## Abstract

The FDTD method was applied to investigate the two-dimensional PhC slabs with periodically-aligned tube-type air holes. By numerical simulations, we found negative refraction behaviors by tuning the incident angle into the narrow negative-refraction angle range, and the collimation effects were analyzed by means of wave vector diagrams.

©2007 Optical Society of America

## 1. Introduction

Photonic Crystals(PhCs) are formed by periodically modulated dielectric structures,
which have the capacity to prohibit the propagation of electromagnetic waves in
certain frequency ranges [1, 2]. At the early time, studies on optical properties of band
gap were the mainstream [3, 4, 5, 6, 7]. However, optical properties in the conduction bands of PhCs
attract much attention recently, since negative refraction and imaging can be
realized for these frequency regions [8]. Negative refraction(NR) phenomenon was firstly investigated
in left-handed materials(LHMs) by Veselago [9], and LHMs are materials with simultaneously negative
permittivity ε(**r**) and negative permeability
μ(**r**), i.e., double-negativity. The vectors of electric
field **E**, magnetic field **H** and wavevector **k**
satisfy left-handed relationship. Due to the alteration of the relationship of
electromagnetic vectors, propagation of light in LHMs should be re-examined. Pendry
theoretically investigated periodically-aligned metallic lines and split-ring
resonator(SRR) structures, and found that in the long wavelength limit and within
some frequencies, the effective ε(**r**) and
μ(**r**) of these two structures possess negative values
respectively, and then, it was found that combining both structures produces
doublely-negative materials under certain conditions [10, 11, 12]. In Ref. [8], Pendry proposed that diffraction limit in conventional
lenses can be overcome by LHMs, which will lead to many novel applications. NR was
found to be an interesting property of dielectric PhCs as well [13, 14, 15, 16]. By far, several groups have proposed some useful methods
to analyze NRs of PhCs. Notomi proposed the grating-based wave vector diagram
analysis to the NRs of 2D strongly-modulated PhCs. He suggested that the mechanism
of NR can be probably understood by diffracted waves propagating alongside with
incident waves compared with the surface normal [17]. Moreover, NRs in 2D PhCs are strongly affected by wedge
types of samples as well [18]. Luo et.al. have studied NRs and subwavelength imaging of
2D PhCs and proposed the bandstructure requirements for all-angle NR (AANR) [13, 19]. In addition, it was found that a partial or deaf gap of
PhCs leads to self-collimation effect [20, 21, 22, 23, 24, 25], which subsequently leads to subwavelength imaging of
acoustic and photonic systems [26, 27]. Although the aforementioned mechanisms require rigid
periodicity of photonic systems, Feng et.al. recently shown NR and subwavelength
imaging in a 12-fold-symmetry photonic quasiperiodic system as well [28], which is due to the interaction of incident waves and
intrinsic Bloch-like waves propagating in quasicrystals [28].

By numerical simulations, it has been suggested that no NR behaviors occur for the incident waves with frequencies within the AANR range [23], which is not totally in accordance with the wave vector diagrams in Ref. [13]. Moreover, it has been proposed that the subwavelength focusing is probably due to the anisotropic properties of the PhC slab considering the collimation effects [21, 23]. In the present work, we study NR and imaging of a 2D square-lattice PhC constructed by periodically-aligned tube-type air holes. By numerical simulations with comprehensive discussions, we confirmed that collimation effects can be analyzed by means of wave vector diagrams and we found that NR behaviors for incident waves with frequencies within AANR range take place under certain conditions. We also calculated some useful parameters of the optical properties of such a PhC slab. The numerical results present here lead to more knowledge about the anisotropic properties of PhC structures.

## 2. Numerical procedures

Consider N straight air holes with tube-type shape distributed in a dielectric medium
with *ε _{m}* = 12.96 (e.g., GaAs or Si at 1.55
microns), the length of the PhC sample along the cylinder’s axis is
infinite, and the air-holes distribution pattern is square-lattice, as shown in the
left inset of Fig. 1(b). The following parameters are used in the
simulations:

*r*

_{3}= 0.36

*a*,

*r*

_{2}= 0.20

*a*and

*r*

_{1}= 0.10

*a*, where

*a*is the lattice constant, and the definitions of

*r*

_{1},

*r*

_{2}and

*r*

_{3}are shown in the inset of Fig. 1(a). A standard plane-wave expansion(PWE) method is used to calculate bands of the PhC, and the calculated H-(TM-) polarization (magnetic field aligned along the cylinder’s axis) bands are shown in Fig. 1(a). We used 961 plane wave series in calculating bands and 289 series in calculating PhC equal-frequency contours (EFCs). In Fig. 1(b), Finite-Difference Time Domain (FDTD) method with absorbing boundary conditions (perfectly matched layers boundary conditions, PML) is carried out to calculate transmission data along Γ

*X*([10] direction) and Γ

*M*([11] direction) directions [29]. Additionally, FDTD with PML boundaries is carried out herein to calculate snapshots of field distributions. By numerical simulations, we found that the present tube-type structures exhibit flexible capacities in modulating the shapes of PhC EFCs by means of changing radii of the inner circles. For brevity, we only investigated TM-polarized modes herein, and similar results can be expected for TE modes as well.

## 3. Photonic bands and equal-frequency contours

The calculated bands for such a 2D PhC structure are shown in Fig. 1(a), notable changes occur for high-order bands
compared with Fig. 3 in Ref. [13]. By comparison with Figs. 1(a) and 1(b), we find a partial gap in the range of reduced
frequencies [0.158; 0.227], and the propagation of light with the frequency in this
gap is forbidden along the Γ*X* direction but allowed
along the Γ*M* direction. In this article, we used a
reduced frequency normalized by *a*/*λ*. As
shown in Fig. 1(b), the transmission data along
Γ*X*, calculated by means of FDTD method for the frequency
range [0.162; 0.233] are nearly zero, which convince the existence of this partial
gap. The second partial gap is in the range of frequency [0.324; 0.346]. We also
found a deaf gap in the range of frequency [0.227; 0.326], and the propagation of
light with the frequency in the deaf gap is forbidden along
Γ*M* but allowed along Γ*X*, due
to the fact that Band II along Γ*M* in Fig. 1(b) is antisymmetric, and the modes of Band II along
Γ*M* are therefore decoupled for any symmetric
incident waves. Besides, band V along Γ*M* in Fig. 1(b) is antisymmetric and the corresponding modes are
inactive as well, thus the second deaf gap is in the range of frequencies [0.424;
0.434]. The above analysis of the band structure in Fig. 1(a) is convinced by the calculated transmission curves
in Fig. 1(b).

It has been theoretically confirmed that in 2D PhC [30], the direction of energy velocity ve (**v**
_{e} = (**v**
_{e}=〈**S**〉/〈U〉), where
〈⋯〉 denotes the spatial averaging within the unit
cell of the time-averaged quantities, **S** is Poynting vector and
*U* is the energy density) is consistent with the direction of
group velocity **v**
_{g} (**v**
_{g} = *∂w*/∂**k**) [14]. Band II along Γ*X* has a
negative slope and therefore **k** ∙ **v**
_{g} < 0, and the diameter of the corresponding PhC EFCs shrinks as
*f* increases. Band III along Γ*M* also
possesses similar shrinking EFCs. In Fig. 2(a), the EFCs of the PhC considered herein are shown.
We chose the frequencies in the ranges of the first partial gap and the first deaf
gap, and examined the NR phenomena and the collimation effects for the corresponding
frequencies.

## 4. Negative Refraction of EM waves through a 2D PhC Slab

#### 4.1. EFCs and Single-beam AANR effects

Luo et.al. suggested that to obtain single-beam AANR, it requires convex and
larger EFCs compared air EFCs to make all angle NRs possible and guarantee a
well defined single beam propagating inside the PhCs, since from the wave vector
diagram analysis in Fig. 2(b), only a sole wave vector in the first Brillouin
zone gives a causal Floquet-Bloch wave which corresponds to a single refracted
beam [13]. Moreover, the incident frequency is required to be
smaller than 0.5 ×
2*πc*/*a _{s}*, where

*a*is the surface-parallel period (

_{s}*a*=

_{s}*a*for Γ

*X*interface and

*a*= √2

_{s}*a*for Γ

*M*interface) to avoid high-order Bragg reflections and guarantee a single reflected beam [13, 18]. Using these three key criteria, we found a frequency range where AANR is possible, and AANR frequency range has an upper limit

*f*= 0.191 and a lower limit

_{u}*f*= 0.185, and both limits are in the range of the first partial gap. The wave-vector diagram analysis of the incident and refracted waves is shown in Fig. 2(b). Due to the positive effective group velocity (

_{l}**k**∙

**v**

_{g}> 0) of Band I and thus no backward-wave effects, AANR in this case is called all-angle right-handed NR (AARNR).

However, due to **k**∙**v**
_{g} < 0 for the modes of band II along
Γ*X*, left-handed NRs (LNR) or backward-wave effects
take place, AARNR is impossible for this band. Caused by the conservation of
surface-parallel wave vector on both sides of the interface, the construction
line intersects PhC EFC at points A and B, as shown in Fig. 2(c). Note that as w increases within this band
range, the diameter of EFC shrinks. Considering the definition of **v**
_{g}, propagation direction is inward for this EFC. Point B corresponds to a
signal that propagate towards the source and therefore does not give rise to a
propagating beam. Only point A corresponds to a propagating refracted beam,
which is alongside with the incident waves compared with the surface normal.

In general, according to the wave vector diagram analysis mentioned in this section, NR behaviors are theoretically possible for incident waves with frequencies within the first partial gap and the first deaf gap.

#### 4.2. Collimation effects

Birefringence is an anisotropic property of biaxial crystals, where an incident
beam is refracted into two non-collinear beams due to different refractive
indices for ordinary and extraordinary axial directions in biaxial crystal.
Since the PhCs are kinds of more complex structures, as aforementioned, it has
been suggested that due to the anisotropic properties of PhCs, the propagation
of refracted beams is concentrated along an allowed direction [23, 25]. This preferential-direction waveguiding phenomenon,
i.e., the collimation effect, has been suggested to lead to subwavelength
focusing in the near-field region in the other side of PhC slabs [20, 21, 23]. In the present work we suggested that collimation
effects studied in Ref. [23] can be analyzed by means of the wave vector diagram and
the EFCs scheme. As shown in Fig. 3(a), a linear region
*A′B′* exists in the PhC EFC for a
frequency *f* = 0.19. Due to the restriction condition of
wave-vector conservation, incident beams with wave vectors in the range from
*A* to *B* in air EFC are refracted at the
air-slab interface to propagating beams in PhCs with wave vectors in the range
from *A′* to *B′* of PhC
EFC. Since the direction of energy velocity *v _{e}* is
identical with that of group velocity

*v*, the propagation directions of refracted waves with wave vectors in the range from

_{g}*A′*to

*B′*are parallel. Therefore, when a point source is located in the left side of the PhC slab considered herein, the emitted waves with wave vectors in the range from

*A*to

*B*are refracted into EM waves propagating parallelly in the PhC slab, which correspond to

*V*in Fig. 3(a). The width of the collimated propagating beam is theoretically limited by the location of the point source and the inclination angle α

_{col}_{AΓB}, as shown in Fig. 3(a). Figure 3(b) shows two cases for

*f*= 0.19 (blue line) and

*f*= 0.28 (red). The circles are the air EFCs for the corresponding frequencies.

Figure 4 shows the PhC slab structures examined herein.
When δ*x* = 0, the distance between the left/right
boundary of the PhC slab and the left/right boundary of the first/last column of
circles is 0.1*a*. Figures 5 show the calculated snapshots of
*H _{z}* field for a point source emitting EM
waves propagating into a PhC slab, and the distance between the point source and
the center of the first column of the PhC slab is denoted by

*d*, as shown in Fig. 4. It has been shown in Fig. 3(b) that the linear part of the corresponding PhC EFC for

*f*= 0.19 is perpendicular to the Γ

*M*direction. In Figs. 5(a) and 5(c), the sample of the PhC slab is truncated along Γ

*M*interface, and comprises 81 layers in the propagation direction and 181 layers in the transverse direction; and the point sources emit EM waves with frequency

*f*= 0.19. It is shown in Fig. 5(a) that collimation effects occur clearly for a point source located at

*d*= 1.0

*a*. Theoretically, the width of the collimated beam is

*w*= 3.32

*a*, which is derived by the inclination angle of α

_{AΓB}(α= 117.9°) and

*d*, i.e.,

However, the averaged width of the refracted collimated beam measured from Fig. 5(a) is *w*′ =
6.0*a*. The deviation from the theoretical results is
probably due to the finite size of the PhC slab and scatterings by the
interfaces. Figure 5(c) shows the snapshot of
*H _{z}* field for a point source located at

*d*= 5

*a*. Compared with Figs. 5(a) and 5(c), it is shown that the width of the collimated beam increases when the distance

*d*increases, which is in accordance with Eq. (1). The analysis of collimation effects by wave vector diagram and EFC scheme in this case, i.e., collimation effects caused by partial gaps in the context of the study in Ref. [23], can be extended to the collimation effects caused by a deaf gap as well. As shown in Fig. 3(b), within the first deaf gap, a linear part exists in the PhC EFC for

*f*= 0.28, which is perpendicular to the Γ

*X*direction. In Figs. 5(b) and 5(d), the sample of the PhC slab is truncated along Γ

*X*interface, and comprises 61 layers in the propagation direction and 91 layers in the transverse direction; and the point sources emit EM waves with frequency

*f*= 0.28. The inclination angle α

_{AΓB}is α = 66.2°. Figure 5(b) shows the calculated snapshot of

*H*field for a point source located at

_{z}*d*= 5

*a*propagating into the sample PhC slab. The point source in Fig. 5(d) is at

*d*= 10

*a*. The increasing of the width of the collimated beams in Figs. 5(b–d) is in accordance with Eq (1) as well, however the theoretically derived widths in these cases are not in good agreement with the numerical results probably due to the stronger boundary effects for high-order bands.

Figure 6(a) shows the calculated snapshot of
*H _{z}* field for a point source with

*f*= 0.19. The size of the square sample is 50

*a*× 50

*a*and is truncated along the Γ

*X*interface, and the point source is located at the center of the sample. As the aforementioned, the corresponding frequency is within the first partial gap, and the propagation is allowed in the Γ

*M*direction and prohibited from the Γ

*X*direction. Moreover, the location of the point source and the angle a limit the width of the collimated beam. Therefore, the propagation of light is along the Γ

*M*direction of the sample and has the behavior of preferential-direction waveguiding, as shown in Fig. 6(a). Figure 6(b) is the calculated snapshot of

*H*field for the

_{z}*f*= 0.28 case. The sample is the same as that in Fig. 6(a). Figure 6(b) clearly shows the preferential-direction waveguiding phenomenon in the Γ

*X*direction as well. The multi-refringence phenomena shown in Figs. 6(a–b) clearly exhibit the strongly anisotropic properties of such PhC slabs for the corresponding frequencies.

The calculated PhC EFCs for the frequencies within the first partial and deaf gap
are not round-shaped but contain linear parts, which imply that the optical
properties of the PhC structure considered herein exhibit strong anisotropy for
the corresponding frequencies. The collimation effects, as one aspect of
anisotropic properties, are shown clearly in the above maps of calculated
snapshots of *H _{z}* field distribution. We also found
that the calculated width of the collimated beam is roughly equal to that
obtained by Eq. (1). From these numerical results, we confirmed that the
wave vector diagram and PhC EFC scheme in the

**k**-space are certainly effective in analysis on the collimation effects in Ref. [20]. In Ref. [23], by numerical simulations, it has been proposed that no NR behavior has been discovered in the frequency regime thought to be the regime for AANR, and they also proposed that the so-called subwavelength imaging in the other side of the slab is due to the strong anisotropy of the PhC slab. Indeed, for the point source emitting waves with maximum inclination angle smaller than α, the imaging within the PhC slab and NR behaviors are impossible since all the propagating refracted waves are parallel along the Γ

*X*or Γ

*M*direction within the slab, and therefore only directed diffractions occur obviously. However, for the point source emitting waves with maximum inclination angle bigger than α, which is to be discussed in the proceeding section, NR behavior is possible according to the wave vector diagram analysis, and the waves propagating scheme for subwavelength imaging within and in the other side of the slab can be achieved under certain conditions. The preferential-direction waveguiding phenomenon discussed in this section may lead to more knowledge of PhCs and manipulation of photonic flows based on collimation effects.

#### 4.3. Negative refraction and focusing with subwavelength resolution

In general, analysis of the band structures of PhC revealed that NR is possible
in two regions: one is right-handed NR region and was demonstrated by Luo et.
al. and discussed in Sec. 4.1, and the corresponding wave vector diagram is
shown in Fig. 2(b); the other is LNR region, i.e., the frequency
range where the group velocity and the phase velocity are anti-parallel, i.e.,
**k**∙**v**
_{g} < 0, as shown in Fig. 2(c). Note that from the analysis of PhC EFCs for
the two frequencies in Fig. 3(b), the allowed incidence angle range for NRs
(> α/2) is quite narrow for the present PhC structure,
which imposes a restriction condition on the positions of point sources for the
purpose to make NRs possible. In Fig. 7, the calculated snapshots of
*H _{z}* field for incident Gaussian beams with

*f*= 0.19 and

*f*= 0.28 are shown in 7(a-c) and 7(b-d) respectively. The respective slanted angles of the slabs in 7(a) and 7(c) are β

_{1}= 7° and β

_{1′}= 40′, and in 7(b) and 7(d) the corresponding slanted angles are β

_{2}= 27° and β

_{2′}= 70°. The surface terminations of all the PhC slabs in Fig. 7 are δ

*x*= 0. The refracted wave vectors are within the bent region of the PhC EFC in 7(a-b) and within the linear region in 7(c-d). The corresponding wave vector diagrams for Fig. 7 are shown in Fig. 3(b). We found NR behaviors in Figs. 7(a–b) and collimation effects in Figs. 7(c–d). Moreover, the refracted waves in Figs. 7(a–b) are single beams with respective refracted angles of γ

_{1}= -57.0o and γ

_{2}= -35.5

*°*. Since most of the propagating power is coupled to a single refracted beam at the air-slab interface, i.e., zero-order diffracted waves, we can apply Snell’s law in the following way

where *n _{e}* is the effective refractive index which is a
function of the frequency

*f*and the propagation direction

**k̂**of refracted waves. The calculated effective refractive indices for

*f*= 0.19 and

*f*= 0.28 herein are

*n*

_{e1}= -1.18 and

*n*

_{e2}= -1.53 respectively. Note that the bent regions just occupy a small portion in the corners of the PhC EFCs considered here and obvious distortions happen to the PhC EFCs due to finite size of the slabs, therefore the incident angles should be carefully tuned into the narrow NR-angle range to realize NR behaviors. Moreover, such distortions also lead to deviations from well-defined single refracted beams in Fig. 7(a).

For a short conclusion, by numerical simulations, we have confirmed that wave vector diagrams and PhC EFC scheme can be used effectively to analyze the propagation properties of light in PhC structures. NR behaviors take place for incident waves with frequencies in the range of the first partial gap (AANR range) and the first deaf gap when the incident angles are carefully tuned into the narrow NR-angle range.

For a triangle-lattice PhC slab superlens, the mechanism of subwavelength
imagings in the frequency range where **k**∙**v**
_{g} < 0 is similar to that of LHMs system. For the round-shaped
EFCs in this range, the perfect PhC is homogeneous and the effective refractive
index *n _{e}* which is also defined as

is negative and uniform within the PhC structure. The incident light with the
frequency in the range is focused both within and in the outgoing side of the
PhC slab according to geometric optics and the diffraction limit in conventional
lenses is overcome by means of amplification of evanescent waves. Moreover, NRs
and subwavelength imagings for such structures are possible in the far-field
region. However, for the superlens system constructed by the present PhC slab,
the EFCs in the aforementioned partial gap are square shaped around the
*M* point in the Brillouin zone, and the mechanism of the
subwavelength imaging for the frequency within the partial gap is different from
that in triangle-lattice systems, because of **k**∙
**v**
_{g} > 0. In the square-lattice case
considered herein, according to the theoretical discussions in Ref. [13], the subwavelength imaging for the frequency within the
AANR region is caused by the amplification of evanescent waves in a manner by
considering evanescent waves coupling to the bound-photon states of the slab,
and therefore the position of the point source should be in the near-field
region of slab, or the evanescent waves will decay away before reaching the slab[21, 22].

The PhC slab in Fig. 8 has 4 layers in the propagation direction and 45
layers in the transverse direction. Figure 8(a) shows the numerical simulation of the
snapshot of *H _{z}* field for a point source emitting
continuous waves at

*f*= 0.190 incident to the slab with surface termination δ

*x*= 0.0

*a*. Figure 8(b) shows the calculated snapshot of

*H*field for the slab with surface termination δ

_{z}*x*= 0.26

*a*. Both point sources are located at

*d*= 1

*a*, and the respective positions of the imagings are

*d*′

_{1}= 2.74

*a*and

*d*′

_{2}= 1.70

*a*from the center of the first column of the right side of the PhC slab.

#### 4.4. The role of surface termination

It has been shown that in the superlens system composed by a triangle-lattice PhC
slab, the surface termination affects the imaging quality strongly [31]. The situation of the PhC slab considered herein is
more complex than that in Ref. [31]. In order to investigate the influences on the
subwavelength imaging by the complex surface, we changed the surface structures
by choosing δ*x*, and the calculated field intensities
along the transverse image plane for *f* = 0.19 with different
δ*x* are shown in Fig. 9. The field intensities are normalized by the
maximum intensity at the image spot. Comparing the numerical simulations, we
found that the minimum of the calculated full width half maximum (FWHM) is WFWHM
= 0.494l when δ*x* equals 0.26*a*. It
has been investigated in Refs. [19, 31] that amplification of the evanescent waves relies on
resonant coupling mechanisms to the intrinsic bound-photon states, which are
greatly affected by the surface structures of the PhC slabs. In addition, we
also found that the position of the focus is very sensitive to the surface
termination, as shown in Fig. 8. The numerical results herein are in good
agreement with those in Refs. [31, 32].

## 5. Summary

The FDTD method is used to investigate the collimation effects, NR behaviors and focusing with subwavelength resolution by a two-dimensional PhC slab. By numerical simulations, it has been shown that the collimation effects in the context of the study in Refs. [20, 23] can been analyzed by means of the wave vector diagrams and PhC EFC scheme. The parameters of the collimated beam were studied as well. It has been shown that NR behaviors occur by this structure under the condition that the incident angle should be carefully tuned into the narrow NR-angle range. It has been shown that focusing with subwavelength resolution can be realized by this structure and the influences on imaging resolution were studied as well.

## Acknowledgments

This work was partially supported by the Natural Science Foundation of China (grant
*Nos. 60538010*, and *10576009*), and the Science
and Technology Commission of Shanghai (grant *No 05SG02*). L. J.
Qian’s email is ljqian@fudan.edu.cn.

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