1. Introduction

In recent years, the propagation of electromagnetic waves in periodically structured materials has attracted extensive attention. The periodic structures can be either realized with meta-materials [1] or with photonic crystals [2]. The theoretical treatment regarding meta-materials is based on the work by Veselago [3] in 1967 who analysed a hypothetical material where both the dielectric constant ε and the magnetic permeability μ are simultaneously negative. Then the index of refraction is negative and the direction of the flow of energy and of the wave vector k (phase velocity v _ph) are opposite to each other. As in this case E, H, and k form a left-handed set, these materials are called “left-handed” (LH).

Photonic crystals (PhC) consist of periodically modulated dielectric or metallic material. In the dielectric case the PhCs have a modulated dielectric constant and μ > 0. Nevertheless, it has been shown [4, 5, 6] that band structure effects can produce effective negative refraction or even a negative index as in the meta-materials. Losses can be much smaller in PhCs when non-conducting dielectric material is used.

Negative refraction (NR) and left-handed (LH) behavior in PhCs have been studied by several authors using numerical calculations(see Ref. [7] and references therein). NR has been reported for the first time in an experiment on a prism of a meta-material [8] at a microwave frequency (10.5 GHz). At a similar frequency (13.698 GHz) NR has been observed on a slab of a two-dimensional (2D) dielectric PhC by measuring the displacement of the transmitted beam while varying the angle of incidence [9, 10]. Negative refraction occured for incident angles larger than 20°. The experimental observation of left-handedness on a wedge-shaped metallic PhC at 9 GHz has been reported by Parimi et al. [11, 12]. The criteria for the occurrence of all-angle negative refraction (AANR) have been determined from numerical calculations [13]. In that work the PhC consists of holes on a square lattice in a dielectric with ε = 12. AANR has been found in narrow frequency ranges in the first and second band. For focusing by a flat lens AANR with refractive index n=-1 has been considered as the necessary criterion [14].

Another interesting point of the wave propagation in PhCs concerns the coupling of the external plane wave to the PhC. A basic question is whether the external wave couples to the internal Bloch waves for incidence on certain interfaces. In a simulation [15] extremely weak coupling has been shown to occur in a triangular-lattice PhC with incidence on the “ΓK surface” for TM polarization (definitions see below). It has been attributed to the characteristics of the Bloch waves. A related phenomenon has been studied by von Freymann et al. [16], viz. the role of the surface termination which results in strongly different diffraction efficiencies in the Littrow geometry.

The purpose of the present work is to study the ranges of negative refraction in the first two photonic bands as well as the coupling of the external waves and criteria for flat-lens focusing in these ranges. We use a PhC with dielectric rods on a 2D square lattice and perform microwave experiments and numerical simulations. The “rod PhC” has larger band gaps (for TM waves, definition see section 2) than the “hole PhC” (for TE waves) [13] with similar dimensions and dielectric constant. It is also easy to fabricate using alumina rods with a high dielectric constant of 9.6. The main results concern: (a) the first observation of the transition from positive to negative refraction as a function of frequency when approaching the first band gap (band edge at 40 GHz for ε = 9.6), (b)coupling of the external waves to the PhC, (c) improved flat-lens focusing by considering a material with an ultra-high dielectric constant as well as by introducing a novel criterion, viz. the constancy of the ratio of the tangents of the incident and refracted angle (R_tan ). The case of a dielectric constant close to 100 has not been considered before but is feasible at microwave frequencies. The usefulness of the novel focusing criterion, more generally applicable than AANR with n=-1, is demonstrated by simulations in the first band.

In this context we make an introductory remark about the use of the notion “refractive index”. In the present work we use an effective index n _beam which takes into account the directions of the group velocties in air and in the PhC reflecting the main flow of energy in the incident and the refracted beam. It is defined via n _beam = sin(Θ _i )/sin(Θ _r ). Θ _i and Θ _r are the angles of the surface normals - the incident and refracted angle, respectively. This definition does not necessarily imply an exactly circular equi-frequency contour (EFC). However, for the flat-lens requirement, viz. n _beam = -1 being independent of the angle of incidence, it is necessary that the EFC is exactly circular and that its radius is equal to that of the air circle k_air = ω/c. The novel R_tan criterion introduced here is not limited to the restrictions on the EFCs and includes n _beam = -1 in the special case R_tan = -1.

2. Results

2.1. Experiment

In the experiment for the 2D square-lattice PhC we use alumina rods in air. The rod radius r is 0.61 mm, the lattice constant a is 1.86 mm, and r/a = 0.328. The length of the rods is 50 mm. The photonic crystal slab has a thickness of 18.4 mm (rod center-to-center) and a width of 118 mm (15 rows of alternatingly 89 and 90 rods). The incident beam hits the PhC on the (11) surface (normal along ΓM), in short notation “ΓM surface”. The polarization of the incident wave is TM in the first band and TE in the second band (E parallel and perpendicular to the rods, respectively). In the simulation (section 2.2) we consider also incidence on the (10) surface (“ΓX surface”) with TE polarization.

As an input parameter for the calculations the refractive index n and the dielectric constant ε of the material of the PhC is needed. For determining it we performed normal-incidence transmission measurements on a bulk plane-parallel alumina disc (diameter = 50 mm and thickness = 24 mm) between 28 and 36 GHz. The average distance between interference fringes gave n _alumina = 3.1 ±0.1. This value is smaller than the one (n = 3.3) used in Ref [17] for f > 20 GHz and confirms the one used in Refs. [9, 10] as being correct. Consequently we use ε=9.6 in the calculations.

With the dimensions of our PhC the upper part of the first band can be investigated by microwaves at Ka-band frequencies (26.5–40 GHz), the second band by millimeter waves in the range 60–70 GHz (in the E-band). As the microwave source we use a Gunn diode and a backward-wave oscillator, respectively. They are connected via a rectangular Ka-band or E-band waveguide to a horn with an opening of 3.5 cm perpendicular the E field. As the detector we use an open-ended waveguide connected to a thermistor. This way we perform transmission experiments with monochromatic TM modes [18] in the first band near its maximum at the M point (the omni-directional gap extends from 40 to 48 GHz), and with TE modes in the second band near its maximum at the Γ point.

For measuring the frequency dependence of the refraction we use incident angles of 45° and 30° in the first band and 15° in the second band. The set-up of the measurement is schematically shown in the inset of Fig. 1. From the displacement of the transmitted beam through the PhC relative to the one without the PhC we determine the angle of refraction. A typical result in the range of negative refraction is shown in Fig. 1. The dependence of Θ_r on frequency in the first band for Θ_i= 45° is presented in Fig. 2 together with the calculated results of section 2.2. Also shown are the experimental data for Θ_i= 30°, in the range near Θ_r corresponding to n _beam = -1.

According to the predictions of the simulations of the next section NR occurs in the upper part of the second band at relatively small incident angles for incidence on the TM surface. This is experimentally demonstrated for an incident angle of 15° at 67 GHz where Θ _r is determined as -43° (see Fig. 3). At this angle and at normal incidence the transmission is measured to be strong, i.e. no indication of weak coupling to the Bloch waves of the PhC is found.

In a control measurement at 35.1 GHz with a positive-index material (bulk PMMA slab) a value of +1.5 is determined for the refractive index. The actual shape of the beam transmitted through the PhC can be influenced by frequency dependent multiple internal reflections. This is likely to be the origin of the weak shoulders in Fig. 1 and of the scatter of the experimental data in Fig. 2. The overall experimental uncertainty of Θ_r is estimated to be ±5°.

2.2. Calculations

We performed calculations regarding the photonic band structure, equi-frequency contours (EFCs), and the wave propagation (FDTD method [19]). The calculations of the band structure [20] and EFCs are based on the plane-wave method (PWM) using the program BandSOLVE [21]. For the first band we choose the TM polarization of the incident wave, for the second one the TE polarization. The notation TM (TE) means that the electric field E is parallel (perpendicular) to the rods. The number of rods in the PhC is given in the figure captions. Rod radius and lattice constant are the same as given in the experimental section. An example of EFCs is shown for the second TE band in Fig. 4.

From the EFC calculations we determine the angle of refraction Θ_r - the angle between v_gr and the normal to the surface (compare the EFCs shown in section 3 for the second TE band). The results are included in Fig. 2 for the first TM band. We find that the transition from positive to negative refraction occurs between 34.2 and 36.3 GHz depending on the angle of incidence (80° - 5°).

 

Fig. 1. Intensity versus position of the TM beam without the PhC and transmitted through the PhC at f = 36.5 GHz and Θ _i = 45°. The dashed curves are low-frequency filtered and used for the determination of the peak positions. The measurement set-up is shown schematically in the inset.

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Fig. 2. Angle of refraction Θ _r as a function of frequency with the angle of incidence Θ _i , as the parameter based on the EFC calculations. Also included are the experimental data points for Θ _i = 45° (diamonds) and Θ _i = 30° (triangles) (details see section 2.1). A variation of the rod radius corresponding to that one of the rods used in the experiments is included for Θ _i = 45° (dashed curve: r = 0.60 mm, all other curves: r = 0.61 mm). The full dots on the calculated curves mark n _beam = -1.

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Fig. 3. Intensity versus position of the TE beam without the PhC (curves peaked near x = 0 mm) and transmitted through the PhC (curves peaked near x = -20 mm) for f = 67 GHz (second band), Θ _i = 15°, and incidence on the ΓM surface of the PhC. Upper diagram experimental results. Smoothed curves are low-pass filtered. Lower diagram: from FDTD calculations.

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To compare the experimental results for the beam shape (Fig. 3) with those of the FDTD calculations the fields E and H are calculated along a line 1.3 mm (spacing of the (11) planes) behind the crystal. The energy flux across this line is the z-component of the Poynting vector S_z = E_yH_z . The mean flux is obtained by averaging the flux values at two points of time a quarter period apart. The source is assumed to have a lateral extension of 2.6λ causing a divergence of the beam corresponding to its Gaussian field distribution (λ is approximately 4.5 mm at 67 GHz in free space). This divergence reproduces the radiation characteristic of the horn antenna of the experiment. For the FDTD calculations of Figs. 8 and 9 in chapter 3 sources with extensions of λ/10 and 1λ,resp., are considered. The divergence is increased as the extension of the source is reduced. For the 2.6λ source the divergence is approximately ±6°, for the λ/10 source it is close to that of a dipole source.

3. Discussion

The ranges of negative refraction are observed in our experiments for the first and second band of the square lattice PhC in accordance with the numerical calculations. In the first band AANR exists between about 36 GHz (sign change of the 5° curve in Fig. 2) and the upper band edge for all incident angles whose corresponding beams can propagate within the PhC (ε = 9.6). The further discussion concerns simulations in certain negative-refraction regimes in the first and second band. In the first TM band the upper part is of particular interest for the investigation of criteria for flat-lens focusing. The criteria involve “all-angle n_beam = -1” and the ratio of the tangents of the incident and refracted angle (R_tan ). In the second TE band the wave propagation is considered at frequencies where the flat-lens criteria are not met, nevertheless a collimation of the transmitted beam is observed.

 

Fig. 4. EFCs of the second TE band with 0.5 GHz steps between 65 and 67.5 GHz. ε = 9.6. Black arrow: incident wave vector k_air at 67 GHz for 15° incidence on the ΓX surface. White arrow: k_phC (direction of the phase velocity). k_phC is obtained using the conservation of k _∥. Dashed arrow: group velocity, obtained from dω/dk in point B. The wave with a k_phC at point A does not exist since the direction of v_gr would not correspond to an energy flow away from the source.

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Before going into the details we discuss whether the strength of the coupling of the external wave is suffuciently strong to observe effects in transmission, which are expected from band-structure and EFC considerations. In this context the work by Ruan et al. [15] on a triangular-lattice PhC is of interest. A main result of it is very weak coupling for incidence on the ΓM surface and TM polarization at a certain frequency where the effective index is -1. We have repeated Ruan’s calculation and found an equivalent result (extremely weak wave propagation within the PhC and on the transmission side). However, when changing the frequency or the ratio r/a by 10 percent the wave amplitude in the PhC and the transmission increase significantly. This agrees with Martinez’ results [17] obtained also for a triangular lattice for incidence of 19 GHz radiation on the ΓK surface. Our experimental and calculated FDTD results on the 2D square-lattice PhC show that in both band regions mentioned above strong transmission, i.e. also strong coupling into the PhC, exists. The conclusion is that Ruan’s results about extremely weak coupling due to the symmetry of the Bloch waves are only of limited relevance for the triangular lattice in a restricted frequency range and are not of relevance for the square-lattice PhC considered here.

 

Fig. 5. Deviations of n _beam from the value -1 in the upper part of the first band of a square lattice PhC with ε= 9.6 and 90 (dashed and full curve, respectively). The frequencies are chosen to give the smallest variations with respect to the incident angle, i.e. 37 GHz for ε= 9.6 and 13.05 GHz for ε= 90.

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For investigating the focusing by a flat lens we consider two criteria: (i) n _beam being -1 for all incident angles(“all-angle n _beam = -1”) and (ii) a novel criterion involving the ratio of the tangent of the incident and refracted beam (R_tan ). Additionally, we perform the calculations with an ultra-high dielectric constant of the rods being about a factor of 10 higher than that of alumina.

Our experimental and calculated results in the negative refraction regime of the first band show that criterion (i) cannot be met with a dielectric contrast (DC) of about 10 (Fig. 2). However, this ideal situation of “all-angle n _beam = -1” can be more closely approached in a 2D square-lattice PhC if the DC is enhanced by a factor of about 10. The relative frequency range with n _beam = -1 between Θ _i = 5° and 80° can be reduced from 10 percent (for DC= 9.6:1) to 1 percent (for DC= 90:1). Considering a certain frequency (37 and 13.05 GHz for ε = 9.6 and 90, resp., with the same size and spacing of the rods) the variation of n _beam with incident angle is strongly reduced for DC= 90:1 (Fig. 5). As stated in the introduction the definition of n _beam does not imply an exactly circular EFC. Since n _beam varies slightly around -1 with incident angle even in the case of DC=90:1, a deviation from the circular shape exists. This ultra-high DC is achievable with certain microwave ceramics. Preliminary experimental and calculated results demonstrate the feasibility of PhCs with such ultra-high DC in the microwave regime.

Variations of n _beam with incident angle generally exist in PhCs apart from very limited regions in the Brillouin zone near points of high symmetry. There, however, the air EFCs (circles) are larger than the EFCs in the PhC, not allowing for n _beam being -1. As a generalized criterion for focusing (criterion (ii) above) which also works for EFCs of different sizes in the air and in the PhC we suggest to consider the ratio of the tangents of the angles of the incident and refracted beams, (R_tan = tan(Θ _i )/tan(Θ _r )). If the ratio is the same for all incident angles or at least for all the beams, that can propagate in the PhC, a focusing by the flat lens can be achieved. It involves the directions of the group velocities and not of the phase velocities as in Snell’s law (for isotropic media). Under this condition all the beams emanating from the source at a distance g from the first surface will focus within the crystal at a distance b = gR_tan . They will be refracted by the second surface of the slab and form an image of the source at a distance b′ = d/R_tan - g. Using the ratio of the tangents implies an EFC shaped as an ellipse with the semi-major axis equal to the radius of the air-EFC. This will not generally be the case. However, the EFCs might closely fit the ellipse on an arc corresponding to a limited range of incident angles (0°to 50° for 12.90 GHz in Fig. 6). This is also shown in Fig. 7 where $R_{\mathit{\text{tan}}}^{-1}$ is plotted versus frequency. At 12.90 GHz up to incident angles near 50° R_tan is independent of the angle. Only for larger incident angles it becomes angle dependent. For the range up to 50° the elliptic approximation is excellent.

 

Fig. 6. EFC in the first TM band for ε = 90 and incidence on the ΓM surface. The air circle (around Γ) and the PhC-EFC (around M) at 12.90 GHz are emphasized. The dashed ellipse corresponds to $R_{\mathit{\text{tan}}}^{-1}$ = 0.5. Size of semi-major axis = k_air , size of semi-minor axis = k_airR_tan for f=12.90 GHz.

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Fig. 7. Ratio of the tangents of the refracted and incident angles ($R_{\mathit{\text{tan}}}^{-1}$) as a function of frequency in the first TM band for ε = 90.

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Fig. 8. Wave pattern (E_y ) for a small (= λ/10) source and propagation (TM polarization) through a PhC slab (surface parallel to ΓM, 79 rows of alternatingly 91 and 90 rods). ε= 90. λ≈23 mm. Left diagram: f = 12.90 GHz (R_tan ≈const.), right diagram: f = 13.05 GHz (n_beam ≈-1). Both frequencies lie near the upper edge of the first band. The limits of the color scale for the left diagram are larger by a factor of three. The lateral size of the source (at z = -7.5 cm for 12.90 GHz, at z = 10.4 cm for 13.05 GHz) is marked by a yellow rectangle (not to scale). Below each diagram showing focusing through the slab is a diagram for the same frequency showing the internal propagation. In the lower diagram reflection by the second surface of the PhC is inhibited by including an absorbing surface. Then, a clear internal focus appears.

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The two criteria for focusing, deduced from band structure and EFC considerations, are checked by FDTD calculations for the square lattice with ε = 90 rods. Special emphasis is put on achieving focusing several wavelengths away from the second surface of the PhC. As will be shown below the optimal frequencies are in the upper part of the first TM band: 13.05 GHz for the “n _beam = -1” criterion and 12.90 GHz for the “R_tan = const” criterion. Fig. 8 (right diagram) shows the wave pattern at 13.05 GHz with the wave emitted from a “point” source (lateral extension λ/10) positioned 1λ away from the first PhC surface. At 13.05 GHz n _beam varies only weakly around the value -1 (Fig. 5). At 12.90 GHz R_tan shows the least dependence on incident angles (Fig. 7). Fig. 8, left diagram, shows the corresponding focusing behavior again for a λ/10 source. For obtaining a steady wave pattern a long computation time is necessary. This leads to several internal reflections in the PhC and disturbs the wave pattern with a clearly visible focus in the interior. In order to make the focus observable, we performed calculations where reflections by the second surface of the PhC are inhibited by placing an absorptive layer there. The internal foci are clearly visible now. Regarding the external focusing, at 12.90 GHz the maximum of the electric field (E_y ) is at a distance of 43 mm (≈ 2λ) from the second surface of the PhC. There the full width at half maximum (FWHM) of the intensity is 0.6λ. At 13.05 GHz the distance of the maximum is 71 mm (≈ 4λ) and the FWHM is 0.8λ. Focusing outside the near field (with spot size ≈ 1.1λ) has been also reported by Guven et al. [22] for the fifth TE band of a hexagonal PhC. By using either the “n _beam = -1” criterion or the R_tan criterion with an ultra-high dielectric constant we get significantly better results (in the first TM band). Since in our case and in Ref. [22] both source and image are several wavelengths away from the PhC the near-field sub-wavelength focusing as treated in Refs. [23], [24], and [25] is not of direct relevance.

 

Fig. 9. Wave pattern from a small (1λ= 4.5 mm) source for incidence on the ΓX surface of a PhC slab (21 rows of 41 rods). The lateral size of the source (at z= -3 cm) is marked by a yellow rectangle. ε = 9.6. f = 66.85 GHz (second TE band, regime of negative refraction). The beam is collimated although neither the condition n_beam = -1 nor R_tan =const is fulfilled.

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An interesting case is observed in the second TE band at frequencies shown in Fig. 4. At 66.85 GHz for incidence on the ΓX surface a collimation is obtained extending over several wavelengths (Fig. 9). Negative refraction (and left-handed behavior [20]) occurs there, however, neither the condition n_beam = -1 nor R_tan =const is fulfilled. A complete understanding of the collimation has still to be found including the possible role of the left-handed behavior of the wave and of the phase index. n_ph = k_PhC /k_air = c/v_ph is negative and varies only weakly for the small angles of incidence where most of the intensity is contained (Fig. 10).

4. Summary

We investigate properties of the 2D square-lattice photonic crystal consisting of dielectric rods in air. We compare the results of band structure calculations and FDTD simulations for finite slab structures with experimental results in the microwave range and identify regions of negative refraction in the first and second photonic band. Regarding criteria for flat-lens focusing, the “all-angle n_beam = -1” criterion is generalized by introducing the ratio R_tan of the tangents of the incident and refracted angles which is required to be constant when varying the incident angle. The effect of increasing the dielectric contrast from 9.6 (with alumina as a standard microwave ceramic) to 90 is to reduce the variations of n_beam around -1 and the deviations of R_tan from a constant value. Consequently the focusing is significantly improved due to this increase and the FDTD simulations show spot sizes below one wavelength at distances several wavelengths behind the PhC.

 

Fig. 10. Phase refractive index n_ph vs. frequency in the second TE band. ε= 9.6. f = 66.85 GHz, marked by a vertical line, is the frequency in Fig. 9.

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