Generalized phase matching condition for &#x2028;lossy periodic photonic structures

Xuhuai Zhang; Stephen R. Forrest

doi:10.1364/OE.18.001151

1. Introduction

Negative refraction in lossless dielectric photonic crystals [1,2] has been well understood using the equi-frequency contour (EFC) approach, which is based on a phase matching condition across an interface of periodicity d, $k_{t}^{B} = k_{t}^{m} + m G$ [3]. Here, $| G | = 2 π / d$ is the magnitude of the interface reciprocal latice vector, m is an integer for denoting different diffraction orders, and $k_{}^{B}$ and $k_{}^{m}$ are real wave vectors of a Bloch wave and a plane wave, respectively. This condition is a corollary of Bloch's Theorem [3], and connects the transverse components (denoted by subscript t) of $k_{}^{B}$ and $k_{}^{m}$ . It has also been used for describing the diffraction of quantum mechanical Bloch waves into free space in angle resolved photoemission spectroscopy experiments [4] that probe the electronic band structure of a solid. In the more general case of lossy periodic media, however, Bloch waves acquire an imaginary component of the wave vector [5]. To our knowledge, diffraction of complex Bloch waves by periodic photonic structures has not been generally treated.

Negative refraction has also been observed in negative index metamaterials (NIMs) [6–8], which are typically periodic structures with subwavelength unit cells containing lossy metal-dielectric composites. Although complex band structure of metamaterials has been routinely calculated, experiments [6–8] that demonstrate negative refraction by composite NIM prisms are generally interpreted as the behavior of a homogeneous NIM with effective permittivity,ε, and permeability, μ.

Recently, we designed a near-infrared NIM with subwavelength unit cells [9]. In that work, assuming zero loss and using the phase matching condition above, we considered diffraction of a real Bloch wave by the periodic hypotenuse of the metamaterial prism, and derived a general relationship between the band structure of a NIM unit cell and the effective index. We applied this relation approximately to the full complex band structure of the NIM unit cell, and achieved reasonable agreement with full-wave simulations of prism refraction [9]. However, substantial differences persist. In addition, it was assumed [9] that the Bloch wave vector is either parallel or anti-parallel to the group velocity, which is not necessarily true in the more general case of unit cells with anisotropy [2]. Moreover, it is not clear whether the neglect of loss is always justified. In fact, the issue of how lossy materials might influence experimental interpretation has been controversial [10] since the first demonstration of negative refraction by NIMs at microwave frequencies [6]. In a subsequent experiment [7], numerical simulations assuming a homogeneous NIM with varying degree of losses have been performed to show independence of the direction of the refracted beam on loss. The effective parameters of the homogeneous NIM [7] have been obtained from the experimental unit cells with the scattering (S) parameter method [11], a homogenization approach that assigns effective ε and μ to inhomogeneous composites. However, these simulations neglect the inherent structure of the unit cell and are therefore somewhat problematic, since the sizes of most NIM unit cells fabricated to date remain a considerable fraction of wavelength (λ). Additionally, the S-parameter method is known to sometimes yield anomalies such as unphysical negative imaginary parts of ε or μ [12].

In this work, we generalize previous treatment of NIM refraction [9] by including the effect of losses and possible anisotropy without using homogenized effective parameters. Based on a generalized phase matching condition with complex transverse wave vectors for periodic media, we describe the diffraction of a complex Bloch wave propagating within a composite prism, and show that the detected light is an inhomogeneous plane wave due to losses in the prism.

2. Diffraction of a complex Bloch wave

Figure 1 shows a schematic of a generalized NIM negative refraction experiment [6–8] that employs bulk prisms made of cubic unit cells of size a. A plane wave is normally incident on the bottom of the prism along $\hat{x}'$ and excites Bloch modes within the composite, where $\hat{z}$ is normal to the plane of incidence. We assume a sufficiently large but finite prism, such that multiple reflections within the prism and diffraction by its corners can be neglected. The wave vector of the Bloch mode in the first Brillouin zone must be along $\hat{x}'$ , due to conservation of its transverse component. The complex dispersion relation of this Bloch mode, $k^{B} = k^{B} (ω)$ , can be calculated [5]. Here, ω is the (real) angular frequency, $k_{r}^{B} = Re (k^{B})$ , and $k_{i}^{B} = Im (k^{B})$ . This choice of real frequency and complex wave vector is consistent with typical experiments that are conducted in the frequency domain with near monochromatic illumination [7]. The electric and magnetic field of this Bloch mode are $E (r') = E_{0} (r') \exp (i k_{r}^{B} x' - k_{i}^{B} x')$ and $H (r') = H_{0} (r') \exp (i k_{r}^{B} x' - k_{i}^{B} x')$ , respectively. Here, $r' = (x', y', z')$ , and $E_{0} (r')$ and $H_{0} (r')$ are both functions periodic on the cubic lattice. The $x'$ component of the time averaged power flow of this Bloch mode per unit cell is given by the surface integral of the Poynting vector $E_{0} (r') \times H_{0}^{*} (r')$ (proportional to momentum density), i.e.,

\begin{array}{l} P (x^{'}) = \frac{1}{2} Re (\int_{S (x^{'})} (E (r') \times H^{*} (r')) \cdot \hat{x}' d y' d z') \\ = \frac{1}{2} Re (\int_{S (x^{'})} (E_{0} (r') \times H_{0}^{*} (r')) \cdot \hat{x}' e^{- 2 k_{i}^{B} x'} d y' d z'), \end{array}

where

S (x')

is the cross section of a single unit cell at

x'

in the

y' - z'

plane. Therefore,

P (x^{'} + a) = P (x^{'}) \exp (- 2 k_{i}^{B} a)

, and has no

y'

or

z'

dependence, due to the periodicity of

E_{0} (r')

and

H_{0} (r')

. Since power must flow away from the surface,

k_{i}^{B} > 0

, corresponding to decay of

P (x^{'})

along

\hat{x}'

due to dissipation within the composite, although in general the localPoynting vector and wave vector may not be in the same direction [13]. Similar considerations of

y'

and

z'

components of the power flow per unit cell show that they are periodic, but do not decay, along their respective axes. These components are not necessarily zero, due to the possible anisotropy of the unit cell [2]. Therefore, we do not assume a total power flow along

\hat{x}'

, although typical unit cells [6–8] are, by design, approximately symmetric with respect to the

z' - x'

plane.

Fig. 1 A homogeneous plane wave is normally incident along $\hat{x}'$ in the x’-y’ plane on a composite prism consisting of subwavelength unit cells and excites a complex Bloch wave that is diffracted by the interface grating at the hypotenuse. The diffracted light is detected in the far field. Constant and exponentially decaying wave amplitudes of phase fronts represent the incident homogeneous and detected inhomogeneous plane waves, respectively. Here, θ and $θ_{0}$ are incidence and refraction angles, a is the unit cell size, d is the interface periodicity, $k^{B}$ is the first Brillouin zone wave vector of the Bloch mode, and $k_{}^{m}$ and $k^{B (m)}$ are the complex wave vector of the m ^th transmitted diffraction order and reflected Bloch wave in the extended Brillouin zone scheme, respectively. Subscripts r and i denote real and imaginary parts, and $G = 2 π / d$ is the magnitude of the surface reciprocal lattice vector.

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For a sufficiently large prism, only the complex Bloch mode with the lowest loss (corresponding to the smallest positive $k_{i}^{B}$ ) contributes to wave propagation within the bulk of the composite [9,14,15]. Modes with $k_{r}^{B} < 0$ correspond to an anti-parallel phase velocity and power flow along $\hat{x}'$ . The figure of merit (FOM) used for gauging the loss of the unit cell is the inverse of the loss tangent, $| k_{r}^{B} / k_{i}^{B} |$ . Typical low-loss NIM unit cells at microwave frequencies [7] are characterized by a high FOM~100 and a narrow negative index band in the first Brillouin zone. This band has opposite phase and group velocities [16] (i.e. $k_{r}^{B} (\partial ω / \partial k_{r}^{B}) < 0$ ), which is consistent with the Bloch mode in Fig. 1 with $k_{r}^{B} sgn (k_{i}^{B}) < 0$ , since the direction of the $x'$ component of group velocity, $\partial ω / \partial k_{r}^{B}$ , is also along that of the power flow ( $sgn (k_{i}^{B})$ ) in the limit of zero loss [17].

Due to the finite size of the unit cell, the hypotenuse of the prism is stepped [18], with prism angle $θ = \arcsin (a / d) = \arcsin (1 / \sqrt{1 + l_{}^{2}})$ . Here, $d = \sqrt{1 + l_{}^{2}} a$ is the interface periodicity, and there are one and l unit cells along $\hat{x}'$ and $\hat{y}'$ per step, respectively. The Bloch wave is impingent on the hypotenuse directed along $\hat{y}$ at an apparent incidence angle, θ, and excites reflected Bloch waves and transmitted plane waves. Obtaining transmission and reflection coefficients requires matching of these waves along the stepped hypotenuse with boundary conditions. In previous work on negative refraction in lossless dielectric photonic crystals [1,2], details of the photonic crystal surface have not been considered, and as a result transmission and reflection coefficients are not available [6–8]. Independent of details of a period on the interface, Bloch’s Theorem dictates [3] the phase matching condition

k_{t}^{B} = k_{t}^{m} + m G,

where

G = (2 π / d) \hat{y}

, and

k_{t}^{m}

is the complex transverse wave vector of the m ^th transmitted diffraction order and reflected Bloch wave in the extended Brillouin zone scheme, as shown in Fig. 1. Since Bloch’s Theorem does not require

k_{t}^{B}

to be real [19], this equation applies even when

k_{i}^{B} > 0

, i.e. in the presence of loss within the prism. In scalar form, this generalized phase matching condition can be written as:

{\begin{cases} {[k_{r}^{B}]}_{y} = {[k_{r}^{m}]}_{y} + m (2 π / d) \\ {[k_{i}^{B}]}_{y} = {[k_{i}^{m}]}_{y} \end{cases},

where the subscript y denotes the scalar component along

\hat{y}

. Both

k_{r}^{B}

and

k_{i}^{B}

are along

\hat{x}'

also as a result of this phase matching condition. Since Bloch’s Theorem derives from the discrete translational symmetry of the physical structure, Eq. (3) is independent of the possible nonzero power flow of the Bloch mode transverse to

k_{}^{B}

due to anisotropy [2]. Equation (3) is also independent of the types of waves involved and may apply to other similarly periodic media.

In the far field, the propagating wave is a solution to the Helmholtz Equation with translational symmetry associated with phase factor, $\exp (i k_{t}^{m} y)$ . This eigenmode in a lossless medium must be an inhomogeneous plane wave [20] with orthogonal planes of constant phase and amplitude. Its wave vector has both real and imaginary components along the normal to these two planes, as indicated by $k_{r}^{0}$ and $k_{i}^{0}$ , respectively, in Fig. 1. The inhomogeneous plane wave amplitude diverges at $y \to \infty$ , but in physical systems the wave is always bounded by a finite aperture. Declercq et al. [21] found that a single inhomogeneous plane wave component dominates the behavior of the bounded wave. Indeed, the time-averaged power flow of a single inhomogeneous plane wave is predicted to be along the real wave (ray) vector $k_{r}^{0}$ [20]. Such predictions, including those based on Eq. (3), have been experimentally verified at acoustical frequencies extensively [22–24].

The dispersion relation of an inhomogeneous plane wave is [20]

{| k_{r}^{m} |}^{2} - {| k_{i}^{m} |}^{2} = k_{0}^{2},

where k₀ = 2π/λ is the vacuum wave number. For m = 0, solutions of

k_{r}^{0}

to Eqs. (3) and (4) are:

\begin{array}{l} {| k_{r}^{0} |}^{2} = \frac{1}{2} {k_{0}^{2} + ({| k_{r}^{B} |}^{2} + {| k_{i}^{B} |}^{2}) \sin^{2} θ \pm \\ \sqrt{{[k_{0}^{2} + ({| k_{r}^{B} |}^{2} + {| k_{i}^{B} |}^{2}) \sin^{2} θ]}^{2} - 4 {| k_{r}^{B} |}^{2} k_{0}^{2} \sin^{2} θ}} . \end{array}

In the lossless case,

k_{i}^{B} = 0

, and Eq. (5) reduces to

k_{r}^{0} = k_{0}

or

k_{r}^{0} = k_{r}^{B} \sin θ

. The former solution corresponds to a homogeneous plane wave (

k_{i}^{0} = 0

) when

k_{0}^{} > k_{r}^{B} \sin θ

, while the latter corresponds to an evanescent wave propagating along the surface in the presence of total internal reflection when

k_{0}^{} \leq k_{r}^{B} \sin θ

. When

k_{i}^{B} > 0

, both

k_{0}^{2}

and

{(k_{r}^{B} \sin θ)}^{2}

are between the two solutions in Eq. (5), and the smaller (negative) solution is discarded, since Eq. (4) requires that

k_{r}^{0} \geq k_{0}

. Higher order waves corresponding to m≠0 can also be excited. Given that typically [6] d~λ/2 and, therefore,

G ~ 2 k_{0}

, Eq. (4) shows

k_{r}^{m} \approx k_{i}^{m} ≫ k_{0}

for m≠0. The imaginary component of Eq. (3) therefore implies that the corresponding rays will be at near-grazing angles and bound to the interface, as indicated by

k_{r}^{- 1}

and

k_{r}^{1}

in Fig. 1.

3. Comparison with NIM refraction experiments

Equation (3) implies that the real part of the effective index follows Snell’s Law, viz:

n_{r} = \sin θ_{0} / \sin θ = (k_{r}^{B} / k_{r}^{0}) sgn (k_{i}^{B}),

where

θ_{0}

is the refraction angle for m = 0. This index corresponds to the far-field power flow direction and is used for explaining experimental results. In the lossless case, Eq. (6) gives:

n_{r} = {\begin{cases} (k_{r}^{B} / k_{0}^{}) sgn (\partial ω / \partial k_{r}^{B}) w h e n (k_{0}^{} > k_{r}^{B} \sin θ) \\ (1 / \sin θ) sgn (\partial ω / \partial k_{r}^{B}) w h e n (k_{0}^{} \leq k_{r}^{B} \sin θ) \end{cases},

where the sgn function ensures that the group velocity is along

\hat{x}'

, consistent with previous results [9]. Assuming parameters (θ, λ/a, and FOM) typical of experimental unit cells [6,7,9,25,26], we calculate n_r as a function of

k_{r}^{B}

, with results shown in Fig. 2 . FOM = 3 typical of unit cells in the optical domain [9,26] leads to a deviation of n_r from Eq. (7) that is dependent on both θ and

k_{r}^{B}

, particularly toward the edge of the Brillouin zone. These observations are consistent with steady state full-wave simulations of

18^{\circ}

and

26^{\circ}

prisms consisting of unit cells of a near-infrared frequency NIM structure [9].

Fig. 2 Observed effective index n_r calculated using Eqs. (5) and (6) in text. The parameters used in the calculation are: $θ = 18^{\circ}$ (Refs. [6,9,25]), $26^{\circ}$ (Refs. [9,25]); λ/a = 6 (Refs. [6,9,25]); FOM = 100 (Ref. [7]), 3 (Refs. [9,26]). The frequency dispersion of the vacuum wave number, $k_{0}$ , is neglected due to the narrow bandwidth of the negative index band. The lossless case (Eq. (7) in text) is nearly identical to that for FOM = 100. Also, $n_{T I R}$ is the onset of total internal reflection in the lossless case for $θ = 26^{\circ}$ .

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In contrast, the lossless case is nearly identical to the case for FOM = 100, which is often observed at microwave frequencies [7]. This suggests that loss can be neglected for typical microwave NIM structures in refraction experiments. We calculated band structures of the unit cells in Refs. [7,8] and derived the effective indices using Eq. (7). The results are plotted together with the experimental data in Fig. 3 . Band structures [16] calculated for the first reported structure are also consistent with the experimentally measured indices [6] using $n_{r} = (k_{r}^{B} / k_{0}^{}) sgn (\partial ω / \partial k_{r}^{B})$ , and are characterized by a dispersion relation $k_{r}^{B} (\partial ω / \partial k_{r}^{B}) < 0$ within the first Brillouin zone. This “backward” Bloch band also accounts for negative refraction in lossless dielectric photonic crystals [1].

Fig. 3 Effective indices derived via Eq. (7) in text from calculated band structures of Refs. [7,8]. Both are consistent with the experimentally measured values for these metamaterials. Solid and dashed lines are guide to the eye.

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4. Discussion

The above general treatment explicitly accounts for the inhomogeneity of the unit cell through a complex Bloch wave, but requires no assumptions about ε and μ. Previous experiments [6–8] have generally been interpreted by assuming a homogeneous NIM, whose optical properties are due to a superposition of the negative ε of metal wires [27] and the negative μ of the split ring resonators [28] that comprise the prism. Although both interpretations can describe the experiments, their predictions may differ in some occasions. For example, Eq. (2) suggests that more than one far-field beam may be excited by the Bloch wave when the unit cell size is sufficiently large but still subwavelength, as is the case for some NIM unit cells. In previous NIM refraction experiments [18,25], such phenomena have been observed and explained as the diffraction by the stepped interface grating between air and a homogeneous NIM, which is equivalent to interpreting the Bloch wave vector $k_{}^{B}$ in Eq. (2) to be that of a plane wave. If the stepped interface is linearized by using partial unit cells, this homogeneous NIM interpretation predicts the transmission of a plane wave through the planar interface between two homogeneous media, where the additional far-field beam disappears. However, the interpretation in terms of Bloch waves suggests that this beam is still present, since Eq. (2) derives from the translational symmetry of the periodic interface only, independent of the details of a period. The observation of two propagating orders from a photonic crystal prism with a linear hypotenuse has been previously reported [29], where the angular positions of the far-field beams are consistent with Eq. (2). This difference is evidence for spatial dispersion, the deviation of the behavior of the NIM from its approximate local homogeneous model.

The results presented here may also suggest the existence of a minimum unit cell size of optical NIMs. The maximum ratio of wavelength to periodicity currently known for optical NIMs is about 7 [9,30]. Tsukerman [31] has recently predicted a minimum unit cell size imposed by a backward Bloch band for photonic crystals consisting of non-dispersive dielectrics. Since negative refraction by metamaterials is shown to be governed by the same Bloch band in the zero-loss limit, whether the minimum periodicity for NIMs containing dispersive metals at optical frequencies is restricted by a similar fundamental limit (apart from technological constraints) remains an open question.

5. Conclusion

In conclusion, based on a generalized phase matching condition, we have shown that the negative refractive behavior of lossless dielectric photonic crystals and lossy metal-dielectric periodic NIMs can be given a unified explanation. We have also discussed its implications for the local homogeneous model of NIMs as well as the possible existence of a minimum unit cell size of optical NIMs.

Acknowledgements

The authors thank the AFOSR MURI contract No. FA 9550-06-01-0279 for financial support. We also thank Prof. Paul Berman for reading the manuscript and useful discussions.

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Generalized phase matching condition for  lossy periodic photonic structures

Abstract

1. Introduction

2. Diffraction of a complex Bloch wave

3. Comparison with NIM refraction experiments

4. Discussion

5. Conclusion

Acknowledgements

References and links

Cited By

Figures (3)

Equations (7)

Optics Express