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FDTD analysis of the optical black hole

Christos Argyropoulos,1,* Efthymios Kallos,1 and Yang Hao1,2

1

School of Engineering and Computer Science, Queen Mary University of London, Mile End Road,London E1 4NS, United Kingdom

2E-mail: [email protected]*Corresponding author: [email protected]

Received May 27, 2010; accepted August 2, 2010;posted August 12, 2010 (Doc. ID 129082); published September 16, 2010

An optical black hole is studied using a parallel radially dependent finite-difference time-domain (FDTD) simu-lation technique. The device requires non-dispersive metamaterial structures and is capable of broadband op-eration, based on transformation optics. Excellent absorption is demonstrated for different angles of wave in-cidence and illumination excitation types. In addition, a practical device, which is made to be matched to freespace, is proposed, and the relevant physics is explored. Finally, peculiar phase distributions of the electro-magnetic waves are observed inside the radially dependent permittivity material of the devices. 2010 Op-tical Society of America

OCIS codes: 230.3205, 160.3918, 050.1755.

1. INTRODUCTIONTransformation optics provides methodologies for engi-neers to manipulate electromagnetic radiation at will[1,2]. Several microwave and optical device designs havebeen proposed, including broadband ground-plane cloak-ing structures [38] and exotic absorbing devices [9,10].More recently, an interesting application that has beenproposed is the design of artificial optical black holes[11,12]. Mimicking their celestial counterparts, these de-

vices bend the electromagnetic waves appropriately andcan almost completely absorb the incident electromag-netic radiation crossing into their region. In addition,

they can operate for all angles of incidence over a broadfrequency spectrum and can be constructed with non-resonant metamaterials. Potential applications of theseexotic devices are perfect absorbers [13], efficient solar en-ergy harvesting photovoltaic systems [14], thermal lightemitting sources, and optoelectronic devices [15].

It should be noted that despite their light-absorbing ca-pabilities, the name black hole is a misnomer in thiscase, because these devices do not possess the main char-acteristic of gravitational black holes, namely, the eventhorizon, the artificial boundary around the device beyondwhich no light can escape. As we also show here, radiationgenerated inside the device does indeed escape into thesurrounding environment. However, the original term

black hole for the device as introduced in [11] will bemaintained here for consistency, despite the fact that aterm like optical attractor might be more appropriate[12].

We investigate the performance of the spherical opticalblack hole embedded in a background medium using afull-wave simulation technique, the well-establishedfinite-difference time-domain (FDTD) method[16]. Due tothe large dimensions of the three-dimensional (3-D) de-

vice, a parallel version of the FDTD technique is utilized,which divides the simulation domain into sub-domainsthat are processed in parallel and significantly decrease

the total simulation time. Different excitation types arechosen to illuminate the device, namely, plane waves andspatially Gaussian pulses. Moreover, the performance ofan alternative black hole, which is not embedded in a par-ticular material [11] but is matched directly to free space,is examined. The latter is a more practical approach tothe optical black hole, in a similar fashion that theground-plane quasi-cloaks operate in free space, as pro-posed in[17] and verified experimentally in [18]. The per-formance of the latter device is found to be similar to theperformance of the embedded structure. Finally, thelosses at the core of the black hole are removed, and a

point source is placed inside. We observe that the cylin-drical wavefronts of the source are disturbed from the ra-dially dependent permittivity of the metamaterial struc-ture, leading to peculiar field phase distributions.

2. PARALLEL RADIALLY DEPENDENT FDTDTECHNIQUE

The FDTD method is used to explore the physics of theabsorbing device. The full-wave numerical technique canefficiently demonstrate the performance of the broadbanddevice. The proposed technique can also model the diffrac-tion effects and near fields, which are neglected with raytracing methods used previously[11]. As a result, a moreclear and complete picture of the behavior of the modeled

device is achieved.The FDTD method is based on the temporal and spatialdiscretization of Faradays and Amperes laws, which are

E= B

t, 1

H =D

t, 2

whereE,H,D, andBare the electric field, magnetic field,electric flux density, and magnetic flux density compo-

2020 J. Opt. Soc. Am. B / Vol. 27, No. 10 / October 2010 Argyropoulos et al.

0740-3224/10/102020-6/$15.00 2010 Optical Society of America

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nents, respectively. Note that harmonic time dependenceexpt of the field components is assumed throughoutthis paper. Faradays and Amperes laws are discretizedwith a common procedure[19], and the conventional up-dating FDTD equations are obtained for the field compo-nents:

Hn+1 =Hn

t

0x,y,zcurlEn+1/2 , 3

En+1 =En + t0x,y,z

curlHn+1/2, 4where 0 and 0 are the permittivity and permeability ofthe space surrounding the device modeled, and n is thenumber of the current time steps. The temporal discreti-zation t has to satisfy the Courant stability criterion

t =x/3c[19]to achieve stable FDTD simulations. xis the uniform spatial resolution of the FDTD cubical Car-tesian mesh. The relative permittivity x ,y ,z and per-meability x ,y ,z can be spatially dependent according

to the geometrical features of the material or metamate-rial structure that is simulated. The derived FDTD equa-tions are similar, and simpler, compared to the previouslyproposed radially dependent numerical technique[20,21]used to model another interesting transformation-baseddevice: the cloak of invisibility. The difference is that un-like most invisibility cloaks, the black hole is composed ofnon-dispersive materials, as it will be shown in the nextsection. Thus, there is no reason to discretize the consti-tutive equations leading to a more complicated FDTD al-gorithm.

In the case of the computational intensive modeling ofthe 3-D spherical black hole, a parallel version of theFDTD algorithm is used. The FDTD updating equationsare the same as before. The only difference is that thecomputational domain is divided into smaller sub-domains based on the space decomposition technique[22],and every domain is then assigned to one processor. Thetangential field components are communicated betweenthe adjacent interfaces at each time step with an appro-priate synchronization procedure, which is provided bythe message passing interface library. After the calcula-tions are completed and the solver has finished, the re-sulting fields from each sub-domain are retrieved and arecombined together to obtain the results in the whole ini-tial domain. The parallel version of the algorithm is idealfor handling complicated and computational intensiveproblems, comprised of huge computational domains andcomplex electromagnetic parameters.

3. PARAMETERS OF SPHERICAL/CYLINDRICAL OPTICAL BLACK HOLE

The spherical 3-D and the cylindrical two-dimensional(2-D) optical black holes will be investigated. Each deviceis divided into two regions: the core (usually absorbing)and the shell. The radially dependent permittivity distri-butions of the spherical and cylindrical black holes aregiven by the formula[11]

r = 0, r Rsh

0Rshr

2

, Rc r Rsh

c + , r Rc, 5

where 0 is the permittivity of the surrounding medium,c is the permittivity of the core, and R sh and R c are the

radii of the shell and core, respectively, of the black hole.The magnetic parameters of the structure are those offree space. The non-magnetic behavior of the structure ishighly desirable, especially at optical frequencies, due tothe lack of physical magnetism at this part of the fre-quency spectrum.

The objective is to achieve a matched device with thesurrounding space, reducing the reflection of the imping-ing electromagnetic waves to a minimum. To obtain thiseffect at the surface of the device, the radius of the de-

vices core has to vary according to the parameters of thesurrounding medium, for a given core material with per-mittivity c. It is given by

r =Rc = cRc =Rsh0

c

. 6

Note that the permittivity in Eq.(5) has a finite range ofvalues 0rc. Hence, the device can be constructedwith non-resonant non-dispersive metamaterial struc-tures consisting of concentric layers with tunable permit-tivity values. We note that in contrast to othertransformation-based devices where their extreme disper-sive parameters allow the control of the waves (such asbending) over subwavelength distances, the designs pre-sented here involve conventional material parametersand require devices that extend several wavelengths inspace. This effect introduces challenges in the simulationof the devices since the number of simulation cells in-creases significantly with the size of the structures. Onthe other hand, the same effect is expected to make ex-perimental realizations easier as larger metamaterialunit cells can be used.

4. NUMERICAL RESULTS OF THESPHERICAL/CYLINDRICAL OPTICAL BLACKHOLE

A. Spherical Optical Black Hole Embedded in DielectricMaterialIn this subsection we investigate a spherical black hole

operating as an electromagnetic concentrator for improv-ing the light capturing capabilities of solar cells. Thespherical black hole is embedded in silica glass SiO2with a relative permittivity of 0 =2.1. The core of the de-

vice is composed of n-doped silicon with relative permit-tivity c + =12+ 0.7, which is a typical material of a thinfilm solar cell. The device is designed to operate at the in-frared section of the spectrum with a central frequency of

f=200 THz. The range of permittivity values, required toconstruct the structure, is between 2.1 and 12, as indi-cated by Eq. (5). The device can have broadband opera-tion, as any frequency dispersion is introduced only by

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whatever material is chosen to build it. Finally, all the sixfield components Ex, Ey, Ez, Hx, Hy, Hz exist at the 3-DFDTD simulations.

The parameters of the device have isotropic values,where =ris given in Eq.(5),and the magnetic param-eters have the permeability of free space. A uniform spa-tial resolution is chosen for the Cartesian FDTD meshgiven by x =y =z =/ 20, where =1.5 m is the wave-

length of the excitation signal in free space. The Courantstability condition is satisfied, and the computational do-main is terminated with Berengers perfectly matchedlayers (PML) [23], modified to be embedded in the silicaglass surrounding material. The excitation field is chosento be a temporally infinite spatially Gaussian pulse with asimilar wave trajectory to the ray tracing results pre-sented in [11]. The FDTD computational domain, re-quired for accurate modeling of the spherical black hole,is equal to 303030. It is divided along thez-axis to60 sub-domains, where individual processors are solvingthe parallel FDTD updating equations, as explained ear-lier. Large amount of memory is required for every simu-lation, roughly 50 Gbytes of random access memory,

which is also distributed between the computing nodes.Each simulation lasts approximately 18 h (30,000 timesteps), until the steady state is reached and the field val-ues no longer evolve significantly with time.

A spatially confined y-polarized Ey Gaussian pulse ischosen to impinge on the spherical black hole from twodifferent angles of incidence to evaluate its omnidirec-tional absorbing performance. Both pulses are / 2 wide inspace [full width at half-maximum (FWHM)], and the fre-quency of operation is chosen as f=200 THz. The excita-tion pulses are infinite in time. After steady state isreached, the 3-D electric field amplitude distributionEyisretrieved. In order to visualize the results, we plot the 2-Damplitude distribution on three different planes, each one

parallel to each of the Cartesian planes of the domain.The results for two different simulation scenarios areshown in Figs. 1 and 2. The locations of the pulse en-trance for each figure are 15 , 15 ,1 and

15 ,7.5 , 4, respectively. It can be clearly seen that theincoming field power is totally absorbed inside the core ofthe device for both cases. The field trajectories rapidlybend toward the core of the device (see Fig. 2), as ex-pected. The device is matched to the surrounding mate-rial and, as a result, the reflections of the device are al-most zero. Furthermore, the absorption of the radiation isexcellent, reaching approximately 95%. Finally, note thatthe 3-D figures were constructed after downsampling thecomputational domain to half its original size, due tomemory constraints in the post-processing.

B. Cylindrical Optical Black Hole Embedded inDielectric Material

In the next subsections of the paper, the study will be fo-cused on the 2-D black hole design due to its simplicity inFDTD modeling compared to the parallel FDTD simula-tions. As it shall be shown, the results of the cylindricaldevice simulations are very similar to the case of the 3-Dspherical black hole studied in the previous subsection.The cylindrical black hole is embedded in SiO2, and thecore of the device is composed ofn-doped silicon. The fre-quency of interest is f=200 THz, and the device can havebroadband operation. The cylindrical coating inner andouter radii areRc =5.6 andRsh =13.33, respectively, thesame with those of the spherical embedded black hole.Transverse magnetic polarization is used throughout the2-D simulations of the optical black hole, without loss of

generality. Only three field components exist Ex, Ey, Hz,and the parameters are isotropic and equal to x =y=rand =0, given in Eq.(5).

The spatial resolution is chosen uniform and equal tox =y =/ 30, where is the wavelength of the excitationsignal at free space. The temporal resolution of the 2-D

simulation is chosen as t =x/2c, according to Courantstability condition [19], where c is the speed of light infree space. The computational domain is terminatedagain with PMLs [23]. During all the 2-D FDTD simula-tions presented here, the steady state is reached after ap-proximately 8000 time steps or 1.5 h. A temporally infi-

Fig. 1. (Color online) Amplitude ofEy field component excitingthe spherical optical black hole embedded in dielectric medium.The spatial Gaussian pulse illuminates the device normal to thex-y surface placed at a central position 15 ,15 , 1. The per-mittivity of the background is 0 =2.1. The intensity of the Eyfield component can be seen from the color bar. The sphericalcoating inner and outer radii are Rc =5.6 and Rsh =13.33, re-spectively. The three projection planes cross the axis at x =15,y=15, and slightly off-center z =13.4.

Fig. 2. (Color online) Amplitude ofEy field component excitingthe spherical optical black hole embedded in dielectric medium.The spatial Gaussian pulse illuminates the device normal to thex-y surface placed at a side position 15 ,7.5 ,4. The permit-tivity of the background is 0 =2.1. The intensity of the Ey fieldcomponent can be seen from the color bar. The spherical coatinginner and outer radii are R c =5.6and R sh =13.33, respectively.The three projection planes cross the axis at x =15, y = 15, andslightly off-center z =13.4to compare with the previous figure.


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nite z-polarized Hz spatially Gaussian pulse is used,2/3 wide in space (FWHM), to illuminate the black hole,in order to mitigate the ray tracing simulations [11]. Fi-nally, the FDTD domain is noticeably large 2828, be-cause the device operates correctly only when its dimen-sions are much larger than the wavelength. The real part

and the amplitude of the magnetic field distribution whena Gaussian beam is impinging on the proposed black hole,after steady state is reached, are reported in Figs. 3(a)and 3(b) (Media 1) respectively. The performance of thedevice is excellent, achieving almost full absorption of thebeam, similar to the performance of the spherical blackhole shown in Fig.2.

Next, az-polarizedHzplane wave illuminates the de-vice. The plane wave is constructed after replacing the topand bottom PMLs with periodic boundary conditions[19].The results are shown in Figs. 4(a) and 4(b), where thereal part and the amplitude of the magnetic field distri-bution are shown. A large shadow is cast at the back ofthe structure, and the reflections are negligible. The larg-

est part of the plane waves energy is absorbed at the core,as can be clearly seen in Fig.4(b). The behavior of the de-

vice is that of an efficient electromagnetic absorber, simi-lar to the one proposed in[10].

C. Cylindrical Optical Black Hole Embedded in FreeSpaceIf we replace the background material of silica glass (usedbefore) with free space, the range of the black holes rela-

tive permittivity values will be wider and between 1 and12. The wider material values are leading to a more com-plicated design of the metamaterial structure and asmaller radius of its core, given by Eq.(6). Hence, the cy-lindrical coating inner and outer radii become Rc =3.85and Rsh =13.33, respectively. However, a device that is

matched directly to free space is always more desirablefor practical applications.

The free space cylindrical black hole is modeled withthe same FDTD method as before and at the same simu-lation scenario shown in Fig.3. The fields are obtained inFigs.5(a) and5(b), and similar results are derived withthe spherical and cylindrical embedded black holes insilica glass. There is slightly increased scattering, be-cause the electromagnetic beam is traveling opticallylonger distance inside the radially dependent permittivitymaterial. Nevertheless, the absorption of the structure isagain excellent. Finally, when the beam is incident from adifferent angle in Figs. 6(a)and 6(b), the performance issimilar to the earlier results (particularly see Fig. 1).

Thus, the optical black hole can be regarded as an omni-directional absorber.

D. Phase Distribution of Source Placed Inside the BlackHoleThe core of the free space optical black hole was initiallycomposed of a lossy material with permittivity c + =12+ 0.7. The imaginary part is now removed from the

Fig. 3. (Color online) (a) Real part and (b)Media 1amplitude ofHz field component at the embedded in silica glass black hole.

Fig. 4. (Color online) (a) Real part and (b) amplitude ofHz field component when a plane wave is impinging at the embedded in silicaglass black hole.

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device, and a softz-polarizedHzpoint source, again at afrequency of f=200 THz, is placed inside the core asym-metrically with a slight deviation from the center of the

y-axis at the point x ,y= 15 ,12.33. This is conductedin order to further explore the wave interactions insidethe radially dependent metamaterial shell. The real partand the phase of the Hz field component are shown inFigs.7(a)and7(b).A peculiar phase distribution of the cy-

lindrical waves is observed, especially in Fig. 7(b).The radially dependent permittivitygiven by Eq.

(5)is substituted in the formulas of phase and group ve-locity to explain the unusual phase distribution of themetamaterial structure. The derived formulas are

vphr =c

r=

c

0

1

Rsh/r2= vph0

r

Rsh,

Fig. 5. (Color online) (a) Real part and (b) amplitude ofHz field component when a temporally continuous spatially Gaussian pulse isimpinging at the matched to free space black hole.

Fig. 6. (Color online) (a) Real part and (b) amplitude ofHz field component at the free space black hole. The pulse is incident with adifferent angle.

Fig. 7. (Color online) (a) Real part and (b) phase distribution ofHz field component at the free space black hole. The wavefronts aretraveling with different speeds at both sides of the device.


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vgr = cr = c0Rshr

2

= vg0Rsh

r, 7

wherevph0and vg0are the phase and group velocities, re-spectively, of light traveling in free space. The radius ofthe black hole Rshis always larger than the radius of thecore:RshRc. As a result, the waves are traveling with ahigher group velocity closer to the core of the black hole,

which is derived straightforwardly from Eq.(7).However,the phase velocity has an opposite behavior inside this ra-dially dependent medium. This is the reason of the moredense population of the wavefronts closer to the core ofthe device as can be clearly seen in Fig.7(b),especially atthe left side.

Note that if the source was placed at the center of thestructure, only perfectly outgoing cylindrical wavefrontswould exist. By placing the source away from the center,the device converts some of the cylindrical wavefrontsinto flat ones on the opposite side, exhibiting a lens-likebehavior. This effect is caused by the longer optical path,where the wavefronts have to travel passing through thewhole length of the dielectric core of the structure. Note

that the radiation generated inside the devices core al-ways escapes to the surrounding environment, even if theimaginary part of the device is not removed (not shownhere).

5. CONCLUSION

To conclude, the spherical (3-D) and the cylindrical (2-D)optical black holes were studied numerically with theFDTD method. Full-wave simulations were performed tostudy the performance of the device under different exci-tations and angles of incidence. The physics of this exoticmetamaterial structure was explored. It behaves as anexcellent omnidirectional absorber achieving absorption

values of approximately 95%. An alternative black hole,matched to the surrounding free space, was proposed, andits excellent performance was demonstrated. Althoughthe devices were tested with infinite in time excitationpulses, similar results are expected if the devices are ra-diated with non-monochromatic pulses. The devices donot have broadband limitations other than the naturaldispersion of the conventional materials chosen to buildthem. Finally, the response of the radially dependent per-mittivity material to cylindrical waves was studied in de-tail. The device can have several potential applications,ranging from excellent absorbers to state-of-the-art solarcells and optoelectronics.

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