## Abstract

Halting and storing light by infinitely decelerating its speed, in the absence of any form of external control, is extremely difficult to imagine. Here we present a theoretical prediction of a controllable optical black hole composed of a planar left-handed material slab. We reveal a criterion that the effective round-trip propagation length in one zigzag path is zero, which brings light to a complete standstill. Both theory and ab initio simulation demonstrate that this optical black hole has flexible controllability for the speed of light. Surprisingly, the ab initio simulations reveal that our scheme has flexible controllability for swallowing, holding, and releasing light.

©2010 Optical Society of America

For a long time, scientists have imagined the creation of artificial black holes in laboratory, like the astronomical black holes anticipated to lurk in distant space. Leonhardt and Piwnieki foretold theoretically an optical analogue of the astronomical black hole, the so-called optical black hole [1, 2]. The key to swallowing light in optical black hole requires that light can be dragged or that the speed of light can be significantly slowed down. Due to its exotic properties and its promising applications, such as in all-optical buffers and variable optical delays, slow light is now attracting great attention [3–15]. Exploiting strong dispersion is a general method for slowing the speed of light. For instance, slow light and even completely halted light have been demonstrated in ultracold gases [7, 8], hot gases [9, 16], and in very cold solids [17], based on the electromagnetically induced transparency. Besides these physical systems, slow light and even completely halted light have also been explored in a great number of schemes, such as coherent population oscillation [10, 18], stimulated Raman scattering [19], stimulated Brillouin scattering [20], surface plasmon polariton [21], spatial nonlocal nonlinearity [22–24], temporal nonlocal Kerr nonlinearity [25], soliton [6], and photonic crystal [11–13].

As is well known, all existing natural matter exhibits single-right-handed character. Therefore, the propagation behavior of electromagnetic (EM) waves is governed by the permittivity alone, because the magnetic component has almost no interaction with the atoms of conventional materials. In contrast, the two-handed property of left-handed (LH) material [26] allows both electric and magnetic field components to interact with the “atoms” of LH material, enabling entirely new EM properties [26, 27] and amazing applications [28, 29]. In addition, LH material can also be involved in the slow-light realm [30–33]. Tsakmakidis *et al*. predicted an “optical clepsydra” [30]. A consummate optical black hole should also have flexible controllability for swallowing, holding, and releasing light, in the absence of any additional control. In the present work, we predict this kind of controllable optical black hole by the use of a planar LH material slab.

To have insight into the physics of the problem of interest, the propagation behaviors of EM waves in a LH slab waveguide configuration, as illustrated in Fig. 1(a), have been investigated. The homogeneous and isotropic planar LH slab, with a thickness of *d*, and simultaneously negative permittivity *ε*
_{1} and permeability *μ*
_{1}, occupies the space ranging from *y* = 0 to *d*. It is sandwiched by two semi-infinite right-handed (RH) media, as the cladding (with *ε*
_{2} > 0 and *μ*
_{2} > 0) and the substrate (with *ε*
_{3} > 0 and *μ*
_{3} > 0). Such a LH slab waveguide supports both oscillatory- and plasmonic-guided modes [31, 34]. The former cannot be supported in single-negative waveguide structures, in contrast, the latter cannot be permitted in conventional dielectric waveguides.

For any guided mode propagating in a waveguide, the EM fields inside the cladding and the substrate are both evanescent, with their amplitudes decaying exponentially away from the boundaries; that is to say, any guide mode is nonradiative along the normal of the boundary. For the oscillatory-guided mode, the intrinsic feature is that the field inside the core is a propagating wave with all three wavevector components being real. In contrast, for the plasmonic-guided mode, the field inside the core belongs to the low-dimensional EM wave, because the wavevector component normal to the boundary is imaginary. The propagation behaviors of an oscillatory-guided mode can be understood by the multiple-wave interference principle.

A plane EM wave inside the core will undergo total reflections at two boundaries in addition to the pure propagation, traveling along a zigzag path, as shown in Fig.1(b)–(d). If an oscillatory-guided mode can be supported, a constructive interference is required, implying that the accumulated phase shift (including three phase shifts, -2*dk*
_{1} due to pure propagation, and *ϕ*
_{12} and *ϕ*
_{13} caused by total reflections at boundaries 1–2 and 1–3) in one zigzag round trip must satisfy -2*dk*
_{1} + *ϕ*
_{12} + *ϕ*
_{13} = 2*pπ* (where *p* is an integer and *k*
_{1} is the normal component of the wavevector in the core). On the basis of this mode equation, by taking the derivative with respect to *β* (*β* is the longitudinal propagation constant in the *z* direction) and substituting *∂ϕ*/*∂β* = *Δz* and -*∂ϕ*/*∂ω* = Δ*t*, the group velocity of an oscillatory-guided mode, *v _{g}* = d

*ω*/d

*β*, is easily yielded as follows:

This is precisely the form of a traditional RH waveguide. 2*d* tan *θ* denotes the pure propagation length inside the core in the *z* direction for one zigzag round trip and (2*d*/*c*)ñ/ cos *θ* represents the corresponding time duration. Δ*z*
_{1m} (where m = 2 and 3) is the Goos-Hänchen (GH) shift caused by the total reflection at boundary 1-m, and Δ*t*
_{1m} is the time delay related to Δ*z*
_{1m}, referred to as the GH time delay.

We introduce a concept of *effective round-trip propagation length*, which is defined as *L _{eff}* = 2

*d*tan

*θ*+ (Δ

*z*

_{12}+ Δ

*z*

_{13}). If the core is dispersionless $(\partial {\epsilon}_{1}{\mu}_{1}/\partial \mathrm{\omega =}0),\xf1=\sqrt{{\epsilon}_{1}{\mu}_{1}}+\omega \partial \sqrt{{\epsilon}_{1}{\mu}_{1}}/\partial \omega $ is equal to the phase refractive index ${n}_{p}=\sqrt{{\epsilon}_{1}{\mu}_{1}}.$ If dispersion is taken into account, ñ is, in fact, the group refractive index

*n*in a bulk material as

_{g}*n*=

_{g}*n*+

_{p}*ω∂*/

_{np}*∂ω*[35]. The group velocity

*v*of an oscillatory-guided mode cannot be solely determined by

_{g}*c/n*, as it also depends strongly on the GH effect. For a TE-polarized wave, we yield Δ

_{g}*z*

_{1m}from Δ

*z*

_{1m}=

*∂ε*

_{1m}/

*∂β*, as follows

where ${k}_{1}=\sqrt{{(\omega /c)}^{2}{\epsilon}_{1}{\mu}_{1}-{\beta}^{2}}\mathrm{and}{\phantom{\rule{.2em}{0ex}}\alpha}_{m}=\sqrt{{\beta}^{2}-{(\omega /c)}^{2}{\epsilon}_{m}{\mu}_{m}}.$

From the view of the energy flux, the group velocity *v _{g}* of an oscillatory-guided mode can also be deduced from the energy flux

*P*and the energy density

*U*by

*v*=

_{g}*P/U*

Surprisingly this expression is completely identical to expression (1), except for the sign function, where sign(*μ*
_{1}) = + 1 if *μ*
_{1} > 0 while sign(*μ*
_{1}) = -1 if *μ*
_{1} < 0. It is paramount that expression (3) allows us to simultaneously determine the direction (with the respect to the *β* vector) and magnitude of the group velocity of an oscillatory-guided mode. Since the denominator in expression (3) is always positive, the sign of the product of *L _{eff}* and sign(

*μ*

_{1}) decides the direction of the group velocity.

The total reflection at the LH-RH interface can exhibit a negative GH shift that is much larger than the wavelength [36, 37]. Therefore, *L _{eff}* is allowed to be positive for relatively small GH shift in Fig. 1(b) and even negative for relatively larger GH shift in Fig. 1(c). In the RH-LH-RH waveguide, due to the negative value of

*μ*

_{1}, the propagation of oscillatory-guided mode is allowed to be backward or forward, as shown in Fig. 1(b) or Fig. 1(c), solely because

*v*can be antiparallel or parallel to

_{g}*β*. Very interestingly, it is possible that

*L*vanishes, implying that the round-trip propagation length 2

_{eff}*d*tan

*θ*is exactly balanced by the total GH shift Δ

*z*

_{12}+ Δ

*z*

_{13}. In the special situation of

*L*= 0, it is deemed that the EM wave is at a complete standstill and is unable to travel further, as shown in Fig. 1(d).

_{eff}To more intuitively recognize the properties of slow light in the LH waveguide, we furnished ab initio simulations based on the finite-difference time-domain (FDTD) method, to emulate the launching, propagating, or trapping and releasing behaviors. Without loss of generality, for all the simulations below, one can judiciously choose a *symmetric* LH waveguide. A slab of thickness *d* = 3 cm, with its permittivity and permeability obeying *ε*
_{1} (*f*) = 1-*f _{pe}*

^{2}/

*f*

^{2}and

*μ*

_{1}(

*f*) = 1-

*f*

_{pm}^{2}(

*f*), respectively, is surrounded by air (ε

_{2}=

*ε*

_{3}=

*μ*

_{2}=

*μ*

_{3}= 1), where

*f*(

_{pe}*f*) is the electric (magnetic) plasma frequency with

_{pm}*f*=

_{pe}*f*= 10 GHz and

_{pm}*f*is the frequency. The slab is left-handed and optically more dense than air in the frequency range of 4.2 to 7 GHz.

From the calculated four dispersion branches of the oscillatory-guided modes, as shown in Fig. 2, we can find that the dispersion of *f* ~ *β* curves exhibits a non-monotonous dependence, implying that the group velocity can indeed be negative or positive and even zero. The four field distributions for the chosen four points (A, B, C and D at *β* = 148 m^{-1}), as illustrated by the insets (La, Lb, Lc and Ld), indicate that the branches 1, 2, 3 and 4 correspond to the first-, second-, third- and forth-order oscillatory-guided modes, respectively. Selecting branch 2 as an example, we illustrate the calculated dependencies of *v _{g}* and

*L*on

_{eff}*β*in the inset R of Fig. 2. Note that the zero points of

*v*and

_{g}*L*are completely identical and both are the same as the zero-dispersion point of the

_{eff}*f*~

*β*curve of branch 2. The signs of

*v*and

_{g}*L*can be changed from positive to negative and vice versa, as

_{eff}*β*increases. The zero-dispersion point occurs at (

*β,f*) = (151 m

^{-1}, 6.04 GHz).

The planar slab waveguide shown in Fig. 1(a) in fact could be considered as a kind of semi-open and semi-closed resonator, because the EM field in the *y* dimension is completely confined (belonging to the nonradiative field), while the EM field in the *x* and *z* dimensions are completely free with no restrictions (vesting in the propagating or radiative field). As is well known, it is impossible for a propagating field in free space to directly excite any guided mode or to be fed directly into a completely decoupled waveguide, from any boundary of the waveguide, due to the wavevector (or moment) mismatching and the nonradiative property of the guided mode in the *y* dimension. However, with the aid of some proper schemes, a propagating field in free space can be continuously fed into a waveguide structure and vice versa. Here we adopt the well-known prism-coupling method based on the attenuation total reflection (ATR), in which a prism with high refractive index is placed above a planar waveguide and is separated from it by a small gap.

Our focus is the dramatically slowing oscillatory-guided modes. Figure 3 furnishes the ab initio simulation results for the forward and backward modes. To effectively launch a guided mode, the direction of the exciting field must be carefully adjusted to satisfy the perfect moment matching. One can explicitly discover that the oscillatory-guided mode travels along the structure, with its phase velocity (*v _{p}* or

*β*) and group velocity

*v*being parallel for the forward mode (in the left column) and antiparallel for backward mode (in the right column), as anticipated by the above theory. As the excitation time increases, the energy fed into the waveguide and the propagation distance inside the structure progressively increases, as shown by (a) and (b) in left and right columns in Fig. 3. When the exciting field is shut off and the coupling prism is also removed, as shown by (c) and (d) in left and right columns in Fig. 3, the EM field fed into the waveguide is always confined to propagate within the structure and has no escape. Of course, the EM energy confined in the structure can be extracted either partially, as shown by (e) in left and right columns in Fig. 3, or entirely by the prism, depending on the extraction time. Both forward and backward oscillatory-guided modes can be dramatically slowed down, with the respective group velocities of 0.057

_{g}*c*and -0.059

*c*.

More interestingly, the halted mode should be intuitively recognized by the ab initio
simulations. The EM field in the normal direction of the waveguide is nonradiative, while it usually has the propagation feature in the in-plane direction of the waveguide. Therefore, to realize the complete standstill of EM field in the structure, the key is that the group velocity can be ultimately slowed down to zero, which requires *L _{eff}* = 0, as based on the above theory. For example, when (

*β, f*) = (145.149 m

^{-1}, 4.386 GHz) or (

*β, f*) = (150.746 m

^{-1}, 6.036 GHz),

*L*vanishes, corresponding to the zero-dispersion point of the

_{eff}*f*~

*β*curve.

To efficiently launch the halted mode in the structure, the *z* component of the wavevector of the exciting field inside the prism should exactly match the longitudinal propagation vector *β*. The ab initio simulation results in Fig. 4 show the launching, keeping, and releasing evolutions of the halted mode for a TE-polarized field at two distinct frequencies of 4.386 GHz (Left column) and 6.036 GHz (Right column). The exciting field begins to be switched on at a moment *t* = 0. As the excitation time increases, the EM energy fed into the structure grows and the local field intensity becomes stronger, as shown in Figs. 4(a) and (b). For the frequency of 4.386 GHz (6.036 GHz), the exciting field is switched off at a moment *t* = 25.810 ns (13.380 ns), and the coupling prism is removed at a moment *t* = 29.732 ns (16.192 ns). The escape of the EM field confined inside the structure from the normal direction is not a concern, because the waveguide is completely decoupled. More importantly, however, it is possible that the EM field may escape from the in-plane dimension, because the waveguide always couples with the environment. Figs. 4(c)–(d) show the field distributions after the coupling prism was removed at two moments *t* = 35.575 and 41.420 ns for *f* = 4.386 GHz (18.290 and 20.391 ns for *f* = 6.036 GHz). As we anticipated, the EM field fed into the structure could be indeed arrested or permanently stored. At a moment *t* = 41.423 ns for *f* = 4.386 GHz
(20.393 ns for *f* = 6.036 GHz), as shown in Fig. 4(e), when the coupling prism is returned to the place above the waveguide (the exciting field is still switched off), the waveguide becomes leaky in the normal direction. We discover that the EM field encaged inside the waveguide can be released. Therefore, by switching the coupling prism on or off, we can easily control how much EM energy will be stored, as well as the keeping time and the extracting fraction of the stored EM energy.

We now would like to explore the plasmonic-guided modes. Unlike an oscillatory-guided mode, for a plasmonic-guided mode as a low-dimensional electromagnetic wave, its longitudinal propagation constant *β* inside the core is always larger than the volume wavevector, due to the imaginary wavevector component in the normal direction. Although the fields inside the core do not undergo any total reflection at any boundary, we have confirmed that the above results regarding the oscillatory-guided mode can be extended into the situation of plasmonic-guided mode, provided that Δ*z*, Δ*t*, *θ*, and ñ are substituted by four *generalized* parameters, Δ*z*′, Δ*t*′, *θ*′, and ñ′, correspondingly where

Of course, the plasmonic-guided modes also exist in forward, backward, and halted modes. The launching, propagating (keeping), and releasing evolution behaviors are not shown here for these modes. So far, we have devoted our efforts to exploring the TE-polarized situation only. The above results can be utilized for the TM-polarized situation, provided that the two symbols of *ε* and *μ* in all the above expressions are interchangeable with each other. We should emphasize that the above results are universal, with no restriction regarding the handedness (RH or LH) of the media.

For the storage of light energy, the slab LH waveguides have some advantages. For instance, the storage of light energy can be achieved in the large area slab LH waveguide by using a micro-prism array, as shown in Fig. 5(a). The light energy stored in the structure can then be partially or entirely extracted by controlling the extraction time through the micro-prism array, as shown in Fig. 5(b).

Above we use a lossless Drude model to describe the permittivity and permeability of the left-handed material. If arbitrarily small imaginary parts (absorption losses) are introduced into the expressions of permittivity and permeability, the waveguide mode with a small amplitude of group velocity would be accompanied with a large propagation loss and zero group velocity mode could be destroyed, as been argued by Reza *et al*. [38], thus the stored light information would be of limited lifetime. This problem could be solved by developing zero-loss left-handed materials, as been proposed by Tsakmakidis *et al*. [39], for example, by using adjacent absorptive and gain resonances [40], by introducing gain mechanism [41], by directly using dielectric-based structures [42], or via electromagnetically induced chirality [43].

In summary, we have shown a controllable optical black hole composed of a left-handed slab, which allows the electromagnetic field to be dramatically reduced in forward or backward direction and even to come to a complete standstill. A simple method for determining the propagation direction of the energy flux (group velocity) and the propagation direction of the wavefront (phase velocity) of the guided modes is then proposed. An expression for the group velocity is developed, which explicitly shows the contribution from the GH shift. This scheme provides an approach for decelerating, storing, and then releasing light, and has some inherent features involving broadband and room-temperature operation, in addition to breaking traditional limitations. This controllable optical black hole opens a way to the control of light energy.

## Acknowledgement

This work is supported by the National Natural Science Foundation of China under Grants 60808003 and 10934003, the Open Project of State Key Laboratory of Functional Materials for Informatics, and the National Basic Research Program of China under Grant 2006CB921805.

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