1. Introduction

Anomalous refraction phenomena in electrodynamics such as negative refraction have a long history, dating back to a speculative discussion by A. Schuster in 1904 [1]. Far later on in 1968, Veselago intensively studied the general properties of wave propagation in negative refractive index media [2]. Especially for applications in the microwave region, Silin described negative refraction in a review [3]. Dated back to 1978, Silin also described the possibility to construct a plane-parallel lens [4] using a material that allows for negative refraction at the straight boundaries to the surrounding medium providing an internal ray crossover, and it was expected that artificial media will also be found for the optical range to construct such a lens.

For optical wavelengths the basics of dispersive and refractive properties of strongly modulated 1D and 2D periodic light-guides near the optical bandgap were investigated in [5, 6]. The fundamentals of anomalous refraction phenomena such as superrefraction and negative refraction with respect to the group velocity directions were explained in detail both theoretically and experimentally using Floquet-Bloch waves and wave-vector diagrams (WVDs). In a later publication [7] further details on refraction, frequency dependent focusing and Floquet-Bloch wave interferences inside a doubly periodic lightguide as well as imaging phenomena including the crossover effect of ray propagation were explained. Since the realization of an omnidirectional photonic bandgap [8], these structures - especially with improved modulation depth - are known as photonic crystals. In more recent publications, Kosaka also describes super- and anomalous refraction [9], which is an interference effect of Floquet-Bloch waves [10]. Wave-vector representations were also used by Notomi [11] especially for the explanation of negative refraction. All-angle negative refraction for imaging with a photonic crystal superlens at very small distances with respect to the photonic crystal structure was described by Luo [12], however, with several constraints on incoming wave-vectors and dispersion contours. Wave focusing inside a negative index photonic crystal slab was also recently reported in [13]. The phenomenon of negative refraction at the boundaries from homogeneous to periodic structures can be obtained in several ways. One way is the usage of the first photonic band and a direction of propagation in wave-vector space being around one of the corners of the first Brillouin-zone. This leads in a relatively small frequency band to the required concave shape of the relevant part of the dispersion contour, as shown in [7, 12]. The other way is to use - in an appropriate frequency band - the nearly circular shape of the dispersion contour around the origin in a higher-order photonic band [11, 13].

Left-handed materials with their inherent negative refraction law allow for a similar ray path to construct a “perfect lens” [14]. Refocusing using negative refraction was also studied recently experimentally [15] in the near infrared using photonic crystal waveguides. As an additional design tool, polarization discrimination using polarization dependent beam-steering in PhCs [16] may be also taken into account. High-resolution superlens imaging in a triangular photonic crystal has been investigated theoretically [17] at abrupt boundaries, however, resulting in reduced transmission efficiency. The influence of measures for improved efficiency which may affect the ideal behavior are not discussed. This strong influence of the surface termination for a good image quality is analyzed in [18]. An intense study of mainly near-field imaging phenomena in photonic crystal slabs is given in [19], and self-focusing in the microwave regime is described in [20].

A number of imaging principles were demonstrated in the past, however, for potential applications they have to be combined with efficiency. Here we present the results of a preliminary - not fully optimized - theoretical study with the main goal of efficient far-field imaging (refocusing) of wide angle (high NA) Gaussian beams where the limits in [12] are not mandatory across the whole photonic crystal. The major problems to solve in this case are the reduction of reflection losses due to mismatch at the boundaries of the photonic crystal slabs and to find a solution of the inherent problem that the real shape of the dispersion contours of a homogeneous photonic crystal usually cause significant aberrations at high NA.

To reduce the reflection problem we use gradual (tapered) transitions. The advantage of such transitions was also already widely investigated in early experiments [7]. For solving the problem of precise imaging we introduce an inhomogeneous photonic crystal structure, with a non-uniform background refractive index of the still homogeneous grating itself in order to get additional degrees of freedom for design. Using these principles, far-field refocusing of a Gaussian beam by a photonic crystal slab is presented for a spot-size of 1.6μm emerging in air, and a more tentative layout is given for refocusing in dielectric “immersion material” to reduce the spot size to 0.8μm, both in the infrared region at 1345nm.

2. Principle of operation and design of the investigated structures

The basic principle of beam refocusing in a photonic crystal structure is negative ray refraction at interfaces to homogeneous media. Here we use a 2D highly modulated square grating structure inclined at 45° with respect to the straight boundaries. For quantitative evaluation we assume a grating consisting of cylindrical dielectric columns (refractive index of 3.2) with a pitch of 282nm, embedded in air. For the case of homogeneous 2D photonic crystals (PhCs) the principle of imaging is described in Fig. 1(a, b), the case of an inhomogeneous photonic crystal structure including a lateral gradient in the background index (characterized by areas with different colors, details see below) is shown in Fig. 1(c, d). We use representations both in real space and in wave-vector (reciprocal) space. For simplicity only the inner parts of the dispersion contours in the wave-vector diagrams (WVDs) are drawn and only TM-polarization is considered. In both cases, a highly divergent Gaussian beam is launched in air at a distance of several micrometers in front of the photonic crystal, using a wavelength of 1345nm. This corresponds to a normalized frequency of 0.21.

After refraction at the first boundary, the path of the transmitted rays are redirected forming some kind of crossover so that - after passing the second boundary of the photonic crystal - the emerging beam is strongly convergent in air with a beam waist (“focal point”) at the same distance to the boundary as for the incident beam. This beam path is due to negative ray refraction and is explained in Fig. 1(a, b) using - besides the central unperturbed ray -some outermost “rays”. Due to boundary conditions, an incident ray represented by the wave-vector “1” will be transformed at the boundary to a set of Floquet-Bloch waves with identical group velocity directions as described in the WVD. The direction of energy flow in the photonic crystal is indicated by the vector “2” resulting in a direction of energy flow in real space as shown by vector “3”. At the second boundary the Floquet-Bloch waves are reconverted into single waves oriented in the original direction “4”, however, laterally shifted, so that the original divergent beam is now convergent with nearly the original beam waist in the focal point. This kind of propagation was in principle demonstrated and explained in [7] for visible light.

The shape of the dispersion contour in the WVD of a homogeneous PhC however is not well suited for precise refocusing of wide-angle beams due to “over-focusing” as will be shown later on. Advanced dispersion engineering however allows for defined local tuning of the dispersion contours as shown in Fig. 1(c, d). This is done by changing the background index of the photonic crystal (i.e., the basic refractive index in the periodic regions to which the index difference of the dielectric columns with respect to air has to be added. So, here it is set to nback = 1.0). In certain segments the background index can be increased by individual values of dnback, resulting in different local forms of the dispersion contours, whereas the periodic modulation of the photonic crystal itself remains still homogeneous (constant index change of dielectric columns with constant diameter). The result in real space shows that with the aid of inclined boundaries between the regions of different background index, ray redirection is also possible to form a convergent beam for refocusing, however with the additional freedom of beam shaping to compensate for angular aberrations, maintaining the advantage of plane boundaries.

 

Fig. 1. Principle of refocusing in real space and in wave-vector space explained in a WVD (only the central and outermost ray paths are drawn). The WVD is normalized to the grating constant. (a, b) homogeneous (c, d) inhomogeneous photonic crystal at a normalized frequency of 0.21. In (c, d) we use three different values of dnback (0.43, 0.45 and 0.5 for the inner contour curve).

Download Full Size | PDF

In the case of a lateral index gradient, one special phenomenon has to be taken into account for optimum power transfer using our initial layout: In real space there exists no longer a crossover for the ray path. The continuity condition with respect to the oppositely slanted boundaries at the second half of the PhC cause only a cross over of the ray directions in the k-space during propagation in this part of the PhC structure, whereas in real space no crossover takes place and the major trace of energy flow remains at its original side with respect to the normal of the transitions (details later on in Fig. 3).

 

Fig. 2. (a) Wave-vector diagram: Detail oriented in the launch direction at different background refractive index differences (dnback = 0.0; 0.2; 0.4) at constant wavelength of 1345nm. (b) Band diagram including the used frequency of 0.21. (c) Shift of the (real) beam waist from z = -20 μm to a virtual beam waist at z = -10μm due to “wide-angle self-collimation” at dnback = 0.2. (d) Shift of the image beam waist to the second transition by negative ray refraction.

Download Full Size | PDF

In Fig. 2 some quantitative details for this new kind of dispersion engineering at constant optical carrier in a normalized WVD are given. They were calculated using algorithms based on a semi-analytical approach and plane-wave expansion [21]. The corresponding band diagram with the frequency of operation in the first photonic band is given in Fig. 2(b). The background index is varied from 1.0 to 1.4. Again, the central and outermost wave-vectors of a highly divergent incident beam are drawn for propagation in air. Depending on the background index in wave-vector space, three different principal orientations for the vectors of energy flow can be stated. For the case of dnback = 0.0, an incident divergent beam remains divergent in the PhC, too. For dnback = 0.2, however, we get a very special solution: For an incident beam covering a total angular range of about 75° , the direction of energy flow remains in the direction of the central ray. Thus self-collimation (“beaming”) can be maintained over a large angular range. This was also verified by numerical simulations using a 2D finite difference time domain (FDTD) method.

In Fig. 2(c) the strongly divergent incident beam emerges after passing the PhC with the same wave-field. Its origin, however, is shifted by the length of the photonic crystal region. For dnback > 0.2, the phenomenon of negative ray refraction takes place and allows for beam focusing in the photonic crystal itself. In Fig. 2(d) the focus is located at the second boundary. Further increase of dnback leads to a stronger angular deflection and shifts the “focus” inside the photonic crystal for imaging applications as explained above. A similar behavior is also possible by changing the optical carrier wavelength.

 

Fig. 3. (a) Negative ray refraction in a homogeneous 2D-photonic crystal, (b) Negative ray refraction and ray redirection in a 2-section inhomogeneous 2D photonic crystal; dashed line: Boundary for different background indices.

Download Full Size | PDF

Another crucial point for wide-angle refocusing is the usable angular range of the incident divergent beam due to the limited angular extension of the inner dispersion contour in the WVD. In [12] the conditions for extension and shape of the dispersion contours for the photonic crystal itself with respect to the diameter of the air contour are given to get “all angle” negative refraction for imaging applications. However, no compensation of aberrations is possible. In our cases, the major restriction - concerning a homogeneous photonic crystal - remains the same: For “all-angle” refocusing the projection of the outermost region of the nearly circular dispersion contour for the photonic crystal with respect to the boundary between the homogeneous medium and the photonic crystal itself must be larger than the corresponding projection for the circle representing the effective index in the homogeneous region outside the photonic crystal. In our “wide-angle” case of Fig. 1(d) the radius of the outer - nearly circular dispersion contour - represents an “effective index” of 0.66 leading to an angular acceptance range of about +/- 40°. As can be seen from the outermost dispersion contour in Fig. 2(a) a locally restricted partly concave shape (for getting group-velocity vectors being directed versus the grating diagonal) is sufficient to accept also larger angles of incidence. Thus, in order to extend the acceptance angle, or to realize “all angle” refocusing also for surrounding homogeneous media with a refractive index nhom > 1 (leading to larger free-space circles instead of the air contour) we can take advantage of our inhomogeneous photonic crystal design. Large acceptance angles of the divergent incident beam can be achieved by using - for the outer angular regions - photonic crystal sections with extended dispersion contours, corresponding to small background indices. Subsequent correction of focusing aberrations is done by specially designed crystal inhomogeneities (see below). Therefore, our concept of using inhomogeneous photonic crystals by changing the background refractive index has two important consequences: Real wide-angle imaging up to “all-angle” imaging using moderate beam-steering angles (for high coupling efficiency) and compensation of inherent aberrations in square photonic lattices by proper layout of the inhomogeneity itself.

As can also be seen from Fig. 2(a), the steering-angle of energy flow in the outer angular parts is too high, leading to overfocusing in outer angular ranges as already explained in [7]. The possibility of compensation by introduction of different background indexes is explained in Fig. 3 by taking only an inclined, slightly divergent Gaussian beam (waist of 5μm) at the same launch position to represent the outer “rays” in Fig. 1. With the large beam waist, only Floquet-Bloch waves in a small angular range in the WVD are excited, corresponding to wave-vector directions around the launch angle of the incident ray.

Fig. 3(a, c) shows the conventional case of negative ray refraction in a photonic crystal with homogeneous background index. In Fig. 3(b, d) the background index at the right hand side with respect to the dashed boundary is larger. Due to the boundary condition the ray is redirected into the direction of the grating diagonal, giving the explanation for the ray trace in Fig. 1(c). Thus, this local dispersion tuning allows for compensation of “imaging aberrations”, however, maintaining the advantage of plane boundaries. Our approach may be replaced by changing the radius of the dielectric columns as explained in section 5.

One remarkable effect caused by the divergence of a Gaussian beam is that the divergence of the original beam (convex wave-fronts) is maintained after reflection, however, due to negative ray refraction the transmitted beam is convergent (concave wave-fronts) and - in the case of change to another background index in Fig. 3(b) - the transmission direction in air is also modified.

3. Effect of tapered transitions

It is well known that strong beam-steering effects occur in PhCs near the edges of the photonic band gap. As there exists a strong dispersion on the group velocity, its absolute value can be highly different from the values of the surrounding homogeneous media, leading to significant mismatch problems and thus low coupling efficiency. Here we use a taper structure which allows for significant reduction of reflections over a wide angular range. For our generalized layout in Fig. 4(a) we use a 10-step bilinearly tapered transition, where the refractive index of the homogeneous section (nhom) decreases linearly in the taper section down to a predefined background index (in the cases presented here: nback = 1.0) in the PhC itself, whereas the refractive index of the grating columns increases linearly to a maximum, maintaining the radii of the dielectric columns.

In order to show the efficiency to reduce reflections over a large angular range using our proposal, the subsequent numerical experiment using 2D - FDTD- simulations was done: We take the worst case of a refractive index nhom = 1.0 for the homogeneous region outside of the periodic structure and its transitions and look at the reflection properties only at one transition. Therefore possible interference phenomena caused by multiple reflections at two boundaries are avoided. For ease of numerical evaluation we take an incident Gaussian beam with a spot size of 5μm, launched shortly in front of the transition at different angles and investigate the reflection properties. As a measure of the reflection properties we take the ratio between the maximum local power in the “reflected” beam and the incident beam, corrected with respect to the intensity degradation along the path of the Gaussian beam, as drawn in Fig. 4(b).

For comparison we investigated three cases: Firstly we consider the case of a step transition with an angular span for the incident beam from normal incidence up to an angle of 50°. The other cases use a 10-step transition with 9 tapered intermediate columns, by changing their refractive index. For the tapered case we use both a linear taper and a nonlinear (exponential) one. The classical step-like transition reveals already at normal incidence a high power reflectivity of 38%. This increases gradually to up to 63% at an angle of incidence of 50°. The linear taper decreases - at normal incidence - the reflection to about only 2% with a gradual increase for angles below 20°. At 30° the residual reflection still remains half of the value for a step-transition. For still larger angles of incidence the reflectivity increases to about the values for a step transition. Using a nonlinear tapering the residual reflection can be reduced to nearly zero in an angular range of up to 20°, however, a better compromise to enhance the angular range is to allow for some residual reflection for small angles. The result is shown in Fig. 4(b) and reveals a reduction of the power reflection by a factor of 10 up to about 20° and still nearly a factor of 3 up to an angle of incidence of 35°. So both tapered transitions are well suited as highly transparent interfaces for transmission of wide-angle beams.

 

Fig. 4. (a) Refractive index profile used for modeling the tapered photonic crystal transitions using 9 rows of dielectric columns: Generalized layout with a refractive index of the surrounding medium nhom ≥ 1 (b) Simulation results for the power reflection at different angles of incidence with respect to the x-axis and nhom = 1 . Three cases are drawn: Step-like transition, linear and nonlinear taper functions with respect to the change of the refractive index of the columns.

Download Full Size | PDF

4. Simulation results

As numerical method of investigation we use 2D - FDTD- simulations, and all our structures will be 2D devices. We investigated two series for refocusing of a Gaussian beam, one with a spot-size of 1.6μm emerging in air (nhom = 1), and a more tentative layout for refocusing in a dielectric medium with nhom = 2 using a spot-size of 0.8μm, both in the infrared region of 1345nm. By scaling down the lattice constant, the principle should be applicable to refocus a beam with a spot-size of about 0.3μm at 500nm. For simplicity we use as definition for focusing efficiency the ratio of maximum power in the focal point with respect to the maximum power at the launch point.

The complete basic layout for all cases of investigation is shown in Fig. 5. This layout includes two tapered transitions from the photonic crystal slab to the homogeneous outside with the refractive index nhom. Additionally we introduce the possibility to verify lateral inhomogeneities in selected areas (see color-coded regions in Fig. 5) using a change in the local background refractive index nback = 1.0 by dnback , both in the tapered regions and in the photonic crystal itself. For the purpose of comparison, we can remove the tapered transitions and also verify a homogeneous background index.

 

Fig. 5. Schematic layout of an inhomogeneous 2D photonic crystal planar lens with tapered transitions to the surrounding homogeneous medium along the z-axis. The background index in the colored regions can be individually changed by different values of dnback , forming lateral index inhomogeneities in the periodic regions.

Download Full Size | PDF

First the focusing properties of a homogeneous photonic crystal with straight boundaries and abrupt (step-like) transitions were investigated (Fig. 6). A strongly divergent Gaussian beam with a spot-size of 1.6μm was launched at z = -20μm in air, the photonic crystal being located between -14μm < z < -4μm. At the desired focal point (equidistant with respect the boundaries) we got a focusing efficiency of 19% in air 6μm behind the second boundary.

 

Fig. 6. Homogeneous photonic crystal lens with straight boundaries and abrupt (step-like) transitions. Focusing efficiency: 19%. (a) Field map (b) Intensity trace along the z-axis at x=0.

Download Full Size | PDF

In order to reduce the influence of reflections at both step discontinuities we used tapered transitions as described in Fig. 4(a) with nhom = 1 and get a focusing efficiency of 57.5% (Fig. 7) outside the photonic crystal. Due to the contribution of over-focused outer rays the maximum local power in the PhC is shifted from the center to the first transition.

 

Fig. 7. Homogeneous photonic crystal lens with tapered transitions. Focusing efficiency: 57.5%

Download Full Size | PDF

For compensation of the significant focusing aberrations in the case of wide-angle beams we additionally tried to improve the performance by introduction of a lateral gradient. The shape of this index gradient should be determined by optimization. Our initial runs showed that a focusing efficiency of 90% is - numerically - possible (Fig. 8).

 

Fig. 8. Inhomogeneous photonic crystal with tapered transitions: Focusing efficiency: 90%

Download Full Size | PDF

In order to verify whether efficient refocusing of smaller spot-sizes is also possible, we embedded the photonic crystal structure into a homogeneous medium with the refractive index nhom = 2.0. In analogy to the numerical sequence above, a homogeneous photonic crystal with straight boundaries and abrupt transitions showed a refocusing efficiency of 19%. Adding tapered transitions increased the focusing efficiency up to 36.5%. Again, the best refocusing was possible with the additional lateral inhomogeneity, leading to a focusing efficiency of 85% (Fig. 9).

 

Fig. 9. Inhomogeneous photonic crystal together with tapered transitions to surrounding media with n = 2.0. Spot-size = 0.8 μm. Focusing efficiency: 85 %

Download Full Size | PDF

5. Alternative methods for realization of inhomogeneous photonic crystals

So far, we used in our numerical simulations a variation of the background refractive index both for the tapered transitions and the laterally graded photonic crystal region, maintaining the diameter of the dielectric columns. Another approach is to leave the background index constant and change the diameter of the dielectric columns, keeping their maximum refractive index fixed. Starting with vanishing background refractive index difference and a given radius of the dielectric columns, we determined the required change of the column radii to get (nearly) equivalent dispersion contours in the WVD with respect to the case of altering the background index. The drawings in Fig. 10 confirm clearly that this equivalence is possible at least in the range of varying background index between 0 and 0.5.

 

Fig. 10. (a) Relevant detail of the WVD oriented in the direction of a grating diagonal. Equivalence between changes in the background index and the radius of the dielectric columns. The solid curves denote changes in the columns radii whereas the dashed curves represent changes in the background index. (b) Relation between changes in background index and variations in the radii of the dielectric columns.

Download Full Size | PDF

6. Conclusions

We proposed and investigated the highly efficient refocusing (imaging) properties of wide-angle Gaussian beams using modified square photonic crystal structures with tapered transitions and precisely defined lateral inhomogeneities. With our tapered transitions the reflection loss can be significantly reduced with respect to a step-transition up to angles of incidence of 35°, for a smaller angular range even with almost vanishing reflection. Using our new form of dispersion engineering we were able - in a first step of optimization - to improve the refocusing efficiency to 90% for a 1.6μm wide Gaussian beam launched in air and at least to 85% for refocusing of a 0.8μm wide Gaussian beam in homogeneous media with a refractive index of 2.0. Compared to a value of less than 20% for a homogeneous photonic crystal structure with step-like transitions, this is an enormous progress in giving design rules for practical applications of photonic crystal lenses.