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Analysis of nonlocal effective permittivity and permeability in symmetric metal–dielectric
multilayer metamaterials
View the table of contents for this issue, or go to the journal homepage for more
2016 J. Opt. 18 065101
(http://iopscience.iop.org/2040-8986/18/6/065101)
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Analysis of nonlocal effective permittivityand permeability in symmetric metal–dielectric multilayer metamaterials
Lei Sun, Xiaodong Yang1 and Jie Gao1
Department of Mechanical and Aerospace Engineering Missouri University of Science and TechnologyRolla, Missouri 65409, USA
E-mail: [email protected] and [email protected]
Received 20 January 2016, revised 26 February 2016Accepted for publication 3 March 2016Published 13 April 2016
AbstractA generalized nonlocal effective medium theory is derived based on the transfer-matrix method todetermine the nonlocal effective permittivity and permeability for the symmetric and periodicmetal–dielectric multilayer metamaterials, with respect to both transverse-electric and transverse-magnetic polarized light at arbitrary angle of incidence. The nonlocal effective permittivity andpermeability tensors are analyzed in detail as functions of the wavelength, the angle of incidence,and the multilayer period. Our generalized nonlocal effective medium theory in consideration ofboth permittivity and permeability can accurately predict the dispersion relation, the transmissionand reflection spectra, and the optical field distributions of symmetric metal–dielectric multilayerstacks with either subwavelength or wavelength-scale period of the unit cell.
Keywords: metamaterials, dispersion, optical nonlocality, effective medium
(Some figures may appear in colour only in the online journal)
1. Introduction
Recently metal–dielectric multilayer metamaterials have beenwidely explored due to their anomalous electromagnetic prop-erties in optical frequencies. Metal–dielectric multilayer meta-materials with hyperbolic dispersion have been used to realizemany unique applications [1], such as negative refraction [2–4],deep subwavelength imaging [5–7], anomalous indefinite cavity[8], photonic density of states enhancement [9, 10], thermalemission design [11], and spontaneous emission engineering[12–14]. By tuning the metallic filling ratio of metal–dielectricmultilayer stacks, epsilon-near-zero (ENZ)metamaterials can beconstructed for the realization of radiation wavefront tailoring[15], displacement current insulation [16, 17], optical non-linearity enhancement [18], and invisible cloaking [19, 20]. Theoptical properties of metal–dielectric multilayer stacks can besimply described by the local effective medium theory (EMT),when the multilayer period is much smaller than the opticalwavelength. In fact, due to the dramatically different
electromagnetic properties of the metal layer and the dielectriclayer, the variation of the electromagnetic field on the scale ofone multilayer period will generate strong spatial dispersion,which will lead to optical nonlocality [21] not considered in thelocal EMT. Recently, in order to take into account the opticalnonlocality in metal–dielectric multilayer stacks, several dif-ferent nonlocal EMT models have been derived, such as thedispersion relation approximation [22–24] and the electro-magnetic field averaging algorithm [25–27]. However, thesemodels only consider the nonlocal effective permittivity asfunctions of frequency and wave vector, but assume unitypermeability. Therefore, the optical properties of metal–di-electric multilayer stacks cannot be completely predicted bythese models, especially when the multilayer period is far awayfrom the deep-subwavelength scale.
According to Herpin’s theorem, any dielectric multilayerstack is equivalent to a two-film combination [28], while asymmetric dielectric multilayer stack is equivalent to a singlefilm with an effective refractive index [29–31] based on thetransfer-matrix method [32]. Therefore, in this work, a gen-eralized nonlocal EMT in consideration of both nonlocal
Journal of Optics
J. Opt. 18 (2016) 065101 (11pp) doi:10.1088/2040-8978/18/6/065101
1 Authors to whom any correspondence should be addressed.
2040-8978/16/065101+11$33.00 © 2016 IOP Publishing Ltd Printed in the UK1
effective permittivity and permeability is derived for the sym-metric metal–dielectric multilayer stack with respect to bothtransverse-electric (TE) and transverse-magnetic (TM) polarizedlight with arbitrary angle of incidence (AOI). The nonlocaleffective permittivity and permeability tensors will be obtainedthrough the transfer-matrix method in order to fully describe thevariation of the electromagnetic field across the metal–dielectricmultilayers with respect to both frequency and wave vector.Compared with the local EMT parameters, the nonlocal effec-tive permittivity and permeability tensors are analyzed in detailas functions of the wavelength, the angle of incidence, as wellas the multilayer period. It is shown that the generalized non-local EMT will converge into the local EMT as the multilayerperiod approaches zero. In general, in contrast to the local EMT,the generalized nonlocal EMT can fully characterize the opticalproperties of the symmetric metal–dielectric multilayer stackswith either subwavelength or wavelength-scale period of theunit cell, and accurately predict the band structures, theiso-frequency contours (IFCs), the transmission and reflectionspectra, as well as the optical field distributions.
2. Development of generalized nonlocal effectivemedium theory
Figure 1(a) illustrates the symmetric and periodic metal–di-electric multilayer stack composed of five-pair (with N = 5)
symmetric unit cells. Each unit cell contains a silver (Ag)layer with the thickness of am sandwiched between two
sapphire (Al O2 3) layers with the thickness of a 2d , under theillumination of the incident light propagating in the x-z planewith an arbitrary AOI q0 in free space. The thickness of oneunit cell is = +a a am d, while the total thickness of themultilayer stack is =a Natot . The electric permittivities of theAg layer and the Al O2 3 layer are denoted as em and ed,respectively. The magnetic permeabilities of the Ag layer andthe Al O2 3 layer are denoted as mm and md, which are equal tounity. In general, the symmetric Ag-Al O2 3 multilayer stackcan be regarded as a bulk of homogenous and anisotropiceffective medium characterized by a set of effective para-meters, that is, a diagonal permittivity tensor and a diagonalpermeability tensor. According to the local EMT, the localeffective permittivity tensor reads ¯ ( )e e e e= diag , ,x y z
loc loc loc loc
with ( ) ( )e e e e= = + +a a a ax y m m d d m dloc loc and e =z
loc
( ) ( )e e e e+ +a a a am d m d m d d m , which only depend on fre-quency. Meanwhile, the local effective permeability tensor isalways equal to unity. In fact, previous studies have shownthat metal–dielectric multilayer stacks possess strong spatialdispersion caused by optical nonlocality, which cannot becharacterized by the local EMT. Furthermore, the previouslyproposed nonlocal EMT models only consider the nonlocaleffective permittivity tensor as a function of both frequencyand wave vector, but the nonlocal effective permeabilitytensor is still treated as unity so that the optical properties ofthe metal–dielectric multilayer stacks cannot completelypredicted yet.
In order to take into account the optical nonlocalityaccurately, a generalized nonlocal EMT is derived through thetransfer-matrix method, and both the nonlocal effectivepermittivity and permeability tensors are introduced as func-tions of frequency and wave vector. In accordance with theboundary conditions, the tangential components of theelectromagnetic field should be continuous across all inter-faces of the Ag-Al O2 3 multilayer stack. Then the transfermatrix for the multilayer stack composed of five-pair unitcells is described as
( · · ) ( )= =M M M M M , 1d m dstack cell5
2 25
in which
for the Ag layer, and
Figure 1. Schematic of the symmetric and periodic Ag-Al O2 3multilayer stack with respect to the AOI q0 in free space.
( ) ( )( ) ( ) ( )
e m q e m q he m q h e m q
=-
-
⎡⎣⎢
⎤⎦⎥M
k a k a
k a k a
cos cos i sin cos
i sin cos cos cos2m
m m m m m m m m m
m m m m m m m m m
0 0
0 0
( · · ) ( · · )( · · ) ( · · ) ( )
e m q e m q he m q h e m q
=-
-
⎡⎣⎢
⎤⎦⎥M
k a k a
k a k a
cos 2 cos i sin 2 cos
i sin 2 cos cos 2 cos3d
d d d d d d d d d
d d d d d d d d d2
0 0
0 0
2
J. Opt. 18 (2016) 065101 L Sun et al
for the Al O2 3 half layer. The parameter hi reads h =ie m qcosi i i for TE polarized light and h =i
m e qcosi i i for TM polarized light. The angle of refractionqi is determined by the AOI q0 as q= =k k sinx 0 0
e m qk sini i i0 with i=m and d for the Ag layer and Al O2 3
layer, respectively. Considering the symmetric and periodicproperties of the Ag-Al O2 3 multilayer stack, the connectionsbetween the transfer matrix for the stack Mstack and the transfermatrix for the unit cell Mcell can be studied. According toequations (1)–(3), Mcell can be expressed as
( ) ( )( ) ( ) ( )
hh
=-
-
⎡⎣⎢
⎤⎦⎥M
k a k a
k a k a
cos isin
isin cos, 4
z z e
z e zcell
in terms of the dispersion relation of the unit cell
( )
( )
j jhh
hh
j j= - +⎛⎝⎜
⎞⎠⎟k acos cos cos
1
2sin sin
5
z d md
m
m
dd m
and the parameter
with j e m q= k a cosd d d d d0 and j e m q= k a cosm m m m m0 .On the other hand, Mstack with N unit cell is determined as
( · ) ( · )( · ) ( · ) ( )
hh
=-
-
⎡⎣⎢
⎤⎦⎥M
k Na k Na
k Na k Na
cos isin
isin cos, 7
z z e
z e zstack
based on the Chebyshev polynomials of the second kind. Bycomparing equations (4) and (7), it is indicated that theelectromagnetic field distribution in the Ag-Al O2 3 multilayerstack can be fully described by that in the symmetric unit cell.This fact implies that the nonlocal effective parameters for thestack are independent of the number of the unit cell N.Therefore, only one symmetric unit cell will be considered inthe determination of the nonlocal effective permittivity andpermeability tensors.
The symmetric unit cell is considered as a homogenousand anisotropic effective medium including the optical non-locality with the nonlocal effective permittivity and perme-ability tensors of ¯ ( )e e e e= diag , ,e x y z and m̄=e
( )m m mdiag , ,x y z . The transfer matrix for the unit cell regardedas a nonlocal effective medium is represented as
( ) ( )( ) ( ) ( )
hh
=-
-
⎡⎣⎢
⎤⎦⎥M
k a k a
k a k a
cos isin
isin cos, 8
z z
z zeff
eff
eff
where the parameter
( )hm
=k
k9z
xeff
0
Figure 2. (a) The real part and (b) the imaginary part of e e eD = -y y yloc, as well as (c) the real part and (d) the imaginary part of
m m mD = -x x xloc, with respect to the variations of wavelength and AOI for TE polarized light.
[ ( ( ) ) ]( ( ) )
( )hh h h j j h h h h j j
h h j j h h h h j j=
+ - + + +
+ - + +
2 sin cos cos sin
2 sin cos cos sin6e
d d m d m d m d m d m
d m d m d m d m d m
2 2 2 2 2
2 2 2 2
3
J. Opt. 18 (2016) 065101 L Sun et al
for the TE polarized light, and
( )he
=k
k10z
xeff
0
for the TM polarized light. Since the nonlocal effectivemedium should represent the unit cell, the relation of h hº eeffshould be satisfied for both TE and TM polarizedlight, according to equations (4) and (8). Meanwhile, the
Figure 3. (a) The real part and (b) the imaginary part of e e eD = -x x xloc, as well as (c) the real part and (d) the imaginary part of
m m mD = -y y yloc, with respect to the variations of wavelength and AOI for TM polarized light.
Figure 4. (a) The real part and (b) the imaginary part of e e eD = -y y yloc, as well as (c) the real part and (d) the imaginary part of
m m mD = -x x xloc, with respect to the variations of the period of the unit cell and wavelength for TEM mode at zero AOI.
4
J. Opt. 18 (2016) 065101 L Sun et al
propagation of the electromagnetic wave in the nonlocaleffective medium can be described by the dispersion relation
( )e m e m
+ =k k
k 11x
y z
z
y x
2 2
02
for the TE polarized light, and
( )e m e m
+ =k k
k 12x
z y
z
x y
2 2
02
for the TM polarized light. Therefore, according toequations (5), (6), and (9)–(12), the nonlocal effectivepermittivity and permeability components for the unit cellcan be expressed as
( )
mh
e hm
m m
=
= +
º =
⎧
⎨⎪⎪⎪
⎩⎪⎪⎪
k k
k
k
k k
1
13
xz
e
yz
ex
z
z z
0
0
202
loc
associated with the TE polarized light, while
( )
eh
m he
e e
=
= +
º
⎧
⎨⎪⎪⎪
⎩⎪⎪⎪
k k
k
k
k k 14
xz
e
yz
ex
z
z z
0
0
202
loc
associated with the TM polarized light. It is noted that threenonlocal effective parameters ( )m e m, ,x y z or ( )e m e, ,x y z needto be solved with only two equations of equations (9) and (11)or equations (10) and (12), for TE or TM polarized light.Therefore, the values of the nonlocal effective parametersalong the z direction, which is normal to the interfaces of themultilayer stack, are approximated as the values of thecorresponding local effective parameters m mºz z
loc and
e eºz zloc in equations (13) and (14). This hypothesis will
not reduce the accuracy of the generalized nonlocal EMT dueto the relatively weak optical nonlocality along the z direction.Furthermore, based on equation (5), the wave vector kz in
Figure 5. (a) The band structure of the Ag-Al O2 3 multilayer stack with the period of unit cell =a 80 nm at zero AOI. The real part of thenormalized wave vector k kz p is denoted as the red curve, while the imaginary part is denoted as the black curve. The IFCs at point C areplotted for (b) TE polarized light and (c) TM polarized light, calculated from the multilayer stack (red-solid curves for the real part and black-solid curves for the imaginary part), the generalized nonlocal EMT (orange-dashed curves for the real part and blue-dashed curves for theimaginary part), and the local EMT (red-dashed curves for the real part and black-dashed curves for the imaginary part).
5
J. Opt. 18 (2016) 065101 L Sun et al
equations (13) and (14) can be expressed as
( )=p j j j j - +h
hhh
⎜ ⎟⎡⎣⎢
⎛⎝
⎞⎠
⎤⎦⎥
k
, 15
z
n
a
2 arccos cos cos sin sind md
m
m
dd m
12
with an arbitrary integer n, which is related to differentbranches of the real part of wave vector ( )kRe z and therestriction for the imaginary part of wave vector ( ) kIm 0z
due to the passive absorption loss. In general, selection of theinteger n must lead to a set of nonlocal effective parametersthat can converge to the local effective parameters when theperiod of multilayer stack becomes much less than thewavelength. Moreover, the nonlocal effective parametersmust vary continuously with respect to the wavelength orfrequency. Additionally, besides the value of the integer n, theplus/minus (±) sign in equation (15) needs to be properlyselected to guarantee the restriction of ( ) kIm 0z accordingto the passive medium condition.
3. Analysis of nonlocal effective permittivity andpermeability
The generalized nonlocal EMT will be utilized to study ametal–dielectric multilayer stack in the subwavelength rangewith the Ag layer thickness of =a 10 nmm and the Al O2 3
layer thickness of =a 70 nmd in the wavelength range from400 to 800 nm. The permittivities of the Ag layer and theAl O2 3 layer, em and ed, are obtained from the experimentallymeasured data [33, 34]. For convenience, the Ag plasmafrequency ·w = ´ -1.37 10 rad sp
16 1 and the correspondingwave vector w=k cp p are applied as the normalization factorin the analysis. According to equation (13), figures 2(a) and(b) display the difference between the nonlocal effectivepermittivity component ey and the local effective permittivitycomponent ey
loc, defined as e e eD = -y y yloc, with respect to
the variations of wavelength and AOI for TE polarized light.While the difference between the nonlocal effective perme-ability component mx and the local effective permeabilitycomponent mx
loc, defined as m m mD = -x x xloc, are shown in
figures 2(c) and (d). Similarly, the differences ofe e eD = -x x x
loc and m m mD = -y y yloc for TM polarized light
are plotted in figure 3 based on equation (14). The resultsclearly show that the nonlocal effective permittivity compo-nents ex and ey and the nonlocal effective permeability com-ponents mx and my are all dependent on both the frequency andthe wave vector ( q=k k sinx 0 0) due to the optical nonlocality.In contrast to the local EMT with e e=x y
loc loc and m m=x yloc loc,
the nonlocal effective parameters e e¹x y and m m¹x y whenthe AOI is not equal to zero. For zero AOI, the TE and TMmodes degenerates into the transverse electromagnetic (TEM)mode and thus equation (13) is identical to equation (14).
Figure 6. (a) The transmission spectra for the =a 80 nm Ag-Al O2 3 multilayer stack (red-solid curve), the generalized nonlocal EMT(orange-dashed curve), and the local EMT (red-dashed curve), as well as the reflection spectra for the multilayer stack (black-solid curve), thegeneralized nonlocal EMT (blue-dashed curve), and the local EMT (black-dashed curve) for the TEM mode at zero AOI. The distributions of(b) electric field Ey, (c) magnetic field Hx, and (d) energy flow Sz at point C for the multilayer stack (red-solid curves) and the generalizednonlocal EMT (blue-dashed curves) are displayed.
6
J. Opt. 18 (2016) 065101 L Sun et al
Moreover, it is noted that the imaginary parts of the nonlocaleffective permeability components mx and my have negativevalues. However, this fact does not contradict the passivemedium condition, since the imaginary part of the wavevector kz is always kept as positive value. In addition, it isshown that the nonlocal effective permeability components mxand my are sensitive to the configuration of the symmetric unitcell in the metal–dielectric multilayer stack. For the currentconfiguration of the unit cell containing Al O2 3-Ag-Al O2 3
structure with the layer thickness of ad1
2-am- ad
1
2, the nonlocal
effective permeability components mx and my have valueslarger than unity, resulting in a paramagnetic response. On theother hand, when the unit cell is switched into Ag-Al O2 3-Agstructure with the layer thickness of am
1
2-ad- am
1
2, the nonlocal
effective permeability components mx and my will have valuesless than unity, giving a diamagnetic response. Unlike thelocal EMT and the previous nonlocal EMT, the generalizednonlocal EMT can predict different paramagnetic and dia-magnetic responses depending on the unit cell configurationin the metal–dielectric multilayer stack. Since the opticalnonlocality is induced by the variation of the electromagneticfield across the scale of the unit cell in the multilayer stack,the nonlocal effective parameters will depend on the period ofthe unit cell. Figure 4 shows the variations of eD y and mD xwith respect to the wavelength and the period of the unit cellfor the TEM mode based on equation (13). It is shown that the
values of eD y and mD x gradually reduce to zero over thewhole wavelength range from 400 to 800 nm as the period ofthe unit cell reduces from 100 to 10 nm. This means that theoptical nonlocal effects are diminished and the nonlocal EMTapproaches the local EMT when the size of the unit cell goesto the deep-subwavelength range.
Figure 5(a) plots the band structure of the multilayerstack with the period of the unit cell =a 80 nm, according tothe diagonalization of the transfer matrix in equation (1) atzero AOI. Four special points are marked in the band struc-ture as point C where ( ) ( )=k k k kRe Imz p z p , and points A1,A2 and A3 where the Bragg condition is satisfied. At thefrequency of point C (417.211 THz or 718.564 nm), the IFCscalculated from the multilayer stack (equation (5)), the gen-eralized nonlocal EMT, and the local EMT are displayed forthe TE mode in figure 5(b) and the TM mode in figure 5(c). Itis clearly shown that for both TE and TM modes, the IFCsbased on the multilayer stack and the generalized nonlocalEMT exactly overlap with each other in both the real part andthe imaginary part, implying that the propagation of electro-magnetic wave in the multilayer stack can be well describedby the generalized nonlocal EMT. In contrast, the IFCs cal-culated from the local EMT possess a large deviation withoutthe consideration of optical nonlocality. Correspondingly,figure 6 presents the transmission and reflection spectra forthe TEM mode with respect to the multilayer stack, the
Figure 7. (a) The transmission spectra for the =a 80 nm Ag-Al O2 3 multilayer stack (red-solid curve), the generalized nonlocal EMT(orange-dashed curve), and the local EMT (red-dashed curve), as well as the reflection spectra for the multilayer stack (black-solid curve), thegeneralized nonlocal EMT (blue-dashed curve), and the local EMT (black-dashed curve) for the TE mode at 45◦ AOI. The distributions of (b)electric field Ey, (c) magnetic field Hx, and (d) energy flow Sz at point C for the multilayer stack (red-solid curves) and the generalizednonlocal EMT (blue-dashed curves) are displayed.
7
J. Opt. 18 (2016) 065101 L Sun et al
generalized nonlocal EMT, and the local EMT, together withthe distributions of electromagnetic field and energy flow forthe multilayer stack and the nonlocal effective medium. Thetransmission and reflection spectra in figure 6(a) clearlyindicate that the generalized nonlocal EMT can predict theexactly same spectra as the multilayer stack, in contrast to thelocal EMT. The generalized nonlocal EMT also accuratelyidentifies the resonance features at the Bragg condition points(A1, A2 and A3) in the spectra. At the frequency of point C,the distributions of electric field Ey (figure 6(b)) and magneticfield Hx (figure 6(c)) are illustrated for both the multilayerstack and the generalized nonlocal EMT. The electromagneticfield oscillates periodically and decays across each unit cell inthe actual multilayer stack, while the electromagnetic fieldcalculated from the generalized nonlocal EMT just smoothlydecays through the effective medium without seeing themultilayer interfaces, following the same trend as the multi-layer stack. The distribution of energy flow (Poynting vectorSz) along the propagation direction is also plotted infigure 6(d). The energy flow in the multilayer stack has aladder-like distribution with constant values in Al O2 3 layersbut exponentially decayed values in Ag layers. Instead, theenergy flow just exponentially decays along the effectivemedium based on the generalized nonlocal EMT, with thesame trend as the multilayer stack. Moreover, the distribu-tions of electromagnetic field in free space are exactly thesame for the multilayer stack and the generalized nonlocal
EMT, which indicates the agreement in the calculated trans-mission and reflection spectra based on both approaches.
Additionally, the generalized nonlocal EMT can alsorepresent the electromagnetic properties of the multilayerstack with respect to the oblique incident light. For instance,figures 7 and 8 display the transmission and reflection spectrarelated to the multilayer stack, the generalized nonlocal EMT,and the local EMT, together with the distributions ofelectromagnetic field and energy flow for the multilayer stackand the nonlocal effective medium, with respect to the TEpolarized and TM polarized incident light at the AOI ofq = 450 , respectively. Figure 7(a) and 8(a) shows that thegeneralized nonlocal EMT can still predict the exactly sametransmission and reflection spectra as the multilayer stack forboth TE mode and TM mode with the oblique incident light,as well as the resonance features at the Bragg condition points(A1 and A2) in contrast to the local EMT. Meanwhile,according to the distributions of electric field Ey (figure 7(b)for TE mode) and Ex (figure 8(b) for TM mode), and magneticfield Hx (figure 7(c) for TE mode) and Hy (figure 8(c) for TMmode) at the frequency of point C, it is shown that theelectromagnetic field oscillating and decaying across themultilayer stack is well represented by the generalized non-local EMT with smooth decay through the effective medium.Correspondingly, the distributions of energy flow (Poyntingvector Sz) with ladder-like profiles in the multilayer stack arealso matched well with the results from the generalized
Figure 8. (a) The transmission spectra for the =a 80 nm Ag-Al O2 3 multilayer stack (red-solid curve), the generalized nonlocal EMT(orange-dashed curve), and the local EMT (red-dashed curve), as well as the reflection spectra for the multilayer stack (black-solid curve), thegeneralized nonlocal EMT (blue-dashed curve), and the local EMT (black-dashed curve) for the TM mode at 45◦ AOI. The distributions of(b) electric field Ex, (c) magnetic field Hy, and (d) energy flow Sz at point C for the multilayer stack (red-solid curves) and the generalizednonlocal EMT (blue-dashed curves) are displayed.
8
J. Opt. 18 (2016) 065101 L Sun et al
nonlocal EMT across the effective medium, as shown infigures 7(d) and 8(d) for the TE mode and TM mode,respectively. Additionally, the distributions of electro-magnetic field in free space also agree well between themultilayer stack and the generalized nonlocal EMT for bothTE mode and TM mode with the oblique incident light.
Besides the multilayer stack with subwavelength period ofthe unit cell, the generalized nonlocal EMT can also be utilizedto study the case with wavelength-scale period of the unit cell.Figure 9 provides the band structure of the multilayer stackwith the period of the unit cell =a 150 nm at zero AOI,together with the nonlocal effective parameters over thewavelength range from 300 to 800 nm. Due to the large size ofthe period, a forbidden band emerges in the band structureapproximately from 400 to 520 nm as shown in figure 9(a),leading to strong resonances in the nonlocal effective permit-tivity ey (figure 9(b)) and the nonlocal effective permeability mx(figure 9(c)) at the boundaries of the forbidden band due to theband transition. Such band transition induced nonlocal effec-tive parameter resonances can be explained via the branchvariation of ( )kRe z according to equation (15). In general,
( )kRe z is determined by the n=0 branch with the plus sign inequation (15) in the lower conduction band, but it is deter-mined by the n=1 branch with the minus sign in the for-bidden band and upper conduction band. The restriction of
( ) kIm 0z should always be guaranteed in all branches tosatisfy the passive medium condition. Moreover, the points that
satisfy the Bragg condition are all marked in the band structure,with P1 to P4 in the lower conduction band and P5 to P7 in theupper conduction band. The corresponding transmission andreflection spectra, as well as the distributions of electric fieldare further presented in figure 10. As shown in figure 10(a), thegeneralized nonlocal EMT can precisely predict the transmis-sion and reflection spectra for the multilayer stack over thewhole wavelength range, while the spectra from the local EMThave great deviations since the period of the unit cell is faraway from the subwavelength range. Meanwhile, due to thewavelength-scale period of the unit cell, the resonance peaks inthe transmission and reflection spectra at the Bragg conditionpoints (P1-P6) are clearly shown. The distributions of electricfield Ey at points P1, P3 and P6 are also displayed infigures 10(b)–(d) for both multilayer stack and the generalizednonlocal EMT. Apparently, the electric field in the actualmultilayer stack oscillates periodically through each unit celland generates resonant modes with different orders. While theelectric field calculated from the generalized nonlocal EMTexhibits resonant standing wave pattern that matches theelectric field of the multilayer stack. These results imply thatthe generalized nonlocal EMT can well describe the electro-magnetic properties for the multilayer stack even in thewavelength-scale range. Meanwhile, the distributions of elec-tric field in free space are exact the same for the multilayerstack and the generalized nonlocal EMT.
Figure 9. (a) The band structure of the Ag-Al O2 3 multilayer stack with the period of unit cell =a 150 nm at zero AOI. (b) The nonlocaleffective permittivity ey and (c) the nonlocal effective permeability mx with the real part in red-solid curves and the imaginary part in black-solid curves.
9
J. Opt. 18 (2016) 065101 L Sun et al
4. Conclusions
A generalized nonlocal EMT in consideration of nonlocaleffective permittivity and permeability has been derivedthrough the transfer-matrix method for symmetric and peri-odic metal–dielectric multilayer stacks, with respect to bothTE and TM polarized light with arbitrary angle of incidence.The nonlocal effective permittivity and permeability tensorsdepending on both frequency and wave vector have beenanalyzed in detail as functions of the wavelength, the angle ofincidence, and the period of the unit cell. The generalizednonlocal EMT will converge into the local EMT as the periodof the unit cell approaches zero. In contrast to the local EMT,the generalized nonlocal EMT can accurately characterize theband structures, IFCs, transmission and reflection spectra, andoptical field distributions for the symmetric metal–dielectricmultilayer stacks with either subwavelength or wavelength-scale period of the unit cell.
Acknowledgments
The authors acknowledge support from the National ScienceFoundation under Grant No.DMR-1552871 and CBET-1402743, and U.S. Army Research Office Award No.W911NF-15-1-0477.
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Figure 10. (a) The transmission and reflection spectra for the =a 150 nm Ag-Al O2 3 multilayer stack, the generalized nonlocal EMT, and thelocal EMT for the TEM mode at zero AOI. The distributions of electric field Ey at (b) point P1, (c) point P3, and (d) point P6 for themultilayer stack (red-solid curves) and the generalized nonlocal EMT (blue-dashed curves) are displayed.
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