Metamaterial Exhibiting Degenerate Band Edge: Reduced Group Velocity and Local Field Enhancement

Y. Caoa, J. Schenka, T. J. Suleskia M. A. Fiddy*a, Jeffrey Raquet b,

J. Ballatoc , K. Burbankd, M. Grahamd, P. Sangerd, aCenter for Optoelectronics and Optical Communications,

bDepartment of Mechanical Engineering, University of North Carolina at Charlotte, Charlotte, NC, 28223;

cThe Center for Optical Material Sciences and Engineering Technologies, Clemson University, 91 Technology Drive, Anderson, SC 29625

dCenter for Rapid Product Realization, Western Carolina University, Cullowee, NC, 28723; *mafiddy@email.uncc.edu; phone 704-687-8594; fax 704-687-8241

Keywords: Anisotropic photonic crystal, resonance effects, degenerate band edge, group velocity

Abstract - We have been studying a novel 1D anisotropic photonic crystal structure which can be designed to have a strong resonant effect, coinciding with a very low group velocity over a specific bandwidth. The structure requires two anisotropic layers and one isotropic layer per period and was first introduced by Figotin and Vitebskiy. By the careful design of the parameters of the structure, we can find a special band edge point which has fourth order degeneracy, and is called degenerate band edge (D.B.E). It was predicted that in the case of a transmission resonance in the vicinity of the D.B.E, the resonant field intensity increases as N4, where N is the total number of periods, while in the case of a regular band edge, the field intensity is proportional to N2.

1. INTRODUCTION

1D isotropic photonic crystal structures have attracted attention due to their potential use in a number of applications [1-4]. The theoretical analysis, fabrication and characterization of 1D anisotropic photonic structures are more complicated not only due to the anisotropy, but, also because of some distinct dispersion characteristics that 1D isotropic photonic crystal structures do not possess. Using band-edge resonances to slow down light in isotropic one-dimensional photonic crystal structures has been investigated by Scalora et al. [5]. It was found that the maximum group delay and the associated maximum field intensity enhancement occur at the transmission peak closest to the band-edge of the forbidden gap. Recently, Figotin and Vitebskiy [6] proposed that in the case of a transmission resonance in the vicinity of the degenerate band edge (DBE) which has degeneracy of the order 4, the resonant field intensity

enhancement is proportional to N4, where N is the total number of periods, while in the case of a. regular band edge (RBE) which has degeneracy of the order 2, the resonant field intensity enhancement is proportional to N2.

Based on their predictions, we have investigated their claim. Materials with the required degree of anisotropy (Δn > 0.1) at optical wavelengths are difficult to find and so we employ form-birefringence to replace the anisotropic layer in the photonic crystal structure design. Real devices have been made for use at microwave frequencies using a rapid-prototyping tool. The photonic crystal structure we studied [7] has a unit cell with two misaligned in-plane anisotropic layers and one isotropic layer. It was found from numerical simulations that the field intensity enhancement strongly depends on the anisotropy of the material, and a larger anisotropy usually leads to a strong resonant effect.

We fabricated our microwave frequency degenerate band edge structure on a rapid prototype machine (Eden 333) using uv curable Fullcure 720 material, and built a simple microwave setup to characterize the structure such as S21 parameters and the electric field intensity distribution inside the structure.

In section 2, the transfer matrix analysis method for the 1D anisotropic photonic crystal is briefly described, and the regular band edge points (RBE) and degenerate band edge points (DBE) are defined. The design and simulation details for a degenerate band edge device using form-birefringent grating layer as the anisotropic layer are presented. In section 3, the experimental set-up for the measurement of the DBE is presented, and some of our preliminary experimental results are discussed.

Conclusions and future directions are presented in section 4.

2. DESIGN OF THE DEGENERATE BAND EDGE STRUCTURE

We consider a multiple layer structure including anisotropic media, which is nonmagnetic and not optically active, and the Cartesian coordinate system is chosen such that the x axis is normal to the interface. The whole structure has a dielectric tensor for each layer, and the dielectric tensor ! for each layer in the xyz coordinate system depends on the orientations of the crystal axes which are described by the Euler’s angle ! ,

! , ! with respect to a fixed xyz coordinate, it is given by

1

3

2

1

00

00

00

ˆ!

"""

#

$

%%%

&

'

= AA

(

(

(

(

(1)

where 1! ,

2! ,

3! are the principal dielectric constants

and A is the coordinate rotation matrix.

From Snell’s law, when a single plane wave propagates through the multiple layer structure, the tangential components of the wave vectors remain the same throughout the layered medium, and all the wave vectors lie in the same plane (the incident plane). Light propagation can be assumed to be in the x-y plane, as a consequence, and the z-component of the wave vector is zero, the electric field can be assumed to have

)(exp 00 tykxki !"# $+ dependence in each crystal layer,

here, ck //20

!"# == , !" cosn= , !" sinn= are implied.

A plane wave propagating in the x-y plane and incident on a single parallel-sided layer of biaxial material, as illustrated in fig.1 will initiate four plane waves in the biaxial layer, two forward-traveling waves and two backward-traveling waves in the same x-y plane. The four waves are linearly polarized, and share a common value of ! with the incident wave. Representing the electromagnetic field in the form of the plane harmonic waves )(

0

trkieEE

!"=

rrrr and )(

0

trkieHH

!"=

rrrr, one can obtain

Maxwell’s equations in the matrix form [1]

Ez

Hsn

HzEsn

rr

rr

!1

ˆ

ˆ

0

0

"=

=

(2)

Figure. 1: An incident pane wave establishes four

traveling waves in a biaxial layer.

where matrix snˆ implements the !)/( 0kk

r operation,

which can be represented by equation (4)

!!!

"

#

$$$

%

&

'

'=

0

00

00

ˆ

()

(

)

sn

(3)

! is the permittivity tensor, and 2/1

000 )/( !µ"z . Six equations are implied in the system (2) but the field components

xE and

xH that are normal to the interface

and not required for boundary condition matching can be eliminated which lead us to an eigenequation having solutions for the four basis fields of [ Ey Hz Ez Hy] which

provide the field matrix F

!!!!!

"

#

$$$$$

%

&

=

'+'+

'+'+

'+'+

'+'+

2211

2211

2211

2211

ˆ

yzyy

zzzz

zzzz

yyyy

HHHH

EEEE

HHHH

EEEE

F

(4)

Here the plus and minus superscripts indicate waves that are positive-going and negative-going with respect to the x-axis. The y and z components of the total anharmonic

field }{ yzzy HEHEm =r

at a point in a layered

medium can be expressed as a linear sum of the four

harmonic traveling wave basis fields, aFmrrˆ= . The

column vector }{ 2211

!+!+= aaaaa

r provides the

complex coefficients for the linear sum as the traveling wave field coefficients. The matrix F has the property of transforming the traveling wave field coefficients a

r to

the total field mr

at the same point in a layered biaxial

medium and, similarly, 1ˆ !F transforms the total field mr

to the traveling wave field coefficients a

rat the same

point.

The four traveling wave fields in a biaxial layer change phase linearly with displacement in the x-direction, but at different rates. Along the same path, which is assumed to be always in the same layer, the absolute values of the field coefficient a ’s remain constant but the phase changes. The phase matrix,

!!!!!

"

#

$$$$$

%

&

'

'

'

'

=

'

+

'

+

)exp(000

0)exp(00

00)exp(0

000)exp(

ˆ

2

2

1

1

(

(

(

(

i

i

i

i

Ad

(5)

where dk±±

=2,102,1

!" transforms the traveling wave field coefficient from one point (at x=x0, say) to the traveling wave field coefficients at another point ( at x=x0-

d ). The transformation property of dA can be written as

00

ˆxddxaAarr

=! , and stated in words as, the phase matrix transforms the field coefficients between two points in the same layer.

The field transfer matrix 1ˆˆˆˆ != FAFM

d transforms the

total field from one point to the total field at another point, such as across the interfaces of a single layer. For N stacked layers the field transfer matrix is the product

NMMMM ˆ...ˆˆˆ

21= .

The propagation of the plane wave in a periodic medium obeys the Floquet theorem [8-9] by which we can get an equation for a unit cell

)()exp()(ˆ xEiKLxEMKKL

rr!=

(6)

The subscript K indicates that the function )(xEK

r

depends on K which is known as the Bloch wave number. L denotes a period of the lattice with a defined

primitive cell and its field transfer matrix L

M . Therefore

the Bloch waves )(xEK

r are the eigenvectors of

LM

and Bloch wave numbers K are related to the

eigenvalues of L

M , i

! by )exp( LiKii

!=" .

Because L

M depends on ω and Snell’s law quantity ! , equation (6) can be considered as dispersion relation

0),( =!" K . For birefringent media, L

M is a 4 × 4 matrix, and one ! corresponds to four wave numbers

iK . Real

iK correspond to propagating Bloch waves

and imaginary iK correspond to evanescent modes.

Based on the degeneracy of the Bloch wave numbers iK

at a specific point 0

!! = , one can find three types of special inflection points in the dispersion curve. The first type is called regular band edge point (RBE), which is the degenerate point of order 2 and one ω corresponds to two equal real K ’s. The second type inflection point is called stationary inflection point (SIP), which has degeneracy of order 3 and one ω corresponds to three equal real K ’s . The third type inflection point is called degenerate band edge point (DBE), which has the fourth order degeneracy and one ω corresponds to four equal real K ’s.

Figure 2 illustrates the dispersion curves around three types of inflection point. Since at these points, the group velocities are all equal to 0, there must be a strong field resonant effect connected with these points, but, because of their different degenerate orders, these resonant effects will differ from one to another. For a stationary inflection point (SIP), a so-called axially frozen mode is connected with this point, and some work has been published on this [2-4]. As mentioned above, Figotin and Vitebskiy [6] suggested that in the case of a transmission resonance in the vicinity of the degenerate band edge (DBE), the resonant field intensity enhancement is proportional to the fourth power of the total number of periods N, which is an extremely attractive band edge resonant effect. This is in contrast to the case of a regular band edge (RBE), for which the resonant field intensity enhancement is only proportional to N2 and it is this effect we have been investigating.

Figure 2: (a) Regular band edge (RBE): corresponds to two equal real K; (b) Stationary inflection point (SIP): corresponds to three equal real K; (c) Degenerate band edge (DBE): corresponds to four equal real K

The existence of the DBE requires that TE and TM modes should be coupled each other inside the structure so that 4×4 matrix has to be always involved to solve the dispersion relation. When TE and TM modes are decoupled, the field transfer matrix will be reduced to the block-diagonal form and the dispersion relation will exist for TE and TM modes separately. Under these conditions there would be no possibility of DBE points occuring. Even if anisotropic layers are present, but the anisotropy axis in all anisotropic layers are either aligned, or perpendicular to each other, TE and TM modes are decoupled. For normal incidence, we therefore need at least two misaligned anisotropic layers in a unit cell with the misalignment angle not equal to 0 and π/2; we have chosen π/4 but with real materials some degree of tenability of this angle for any given structure is likely to be necessary.

Figure 3(a) shows a cartoon of a simple structure with a three-layer unit cell each having two equal-thickness misaligned in-plane anisotropic layers A1, A2, and one isotropic layer B. In 3(b), a two-layer unit cell structure is shown with only two misaligned in-plane anisotropic layers A1 and A2 which have different thicknesses or are made of different anisotropic materials. In Fig 3(c), the unit cell has one in-plane anisotropic layer A and one isotropic layer B. In (c) the DBE only exists under oblique incidence [10]. The DBE can exist for both (a) and (b) under normal incidence, but our simulations indicate that the three-layer unit cell structure has a better DBE resonance performance than the two-layer unit cell structure assuming both structures possess similar degrees of anisotropy [7].

In the effective medium theory (EMT), our form-birefringent anisotropic structure will behave like homogeneous uniaxial thin layer in the long wavelength limit [11[ with principal dielectric constants (no

2 and ne2)

which can be readily calculated.

(a) (b) (c)

Figure 3: Periodic stack of N cells of length L, in which A1 and A2 are of equal-thickness and represent misaligned anisotropic layers with in-plane anisotropy, and B are isotropic layers. (b) Periodic stack of N cells of length L, with two misaligned in-plane anisotropic layers A1 and A2 having different thicknesses or different anisotropic materials. (c) Periodic stack of N cells L, with A layers having in-plane anisotropy, and with B isotropic.

3 STRUCTURE DESIGN

To determine the grating period and duty cycle of the anisotropic grating layer, we calculate both zero-order and second-order effective index by using the following formulae [11]. When the grating-period-wavelength ratio is small enough for the zero-order and second results to be within our tolerance level, we fix the grating period. The optimum duty cycle is chosen to maximize the anisotropy (n0 /ne) of the effective uniaxial thin film.

For the three-layer unit cell structure, once the anisotropic material and isotropic material are chosen, the misalignment angle between A-layers ,φ and the thickness of the B-layer DB have to be selected while searching for the DBE point. The thickness of two A layers can be calculated using formula DA1 = DA2 = (L-DB)/2. First the field transfer matrix is calculated for the unit cell which depends φ, the dielectric permittivity, the thickness of each layer, the incident frequency ω, and incident direction θ (θ, = 0 and hence β = 0 for normal incidence) Using equation (6), we can get the dispersion curves and adjusting the misalignment angle and the thickness of the air layer DB, we check the dispersion curves. One can make φ equal to π/4, and adjust the other parameter until the DBE condition is satisfied, i.e. with four equal real values for k of either 0 or 3.142. Fig. 5 shows the dispersion curve around the lowest order spectral branch DBE point when φ is equal to π/4, and DB adjusted to three different values.

A1 B A2

A1 B A2

L

A1 A2

A1 A2

L

B A B A

L

Figure 5: (a) DB = 0.4L, regular band edge. (b) DB = 0.26L, degenerate band edge. (c) DB = 0.1L, double band edge

Once the DBE point is identified, we calculate the transmission curves for the N-period structure, find the transmission peak frequency closest to the DBE point, and then calculate the field intensity distribution inside the structure at this resonant frequency, and determine the intensity peak values.

We use matrices Car

and Sar

to represent the field

coefficients and the system matrix SCFMFA ˆˆˆˆ 1!

= transforms the traveling wave field coefficients hence we

can use SCaAa ˆˆ=

r to calculate the relationship between

the input and output field coefficients, and hence the corresponding reflectance and transmittance coefficient. Once the transmittance coefficient for a specific frequency is found, we can calculate the corresponding output field coefficient and the total output field and the field distribution inside the structure can be calculated using the field transfer matrix. Fig. 6(a) shows the transmission versus frequency around the DBE frequency (0.878 c/L). The transmission peak frequency closest to the DBE is 0.877(c/L). Fig. 6(b) shows the field intensity distribution inside the structure at the transmission peak frequency 0.877 (C/L). The incident polarization for both figures is TM. A strong field enhancement occurs at the transmission peak frequency closest to the DBE.

3. EXPERIMENT AND RESULTS

We fabricated the DBE structure to operate in the X band (8~12 GHz) on a rapid prototype machine (Eden 333 at Western Carolina University) using uv curable Fullcure 720 material. Permittivity control is possible using dopants developed at Clemson University. One unit is shown in figure 7, alongside the experimental set-up.

Figure 6 (a) The transmission curves around the DBE frequency. (b) The field intensity distribution inside the structure at the transmission frequency closest to the DBE

The measurement set up comprises two X-band horns separated by 72 cm. A test first measures the free space transmission intensity at each point between the horns. Each point is spaced by 1cm, which correlates to the spacing ports on the test structure. These data are used as a reference to compare and normalize the measured data obtained from the coaxial probe antenna. A signal generator scans discrete frequencies between 8GHz and 12GHz. A spectrum analyzer acquires the intensity measurements and identifies the bandgap from which the DBE location is found. Field intensity data are taken over the entire span of the structure noting where the highest intensity observations occur. The incident wave may not be precisely collimated which can be an issue. Also, at present, the structure is not shielded which would prevent unwanted outside interference. Finally, absorption in the structure has not been explicitly measured as of yet but is known to be extremely low.

Figure 7 Experimental setup and one unit cell of the DBE structure.

Ideally the incident and transmitted waves should be approximately plane waves, which is required by the transfer matrix model. S21 parameters (transmission coefficient) were measured by both a vector network analyzer and spectrum analyzer. The vector network analyzer is also used to measure the group delay in the frequency domain and the spectrum analyzer used to measure the field intensity distribution inside the structure by using a probe.

The index of refraction of the bulk material used for this DBE sample is 1.64. The whole sample consists of up to 54 unit cells as described and shown in figure 7. Layers A1 and A2 are identical subwavelength 1-D gratings with a φ of 45 degrees and the isotropic B layer is air here. The total thickness of one unit cell L is 1.18 cm, the thickness of each anisotropic layer DA1 = DA2 = 0.493cm, and the thickness for the air layer DB = 0.198cm. The period of the subwavelength grating providing the anisotropy is 1 mm, with a duty cycle of 0.5 which is much smaller than the incident wave wavelength, and close to the quasi-static limit; effective indices are 1.36 and 1.21.

Figure 8 illustrates the presence of the measured bandgap in this structure for two difference values of N, the number of periods.

Figure 8: top graph is for N = 30 and lower graph for N = 54. Note the scales are not identical in order to better illustrate the differences.

It can be seen that there is some broadening and deepening of the bandgap with increasing N. The DBE point is anticipated to be close to10GHz by design and

the peaks to the low frequency side of the bandgap are not consistent for the two cases, appearing to be at 10.5GHz and 10.1GHz respectively. These finite size effects need to be more carefully modeled in the context of either a lossy or shielded waveguiding structure in order to properly identify the DBE location in a single or low moded structure of this type. Field intensity localizations are observed in this structure but not the large enhancements predicted by the model, which assumes a plane wave (i.e. of large transverse extent) incident.

An example of the measured field distribution inside the structure is shown below in figure 9.

Figure 9: experimental measurement of internal field with 10.25GHz illumination. In the ideal case, and with low loss material, one would expect a field intensity enhancement of ~ 3 for TE illumination and ~ 8 for TM illumination.

4 CONCLUSIONS

The goal of this research is to realize the predicted N4 field intensity enhancement expected for a specific type of one dimensional periodic crystal having anisotropic layers in each period. Our ultimate goal is to fabricate a structure of this type for use at optical frequencies. Given this, and since there are few materials available that meet the anisotropy requirements at optical frequencies, we have focused on making a form-birefringent structure which can emulate the desired degree of anisotropy. This has the advantage that we can stamp or mold these unit cells at optical frequencies (using our Molecular Imprint nanoimprint tool) or directly write them using the extensive rapid prototyping facility at Western Carolina University. Moreover, we can realize a range of host or inherent material permittivities with either fabrication method, by doping these uv curable materials with specialized compounds developed at Clemson University.

We designed a degenerate band edge structure using a transfer matrix method and built an experimental set up to verify our design. We have still to verify the DBE effect

due to various factors such as uncertainties in the uniformity of the illuminating plane wave and, more importantly, the mode coupling that can and will occur in our finite cross sectional area low-but-not-single-moded DBE structure. We may also have some reduction of the N4 enhancement due to inherent absorption losses in the structure but we believe this to be a secondary concern for us at this time. Also, it is important to note that numerical simulations monitoring the DBE point as a function of the host material permittivity, when using form-birefringence to satisfy the anisotropy requirement, indicates considerable robustness to fabrication errors at least to critical dimensional errors of the order of 5% or smaller.

Most important in adjusting and adapting this structure to realize the gigantic field intensity enhancements predicted, is to optimize the waveguiding structure which we are able to actually fabricate. Minimizing losses suggests shielding the structure while still somehow allowing measurements to be made of the internal field intensity. Recent simulations [12] suggest that some absorption losses can shift the intensity maximum from the center and toward the input face. Lack of shielding can lead to a complex distribution of coupled modes which can diminish the intensity build up in the DBE resonant mode. A more careful analysis of the present structure is underway to analyze the experimental result obtain and interpret them in order to move close toward our goal of realizing the large predicted enhancement. Applications for this include enhanced sensitivity detectors, more readily obtained nonlinear phenomena and the exploitation of the high fields for sensing.

5 ACKNOWLEDGEMENTS

The authors acknowledge the support of both DARPA/ARL through grant W911NF-04-1-0319 and the Charlotte Research Institute. We also acknowledge many discussions with A. Figotin and I. Vitesbskiy (UC Irvine) whose theoretical work, referenced in the paper, provided the framework for our simulations and fabrication efforts.

6 REFERENCES

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