Abstract

In many research areas, the reflective properties of a bulk medium are characterized by its impedance or an impedance-like quantity. Such a quantity is essential for the efficient design of stacked structures such as antireflection coatings and thin-film filters. For 2D photonic crystals and metamaterials, the literature contains multiple definitions of impedance, not all of which are consistent. We review these proposed definitions, evaluate their regions of applicability, and numerically test their accuracy in a variety of salient photonic crystal examples.

© 2013 Optical Society of America

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2013 (1)

K.-H. Kim and Q-Han Park, “Perfect anti-reflection from first principles,” Sci. Rep. 3, 1062 (2013).

2012 (7)

R. V. Craster, J. Kaplunov, E. Nolde, and S. Guenneau, “Bloch dispersion and high frequency homogenization for separable doubly-periodic structures,” Wave Motion 49, 333–346 (2012).
[CrossRef]

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2011 (4)

C. R. Simovski, “On electromagnetic characterization and homogenization of nanostructured metamaterials,” J. Opt. 13, 013001 (2011).
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2010 (9)

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P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
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2006 (4)

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A. Talneau, M. Mulot, S. Anand, and P. Lalanne, “Compound cavity measurement of transmission and reflection of a tapered single-line photonic-crystal waveguide,” Appl. Phys. Lett. 82, 2577–2579 (2003).
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D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
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M. Palamaru and P. Lalanne, “Photonic crystal waveguides: out-of-plane losses and adiabatic modal conversion,” Appl. Phys. Lett. 78, 1466–1468 (2001).
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Figures (28)

Figure 1
Figure 1

Lee et al.’s antireflection coating, which consists of a grating placed at a variable distance from the front of the PC. Reprinted with permission from [38]. Copyright 2008, Optical Society of America.

Figure 2
Figure 2

Hugonin et al.’s PCW antireflection coating, consisting of two layers with modified lattice constant. Reprinted figure with permission from [41]. Copyright 2007, Optical Society of America.

Figure 3
Figure 3

Li et al.’s antireflection coating, which consists of a layer of uniform dielectric and an air slot, each of variable width. Reprinted with permission from Z. Li et al., J. Phys. D 40, 5873–5877 (2007) [43]. Copyright 2007, IOP Publishing Ltd. http://iopscience.iop.org/0022-3727/40/19/012/.

Figure 4
Figure 4

Baba et al.’s antireflection coating for a superprism, consisting of one layer of teardrop-shaped projected holes. Schematic, left, reprinted with permission from [45]. Copyright 2004, Optical Society of America. SEM image, right, reprinted with permission from T. Matsumoto et al., Appl. Phys. Lett. 91, 091117 (2007) [44]. Copyright 2007, AIP Publishing LLC.

Figure 5
Figure 5

Truncated row of holes at the edge of a PC. Reprinted with permission from [51]. Copyright 2007, Optical Society of America.

Figure 6
Figure 6

Zhang and Li’s antireflection coating of truncated holes with air slots. Reprinted from B. Zhang and M. Li, Eur. Phys. J. D 45, 321–323 (2007) [54], with kind permission of The European Physical Journal. Copyright 2007, Springer.

Figure 7
Figure 7

Śmigaj et al.’s antireflection grating. Reprinted with permission from W. Śmigaj and B. Gralak, Phys. Rev. B 85, 035114 (2012) [53]. Copyright 2012 by the American Physical Society. http://prb.aps.org/abstract/PRB/v85/i3/e035114.

Figure 8
Figure 8

Forward and backward plane wave amplitudes defined at two phase origins separated by d.

Figure 9
Figure 9

Incoming (f1+ and f2) and scattered (f2+ and f1) plane wave amplitudes at the interface between two isotropic uniform media.

Figure 10
Figure 10

One-layer coating consisting of a slab of material 2 with thickness d inserted between semi-infinite media 1 and 3.

Figure 11
Figure 11

Part of a PC band structure, which relates normalized frequency a/λ (a is the lattice constant) to the propagation constant ky. This band structure is for a triangular-lattice PC with air holes of radius r=0.3 in an n=3 background dielectric, for normally incident light (i.e., along ΓM in the Brillouin zone). This band structure was generated by using the multipole method with lattice sums [82] and a transfer matrix method [83].

Figure 12
Figure 12

Two representations of a PC’s complex band structure, which encapsulates how its Bloch modes propagate or decay. The band structure is for the same PC and normally incident light (k=0) as in Fig. 11, and is calculated by the same technique. (a) Real and (b) imaginary parts of the wave-vector component ky, relative to normalized frequency a/λ. (c) Complex band structure in one figure. The color of each point is arbitrary; the only purpose of using multiple colors is to link the value of Re(ky) in (a) to Im(ky) in (b) at each frequency. For clarity, the magenta and cyan ky values are not shown in (a) or (c). The two black lines in (b) mark Wood anomalies.

Figure 13
Figure 13

Forward and backward Bloch mode amplitude vectors defined at two phase origins separated by e2. Here, =5.

Figure 14
Figure 14

Incoming (c1+ and c2) and scattered (c2+ and c1) Bloch mode amplitude vectors at the interface between two PCs.

Figure 15
Figure 15

(a), (b) Impedance of PC 1, a square-lattice PC, for normally incident light polarized with the E field out of the plane of periodicity. (c), (d) Reflection coefficient at an interface between an n=3 dielectric (PC 1’s background material) and PC 1, incident from the dielectric. N.B.: Results from Z are only shown for |r|2: see the comment in the introduction of Section 9.

Figure 16
Figure 16

Impedance of PC 2, a rectangular-lattice PC, for normally incident light polarized with the E field out of the plane of periodicity.

Figure 17
Figure 17

Reflection coefficient at an interface between the rectangular-lattice PC 2 and the square-lattice PC 1, incident from PC 2, for normally incident light in a propagating Bloch mode, polarized with the E field out of the plane. Both PCs are semi-infinite, each consisting of a half-plane. The frequency range just below a/λ=0.3, for which no data are presented, corresponds to the bandgap of the incident medium, PC 1.

Figure 18
Figure 18

Reflectance of a coated PC structure for normally incident light polarized with the E field out of the plane, consisting of 10 periods of the rectangular-lattice PC 2 sandwiched between two 1-period layers of the square-lattice PC 1. This structure is embedded in an n=3 dielectric, which is the incident medium. The colors of the series are the same as in Fig. 17.

Figure 19
Figure 19

PC 1–PC 2–PC 1 stack simulated in Subsection 9.1c. White represents the n=3 background dielectric; the black circles are air holes. The structure is periodic in the x direction and is surrounded by n=3 dielectric on its left and right sides. Light is incident from the left of the figure. The dashed lines mark the (conceptual) interfaces between PCs.

Figure 20
Figure 20

(a), (b) Impedance of PC 1, a square-lattice PC, for normally incident light polarized with the H field out of the plane of periodicity. (c), (d) Reflection coefficient at an interface between an n=3 dielectric (PC 1’s background material) and PC 1, incident from the dielectric.

Figure 21
Figure 21

(a), (b) Impedance of PC 1, a square-lattice PC, for light polarized with the E field out of the plane of periodicity and with kx=3kcos(30°). (c), (d) Reflection coefficient at an interface between an n=3 dielectric (PC 1’s background material) and PC 1, incident from the dielectric at an angle of 30° to the normal.

Figure 22
Figure 22

(a), (b) Impedance of PC 3, a triangular-lattice PC, for normally incident light polarized with the H field out of the plane of periodicity. (c), (d) Reflection coefficient at an interface between an n=3 dielectric (PC 3’s background material) and PC 3, incident from the dielectric. For all impedance definitions the square unit cell (Fig. 24 below) is used, except for Z, which is explicitly defined as using the parallelogram unit cell (see discussion in Subsection 9.5a for the significance of this statement).

Figure 23
Figure 23

(a), (b) Impedance of PC 3, a triangular-lattice PC, for light polarized with the H field out of the plane of periodicity and with kx=3kcos(20°). (c), (d) Reflection coefficient at an interface between an n=3 dielectric (PC 3’s background material) and PC 3, incident from the dielectric at an angle of 20° to the normal. For all impedance definitions the square unit cell (Fig. 24) is used, except for Z, which is explicitly defined as using the parallelogram unit cell.

Figure 24
Figure 24

These are two sensible choices for the unit cell of a triangular lattice. The rectangular unit cell (a) is perhaps simpler to represent numerically, but two of its edges each overlap two adjacent unit cells’ edges. The parallelogram tiling (b) avoids this problem. In each case, the domain of integration for the relevant impedance definitions is indicated by a thick line. The phase origin for the unit cell is defined to be at this line’s center.

Figure 25
Figure 25

Impedances and calculated reflection coefficients for the same interface as in Fig. 23, except that all impedance definitions are calculated along the edge of the parallelogram-shaped unit cell.

Figure 26
Figure 26

Impedance of PC 4, a triangular-lattice PC, for light polarized with the H field out of the plane of periodicity and with kx=3kcos(20°). For all impedance definitions the square unit cell (Fig. 24) is used, except for Z, which is explicitly defined as using the parallelogram unit cell.

Figure 27
Figure 27

Reflection coefficient at an interface between PC 4 and PC 3, incident from PC 4, for incident modes polarized with H field out of the plane and with kx=kcos(20°). Both PCs are semi-infinite, each consisting of a half-plane.

Figure 28
Figure 28

Reflectance of a coated PC structure for light incident at 20° to the normal and polarized with the H field out of the plane. The structure consists of ten-period PC 4 sandwiched between two one-period layers of PC 3. This structure is embedded in an n=3 dielectric, which is the incident medium. The series colors are as in Fig. 27.

Tables (3)

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Table 1. Four PCs Simulated in Section 9a

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Table 2. Summary of Cases Considered in Section 9a

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Table 3. Summary of Impedance Definitions Evaluated in Section 9a

Equations (62)

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E(r)=sEsexp(ik(s)·r),
H(r)=sHsexp(ik(s)·r),
β=±(nk0)2k·k.
Zc=|E|/|H|=μ/ε.
Z=EH,
E(r)=(f+eiβr+feiβr)eik·r,
H(r)=(f+eiβrfeiβr)eik·r/Z.
f+=exp(+iβd)f+,
f=exp(iβd)f.
f1=r12f1++t21f2,
f2+=t12f1++r21f2.
r12=Z2Z1Z2+Z1,
t12=2Z2Z2+Z1,
rnet=r12+r23exp(2iβd)1+r12r23exp(2iβd),
tnet=t12t23exp(iβd)1+r12r23exp(2iβd),
kx(s)=kx+2πsax,
ky(s)=±k2kx(s)2,
k=|kx+2πsax|
ax/λ=1n||kx|2π1ax|,
Ψm,k(r)=um,k(r)eik·r,
Em(r)=em±(r)eik·re±iβmr,
Hm(r)=±hm±(r)eik·re±iβmr.
E(r)=(mcm+em+(r)eiβmr+cmem(r)eiβmr)eik·r,
H(r)=(mcm+hm+(r)eiβmrcmhm(r)eiβmr)eik·r,
c+=Λ+c+,
c=Λc,
Λ±=diag(e±iβma),
Λ±=diag(eik·a2)=diag(ei(k·a2±βma)).
c1=R12c1++T21c2,
c2+=T12c1++R21c2.
Rnet=R12+T21Λ2R23Λ2+(IR21Λ2R23Λ2+)1T12,
Tnet=T23(IR21Λ2R23Λ2+)1T12.
Zvg=1/vg={2|ε(r)Ez(r)|2SEzpolarization2|μ0Hz(r)|2SHzpolarization,
Zvg=vgZ0,
1vg=dkdω.
εeff=εandμeff=μ.
η(x,y)=E(x,y)/H(x,y).
ZBoscolo=η(x,0)Re(S(x,0))dxRe(S(x,0))dx.
YUshida=1/ZUshida=±1/η(0,0),
η=|Ez|2Ez*Hx,
ZBiswas=|E|2dxdz∫∫E*×Hdxdz=|Ez|2+|Ex|2dxdz∫∫Ez*HxEx*Hzdxdz,
ZMomeni=|E|22S,(Ezpolarization),
ZMomeni=2S|H|2,(Hzpolarization),
Z1PW=eikxxE(x)dxeikxxH(x)dx,
ZB=E(x,0)dxH(x,0)dx.
ZRE=Zvac1+r1r,
R12=(A12A12T+I)1(A12A12TI),
T12=(A12TA12+I)12A12T,
δij=1axHi(x)Ej(x)+Ei(x)Hj(x)dx,
δij=2axHi(x)Ej(x)dx.
Zi,j=2axHi(0)(x)Ej(x)dx,
Zi,j=1axHi(0)(x)Ej(x)+Ei(0)(x)Hj(x)+[Hi(0)(xax/2)Ej(x)Ei(0)(xax/2)Hj(x)]eikxax/2dx.
t12=2Z1Z2Z2+Z1[modes normalized byEq.(36)],
(ei+(x,0)×hj+(x,0)ej+(x,0)×hi+(x,0))ydx=δij,
(ei+(x,0)×hj(x,0)ej(x,0)×hi+(x,0))ydx=0,
em(x)=em+(x),
hm(x)=hm+(x)
em+(x)=em(x)=em+(x),
hm+(x)=hm(x)=hm+(x).
(ei+(x)×hj+(x)+ej+(x)×hi+(x))ydx=δij,
(ei+(x)×hj+(x)ej+(x)×hi+(x))ydx=0,
δij=2axHi(x)Ej(x)dx,

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