In- And Out-Coupling Devices For Subwavelength Waveguides. Part 1
ptical communication systems have proven their advantages compared to electronic ones owing to a wider modulation bandwidth and low losses in optical fibers as opposite to the metallic waveguides. There is a trend of using optical waveguides for communication not only at large distances on a scale of a house, a city, a country or the whole planet, but also on the small distances within a single electronic device, a printed circuit board or even within an integrated circuit . Such technologies would allow for considerable reduction in energy consumption and increase of computer performance.
Despite of advantages of using optical waveguides inside the integrated circuits, the possibilities of miniatiruzation of a dielectric (for example, glass or silicon) waveguide are limited by its minimal size, at which at least a single mode is guided, that is roughly λ/2n, where λ is the light wavelength in vacuum and n is the refractive index of the core material. Typical minimal size of the silicon integrated waveguides are 200 Ч 500 nm 2  that is more than one order of magnitude larger than the size of the electronic components of the modern integrated circuits (for example, from 2015 the Intel integrated circuits produced with 14-nm technology are commercially available ). Smaller size of the waveguide can be achieved by using plasmonic metallic waveguides and in this case the size of the mode can be smaller than 100 nm . Even though the plasmonic waveguides have got larger losses compared to the dielectric ones, transmission through them can be acceptable for using on the small on chip distances or if using additional loss compensating medium .
In this article we consider the physical principles of in- and out-coupling devices (IODs) for optical and near-infrared radiation (let us call both ranges the optical radiation for brevity) into subwavelength waveguides. The main characteristic of the IODs is the coupling efficiency η = Pin/P0 that is the ratio of the power Pin coupled to the subwavelength waveguide to the power P0 incident from the free space or a wide waveguide. Obviously, maximization of the coupling efficiency requires minimization of absorption, reflection or scattering losses.
It is worth mentioning that IOD is not equivalent to a microscope. A high transmission through the optical of system is desirable, but not mandatory for a microscope, while the image of an object formation is needed. At the same time, for IOD a high transmission is a must, but it is sufficient to concentrate the radiation in a single spot without obtaining a high-quality image.
IODs can be divided into the following classes on the basis of the physical principles they are based on (Fig. 1):
Coupled waveguides IODs,
Discrete scatterers IODs.
We consider each of them separately.
Lens based in-
and out-coupling devices
Before considering the lens-based IODs, let us consider the physical reasons for the natural diffraction limit. Dispersion diagram (see Fig. 2) in the form of an isofrequency contour (2D case) or an isofrequency surface (3D case) is a useful tool for the description of optical radiation behavior in a material. Dispersion equation for an isotropic dielectric with the refractive index n is
where k0 = ω/c is the wavenumber in vacuum, kn and kt are the normal and tangential components of the wavevector, correspondingly. This equation describes a circle and its radius is proportional to the refractive index n (Fig. 2).
The electromagnetic field of a point source can be decomposed into the spatial Fourier harmonics and in the wavevector spectrum contains the spatial harmonics with arbitrarily large wavevector components kt (to check this it is sufficient to make a Fourier decomposition of the Dirac delta-function). It follows from the dispersion equation in vacuum that the propagating modes with tangential component kt < k0 and the real normal wavevector component can propagate well in the free space, while the evanescent modes with kt > k0 have got the imaginary and they exponentially decay with the distance from the source. The same effect occurs with the evanescent harmonics with the tangential wavevector component kt > k0n not only in vacuum, but also in any dielectric with the refractive index n. The evanescent harmonics decay occurs on the distance scale of a wavelength λ, thus the typical optical systems for imaging use only propagating modes and only the part of them that propagates within the optical system aperture.
This situation is illustrated in the Fig.3. The image of a point source, obtained with an optical system, is not a point but a set of rings, since the optical system filters out the evanescent harmnonics and propagating harmonics that are not captured by the system. The ideal point source image would be possible if all the propagating and evanescent harmonics are collected.
Let us consider now various lenses that can be used for the in- and out- coupling devices.
A dielectric lens (Fig.1a) is know from the ancient times as a device for light focusing. Quite bulky convex or plano-convex lens can be lightened by removing part of its material in the configuration of the Fresnel lens or Fresnel zone plate. The lens can be a separate device as well as integrated with an optical fiber. For example, it can be formed on the end of the fiber by melting or made inside the fiber with a non-homogenous refractive index distribution (graded-index lens or GRIN lens). Since the dielectric lenses are made of transparent materials, the transmission through them is very high, especially in case of using antireflection coatings, but such lenses cannot beat the diffraction limit due to the reasons mentioned above. For example, a typical 100x microscope objective with an oil immersion has the numerical aperture NA = 1.25 that allows for focusing light with the wavelength λ = 1.55 µm into a spot of the size λ/2NA = 620 nm. Consequently, the dielectric lenses can be used only for light coupling into the waveguide with the mode size larger than λ/2NA.
Photonic crystal lens
Photonic crystal is a non-homogenous dielectric structure (see Fig.4) with the period size comparable to the wavelength λ. Interestingly, there exist a photonic band gap in the photonic crystals, that are the conditions when the light of a certain wavelength cannot propagate in the material. An isofrequency contour of the photonic crystal close to the edge of the Brillouin zone can be very different from the circular one and can even be of a negative curvature (Fig.4) that leads to the negative light refraction. A slab of a photonic crystal material thus can work as a collecting lens.
Focusing with the help of a photonic crystal lens to the spot of the size 0.38 λ was demonstrated in the work  using and InP/InGaAsP photonic crystal. Such focusing is not much better than focusing with dielectric lenses. An additional problem is the light coupling into the photonic crystal, than requires using anti-reflection coating to reduce the reflection of the boundary.
A thin layer of metal with slits is partially transparent for optical radiation. When the wave passes through the structured metal it diffract and excite surface plasmon polaritons (Fig.5) which typically have got a larger wavevector in the metal plane (at the certain conditions, for example, in mid-infrared range in graphene the wavevector of the surface plasmon polaritons can be tens and hundreds times larger) than the wavevector in the free space. The period of the slits should be such that the plasmons from different slits have the same phase when reaching a certain point. The slits can have various shape in the metal plane, for example, semi-circular , then the plasmons, excited at different points reach the center of the structure in phase and in the center of such plasmonic lens the field maximum is observed. In other words, such lens is the zone plate for plasmons.
Focusing with the help of plasmonis lenses was demonstrated theoretically as well as experimentally , using the scanning near-field optical microscope for characterization. There was demonstrated focusing to the spot of 0.74 λ on the wavelength λ = 633 nm in the work . In the work  the experimentally measured size of the focal spot was smaller than 0.2 λ at the wavelength 523 nm upon surface plasmon-polaritons focusing in the near field. In the theoretical work  focusing on the same wavelength into a spot of 0.1 λ was demonstrated. The theoretically calculated transmittance in the latter case was 30% and in order to reach such transmittance (relatively high for plasmonic lenses) an additional resonator was used.
The plasmonic lenses can overcome the diffraction limit since the surface plasmon polaritons wavevector is larger than the wavevector in the free space. At the same time, the presence of metal leads to the optical losses upon plasmons propagation as well as to partial reflection of the incident wave. Small coupling efficiency would hardly allow for using the plasmonic lens as an efficient IOD.
Negative index material lens
The idea to use a parallel slab of a negative index material (NIM) as a lens was suggested by V. Veselago almost 50 years ago . In fact the Snell’s law of refraction n1 sin φ1 = n2 sin φ2 is inverted upon the wave refraction into the NIM and, having a sufficiently thick layer of a NIM, one can obtain an optical image. Interestingly, the NIM lens has 2 focal points: one inside the lens and one outside.
It was believed for a long time that such materials are unrealistic, but then the idea came out how to make such materials artificially creating metamaterials, that are the artificial composite materials with size of the constitutive elements, meta-atoms, much smaller than the wavelength λ . The difference between metamaterials and photonic crystals is that in the former case the period is smaller. As a consequence, the light wave does not "feel" the internal discrete structure and propagates in the metamaterials almost as in a homogenous medium. NIM requires for simultaneously negative dielectric permittivity ε and magnetic permeability µ and this condition can be fulfilled by using the metal-dielectric meta-atoms with electric and magnetic resonances.
In 2000 J.Pendry demonstrated  that NIM lens allows not only for focusing the radiation, but also for amplification of evanescent harmonics, thus such lens is able theoretically to create ideal image with the finest details. Such lens was named the superlens. There was later demonstrated  that with the help of a superlens it is possible to match perfectly two identical waveguides (Fig.6). The evanescent modes decaying in the free space are amplified in the in the NIM slab to the required value to re-create the initial field distribution of the wave coming out of the waveguide. It is natural to assume that using the NIM lens of a more complex shape than the flat slab it is possible to match two waveguides of different cross-section.
Nevertheless, the NIM lens in optical range is still an interesting theoretical model rather than a practical device. In reality, there are strong limitations for obtaining a perfect image with such lens. Optical losses, meta-atoms inhomogeneity and even transverse size of the NIM lens decrease the resolution considerably  and the suggested NIMs in the optical range are based on the resonant meta-atoms which inevitably introduce strong losses.
Moreover, in order to achieve an ideal image it is necessary to transmit through the lens the waves with the whole set of the wavevectors that is with the tangential wavevector component ranging from 0 to infinity. At a certain wavevector kt=2π/Λ, where Λ is the period of the metamaterial the waves do not feel the metamaterial as quasi-homogeneous and some waves cannot propagate at all as this happens in photonic crystals, where the period is comparable to the wavelength and a photonic band gap is formed. In addition to this even for the small values of the tangential wavevector component even in cubically symmetric metamaterial strong spatial dispersion (dependence of the wave speed on the direction of propagation) can happen  and this also prevents getting a perfect image. Last, but not the least, experimental realization of NIMs, which are suitable for the superlens, is very problematic since it requires fabrication of fine metallic nanostructures with high precision.
Snell’s law of refraction on the flat boundary between two media with the refractive indices n1 and n2 relate the angle of incidence φ1 to the angle of refraction φ2, so that n1 sin φ1 = n2 sin φ2.
In a general case the phase of the plane wave changes upon refraction, but the phase change is equal along the whole boundary due to translation symmetry. However, in the case of the inhomogeneous phase change ΔΦ along the boundary (for example, in the direction x), so ΔΦ ≠ const, the refraction law breaks down 
Such translational symmetry breaking can be created with a two-dimensional metamaterial (metasurface) with gradually changing geometrical parameters of the meta-atoms. If the transmission phase is proportional to the coordinate square, the refracted wave changes from a flat wave to a converging wave. Thus the gradient metasurface can focus the radiation  (Fig. 7). The single layer metasurface is actually a very thin Fresnel lens or a GRIN lens. The gradient metasurface lens, however, has got the same limitations as the common dielectric lens. Only the waves with kt < k0 can propagate in the air and they form the diffraction-limited image.
Hyperbolic metamaterial lens
The problems of the NIM lens realization made the scientific community looking into more realistic systems, which would allow obtaining an image close to ideal. Such system was discovered and it was named "the poor man’s superlens’ . It came out that for one of light polarizations it is sufficient to have a negative dielectric permittivity ε only, and this is exactly the case for metals. There was experimentally demonstrated a sub-diffraction image with the help of a silver layer, but such lens work in the near-field zone only.
More interesting from the point of view of focusing to the subwavelength spot size would be a lens that transformed evanescent waves into propagating ones. Such system was suggested and was named the hyperlens. The name hyperlens means only that the lens is made of a metamaterial with the hyperbolic dispersion. In an isotropic dielectric the isofrequency surface is spherical and in an anisotropic dielectric it is elliptical (the main components of the permittivity tensor εx, εy, εz are positive and let us assume the non-diagonal components equal to zero). It can be demonstrated that if one or two main components of the permittivity tensor are negative and the other ones are positive, than the isofrequency surface is hyperbolic (Fig.8). This fact leads to the interesting consequences.
Ellipsoid is the surface of a finite volume. In other words the components of the wavevector (kx, ky, kz) lying on the ellipsoid surface cannot be arbitrarily large and, as we discussed before, this is the reason of the diffraction limit obtained with a lens. Opposite to this, the components of the wavevector lying on the surface of a hyperboloid can be arbitrarily large. This phenomenon is used for image acquisition with the hyperlens.
The hyperlens is most often a set of thin metallic wires in a dielectric matrix (and such hyperlens can work nearly in any frequency range of electromagnetic radiation) or a set of thin metal and dielectric layers (hyperlens for optical and ultraviolet radiation) . The isofrequency surface is hyperbolic only up to a certain limit, since at the wavevector component kt, comparable to 2π/Λ, where Λ<<λ is the structure period, the wave feel the discrete structure of the metamaterial and it starts working in the regime of a photonic crystal. Nevertheless, even if considering a conservative from the point of view of modern technology value of the period 40 nm and the wavelength 400 nm, such material can guide the waves with the tangential component kt up to 10 k0 that is absolutely impossible for any material. A flat slab of the hyperbolic material can transmit the image in the near field zone without magnification.
Even more interesting situation occurs when the surface of the hyperbolic metamaterial is curved to the shape of cylinder or a sphere . In this case there happens not only the transmission of the evanescent waves with kt > k0, but also their conversion into propagating waves which can be further detected with, for example, a microscope. Contrary to the NIM lens, the hyperlens has been experimentally demonstrated . It will probably become a kind of add-on to an optical microscope increasing its resolution.
Nevertheless, the hyperlens is not the best option for matching waveguides, first of all due to optical losses inevitably present due to the fabrication inhomogeneities (metallic layers or wires should be identical) as well as due to the metal employment for obtaining the hyperbolic dispersion. The experimentally demonstrated resolution in optical – infrared range is most often not better than λ/4 . Transmittance of the optical radiation through the hyperlens is not high. Experimentally measured transmittance through 3 periods of the hyperlens was only 52% at the wavelength 900 nm .
To be continued.