Experimental results showing “negative refraction” and some kind of “lensing” −in the microwave-infrared range− are often presented in the literature as undisputable evidence of the existence of composite left-handed materials. The purpose of this paper is to present experimental results on “negative refraction” and “lensing” at visible wavelengths involving a waveguide array formed by a tight-packed bundle of glass fibers. We will demonstrate that the observed phenomena are not necessarily evidence of the existence of left-handed materials and that they can be fully explained by classical optic concepts, e.g. light propagation in waveguides.
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It has been suggested that could exist certain unusual materials −called left-handed materials (LHM)− in which light deviates towards the opposite side as it does in normal matter, so that a slab-shaped LHM could work as a lens [1–10]. A LHM-lens is often called a “superlens”, where the prefix “super” refers to the hypothetical ability of LHM-lenses to amplify evanescent waves, which could help to overcome the resolution limit of standard lenses.
Although there is consensus about that natural LHM do not exist, experimental results recently presented in the literature [4–8] showing “negative refraction” and some kind of “lensing” −from the microwave to the infrared spectral range− appear to be irrefutable evidence of the existence of certain structured (engineered) LHM denominated metamaterials. [Since the amplification of evanescent waves has not been experimentally demonstrated convincingly yet (see e.g .), we prefer to use the term “lensing” instead of “superlensing” to refer to the observed phenomena.]
Most metamaterials are periodic arrays of resonators assembled into a unit to form a “composite material”. This is precisely the key issue that generates discussion (see e.g [12,13].): Are the observed phenomena indisputable evidence of the existence of metamaterials? or actually, the observed phenomena are simply a consequence of the structured nature of the composite material?
Refraction of light in absorbent materials (e.g., metamaterials) is a complicated phenomenon with unusual characteristics. Concepts such as negative refraction, negative phase velocity and counterposition are connected to each other and deserve a close examination [14,15]. Also, negative refraction, superlensing and LHM are not necessarily associated concepts. In fact, there are theoretical models and simulations of photonics crystals [16,17] that predict the existence of “superlensing” (in the sense of “superresolution”) without involving negative refraction and amplification of evanescent waves, in contrast to LHM-lenses. The predicted lensing effect can be understood as a phenomenon of “canalization” of images through the photonic crystal.
The purpose of this paper is not of theoretical nature, but it is to present experimental results on “negative refraction” and “lensing” at visible wavelengths using a waveguide array formed by a tight-packed bundle of glass fibers. Of course our composite material is not a metamaterial, and thus, in consequence, we are providing an experimental demonstration that “negative refraction” and “lensing” are not necessarily evidence of the existence of metamaterials. In fact, the observed phenomena can be fully explained by classical optic concepts as light propagation in waveguides, i.e. a sort of “canalization” process through the fiber bundle.
In the next section we describe the theory underlying our experiments. The experimental results are presented in Sect. 3.
Consider a light wave (with some lateral broad) characterized by a wavevector incident under an angle θ on the entrance surface of a two-dimensional waveguide, as shown in Fig. 1 . We will suppose that the waveguide is homogeneous and its axis is along the z-coordinate. Also, we will consider that the waveguide diameter (a) is much larger than the light wavelength (λ), and that the distances along the z-direction verify , so that the light propagation will be essentially governed by geometrical optic laws, i.e. in the next diffraction effects can be disregarded.
After refraction in the first interface, the light will suffer multiple reflections at the waveguide walls. Since the incident light beam has certain lateral broad, across a given waveguide cross section it can coexist waves traveling in “positive” and “negative” x-direction, but the z-component of the wavevectors of the light propagating inside the waveguide will remain constant. A certain amount of light could eventually escape through the top and bottom wall of the waveguide, but we are interested in the waves propagating inside it.
When the light comes out of the waveguide, in addition to a wavevector parallel to , there will be eventually another lightwave that forms an angle -θ with the z-direction, i.e. . Thus, one has a sort of “negative” refraction at the waveguide exit surface that is originated by the light reflections inside the waveguide. [Clearly, the intensity of the light refracted in the “negative” direction does not need to be equal to that refracted in the “positive” direction.]
Now, consider a large number of tight-packed parallel waveguides (of length d) forming an array in the x-direction, and a point-like light source P placed at a distance in front (on the left-side) of the array, as shown in Fig. 2 . By simple geometrical considerations, it is clear that the “negative” refracted rays at the exit surface of the waveguide will converge on a point P’ at a distance on the right-side of the array. Also, in addition to the convergent light wave, we will have another wave that appears to diverge from a virtual point O at a distance from the array exit surface.
Thus, we have shown that a waveguide array can work as a lens. Since there is translational symmetry along the x-direction (i.e., there is not “optical axis”), the images will not be inverted as occurs with usual optical lenses. Also, the (arbitrary) distance from the object to the bundle entrance surface will be identical to the distance from the bundle exit surface to the image.
3. Experimental results
We have performed a series of experiments using a fiber bundle as waveguide array. We have used a commercial fiber bundle (“optical conduit” provided by Edmund Optics), from which we cut a “slab”-shaped piece (a cylinder) of length d = 18 mm. [In fact, the length of the bundle does not play an important role, the experimental results shown below are representative for a series of experiments performed with cylinders of different lengths.] The separation between fibers was of the order of 200μm (see Fig. 3 ) and the external bundle diameter was 6.4 mm.
Since our array has cylindrical symmetry, when a light beam is incident from the left, on the right-side one has a conical lightwave [18,19] and not simply two wavevectors, as in the case of a two-dimensional waveguide discussed above. Despite of the conical pattern distribution, across each plane containing an incident wavevector and the z-axis, at the exit we will have two wavevectors that correspond to the “positive”- and “negative”-refraction. [The intensity distribution in the cone wall does not need to be homogeneous, and thus, the intensity of the light refracted in the “negative” direction could be only a small fraction of the incident intensity. Also, in addition to the lightwave propagating inside the waveguide, we will have a certain amount of scattering at the outer surface of the fibers, which could produce a spurious background illumination on the screen used to project the images [18,19].]
Figure 4 shows the light ray paths in a plane containing a light beam incident on the fiber bundle (a He-Ne laser beam incident from the left) and the z-axis. The picture was acquired placing a sheet of black paper containing the laser beam and orthogonal to the entrance and exit surfaces of the fiber bundle. On the right-side of the image are clearly seen the “positive”- and “negative”-refracted beams (see also Fig. 1).
In another series of experiments, we have utilized a He-Ne laser beam to illuminate a pinhole placed in front of the fiber bundle at = 9 cm, and observed the resultant pattern on a semi-transparent screen placed at the same distance (9 cm) from the exit surface of the bundle. Figures 5(a) and 5(b) show two different views of the experiment: The image of the point-like source (P) on the screen (S) is clearly visible. Around the central light-spot on the screen there is a relatively illuminate zone that corresponds to the divergent wave mentioned in Sect. 2 (see also Fig. 2).
Figure 6 shows the intensity distribution across the image of a point-like light source (P) placed at a distance = 3.5 cm: The picture was acquired placing a screen (a sheet of black paper) orthogonal to the bundle surface through the image of P. Clearly, the light intensity concentrates on the center of the pattern at a distance ~3.5 cm from the exit surface of the fiber bundle.
As mentioned before, the distance between consecutive fibers of the bundle is of the order of 200μm, but the fiber bundle was not originally manufactured to form an exact periodic array. Thus, it is reasonable to assume that the superposition of the electromagnetic fields that form the images can be considered as incoherent. Then, it is clear that the coherent nature of the light source (i.e. the He-Ne laser) used in our previous experiments does not play an important role in the experimental results presented above.
In order to demonstrate that the “lensing” effect works also with incoherent illumination, we have performed a series of experiments using a high-power UV-LED (Nichia, NCSU034, λ ~388nm, 300 mW). We have utilized as test object a chrome-on-glass target with the symbol “0” and three vertical bars graved on its surface. Since the LED has a wide illumination angle, we used a × 10 microscope objective to concentrate the illumination on the test object.
In Figs. 7(a) and 7(b) are shown the symbols on the target illuminated by the UV-LED. The size of the symbols graved on the test target was of the order of 1-2 mm, and they were placed at a distance = 3.5 cm from the bundle entrance surface.
Figures 7(c) and 7(d) show the corresponding images obtained on a screen placed at 3.5 cm from the bundle exit surface. Clearly, the obtained images are not perfect images with optimum resolution. However, taking in account the different factors that degrade the images −e.g. small diameter of the available bundle (~6.4 mm), low resolution of the bundle due to large fiber separation (~0.2 mm), and relatively large distances (~7 cm)− the obtained images can be considered satisfactory.
4. Discussion and conclusions
The purpose of the present paper was to demonstrate that experimental results showing “negative refraction” and some kind of “lensing” are not necessarily undisputable evidence of the existence of metamaterials. Our demonstration was not theoretical, but it was purely of experimental nature: We have performed a series of experiments showing “negative refraction”and “lensing” at visible wavelengths. The observed phenomena can be considered a sort of “canalization” of images that can be fully explained using geometrical optic laws.
Of course, not all experiment on “negative refraction” and “lensing” reported in the literature may be explained on the basis of geometrical optics. It is clear that when the diameter of the waveguide decreases and becomes comparable to (or lower than) the wavelength of the fields, the consideration of the full wave theory of light propagation is necessary.
In general, the reported metamaterials are assemblies of resonant waveguides disposed periodically in an array with period comparable (or lower than) the wavelength [4,5,7,8]. So that, in addition to the diffraction, we have to consider the interference of wavelets originated at the different waveguides. Actually, this suffices to give an alternative explanation of most experimental results on metamaterials, as discussed in [12,13].
The authors thank the financial support from PEDECIBA (Uruguay) and the Comisión Sectorial de Investigación Científica (CSIC, UdelaR, Uruguay).
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