In this vein, systems like the Bose-Einstein vice versa [12]. The extrinsic (or mean) curvature H,on condensate [4] and optical solitons in glass fibers [5] are the other hand, is not an invariant with respect to the interpreted as dynamical systems on a background repre- topology. senting an effective space-time metric. When neglecting polarization effects, the evolution of On the other hand, the formalism of general relativity monochromatic light bound to an arbitrarily shaped two- (GR) has become an appealing tool to cope with the dimensional layer can be described by the scalar manipulation of light. Transformation optics utilizes the Helmholtz equation [11,13] covariant formalism to design novel devices by pointing ðÁ þ k2ÞÉ ¼ðH2 À KÞÉ: (1) out the equivalence in Maxwell’s equations between space- g ¼ time curvature and a modulation of the refractive index of Here k 2n0= is the wave number of light with wave- the transmitting medium [6–9]. This modulation is length , propagating in a material with refractive index ¼ achieved experimentally by artificial metamaterials, which n0. All the experiments in this work were performed at until now are very hard to assemble and are often con- 633 nm. The scalar formulation is justified by the experi- nected to severe losses during the propagation. It was ments, which show no dependencepffiffiffi onp theffiffiffi polarization. The ij recently proposed to use metamaterials to mimic a nonflat covariant Laplacian Ág ¼ð1= gÞ@i gg @j, with i; j ¼ space-time [10]. x; z, is given as a function of the metric tensor elements In our work we refrain from refractive index modulation. gij and g ¼ detðgijÞ, and thus depends on the Gaussian We choose a more direct, geometric approach to study the curvature only. [The explicit dependence for macroscopi- consequences of space curvature on the evolution of light. cally curved surfaces can be derived from Eq. (2) or found To this end we abandon one spatial dimension and inves- in [11].] tigate light propagation on specifically shaped two- Influence of the extrinsic curvature.—As an example for dimensional surfaces, as introduced in [11]. the influence of the topology of the surface, represented by Geometry of surfaces.—Picture a surface S embedded in the extrinsic curvature H, we studied light propagation in a three-dimensional Euclidean space. For every regular point microscopically undulated waveguide [see Fig. 1(a)]. It 0031-9007=10=105(14)=143901(4) 143901-1 Ó 2010 The American Physical Society week ending PRL 105, 143901 (2010) PHYSICAL REVIEW LETTERS 1 OCTOBER 2010 was realized via the femtosecond direct-writing method Eq. (1) can be neglected, since k2 dominates over H2 and [14]. High power femtosecond pulses are focused into bulk K. The Gaussian curvature K still affects the beam evolu- fused silica yielding an increase of the material’s refractive tion via the covariant Laplacian on the left-hand side. For index in the focal region [15,16]. Because of the use of a convenience and as a natural generalization of flat special fused silica with a high content of hydrogen, the Euclidean space, we restrict ourselves to rotationally sym- writing process causes a massive foundation of nonbridg- metric surfaces of constant Gaussian curvature now. From ing oxygen-hole color centers in the waveguide region. the point of view of confined light beams, these surfaces When excited by the guided beam, their fluorescence is are highly symmetric, namely, locally invariant under imaged by scanning a microscope objective along the translation and rotation. Therefore, they are an optical sample. For a detailed description of excitation and detec- realization of the cosmological principle, which states tion procedures, see, e.g., [17]. that space is homogenous and isotropic. This feature makes The manufactured waveguide has a sinusoidal cross optics on curved surfaces a powerful tool for beam shap- section and consequently a vanishing Gaussian, but notice- ing, since it is in strong contrast to photonic crystals or able extrinsic curvature varying periodically in the trans- structures with graded refractive index, where there is al- verse direction. With an undulation amplitude of 4 m,an ways a preferred direction involved in the design. undulation period of 30 m, and a thickness of 10 m, the One has to be cautious when using the covariant formal- extrinsic curvature H is 2 orders of magnitude smaller than ism of GR, because any statement must be treated in the k. This is comparable with a refractive index change on the light of the chosen coordinates. Since we are dealing with order of 10À5, which as known from nonlinear optics rotationally symmetric surfaces only, we conveniently influences the diffraction behavior considerably [18]. chose an orthogonal set of coordinates varying in proper Having no Gaussian curvature, the metric of flat space is lengths along the contour and the rotation angle of the reproduced and Ág becomes the common Laplacian. Only solids of revolution (see [11]). However, while all equa- H2 remains acting like an effective potential in the wave tions must be understood for propagation along a certain equation. The experimentally observed diffraction pat- direction, the physics is rotationally invariant and the tern [Fig. 1(c)] is in agreement with simulations based on stated evolution holds for other directions, too. Two solids Eq. (1) [Fig. 1(d)]. It reveals that light does not smoothly with negatively curved surfaces were fabricated to allow diffract as in flat space [Fig. 1(b)]. The diffraction pattern for equal propagation lengths in different directions and resembles that occurring in a periodic lattice [19], which to emphasize that equivalence [see insets of Figs. 4(a) and now arises from the periodicity of H2 only. The extrinsic 4(c)]. curvature realizes thus a kind of topological photonic Depending on the sign of Gaussian curvature, the met- crystal [20]. rics in the chosen set of coordinates read [11] Influence of the intrinsic curvature.—For macroscopic ds2 ¼ dx2 þ cos2ðx=RÞdz2; (2) radii of curvature the influence of the right-hand side of q where we introduced the notation cos1 ¼ cos, cosÀ1 ¼ (a) (b) 1.2 0.020 -2 cosh, with q ¼ sgnðKÞ (analog for sin ). Here R ¼ m q µ pffiffiffiffiffiffiffi [ ] 2 ude 1.0 0.015 1= jKj is the geometric mean radius defined by the 0.8 eH Gaussian curvature K. As long as a paraxial approximation 0.6 0.010 vatur . field amplit 0.4 can be applied, Gaussian beams maintain a Gaussian shape 0.005 0.2 even in the presence of curvature. Analytical expressions norm trinsic cur 0 0 ex -60 -40 -20 020 40 60 describing the evolution of the beam width with respect µ transverse coordinate [ m] to the propagation length z are [11] (c) (d) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ¼ À ð À 4 4Þ 2ð Þ q z 0 1 q 1 qS=0 sinq z=R ; (3) pffiffiffiffiffiffiffiffiffi where 0 is the initial beam width and S ¼ R=k is a characteristic beam width defined by the surface itself.
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