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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 16 — Aug. 2, 2010
  • pp: 16646–16662
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Cylinder light concentrator and absorber: theoretical description

Alexander V. Kildishev, Ludmila J. Prokopeva, and Evgenii E. Narimanov  »View Author Affiliations


Optics Express, Vol. 18, Issue 16, pp. 16646-16662 (2010)
http://dx.doi.org/10.1364/OE.18.016646


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Abstract

We present a detailed theoretical description of a broadband omnidirectional light concentrator and absorber with cylinder geometry. The proposed optical “trap” captures nearly all the incident light within its geometric cross-section, leading to a broad range of possible applications – from solar energy harvesting to thermal light emitters and optoelectronic components. We have demonstrated that an approximate lamellar black-hole with a moderate number of homogeneous layers, while giving the desired ray-optical performance, can provide absorption efficiencies comparable to those of ideal devices with a smooth gradient in index.

© 2010 OSA

1. Introduction

Recent advances in the development of optical metamaterials, artificial structures with the desired local electromagnetic response determined by artificial subwavelength patterning of the media, have opened a wide range of new concepts and applications, from super-resolution imaging to optoelectronics and photovoltaics [1

1. V. M. Shalaev, W. S. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30(24), 3356–3358 (2005). [CrossRef]

8

8. E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009). [CrossRef]

]. In particular, the recently proposed [8

8. E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009). [CrossRef]

] “optical black hole”, an omnidirectional light concentrator and absorber, could significantly enhance the performance of solar energy harvesting systems and lead to novel nonlinear-optical devices. In this work, we present a detailed theoretical description of this device, and develop appropriate numerical methods, which – in addition to the accurate analysis of the optical black hole, - could be applied to a broad range of metamaterial-based optical systems with centrally symmetric gradient index. We note that in the context of this paper, the term “black hole” simply refers to the highly efficient “capture” of the light incident on the device, and does not imply any profound analogy to General Relativity.

The remaining part of the paper is organized as follows. In Section 2, we introduce the initial assumptions, state the geometry of the cylindrical optical system with an axially symmetric graded index, and show basic examples of ray-tracing inside such devices, including the fundamental optical black-hole [8

8. E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009). [CrossRef]

]. Section 3 deals with the ray optical description of the system followed by its full-wave description in Section 4. The analytical and numerical apparatus for approximating the ideal fundamental black-hole with concentric lamellar systems of uniform isotropic layers is then discussed in Section 5. Section 5 also covers derivations of the scattering and absorption cross-sections, which are then compared with their semiclassical approximations in Section 6. Finally, we summarize our work in Section 7.

2. The system

In its original proposal [8

8. E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009). [CrossRef]

], the optical “black hole” (see Fig. 1(a)
Fig. 1 (a) the schematics of the system; panels (b), (c), (d), and (e) show light ray trajectories for p equal to −1, 1, 2, and 3 respectively, where core radius r c much smaller than shell radius r s (r c = r s).
) consists of the “payload” core with the radius rc, supporting the desired functionality (such as e.g. a solar cell for concentrated photovoltaics applications), surrounded by the concentrating shell of radius rswith an axially symmetric dielectric function, ε(r), changing in the radial direction as
ε(r)εsΔε(r)
(1)
where
Δε(r)={1,r>rs(rs/r)p,rc<rrs(εc+ιγ)/εs,rrc
(2)
and p>0 is a constant. In order to match the real parts of relative permittivity at the interface r=rc the radius of the core is defined by the ratio of the permittivity of the surrounding medium (εs) to that of the core (εc),
rc=rs(εs/εc)1/p.
(3)
Note that we assume that the metamaterial forming the shell, is essentially non-magnetic, so that the refractive index is n=ε .

3. Ray-optical description

In the semi-classical limit, the radial variation of the refractive index of the shell (where n=ε) corresponds to the effective potential
Veff(r)=12(ω/c)2[εsε(r)],
(4)
which for the dielectric permittivity defined by (2) with p2 traps all the incident light trajectories – see Figs. 1(b)-1(e). For these ray trajectories, a straightforward integration of the Hamiltonian equations with the effective potential defined by (4) yields in the polar coordinates
ϕ(r)=ϕ0+mrsmrdξC0ε(m/ξ)ξ2,
(5)
where m is the (conserved) angular momentum, C0=(m/y0)2/εs (with y0 being rssinϕ0), and the constants rs and ϕ0 are set by the initial conditions of the given trajectory. In particular, for the permittivity defined in the shell (2), Eq. (5) after integration gives
r(ϕ)=rs×{|cscϕ0cos[p22(ϕϕp)]|2p2,p2exp[cotϕ0(ϕ0ϕ)],p=2,
(6)
where ϕpis yet another constant defined by ϕp=ϕ0(12p2)+πp2.

Case p=1 corresponds to the gravitational potential (or electrostatic potential) and emulates the motion of a non-accelerated particle in central gravitational field; so that the parabolic trajectories (Fig. 1c) are expected. Case p=2 gives the potential of fatal attraction [9

9. L. D. Landau, and E. M. Lifshitz, Mechanics, Pergamon Press, Oxford, (1976).

]. In fact for all p2 (6) corresponds to the ray trajectories “falling” into the core of the “black hole” – see Fig. 1(b). Figures 1(b)-1(e) show the traces of light rays for p equal to −1, 1, 2, and 3 respectively; the trivial case of the beam propagating in free-space (p=0) is omitted.

This concept can be realized in both the spherical and cylindrical geometries, with the latter suitable for optoelectronic devices where the light propagation is generally realized in a waveguide geometry. In the present paper we will focus on this cylindrical realization of the optical black hole, leaving its spherical counterpart to a forthcoming work.

4. The wave optical description

In the present section of the paper, we develop the wave-optical description of the “black hole” light concentrator. We define a given monochromatic electromagnetic vector field by the vectors of electric field Eexp(ιωt) and magnetic field Hexp(ιωt), and by its angular frequency ω. We start with the monochromatic Maxwell’s equations ×E=ιωB, ×H=ιωD arriving first at [10

10. R. K. Luneburg, Mathematical Theory of Optics, University of California Press, Berkeley, 1964, p. 12.

]:
k02εμH=××Hlnε××H,
(7)
k02εμE=××Elnμ××E.
(8)
Getting H=lnμH, E=lnεE from B=D=0, we then rewrite (7)-(8) as
k2H+2H=(lnμH)lnε××H,
(9)
k2E+2E=(lnεE)lnμ××E.
(10)
here k2=εk02, k02=ω2/c2, and c2=1/(ε0μ0), where ε=ε(x,y) is a smooth spatially-dependent dielectric function, the free-space parameters k0, c, ε0, and μ0 respectively denote the wavenumber, the velocity of light, the permittivity, and the permeability.

Here we assume only transverse electric (TE, E=z^e(x,y)) and transverse magnetic (TM, H=z^h(x,y)) waves within a nonmagnetic medium (μ=1), so we reduce the above vector equations and use in-plane scalars e and h, which are further on collectively denoted as f.

4.1 Scalar wave equations and separation of variables

Hence, both TE and TM wave Eqs. (9) and (10) reduce to a common scalar wave equation
2f+k2f+Lf=0.
(11)
For any TM-wave (f=h) the operator L is given by L=lnεand does not vanish for spatially dispersive ε, while it vanishes for any TE-wave (f=e).

The cylindrical symmetry of the device allows a straightforward separation of radial and angular variables in the cylindrical coordinates, (r, ϕ, z), linked to the Cartesian coordinates (x, y, z) through x=rcosϕ, y=rsinϕ. Exploiting the cylindrical symmetry of the dielectric function, ε=ε(r), and decomposing the solution of (11) into the cylindrical waves, f=m=+fm(r)expιmϕ, we arrive at equation for scalar modes fm
r2fm+[r+q(r)]fm+[k2r2m2]fm=0,
(12)
where the prime denotes the radial derivative, and q(r)=0 or q(r)=r2lnε for TE or TM modes respectively.

4.2 TE and TM solutions for ε(r)=const

The solvable cases that we are using here include the trivial instance of ε(r)=const for both TE and TM modes. In this case, (12) reduces to the Bessel equation for r¯=kr, and the general solution is a linear combination of two linearly independent Bessel functions [11

11. E. J. Post, Formal Structure of Electromagnetics: General Covariance and Electromagnetics, 1962, p. 152.

].

In most of the cases it is more convenient to use the following combination
fm(r)=amJm(r¯)+bmHm+(r¯),
(13)
since it represents the unique decomposition (usually found in scattering problems) into the incident and scattered field if the domain is infinite. On the other hand if the solution is written in the domain, which contains the origin, then the additional boundary condition should be provided to avoid singularity (bm=0).

For simulations we use the normalized variant of (13), which improves the condition number of the linear system matrix (Section 5.1):
fm(r)=τmJm(r¯)/Jm(r¯s)+ρmHm+(r¯)/Hm+(r¯s).
(14)
However in some applications, including the calculation of cross sections, a better option is an equivalent combination of the Hankel functions Hm±(r¯)=Jm(r¯)±ιYm(r¯)
fm(r)=cmHm(r¯)+cm+Hm+(r¯),
(15)
which gives expressions that are easier to handle for mathematical analyses (Section 5.3). Back and forth translations between formulas (13) and (15) are performed by cm=12am, cm+=12(am+2bm),and am=2cm, bm=cm+cm.

4.3 TE and TM solutions for the ‘quadratic decay’ of permittivity, r2ε(r)=const

For the cylindrical realization of the black hole, which in its simplest form corresponds to an infinite cylinder with a radial distribution of the dielectric permittivity given byr2ε(r)=constor k2r2=ν2=const, all optical modes of the system can also be separated into the TE and TM categories. For the TM waveq(r)=r2lnε=2r, while for TE we always have q(r)=0.

For a TM mode, (12) can be therefore reduced as
r2fm+3rfm+[ν2m2]fm=0,
(16)
and for a TE mode as
r2fm+rfm+[ν2m2]fm=0.
(17)
Both (16) and (17) are the Euler differential equations, which are solved by substitution s=lnr. Then, since r2f+rf=s2f and rf=sf we arrive at the characteristic equation for f(exps) with roots ξ1,2, where the general solution reads
fm(r)={amrξ1+bmrξ2,ifξ1ξ2(am+bmlnr)rξ1,otherwise
(18)
and the characteristic roots ξ1,2 in (18) are
ξ1,2={1±ιν2m21,(TE)±ιν2m2,(TM)
(19)
or equivalently, ξ1,2=α±ιβm, where α=1 for TM, or α=0 for TE, and βm=ν2m2αin both cases.

4.4 Boundary conditions and the mode matching

In particular for a nonmagnetic, cylindrically symmetric medium without any free charges the condition of continuity of electromagnetic fields at a given interface r=ri>0 reads

TE:
[e]r=ri=[e]r=ri=0,
(20)
TM:
[h]r=ri=[ε1h]r=ri=0,
(21)
where [f(r)]r=ri=lima0f(ri+a)lima0f(ria). Since we assume that both the inner and outer sub-domains are of constant permittivity, with the general solution given by Eq. (13), which should have a physical solution at r=0, then bm=0 and

fm(r)=amJm(kr).
(22)

If the function f describes the scattered field (i.e. does not include the incident wave), it should obey the Sommerfeld radiation condition, limrr(fιkf)=0. The asymptotic behavior of the Hankel functionsHm±(kr)r1/2e±ιkr for r forces us to choose only one term in (13); hence, for the scattered field we have

fm(r)=bmHm+(kr).
(23)

4.5 Incident light: the Gaussian beams and plane wave

While the angular momentum formalism introduced above, represents the “natural” choice for a system with rotational symmetry, to complete the description of the scattering problem, we also need to develop the angular momentum expansion of the incident field. Here, we will consider the most common examples of the illumination field: the plane wave and the Gaussian beam.

Starting from the classical generating function for the Bessel functions [12

12. S. Gradshteyn, and I. M. Ryzhik, Tables of integrals, series and products, Academic Press, New York, CD-ROM Edition, 1994, Eq. (8).511.4.

]:
exp(ιzcosφ)=ιmeιmφJm(z),
(24)
for the incident plane wave we obtain (am=ιm)

f(x,y)=eιkx=m=+amJm(kr)eιmφ.
(25)

While plane wave illumination represents perhaps the most common setup of a scattering problem, it does not convey the case when the target is illuminated by a focused beam. The latter is generally adequately described by the Gaussian beam [13

13. M. Born, and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, New York, 1999).

], which yields an accurate description of the beam focused by linear optical elements. However, one needs to keep in mind that the Gaussian beam in its standard form is not an exact solution of Maxwell’s equations in the free space, but instead corresponds to the paraxial approximation [13

13. M. Born, and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, New York, 1999).

]. To avoid the approximate nature of this approach and resulting inaccuracy, we consider that the field, while following the Gaussian profile at the waist cross section, is in fact the exact solution of the Helmholtz wave equation with the given direction of propagation [14

14. G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69(4), 575 (1979). [CrossRef]

]:
{2f+k2f=0f(0,y)=exp[(y/w)2]
(26)
The resulting solution of (26) in each of half-spaces y0, y0 can be written as a composition of y-plane waves
f(x,y)=11aqexp[ιk(qy+1q2x)]dq+|q|>1aqexp[ιkqykq21|x|]dq,
(27)
where the amplitudes aq=12πkwexp[14(kwq)2] satisfy the condition ey2/w2=aqeιkyqdq. Here, the field is separated into the contributions of the propagating (the first integral in (27)) and the evanescent waves (the second term in (27)). The contribution of evanescent waves, can be estimated as
σ=||q|>1aqexp[ιkqykq21|x|]dq|kwπ1+e14k2w2q2dq=erfc(12kw).
(28)
As a result of (28), for a Gaussian beam with waist w2λ propagating in media with ε1 neglecting the evanescent waves integral in (27) gives a reasonable approximation, since the relative error of the field magnitude is less than erfc(12kw)erfc(2π)6×1019.

Adding the arbitrary shift of the focus location by straightforward translation, xxx0,yyy0, the final expression of a focused beam in Cartesian coordinates is
f(x,y)=kw2π11e14k2w2q2+ιk[q(yy0)+1q2(xx0)]dq,
(29)
or (29) can be equivalently written in the cylindrical coordinates with x=rcosϕ and y=rsinϕ giving
f(r,ϕ)=kw2π11e14k2w2q2ιk(qy0+1q2x0rcos[ϕψ(q)])dq,
(30)
where ψ(q)=sin1q.Using the generating function (24) for z=kr and φ=ϕψ(q) we obtain from (30) the final series
f(r,ϕ)=m=+amJm(kr)eιmϕ,
(31)
where am=kw2πιm11exp[14k2w2q2ιk(qy0+1q2x0)ιmsin1q]dq.

5. Layered Systems

Practical realizations of concentric cylinder optical and optoelectronic devices often involve a design that includes a number of (homogeneous) layers. In particular, external layers may be incorporated as a protective cover, while an internal layer between the absorber and the shell of the optical concentrator provides additional mechanical support or works as a channel for a liquid absorber. In this Section, we develop the theoretical description of light scattering and absorption by such layered systems.

5.1 Cascading the cylindrical layers

The boundary value problem of the scattered cylindrical modes inside a set of concentric cylindrical layers leads to a system of linear equations. Utilizing a direct linear algebra method for obtaining the scattered cylindrical modes inside layers with losses or axially symmetric distribution of the dielectric function could be problematic due to typically large condition numbers of the system matrices. The algorithm that we use includes a normalization step, making it possible to improve poor conditioning. However, if the algorithm were used for getting analytical formulae the normalization would not be necessary.

We consider a concentric cylindrical device having l1 layers, ri<r<ri+1, i=1,l1¯, with the “outer” (r>rs) layer filled with a host media, as shown in Fig. 2
Fig. 2 Schematic geometry of the layered structure.
.

The general solution of the wave equation in i th layer can be represented as a sum of angular harmonics, fi(r,ϕ)=m=+fmi(r)expιmϕ where the radial part fmi(r)consists of two partial solutions Rmi(r) and Tmi(r)with yet undefined coefficients. These coefficients can be obtained from appropriate 2l boundary conditions. All cylindrical modes are linearly independent, thus each of the boundary conditions can be separated into the individual equations for each m th mode, similar to that shown in (14) presented in a more general form,
f(m)i(r)=τiTi(r)/Ti(ri)+ρiRi(r)/Ri(ri),i=1,l¯,r1=r2,
(32)
where for the simplicity of notation (and from now on in this section) we suppress the index m. To satisfy the condition (22) in the core layer we put ρ1=0 and T(m)1(r)=Jm(k1r).

Furthermore, the total field in the “outer” layer can be separated into the “incident” and “scattered” parts. The Sommerfeld radiation condition R(m)l(r)=Hm+(k0r) is imposed on the scattered field (cf. Eq. (23)), while for the incident wave τ(m)l=amJm(r¯s), with the values of the coefficients am taken from the expansion of the incident wave (25), (31).

To obtain the rest of 2l2 equations we use the standard boundary condition (20), (21) for TE polarization at each interface r=ri+1,
[11LTi,1LRi,1][τi+1ρi+1]=[11LTi,0LRi,0]diag(FTi,FRi)[τiρi],
(33)
where FXi=Xi(ri+1)Xi(ri), LXi,q=Xi+qXi+q(ri+1), X=R,T, q=0,1.

For the TM polarization the only change is that LXi,0(εi+1/εi)(ri+1)LXi,0. Inverting the matrix at the left-hand-side of (33) and evaluating its determinant as di+1=LRi,1LTi,1=W[Ti+1,Ri+1]Ti+1Ri+1(ri+1) we obtain
[τi+1ρi+1]=wmi[τiρi],i=1,l1¯,
(34)
where
wmi=1di+1[FTi(LRi,1LTi,0)FRi(LRi,1LRi,0)FTi(LTi,1LTi,0)FRi(LTi,1LRi,0)],
(35)
and W[T,R]=TRTR is the Wronskian [12

12. S. Gradshteyn, and I. M. Ryzhik, Tables of integrals, series and products, Academic Press, New York, CD-ROM Edition, 1994, Eq. (8).511.4.

], which is taken for the partial solutions of the i+1 layer. In particular, for homogeneous layers we exploit the known value of the Wronskians for the Bessel functions [15

15. M. Abramowitz, and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York, Dover, (1972).

],
W[Jm,Ym](x)=2/πx;W[Jm,Hm±](x)=±ι2/πx;W[Hm+,Hm](x)=ι4/πx.
(36)
For a layer with the inverse-square radial dependence of the dielectric we have T=rα+ιβm, R=rαιβm which yields TRW1[T,R]=ιr/(2βm). This approach is also known as the ‘cascading layers’ method [16

16. V. Kildishev, U. K. Chettiar, Z. Jacob, V. M. Shalaev, and E. E. Narimanov, “Materializing a binary hyperlens design,” Appl. Phys. Lett. 94(7), 071102 (2009). [CrossRef]

], since the solution in each layer (34) is then a composition of transfer matrices applied to the first layer [τi+1,ρi+1]T=(k=i1wk)[τ1,ρ1]T) [13

13. M. Born, and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, New York, 1999).

].

The resulting system obtained after stating all 2l boundary conditions has the form:

{ρ1=0τi+1=w11iτi+w12iρiρi+1=w21iτi+w22iρiτlknown,e .g .,a(m)incJm(r¯l),(i1,l1¯)
(37)

The system (37) can be solved in two sweeps, similar to the standard Thomas algorithm [17

17. S. D. Conte, and C. W. Boor, “Elementary Numerical Analysis: An Algorithmic Approach, 3rd edition,” 1980.

]. In this approach, the auxiliary normalized coefficients τ¯i=τi+1/τi, and ρ¯i=ρi/τi, (i1,l1¯) are calculated by forward sweep using the recurrence (38)
{ρ¯1=0τ¯i=w11i+w12iρ¯iρ¯i+1=(w21i+w22iρ¯i)/τ¯i,(i1,l1¯)
(38)
followed by the backward sweep of the reflection and transmission coefficients (39), while using the initial condition at the external interface:
{τi=τi+1/τ¯iρi=ρ¯iτi,(i1,l1¯)
(39)
where τl is known and ρl=ρ¯lτl.

The above algorithm includes the normalization of the partial solution in each layer by its value on the inner interface (the field in the core layer is an exception and is normalized by its value on the outer interface r=r2, the latter is achieved by substitution r1=r2 in Eq. (32)). The normalization significantly improves the accuracy of the mathematical operations, performed numerically with a limitation of floating-point representations and arithmetic. Finally, the procedure yields the m th mode of the field by using (32); the solution to the entire problem in the i th layer is given as a sum of the angular momentum modes, fi(r,ϕ)=m=fmi(r)eιmϕ.

5.2 Scattering and absorption efficiencies

For any optical system, scattering and absorption efficiencies have the physical meaning of the relative size of the equivalent “black body” target which results in the same amount of scattered and absorbed power [18

18. H. Hulst, Light scattering by small particles, Dover, 1981, p. 309.

].

First, we recapitulate that the total field E=z^f (TE) or H=z^f (TM) outside the device at the external boundary of the shell, r=rs is defined as (eιωt is omitted)
f=m=+fm(r¯s)expιmϕ.
(40)
Then, we assume that the m th mode of the total field is given by,
fm=Hm+cm+Hm+,
(41)
or
fm=2Jm+(cm+1)Hm+,
(42)
so that the total field (40) is decomposed into a plane wave illumination source
finc=2mJm=2eιky,
(43)
and the scattered field
fsca=m=(cm+1)Hm+expιmφ,
(44)
where coefficients cm+ are determined from the boundary value problem. In (41)-(44) and further on we use abbreviations Jm, Hm± for Jm(r¯s), Hm±(r¯s).

Equations (40)-(43) are used for the time-averaged Poynting vector Sabs=12Re(E×H*), and its normalized radial component Srabs=r^S/W, where W is the total power transmitted by the incident wave through 2rs–wide strip per its unit length:
Srabs=18r¯sr^Re[z^f×(ιf×z^)*]=18rsIm(f*fr¯s),
(45)
and since the absorption efficiency per unit length is taken as a normalized flux of power flowing into the device, Qabs=02πSrabsrsdϕ, fm=dfm/dr¯swe arrive at

Qabs=π4m=Im(fm*fm).
(46)

After applying (41) to (46), we note that Im(fm*fm) includes a vanishing sum of a complex number with its conjugate, Im[(cm+Hm+Hm+)*+cm+Hm+Hm+]=0, and another term, Im[(HmHm+)*+|cm+|2HmHm+]=(1|cm+|2)Im(HmHm+), so that
Qabs=12r¯sm=(1|cm+|2);
(47)
which is obtained by substituting Im(HmHm+)=W(Jm,Ym) with 2/(πr¯s) (cf. Eq. (36)).

Similarly the scattering efficiency per unit length is calculated by taking the flux of normalized power of the scattered field (44) moving away from the device, Qsca=02πSrscarsdϕ, where likewise (45), Srsca=18rsIm(fsca*fscar¯s); and we obtain
Qsca=12r¯sm=|1cm+|2.
(48)
Note that if the solution of the same scattering problem is given for an arbitrary source of illumination as fm=c˜mHm+c˜m+Hm+, then due to the linearity of the problem the coefficients relate to those from (41) as cm+=c˜m+/c˜m. In particular this gives immediate expressions for the absorption and scattering efficiencies, if the total field is given as fm=amJm+bmHm+ for which c˜m=12am,     c˜m+=bm+12am; then substitution cm+=c˜m+/c˜m=1+2am1bm and the use of am=ιm gives

Qabs=2r¯sm=(Re[ιmbm]+|bm|2),
(49)
Qsca=2r¯sm=|bm|2.
(50)

5.3 The ideal electromagnetic black hole

We consider the ideal “fundamental” (p = 2) electromagnetic black hole, see (1), (2), (3). It is a three layered system with a gradient-index shell and absorbing core, for which the formalism of Section 5.1 can be applied to calculate the exact full-wave solution. Fig. 3
Fig. 3 (Media 1) Ideal black hole with εs = 2.1, εc = 12, γc = 0.7, r s = 20 µm, and r c = r ssc)½ = 8.367 µm. The Gaussian beam with free-space wavelength λ = 1.5µm and full width w = 2λ is focused at x0 = 0, and (a) y0 = 1.5r s; (b) y0 = r s; (c) y0 = 0.75r s, and (d) y0 = 0. Range of modes used in series from (a) to (d): [-50;50], [-140;-40], [-170;-70], [-230;-130]
shows an example simulation of an ideal black hole with εs=2.1, rs=20µm, andrc=8.367µm. The device is illuminated with a Gaussian beam (free-space wavelength λ=1.5µmand full width w=2λ), which is focused at fixed x0, and different y0 (where x0=0, and y0/rs=1.5,1,0.75, and 0).

Furthermore the approach is also used to determine the final form of the absorption efficiency. For this we use Eqs. (47), (48) and derive the scattering coefficients. We write the total field outside the device asfm3(r¯)=Hm(r¯)+cm+Hm+(r¯)=τm3Hm(r¯)/Hm+ρm3Hm+(r¯)/Hm+, where τm3=Hm, ρm3=cm+Hm+. Then the exact formula for the scattering coefficients is given as

cm+=ρm3Hm+=(Hm/Hm+)[(w212w111+w222w211)/(w112w111+w122w211)]m.
(51)

The coefficients wijk are readily available from (35), then writing the explicit expressions for the matrix elements to (51) we obtain:
cm+=Hm+ηmHmHm++ηmHm+,
(52)
where
ηm=βmr¯s[αβmι1X1+X],X=1+ιY1ιY(rcrs)ι2βm,Y=αβm+Ar¯cJm(r¯c)βmJm(r¯c).
The coefficients α±ιβmare the characteristic roots of the Euler equation (cf. Eq. (19)), and factor A is obtained from the boundary conditions,

αβmATE0r¯s2m21TM1r¯s2m21εc/(εc+ιγc)
(53)

For reshaping ηm we use the identity ι(1X)/(1+X)=tan(ι2lnX). Similarly in the expression for X, we use the identity (1+ιY)/(1ιY)=exp(ι2tan1Y) so that coefficients ηm can be written for both polarizations as

ηm=βmr¯s[αβmtan(βmlnrcrs+tan1[αβm+Ar¯cJm(r¯c)βmJm(r¯c)])].
(54)

5.4 The lamellar electromagnetic black hole

The formalism of a layered system for example can be used to study a non-ideal lamellar “black hole” optical concentrator and absorber, which approximates an ideal device (εs=2.1, rs=20µm, andrc=8.367µm) with a different number of homogeneous layers, l. The device is illuminated by a plane wave with free-space wavelength λ=1.5µm; for l equal to 3, 5, 9, and 17. Scattering and absorption efficiencies are given in (49)-(50), the latter, Qabs, yields 72%, 84%, 90%, and 94% respectively. Fig. 4(e)
Fig. 4 Simulated field maps for a lamellar “black hole” optical concentrator and absorber (εs = 2.1, r s = 20 µm, and r c = 8.367 µm) vs. the total number of layers, l. The device is illuminated by a plane wave with free-space wavelength λ = 1.5µm. (a) l = 3, and 72% absorption efficiency, (b) l = 5, and 84% absorption efficiency, (c) l = 9, and 90% absorption efficiency, (d) l = 17, and 94% absorption efficiency. Panel (e) depicts the reference case of the ideal black hole with smooth gradient, and 99% absorption efficiency.
depicts the field map of the ideal black hole with smooth gradient, where 99% absorption efficiency is achieved.

6. Semiclassical description

The primary objective of the present section is the derivation of the semiclassical expansion (r¯s1) of the scattering and absorption cross sections of the “fundamental” (p = 2) electromagnetic black hole, introduced in Section 2.

Intuitively, from the ray dynamics of Section 2, we expect the absorption cross section to be close to the “geometrical” limit, so that Qabs1. The ray-optical treatment of Section 5 however gives no quantitative measure of how closely this limit is actually approached in a practical device. On the other hand, the exact description of Section 5, while giving a precise quantitative account, provides no compact lucid description of the system behavior. In contrast to this behavior, the semiclassical approach, mathematically corresponding to taking the formal limit r¯s1 of the exact solution, combines the clear physical picture of the actual dynamics in terms of the underlying ray trajectories, with the quantitative description of the interference and diffraction in the system.

The semiclassical approximation for the reflection coefficients and the absorption cross-section can be obtained by taking the limit r¯s1 in (52) and (54). Mathematically, this corresponds to the famous Debye expansion [12

12. S. Gradshteyn, and I. M. Ryzhik, Tables of integrals, series and products, Academic Press, New York, CD-ROM Edition, 1994, Eq. (8).511.4.

] of the Bessel functions
Hm±(x)=2πx2m2[1+O(1x2m2)]exp[±ισm(x)],
(55)
where the “phase” σm(x)=x2m2mcos1(m/x)π/4.

Physically, the angular momenta summation in absorption efficiency (47) can be separated into two parts – the sum over |m|<r¯s and the sum over |m|>r¯s. As the dimensionless angular momentum m has physical meaning of the product of the (fixed) wavenumber and the impact parameter – see Fig. 2 – the sum over |m|<r¯s corresponds to the rays that actually hit the target and as a result experience strong scattering. On the other hand, the sum for |m|>r¯s takes into account the rays that miss the object. In the geometrical-optics limit, those rays experience no scattering and thus do not contribute to the cross-section. Beyond ray optics, the waves with |m|>r¯s, do experience some amount of scattering normally associated with diffraction, however in the limit r¯s1 the resulting contribution is exponentially small. Therefore, in the semiclassical limit r¯s1the sum over large angular momenta (|m|>r¯s) can be neglected, and we obtain

Qabs12r¯s|m|r¯s(1|cm+|2).
(56)

Furthermore, for r¯s1 the relative change in |cm+|, when one goes from m to m+1, is reasonably small – leading to the relative error 1/r¯s– and thus the summation in Eq. (56) can be approximated by the integral

Qabs12r¯sr¯sr¯sdm(1|cm+|2).
(57)

Note that in this approximation the values of the parameter βm that are different for the TE and TM polarization (cf. Eq. (53)), should be considered equal, i.e. r¯s2m21r¯s2m2, then, applying Debye approximation to (52) and using identity (1+ιY)/(1ιY)=exp(ι2tan1Y) we obtain,
cm+=exp{ι2[tan1(r¯sβmηm12r¯s2βm3)σm(r¯s)]}.
(58)
Furthermore, for r¯s1 the term 12r¯s2/βm3 can be neglected, in addition, from (54) we have
r¯sβmηm=αβmtan(βmlnrcrs+tan1[αβm+Ar¯cJm(r¯c)βmJm(r¯c)]),
(59)
and since the fraction α/βm is very small for the TM polarization and is exactly equal to zero for TE polarization, using the Debye expansion for Bessel function in the limit r¯s1 and low losses in (59) we obtain,
tan1(r¯sβmηm)=βmlnrcrstan1[Ar¯cJm(r¯c)βmJm(r¯c)].
(60)
Using a similar approximation for Jm(r¯c)12πr¯c2m2cos[σm(r¯c)]in (60) we write,
tan1(r¯sβmηm)=βmlnrcrstan1[A(12r¯c2βm(r¯c2m2)+r¯c2m2βmtan[σm(r¯c)])],
(61)
where phase factor σm(r¯c) is already defined in (55), and r¯c=kcrc.

Then, dropping the small term 12r¯c2/[βm(r¯c2m2)] and taking Ar¯c2m2/βm1 (due to small losses) in (61), we are back to (58) and obtain,
cm+=exp{ι2[βmlnrcrsσm(r¯c)+σm(r¯s)]},
(62)
with the magnitude of (62) being
|cm+|=exp{γcεcr¯s2m2},
(63)
where we used, σm(r¯c)σm(r¯s)ι2γcεcr¯s2m2.

For the absorption efficiency (57) after substitution (63) we then obtain,
Qabs12r¯sr¯sr¯s(1exp(2γcεcr¯s2m2))dm,
(64)
and after variable substitution (64) can be written as a standard integral
Qabs=1F(2r¯sγcεc)=π2[I1(2r¯sγcεc)L1(2r¯sγcεc)],
(65)
where I1 and L1 are the modified Bessel and Struve functions of the first order, and

F(x)=0π/2cosθexp(xcosθ)dθ=1π2[I1(x)L1(x)]={112x+O(x2),x11x2+O(1x4),x1
(66)

Equations (65) and (66) finalize the derivation of the semiclassical expansion (r¯s1) of the scattering and absorption cross sections of the “fundamental” (p=2) electromagnetic black hole, which is the specific aim of this Section. As expected, for both polarizations in the absence of losses we find zero absorption cross-section. However, even for moderate losses γc/εc1/r¯s the absorption efficiency is close to one. Thus, as predicted by the semiclassical theory, the device does indeed absorb nearly all the incident light from every direction.

To further illustrate our results we first qualitatively portray the decay of performance as we move away from the semiclassical limit. Fig. 5
Fig. 5 Field patterns of the black hole device (ε s = 2.1, r s = 20 µm, and r c = 8.367 µm) illuminated by a TE-polarized plane wave with a free-space wavelength of (a) 1.5 µm, (b) 3.0 µm, (c) 4.5 µm, and (d) 6.0 µm.
shows field patterns of the black hole device (εs=2.1 ,  rs=20µm, r s = 20 µm, and rc=8.367µm) illuminated by a TE-polarized plane wave for a free-space wavelength changing from 1.5 to 6.0 µm, where increasing scattering is observed with increase of wavelength. Then, we perform a quantitative comparison of the absorption efficiency, Qabs, versus the ratio λ/rs.

The efficiencies, shown in Fig. 6a
Fig. 6 (a) Absorption efficiency Q a vs. the ratio λ/r s obtained for the TE- and TM-polarized plane wave; (b) a more detailed plot for the ratios from 0.075 to 0.15.
, are separately calculated for the TE- and TM-polarized plane wave using the exact method of Section 5, and the semiclassical result of Section 6, which is valid for both polarizations. Figures 6a also indicates good quality of the semiclassical approximation of the absorption cross-section even far beyond the semiclassical limit. Figure 6b depicts a more detailed plot for λ/rschanging from 0.075 to 0.15.

7. Summary

In summary, we have developed a theoretical description of wave propagation in cylindrically-symmetric gradient-index systems, and applied this approach to light “trapping” in the recently proposed electromagnetic black hole concentrator and absorber.

In particular we have demonstrated that an approximate lamellar black-hole with a relatively limited number of homogeneous layers, while giving the desired ray-optical performance, can provide absorption efficiencies comparable with those of ideal devices with smooth gradients in index. It is forecasted that the use of non-uniform layers could further improve the performance of the lamellar device with a given number of homogeneous layers. The numerical validation of the mathematical apparatus for designing the black-hole devices opens the path for the development of test beds that could further advance the application of this light-trapping technique into a viable commercial option.

Acknowledgments

This work was partially supported by ARO MURI56154-PH-MUR, 50342-PH-MUR, and also through IIP SB RAS No 113, and RFFS No 09-01-00352-а.

References and links

1.

V. M. Shalaev, W. S. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30(24), 3356–3358 (2005). [CrossRef]

2.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

3.

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006). [CrossRef] [PubMed]

4.

A. Salandrino and N. Engheta, “Far-Field Subdiffraction Optical Microscopy Using Metamaterial Crystals: Theory and Simulations,” Phys. Rev. B 74(7), 075103 (2006). [CrossRef]

5.

N. M. Litchinitser and V. M. Shalaev, “Metamaterials: transforming theory into reality,” J. Opt. Soc. Am. B 26(12), B161–B169 (2009). [CrossRef]

6.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008). [CrossRef] [PubMed]

7.

T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics 2(5), 299–301 (2008). [CrossRef]

8.

E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009). [CrossRef]

9.

L. D. Landau, and E. M. Lifshitz, Mechanics, Pergamon Press, Oxford, (1976).

10.

R. K. Luneburg, Mathematical Theory of Optics, University of California Press, Berkeley, 1964, p. 12.

11.

E. J. Post, Formal Structure of Electromagnetics: General Covariance and Electromagnetics, 1962, p. 152.

12.

S. Gradshteyn, and I. M. Ryzhik, Tables of integrals, series and products, Academic Press, New York, CD-ROM Edition, 1994, Eq. (8).511.4.

13.

M. Born, and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, New York, 1999).

14.

G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69(4), 575 (1979). [CrossRef]

15.

M. Abramowitz, and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York, Dover, (1972).

16.

V. Kildishev, U. K. Chettiar, Z. Jacob, V. M. Shalaev, and E. E. Narimanov, “Materializing a binary hyperlens design,” Appl. Phys. Lett. 94(7), 071102 (2009). [CrossRef]

17.

S. D. Conte, and C. W. Boor, “Elementary Numerical Analysis: An Algorithmic Approach, 3rd edition,” 1980.

18.

H. Hulst, Light scattering by small particles, Dover, 1981, p. 309.

OCIS Codes
(260.0260) Physical optics : Physical optics
(070.7345) Fourier optics and signal processing : Wave propagation

ToC Category:
Physical Optics

History
Original Manuscript: March 30, 2010
Revised Manuscript: June 19, 2010
Manuscript Accepted: June 29, 2010
Published: July 23, 2010

Citation
Alexander V. Kildishev, Ludmila J. Prokopeva, and Evgenii E. Narimanov, "Cylinder light concentrator and absorber: theoretical description," Opt. Express 18, 16646-16662 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-16646


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References

  1. V. M. Shalaev, W. S. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30(24), 3356–3358 (2005). [CrossRef]
  2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]
  3. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006). [CrossRef] [PubMed]
  4. A. Salandrino and N. Engheta, “Far-Field Subdiffraction Optical Microscopy Using Metamaterial Crystals: Theory and Simulations,” Phys. Rev. B 74(7), 075103 (2006). [CrossRef]
  5. N. M. Litchinitser and V. M. Shalaev, “Metamaterials: transforming theory into reality,” J. Opt. Soc. Am. B 26(12), B161–B169 (2009). [CrossRef]
  6. N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008). [CrossRef] [PubMed]
  7. T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics 2(5), 299–301 (2008). [CrossRef]
  8. E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009). [CrossRef]
  9. L. D. Landau, and E. M. Lifshitz, Mechanics, Pergamon Press, Oxford, (1976).
  10. R. K. Luneburg, Mathematical Theory of Optics, University of California Press, Berkeley, 1964, p. 12.
  11. E. J. Post, Formal Structure of Electromagnetics: General Covariance and Electromagnetics, 1962, p. 152.
  12. S. Gradshteyn, and I. M. Ryzhik, Tables of integrals, series and products, Academic Press, New York, CD-ROM Edition, 1994, Eq. (8).511.4.
  13. M. Born, and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, New York, 1999).
  14. G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69(4), 575 (1979). [CrossRef]
  15. M. Abramowitz, and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York, Dover, (1972).
  16. V. Kildishev, U. K. Chettiar, Z. Jacob, V. M. Shalaev, and E. E. Narimanov, “Materializing a binary hyperlens design,” Appl. Phys. Lett. 94(7), 071102 (2009). [CrossRef]
  17. S. D. Conte, and C. W. Boor, “Elementary Numerical Analysis: An Algorithmic Approach, 3rd edition,” 1980.
  18. H. Hulst, Light scattering by small particles, Dover, 1981, p. 309.

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