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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 11 — May. 23, 2011
  • pp: 10088–10101
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Slow-light-enhanced codirectional couplers with negative index materials

L. Zhao and Wenhui Duan  »View Author Affiliations


Optics Express, Vol. 19, Issue 11, pp. 10088-10101 (2011)
http://dx.doi.org/10.1364/OE.19.010088


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Abstract

Optical codirectional coupling structures consisting of two parallel planar waveguides with negative index materials (NIMs) are systematically studied in different configurations using coupled-mode theory under the weak-coupling condition. As a result, we find that the coupling strength between copropagating optical modes can be enhanced in such structures. More importantly, both our analytical derivations and numerical simulations clearly indicate that the slow-light effect in the waveguides with NIMs plays an essential role in such enhancement. The configuration with two conventional positive-index-material cores embedded in NIM claddings (or vice versa) can lead to the strongest enhancement because it can give rise to the slowest light in our scheme. Therefore, as well as offering a fundamental understanding of the slow-light effect in codirectional coupling structures with NIMs for constructing compact photonic devices, our investigations suggest a useful guideline for optimizing the design of codirectional couplers using slow-light systems for both the classical and quantum information processing and communication networks.

© 2011 OSA

1. Introduction

Negative index materials (NIMs) with simultaneously negative permittivity and permeability have attracted rapidly increasing interest due to their important applications in far-field imaging with subwavelength resolution [1

1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).

4

4. N. Fang, Z. Liu, T.-J. Yen, and X. Zhang, “Regenerating evanescent waves from a silver superlens,” Opt. Express11, 682–687 (2003), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-7-682. [PubMed]

]. Since the experimental verification in the microwave regime [5

5. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [PubMed]

, 6

6. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). [PubMed]

], significant efforts have been made to fabricate various types of NIMs. For example, tunable NIMs for terahertz waves [7

7. H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics 2, 295–298 (2008).

] and loss-free NIMs for visible and infrared light [8

8. S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature (London) 466, 735–738 (2010).

10

10. A. Fang, Th. Koschny, and C. M. Soukoulis, “Self-consistent calculations of loss-compensated fishnet metamaterials,” Phys. Rev. B 82, 121102(R) (2010).

] have been accomplished successfully, thus providing a solid foundation for the development of new functional optical devices. In recent years, waveguide structures, as a key element in integrated photonics, have been studied extensively on the basis of NIMs [11

11. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).

19

19. Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. USA 108, 5169–5173 (2011). [PubMed]

]. In particular, the “trapped rainbow” effect in the left-handed waveguides with NIMs has received massive attention both in theory [15

15. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature (London) 450, 397–401 (2007).

, 16

16. K. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, “Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures,” Appl. Phys. Lett. 89, 201103 (2006).

] and in experiment [17

17. X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95, 071111 (2009).

19

19. Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. USA 108, 5169–5173 (2011). [PubMed]

] because it could lead to broadband slow and even trapped light in solid-state optical systems for potential applications in optical data processing and quantum optical memories. Nevertheless, the functionality of a single left-handed waveguide is too limited to satisfy diverse demand in photonics.

To extend the performance of individual waveguides, a codirectional coupler composed of two parallel planar waveguides is a promising solution. As a matter of fact, based on conventional positive index materials (PIMs), such as GaAs, devices of this kind have been extensively investigated both theoretically and experimentally using the coupled-mode theory in guided-wave optics [20

20. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1989), Chap. 22.

, 21

21. K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Elsevier Academic Press, 2000), Chap. 4.

]. And, many important functions, such as optical signal switching, splitting, filtering, multiplexing, and demultiplexing for two copropagating light fields confined in the waveguides, can be realized in integrated photonic circuits. Recently, quantum photonic circuits based on codirectional couplers on a silicon chip have been experimentally demonstrated [22

22. A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008). [PubMed]

24

24. A. Politi, J. C. F. Matthews, and J. L. O’Brien, “Shor’s quantum factoring algorithm on a photonic chip,” Science 325, 1221 (2009). [PubMed]

]. In these integrated circuits, various quantum manipulation and operations, such as quantum interference, entanglement, controlled-NOT quantum logic gates, and Shor’s quantum factoring algorithm, can be performed with high fidelity. Therefore, it can be seen that codirectional couplers are important functional devices in integrated photonic circuits for both the classical and quantum information processing.

Considering the successful fabrication of NIMs and following the idea that NIMs can amplify evanescent waves, in some earlier theoretical work, a codirectional coupler having the particular configuration of a NIM layer inserted between two parallel conventional waveguides was proposed and the coupling strength enhancement between two guided copropagating signals was found [25

25. S. Xiao, L. Shen, and S. He, “A novel directional coupler utilizing a left-handed material,” IEEE Photon. Technol. Lett. 16, 171–173, (2004).

]. However, in that work, the possible slow-light effect in the waveguides with NIMs was not considered. Motivated by the fact that the slow-light effect can usually substantially modify certain electromagnetic properties of optical systems [26

26. T. F. Krauss, “Why do we need slow light?” Nat. Photonics 2, 448–450 (2008).

, 27

27. J. B. Khurgin, “Slow light in various media, a tutorial,” Adv. Opt. Photon. 2, 287–318 (2010).

], we should undertake in-depth investigations involving the slow-light effect to fully understand and thus optimize the codirectional coupling structures with NIMs.

In this paper, we thus systematically study codirectional coupling structures composed of two parallel planar waveguides with NIMs in different configurations using coupled-mode theory in the weak-coupling approximation. The presence of NIMs in the structures can greatly enhance the coupling strength between the two waveguides. However, more importantly, both our analytical derivations and numerical calculations clearly indicate that such enhancement can be essentially attributed to the slow-light effect in the waveguides with NIMs. The configuration with two conventional PIM cores embedded in NIM claddings (or vice versa) can lead to the strongest enhancement because it can give rise to the slowest light in our scheme. Based on this enhancement, the size of the coupling devices can be greatly shrunk for signal transfer between the two waveguides, and thus compact devices for signal switching, splitting, filtering, and multiplexing/demultiplexing can be accomplished for optical logic elements and large-scale integrated photonic circuits. Therefore, our findings could offer a helpful guideline for optimizing the design of codirectional coupling devices using slow-light systems. Additionally, the “trapped rainbow” effect in the left-handed waveguides may also provide the possibility of broadband slow-light coupling devices for high-speed optical information processing.

2. Theoretical model

In our scheme, for simplicity, we consider a codirectional coupler with the symmetric structure shown in Fig. 1, where we assume |n 2| > {|n 1|, |n 3|} for the total internal reflection at the interfaces for waveguiding. Following the coupled-mode theory in guided-wave optics [20

20. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1989), Chap. 22.

, 21

21. K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Elsevier Academic Press, 2000), Chap. 4.

], we can first formally derive the coupling properties of guided modes in this two-waveguide structure, where both the permittivity and permeability of the waveguide materials are taken into account.

Fig. 1 (Color online) Schematic diagram of a codirectional coupler consisting of two parallel planar waveguides in a symmetric manner, where the cores and the claddings can be made of either PIMs or NIMs. The two waveguide cores have the same width w and the separation distance between them is s. The shortest coupling length for a complete signal transfer between the two waveguides is Lc.

For the isolated waveguide A, the spatial distributions of its relative permittivity and permeability are given by
[εrA(x),μrA(x)]={(εr1,μr1),x<0,(εr2,μr2),0xx1,(εr3,μr3),x>x1.
(1)
Similarly, for the isolated waveguide B, we have
[εrB(x),μrB(x)]={(εr3,μr3),x<x2,(εr2,μr2),x2xx3,(εr1,μr1),x>x3.
(2)
The propagation of the guided waves in the two isolated waveguides is thus described by Maxwell’s equations,
{×Em=iωμ0μrmHm,×Hm=iωε0εrmEm,
(3)
where ω is the frequency of the guided waves, ε 0 (μ 0) is the vacuum permittivity (permeability), and m = A or B denoting the two isolated waveguides, respectively. Note that the time-dependent term exp(–iωt) in (E m, H m) has been removed.

When the two parallel waveguides are very close to each other, codirectional coupling between the two guided modes will occur in the x direction due to the evanescent wave tunneling. This coupling can affect the propagation of the guided modes in the z direction. According to the coupled-mode theory [20

20. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1989), Chap. 22.

, 21

21. K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Elsevier Academic Press, 2000), Chap. 4.

], the total fields in the coupled-guided structure can be given by a linear combination of the guided fields in each isolated waveguide, i.e.,
{E=U(z)EA+V(z)EB,H=U(z)HA+V(z)HB,
(4)
where U(z) and V(z) characterize the propagation of the total field in the z direction. Furthermore, the total field should also satisfy Maxwell’s equations, yielding
{×E=iωμ0μrH,×H=iωε0εrE,
(5)
where εr = εrA + εrBεr 3 and μr = μrA + μrBμr 3 are the spatial profiles of the relative permittivity and permeability of the two-waveguide structure. Substituting Eqs. (1)(4) into Eq. (5) and using the vector analysis in Refs. [21

21. K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Elsevier Academic Press, 2000), Chap. 4.

, 28

28. W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).

], one can obtain
Uzz^×EA+Vzz^×EB=iωμ0[U(μrBμr3)HA+V(μrAμr3)HB],
(6)
Uzz^×HA+Vzz^×HB=iωε0[U(εrBεr3)EA+V(εrAεr3)EB],
(7)
where represents the unit vector in the z direction (see Fig. 1). We can take the scalar product of both sides of Eq. (6) with HA* and use the vector analysis in Refs. [21

21. K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Elsevier Academic Press, 2000), Chap. 4.

, 28

28. W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).

] again, which yields
Uzz^(EA×HA*)+Vzz^(EB×HA*)=iωμ0[U(μrBμr3)HAHA*+V(μrAμr3)HBHA*].
(8)
Similarly, taking the scalar product of both sides of Eq. (6) with HB* yields
Uzz^(EA×HB*)+Vzz^(EB×HB*)=iωμ0[U(μrBμr3)HAHB*+V(μrAμr3)HBHB*].
(9)
Taking the scalar product of both sides of Eq. (7) with EA* yields
Uzz^(EA*×HA)+Vzz^(EA*×HB)=iωε0[U(εrBεr3)EA*EA+V(εrAεr3)EA*EB].
(10)
Taking the scalar product of both sides of Eq. (7) with EB* yields
Uzz^(EB*×HA)+Vzz^(EB*×HB)=iωε0[U(εrBεr3)EB*EA+V(εrAεr3)EB*EB].
(11)
Therefore, combining Eqs. (8) and (10), we have
Uzz^(EA×HA*+EA*×HA)+Vzz^(EB×HA*+EA*×HB)=iω[Uμ0(μrBμr3)HAHA*+Vμ0(μrAμr3)HBHA*+Uε0(εrBεr3)EA*EA+Vε0(εrAεr3)EA*EB].
(12)
Combining Eqs. (9) and (11), we have
Uzz^(EA×HB*+EB*×HA)+Vzz^(EB×HB*+EB*×HB)=iω[Uμ0(μrBμr3)HAHB*+Vμ0(μrAμr3)HBHB*+Uε0(εrBεr3)EB*EA+Vε0(εrAεr3)EB*EB].
(13)
Furthermore, we should consider the weak-coupling condition in the coupled-mode theory. Under this condition, the guided electromagnetic waves [i.e., (E A, H A) and (E B, H B)] are strongly confined in each individual waveguide, and thus the evanescent-wave tunneling between the two waveguides is very weak. Therefore, one can define the following parameters to describe the weak-coupling condition
R1=|+z^(EB×HA*)dx+z^(EA×HA*)dx|1,and R2=|+(μrBμr3)HAHA*dx+(μrAμr3)HBHA*dx|1,
(14)
R3=|+z^(EA*×HB)dx+z^(EA*×HA)dx|1,and R4=|+(εrBεr3)EA*EAdx+(εrAεr3)EA*EBdx|1,
(15)
R5=|+z^(EA×HB*)dx+z^(EB×HB*)dx|1,and R6=|+(μrAμr3)HBHB*dx+(μrBμr3)HAHB*dx|1,
(16)
R7=|+z^(EB*×HA)dx+z^(EB*×HB)dx|1,and R8=|+(εrAεr3)EB*EBdx+(εrBεr3)EB*EAdx|1.
(17)
All these equations for the weak-coupling condition will be verified in our numerical calculations later in Section 3. Using the approximations in Eqs. (14) and (15), Eq. (12) can be simplified to
Uz=κABV,
(18)
where
κAB=iω+[ε0(εrAεr3)EA*EB+μ0(μrAμr3)HA*HB]dx+z^(EA×HA*+EA*×HA)dx.
(19)
Likewise, using Eqs. (16) and (17), Eq. (13) can be simplified to
Vz=κBAU,
(20)
where
κBA=iω+[ε0(εrBεr3)EAEB*+μ0(μrBμr3)HAHB*]dx+z^(EB×HB*+EB*×HB)dx.
(21)
Note that we integrate both sides of Eqs. (12) and (13) over x in the transverse direction of the codirectional coupler (see Fig. 1). The parameters κAB and κBA in Eqs. (19) and (21) give the coupling coefficients between the two waveguides. It can be clearly seen that the numerators of Eqs. (19) and (21) represent the spatial overlap of the guided electromagnetic waves (E A, H A) and (E B, H B). More importantly, the denominators in Eqs. (19) and (21) actually represent the energy fluxes along each individual waveguide, respectively. Based on the investigations in Refs. [11

11. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).

, 12

12. A. C. Peacock and N. G. R. Broderick, “Guided modes in channel waveguides with a negative index of refraction,” Opt. Express11, 2502–2510 (2003), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-20-2502. [PubMed]

, 15

15. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature (London) 450, 397–401 (2007).

, 16

16. K. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, “Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures,” Appl. Phys. Lett. 89, 201103 (2006).

], we know that the left-handed waveguide with NIMs can usually lead to slow and even trapped light having a very small energy flux along the waveguide. This fact intuitively inspires us that the slow-light effect could greatly enhance the coupling coefficients κAB and κBA between the two waveguides.

In our symmetric configuration, we can obtain that κAB=κBA*=κ. Furthermore, for simplicity, we only consider the coupling between the two modes with equal propagation wave vector kz along the waveguides (i.e., phase-matching). Besides, we assume that the optical signal is initially injected into waveguide A (i.e., U(0) = 1 and V(0) = 0). Hence, the solutions of Eqs. (18) and (20) are given by
U(z)=cos(|κ|z),and V(z)=|κ|κsin(|κ|z),
(22)
which indicate a sinusoidal oscillation of the optical signal between the two waveguides. The shortest coupling length for a complete signal transfer is thus given by
Lc=π2|κ|.
(23)
At this length, the optical signal initially injected into waveguide A can be completely switched into waveguide B. Apparently, If the propagation length of the incident signal is only half of the shortest coupling length Lc, the signal will be equally split into the two waveguides.

3. Numerical simulations and results

In order to numerically evaluate our scheme, for simplicity, we assume that the guided electromagnetic waves in waveguides A and B are the TE modes with the electric fields E A and E B polarized in the y direction. For the TM modes, we can carry out a similar treatment. Therefore, the electric field in the isolated waveguide A can be expressed by
EA=EAyy^={aexp(αx+ikzz)y^,x<0,[bsin(kxx)+ccos(kxx)]exp(ikzz)y^,0xx1,dexp[α'(xx1)+ikzz]y^,x>x1.
(24)
where x 1 = w is the width of waveguide A, ŷ is the unit vector in the y direction, kx is the transverse wave vector perpendicular to the waveguide in the core region 0 ≤ xx 1, kz=(k02n22kx2)1/2 is the propagation wave vector along the waveguide, k 0 is the wave vector of the incident optical field in vacuum, α=(kz2k02n12)1/2=[k02(n22n12)kx2]1/2 is the decay factor of the evanescent field in the cladding region x < 0, α'=(kz2k02n32)1/2=[k02(n22n32)kx2]1/2 is the decay factor of the evanescent field in the cladding region x > x 1, and a, b, c, d are the normalization coefficients. According to the symmetry, the electric field in the isolated waveguide B can be expressed by
EB=EByy^={dexp[α'(xx2)+ikzz]y^,x<x2,(bsin[kx(xx3)]+ccos[kx(xx3)])exp(ikzz)y^,x2xx3,aexp[α(xx3)+ikzz]y^,x>x3.
(25)
where x 2 = w + s, x 3 = 2w + s, and s is the separation distance between the two waveguides. Moreover, using the Maxwell’s Eqs. (3), one can find the expressions of H A and H B. For simplicity, we here omit the expressions. Additionally, to determine the eigenmodes in waveguide A, we can use the continuity conditions at x = 0 and x = w, and find
tan(kxw)=kxμr2(α'μr1+αμr3)kx2μr1μr3αα'μr22,
(26)
which determines the transverse wave vectors kx for the discrete eigenmodes guided in waveguide A. Likewise, for waveguide B, we can obtain the same expression.

In Fig. 2, we consider four symmetric two-waveguide configurations with different refractive index profiles as follows, where waveguides A and B have the same width w = 5.5 cm and the separation distance between the two waveguides is s = 5 cm.

Fig. 2 (Color online) Four symmetric two-waveguide configurations with different refractive index profiles, corresponding to the codirectional coupling structure in Fig. 1, where waveguides A and B have the same width w = 5.5 cm and the separation distance between the two waveguides is s = 5 cm. We employ three different materials to construct the configurations, which include teflon with nteflon 1.4, εrteflon=1.96, and μrteflon=1, air with nair = 1, εrair=1, and μrair=1, and the artificial NIM mentioned above with nNIM = −1, εrNIM=2.70, and μrNIM=0.37. For simplicity, we always use the teflon in the core regions 0 ≤ xx 1 and x 2xx 3 and place the air or NIM in the different locations of the cladding regions. (a) There is only the NIM in the cladding regions x < 0, x 1 < x < x 2 and x > x 3. (b) The NIM is in the bilateral cladding regions x < 0 and x > x 3 and the air is in the central cladding region x 1 < x < x 2. On the contrary, in (c), the NIM is in the central cladding region and the air is in the bilateral cladding regions. (d) For comparison, we consider a conventional codirectional coupling structure with only the air in the cladding regions.
  1. For the configuration shown in Fig. 2(a), we have n 1 = n 3 = −1 with εr 1 = εr 3 = −2.70 and μr 1 = μr 3 = −0.37, and n 2 = 1.4 with εr 2 = 1.96 and μr 2 = 1 for the spatial distributions of the relative permittivity and permeability of the two waveguides given by the Eqs. (1) and (2). In brief, this configuration means that the two teflon cores are embedded in the NIM claddings.
  2. For the configuration shown in Fig. 2(b), we have n 1 = −1 with εr 1 = −2.70 and μr 1 = −0.37, n 2 = 1.4 with εr 2 = 1.96 and μr 2 = 1, and n 3 = 1 with εr 3 = 1 and μr 3 = 1. In brief, this configuration means that the NIM is placed on both outer sides of the two teflon cores and the central region between the cores is left empty with air.
  3. For the configuration shown in Fig. 2(c), we have n 1 = 1 with εr 1 = 1 and μr 1 = 1, n 2 = 1.4 with εr 2 = 1.96 and μr 2 = 1, and n 3 = −1 with εr 3 = −2.70 and μr 3 = −0.37. In brief, this configuration means that the NIM is inserted between the two teflon cores, which can be directly analogous to the proposed codirectional coupling structure (i.e., a NIM layer inserted between two parallel conventional waveguides) in Ref. [25

    25. S. Xiao, L. Shen, and S. He, “A novel directional coupler utilizing a left-handed material,” IEEE Photon. Technol. Lett. 16, 171–173, (2004).

    ].
  4. For comparison, a conventional codirectional coupler with only PIMs (i.e., the teflon cores and the air claddings) is shown in Fig. 2(d). In this configuration, we have n 1 = n 3 = 1 with ε 1 r = εr 3 = 1 and μr 1 = μr 3 = 1, and n 2 = 1.4 with εr 2 = 1.96 and μr 2 = 1.

Based on the derivations in Section 2 and the symmetry of the configurations, we can numerically calculate the coupling coefficients |κ| = |κAB| = |κBA| given by Eq. (19) or (21) and the shortest coupling length Lc for a complete signal transfer given by Eq. (23) in all four cases in Table 1. Furthermore, in order to characterize the slow-light effect in the four cases, we calculate the normalized energy flux P of the guided mode in each isolated waveguide [11

11. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).

, 12

12. A. C. Peacock and N. G. R. Broderick, “Guided modes in channel waveguides with a negative index of refraction,” Opt. Express11, 2502–2510 (2003), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-20-2502. [PubMed]

]. Considering the denominator in Eq. (19), we first calculate the normalized energy flux PA of the guided mode in isolated waveguide A, which is defined as [11

11. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).

, 12

12. A. C. Peacock and N. G. R. Broderick, “Guided modes in channel waveguides with a negative index of refraction,” Opt. Express11, 2502–2510 (2003), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-20-2502. [PubMed]

]
PA=PA,core+PA,cladding|PA,core|+|PA,cladding|=0SAzdx+0x1SAzdx+x1+SAzdx|0SAzdx|+|0x1SAzdx|+|x1+SAzdx|,
(28)
where SAz=Re[z^(EA×HA*)]=Re[z^(EA*×HA)] is the z component of the Poynting vector along waveguide A, E A is given by Eq. (24), and H A can be derived from the Maxwell’s Eqs. (3) and Eq. (24). Due to the symmetry of the configurations, for isolated waveguide B, one can obtain PB = PA = P. Moreover, we also numerically verify the weak-coupling parameters R 1R 8 given by Eqs. (14)(17) in Table 1, where, due to the symmetry, we have R 1 = R 3 = R 5 = R 7, R 2 = R 6, and R 4 = R 8. In case (iv) with the teflon in the cores and the air in the claddings, the relative permeability of the structure is always unity, yielding μrAμr 3 = μrBμr 3 = 0. Thus, R 2 and R 6 are not applicable (N/A) in this case, and the expressions of κAB and κBA given by Eqs. (19) and (21) can be reduced to those of the conventional codirectional couplers given by Refs. [20

20. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1989), Chap. 22.

, 21

21. K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Elsevier Academic Press, 2000), Chap. 4.

]. Additionally, note that, for simplicity, we only consider the coupling between the eigenmodes in the individual waveguides having the minimum transverse wave vectors kx given by Eq. (26).

Table 1. Numerical results of the Coupling Coefficient |κ|, the Shortest Coupling Length Lc for a Complete Signal Transfer in Terms of the Vacuum Wavelength λ = 5.77 cm, the Normalized Energy Flux P in Each Isolated Waveguide, and the Weak-Coupling Parameters R 1R 8 for the Four Different Configurations1

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In Table 1, one can clearly see that, with respect to the same geometric parameters w and s, case (i) has the smallest energy flux (i.e., the slowest group velocity of the guided mode), thus resulting in the largest coupling coefficient |κ| which is more than fivefold larger than that of the conventional configuration in case (iv). Case (ii) leads to the second largest coupling coefficient |κ|, while the coupling coefficient |κ| in case (iii) is only slightly larger than that of the conventional configuration. However, it is also seen that cases (ii) and (iii) have the same energy flux, which means that, in these two cases, one should consider more factors to determine why their coupling coefficients |κ| are different. To answer this problem in detail, let us return to Eqs. (19) and (21). As mentioned earlier, the numerators represent the spatial overlap between the electromagnetic fields guided in the two isolated waveguides [i.e., (E A, H A) and (E B, H B)]. Therefore, for the same energy flux in the denominators of Eqs. (19) and (21), we should further examine the spatial overlap of (E A, H A) and (E B, H B). It is well-known that, for an electromagnetic wave, its electric field usually dominates over its magnetic field. Hence, we plot the transverse spatial distributions of the normalized electric fields E A [i.e., EAy(x)] given by Eq. (24) and E B [i.e., EBy(x)] given by Eq. (25) for both cases (ii) and (iii) in Fig. 3, where the common z-dependent propagation term exp(ikzz) in the expressions is ignored. As can clearly be seen in this figure, the refractive-index profile in case (ii) causes that E A and E B are located close to each other [Fig. 3(a)], whereas the refractive-index profile in case (iii) causes that E A and E B are located far away from each other [Fig. 3(b)]. As a result, the spatial overlap between E A and E B in case (ii) is larger than that in case (iii). Hence, case (ii) has a larger coupling coefficient |κ| than case (iii) although both cases have the same energy flux in each isolated waveguide.

Fig. 3 (Color online) Transverse spatial distributions of the normalized E A [i.e., EAy(x)] (solid curve) and E B [i.e., EBy(x)] (dot-dashed curve) with the same kx = 51.61/m given by Eq. (26) in the isolated waveguides A and B, respectively. The vertical dotted lines denote the positions of waveguides A and B. The refractive-index profile in (a) is corresponding to the configuration in Fig. 2(b) [i.e., case(ii)]. The refractive-index profile in (b) is corresponding to the configuration in Fig. 2(c) [i.e., case(iii)]. The different refractive-index profiles cause that E A and E B in (a) are located close to each other, whereas those in (b) are far away from each other. Consequently, one can clearly see that the spatial overlap between E A and E B in case (ii) is larger than that in case (iii).

Note that our case (iii) can be directly analogous to the proposed codirectional coupling structure (i.e., a NIM layer inserted between two parallel conventional waveguides) in Ref. [25

25. S. Xiao, L. Shen, and S. He, “A novel directional coupler utilizing a left-handed material,” IEEE Photon. Technol. Lett. 16, 171–173, (2004).

]. Based on our results, such structure cannot be the best option to obtain the strongest enhancement. Moreover, in case (ii), because there is no NIM in the central cladding region between the two PIM cores, the coupling enhancement in this case apparently cannot be explained by the idea that NIMs can amplify evanescent waves used in Ref. [25

25. S. Xiao, L. Shen, and S. He, “A novel directional coupler utilizing a left-handed material,” IEEE Photon. Technol. Lett. 16, 171–173, (2004).

], but should be attributed solely to the slow-light effect.
Based on our analysis, one can clearly see that case (i) with two PIM cores embedded in the NIM claddings can give rise to the slowest group velocity, thereby leading to the strongest coupling between the two copropagating electromagnetic modes guided in the waveguide-based codirectional coupler [i.e., more than fivefold larger than that in the conventional case (iv)]. Note that because we use the experimentally available parameters to evaluate our scheme, it is reasonable to believe that even stronger enhancement can be achieved in case (i) using better parameters. Additionally, all the weak-coupling parameters R 1R 8 are much smaller than 1, which verifies the validity of the weak-coupling condition in our scheme.

In order to give more examples to indicate the slow-light-enhanced codirectional coupling effect, we also investigate a codirectional coupling structure with two NIM cores surrounded by a normal dielectric PIM [see case (v) in Fig. 4(a)]. To obtain the guided TE eigenmodes, we can design the two-waveguide structure as follows. Using Eq. (27), one can assume that, in case (v), the frequency of the incident field is ω/2π = 4.88 GHz, the wavelength in the vacuum is λ = 6.14 cm, the relative permittivity is εr 2 = −3.20, the relative permeability is μr 2 = −0.70, and thus n2=εr2μr2=1.5. Besides, one can also assume the normal dielectric PIM to be the air with n 1 = n 3 = 1. For comparison, we also construct a conventional codirectional coupler with two PIM cores (n 2 = 1.5, εr 2 = 2.55, and μr 2 = 1) surrounded by the air claddings in case (vi) [Fig. 4(b)]. Both the cases have the same waveguide width w = 8 cm and the same separation distance s = 4 cm. Our numerical results are given in Table 2, where we can clearly see the strong enhancement of the coupling strength (i.e., more than eightfold larger). More importantly, similar to case (ii) studied earlier, because there is only the PIM (i.e., the air) in the central cladding region in case (v), the coupling enhancement in case (v) can again be attributed solely to the slow-light effect.

Fig. 4 (Color online) (a) Refractive index profile of case (v) with two NIM cores surrounded by a normal dielectric PIM, where the NIM cores have εr 2 = −3.20, μr 2 = −0.70, and n2=εr2μr2=1.5, and the normal dielectric PIM is the air with n 1 = n 3 = 1. (b) For comparison, we also consider a conventional codirectional coupling structure corresponding to case (vi), where the PIM cores have εr 2 = 2.55, μr 2 = 1, and n 2 = 1.5, and the claddings are the air. Note that both the cases have the same waveguide width w = 8 cm and the same separation distance s = 4 cm.

Table 2. Numerical Results of the Coupling Coefficient |κ|, the Shortest Coupling Length Lc for a Complete Signal Transfer in Terms of the Vacuum Wavelength λ = 6.14 cm, the Normalized Energy Flux P in Each Isolated Waveguide, and the Weak-Coupling Parameters R 1R 8 for the Two Different Configurations1

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Moreover, since the dielectric and magnetic response of the NIM is dispersive and most of the electromagnetic fields are confined in the NIM cores in case (v), we also consider the influence of the NIM dispersion on the coupling strength in case (v) where the air claddings are assumed to be dispersiveless. In Fig. 5, we indicate the relationship between the coupling length Lc and the frequency ω/2π at the vicinity of 4.88 GHz. By adjusting the frequency of the incident microwave field in case (v) from 4.85 GHz to 4.90 GHz, the coupling length can be reduced by up to nearly two thirds (∼ 64%). This significant reduction in size can be used to further optimize the design of the codirectional couplers with NIMs, thereby constructing more compact devices.

Fig. 5 (Color online) The influence of the NIM dispersion on the coupling length in case (v) shown in Fig. 4(a). The coupling length Lc can be dramatically reduced from 388λ to 140λ when the frequency of the incident wave is increased from 4.85 GHz to 4.90 GHz, where λ = 2π c is the vacuum wavelength and c is the light speed in vacuum. Note that the vacuum wavelength λ is 6.12 cm at 4.90 GHz slightly smaller than 6.19 cm at 4.85 GHz. Therefore, the coupling length can be reduced by up to nearly two thirds (∼ 64%).

4. Discussion

Based on the coupled-mode theory in the weak-coupling approximation, it is clearly shown that the slow-light effect can greatly enhance the coupling strength in the codirectional coupling structures. This finding suggests that the slow light can effectively prolong the interaction time of two guided modes, thus enhancing the coupling strength between the two waveguides. The result in the weak-coupling regime may also inspire the work in the strong-coupling regime. In principle, the strong waveguide coupling in the slow light regime can not only prolong the interaction time between the guided modes, but also result in their large spatial overlap. Therefore, these two factors for the strong coupling in the slow light regime can doubly enhance the coupling strength and thus minimize the coupling length Lc. In order to correctly investigate this situation, the improved coupled-mode formulations could be employed to describe the strong coupling regime [28

28. W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).

].

Moreover, let us consider an extreme case in the weak-coupling regime, i.e., the coupling between two trapped modes in the waveguides. For example, similar to the structures given in Refs. [11

11. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).

, 12

12. A. C. Peacock and N. G. R. Broderick, “Guided modes in channel waveguides with a negative index of refraction,” Opt. Express11, 2502–2510 (2003), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-20-2502. [PubMed]

, 15

15. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature (London) 450, 397–401 (2007).

, 16

16. K. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, “Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures,” Appl. Phys. Lett. 89, 201103 (2006).

], we can construct a codirectional coupling structure with two NIM cores surrounded by a normal dielectric PIM to obtain the trapped modes. Using Eq. (27), one can assume that the frequency of the incident microwave field is ω/2π = 5.03 GHz, the wavelength in the vacuum is λ = 5.97 cm, the relative permittivity of the NIM core is εr 2 = −2.96, the relative permeability of the NIM core is μr 2 = −0.53, and thus n2=εr2μr2=1.25. Besides, one can also assume the normal dielectric PIM in the claddings to be the air with n 1 = n 3 = 1. When the waveguide width is w = 3.714 cm, we can obtain the trapped mode in each individual left-handed wavguide. The total energy flux in each waveguide [i.e., the denominators in Eqs. (19) and (21) and the numerator in Eq. (28)] is close to zero. Thus, the parameters R 1 , 3 , 5 , 7 in Eqs. (14)(17) are not applicable. When the separation distance is s = 11 cm, we can have R 2 , 4 , 6 , 8 ≪ 1 to satisfy the weak-coupling condition. Then, let us return to the original coupled Eqs. (12) and (13). One can find that the terms containing U (or V) in Eq. (12) [or (13)] can be ignored, and thus the two equations for the trapped modes are decoupled in the weak-coupling regime. This result means that the coupled-mode theory in the weak-coupling regime can only describe two spatially separated trapped modes without coupling. In other words, our finding also implies that, if the two waveguides are placed more closely and thus there exist some interactions (i.e., spatial overlap) between the two trapped modes, the zero group velocity will lead to strong coupling between the modes because the interaction time becomes infinite. This goes beyond the weak-coupling approximation. Therefore, as mentioned above, the improved coupled-mode formulations or more advanced numerical methods, such as the finite-difference time-domain methods, could be adopted to precisely describe the strong coupling between two trapped modes.

In our numerical calculations in Section 3, we only consider the coupling between two phase-matched guided modes. However, our theoretical derivations in Section 2 are more general because Eqs. (18)(21) can actually cover the phase-mismatching effect. If the two electromagnetic modes (E A, H A) and (E B, H B) have different propagation constants kz (i.e., δ = kz,Akz,B where δ is the phase-mismatch factor), based on the theoretical derivations in Refs. [20

20. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1989), Chap. 22.

, 21

21. K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Elsevier Academic Press, 2000), Chap. 4.

], the maximum power-coupling efficiency in the phase-mismatched cases can be equivalently defined by
F=|κ|2|κ|2+δ2,
(29)
where |κ| is the coupling coefficient given in Eqs. (22). This equation tells us that the maximum coupled power will decrease with the increase of the phase mismatch as usual. When the phase mismatch is too large (i.e., δ ≫ |κ|), no power will transfer between the two waveguides.

As mentioned earlier, tunable NIMs for terahertz waves [7

7. H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics 2, 295–298 (2008).

] and loss-free NIMs for visible and infrared light [8

8. S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature (London) 466, 735–738 (2010).

10

10. A. Fang, Th. Koschny, and C. M. Soukoulis, “Self-consistent calculations of loss-compensated fishnet metamaterials,” Phys. Rev. B 82, 121102(R) (2010).

] have been accomplished successfully. Our scheme in the microwave regime can be extended to optimize the design of micron-scale optical codirectional coupling devices in integrated photonic circuits. Moreover, in our scheme, we only consider the symmetric configurations for signal switching and splitting. If a small tunable refractive-index difference is introduced into the two cores as discussed in Ref. [20

20. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1989), Chap. 22.

], compact optical multiplexing and demultiplexing devices could also be implemented. More interestingly, the strongly enhanced coupling in our scheme could also be useful for constructing compact waveguide quantum circuits for efficient quantum operations and manipulation directly on-chip [22

22. A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008). [PubMed]

24

24. A. Politi, J. C. F. Matthews, and J. L. O’Brien, “Shor’s quantum factoring algorithm on a photonic chip,” Science 325, 1221 (2009). [PubMed]

]. Additionally, because the “trapped rainbow” effect inherently allows broadband function for slow and trapped light [15

15. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature (London) 450, 397–401 (2007).

19

19. Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. USA 108, 5169–5173 (2011). [PubMed]

], our scheme may have potential applications in high-speed signal processing.

As it is known, the effects of slow and trapped light have been observed in other optical systems, such as electromagnetically induced transparency (EIT) media [29

29. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).

, 30

30. M. D. Lukin, “Colloquium: Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. 75, 457–472 (2003).

]. In particular, significant slowing of light with a group index of 1200 in EIT-based waveguides on a chip has been experimentally demonstrated [31

31. B. Wu, J. F. Hulbert, E. J. Lunt, K. Hurd, A. R. Hawkins, and H. Schmidt, “Slow light on a chip via atomic quantum state control,” Nat. Photonics 4, 776–779 (2010).

]. Inspired by our theoretical studies in this article, it would be certainly intriguing to investigate the phenomenon of coupling enhancement between the slow-light signals guided in two parallel EIT-based waveguides because such enhancement could also facilitate the realization of EIT-based nonlinear quantum information processing and networking at ultralow power levels in integrated photonic structures.

5. Conclusion

Based on the coupled-mode theory under the weak-coupling condition, we have systematically investigated codirectional couplers consisting of two parallel planar waveguides with NIMs in different configurations. Using the experimental setups and parameters already realized in the microwave regime, both our analytical and numerical results clearly indicate that the slow-light effect plays an essential role in the enhancement of the coupling between two copropagating electromagnetic modes guided in the waveguides. The best option to obtain the strongest coupling enhancement is the configuration with two PIM cores embedded in the NIM claddings [i.e., case(i) in Section 3] and vice versa [i.e., case(v) in Section 3] because these configurations can generate the slowest light (i.e., the smallest energy flux) in the waveguides. Such enhancement can greatly shrink the size of codirectional couplers, thus leading to compact devices for optical signal switching, splitting, filtering, multiplexing, and demultiplexing for potential applications in large-scale integrated photonic circuits for both the classical and quantum signal processing and computing. Moreover, due to the inherent broadband properties of the “trapped rainbow” effect in the waveguides with NIMs, high-speed optical devices could also be achieved for signal processing. Additionally, our theoretical studies also imply that the coupling enhancement could be accomplished in other slow-light waveguide systems, such as EIT-based waveguides. The extension of our studies to these systems might lead to compact optical devices with implications for EIT-based nonlinear quantum information processing and networking at ultralow power levels in integrated photonic structures.

Acknowledgments

L. Z. thanks F. Peng for helpful discussions. We acknowledge the financial support from the Ministry of Science and Technology of China (Grants No. 2011CB921901 and 2011CB606405), the National Natural Science Foundation of China, and the China Postdoctoral Science Foundation (Grant No. 20100470362).

References and links

1.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).

2.

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

3.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005). [PubMed]

4.

N. Fang, Z. Liu, T.-J. Yen, and X. Zhang, “Regenerating evanescent waves from a silver superlens,” Opt. Express11, 682–687 (2003), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-7-682. [PubMed]

5.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [PubMed]

6.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). [PubMed]

7.

H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics 2, 295–298 (2008).

8.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature (London) 466, 735–738 (2010).

9.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010). [PubMed]

10.

A. Fang, Th. Koschny, and C. M. Soukoulis, “Self-consistent calculations of loss-compensated fishnet metamaterials,” Phys. Rev. B 82, 121102(R) (2010).

11.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).

12.

A. C. Peacock and N. G. R. Broderick, “Guided modes in channel waveguides with a negative index of refraction,” Opt. Express11, 2502–2510 (2003), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-20-2502. [PubMed]

13.

K. Halterman, J. Elson, and P. Overfelt, “Characteristics of bound modes in coupled dielectric waveguides containing negative index media,” Opt. Express11, 521–529 (2003), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-6-521. [PubMed]

14.

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,” Phys. Rev. B 73, 085104 (2006).

15.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature (London) 450, 397–401 (2007).

16.

K. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, “Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures,” Appl. Phys. Lett. 89, 201103 (2006).

17.

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95, 071111 (2009).

18.

V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Experimental observation of the trapped rainbow,” Appl. Phys. Lett. 96, 211121 (2010).

19.

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. USA 108, 5169–5173 (2011). [PubMed]

20.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1989), Chap. 22.

21.

K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Elsevier Academic Press, 2000), Chap. 4.

22.

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008). [PubMed]

23.

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009).

24.

A. Politi, J. C. F. Matthews, and J. L. O’Brien, “Shor’s quantum factoring algorithm on a photonic chip,” Science 325, 1221 (2009). [PubMed]

25.

S. Xiao, L. Shen, and S. He, “A novel directional coupler utilizing a left-handed material,” IEEE Photon. Technol. Lett. 16, 171–173, (2004).

26.

T. F. Krauss, “Why do we need slow light?” Nat. Photonics 2, 448–450 (2008).

27.

J. B. Khurgin, “Slow light in various media, a tutorial,” Adv. Opt. Photon. 2, 287–318 (2010).

28.

W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).

29.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).

30.

M. D. Lukin, “Colloquium: Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. 75, 457–472 (2003).

31.

B. Wu, J. F. Hulbert, E. J. Lunt, K. Hurd, A. R. Hawkins, and H. Schmidt, “Slow light on a chip via atomic quantum state control,” Nat. Photonics 4, 776–779 (2010).

OCIS Codes
(130.2790) Integrated optics : Guided waves
(130.3120) Integrated optics : Integrated optics devices
(230.7390) Optical devices : Waveguides, planar
(350.3618) Other areas of optics : Left-handed materials

ToC Category:
Optical Devices

History
Original Manuscript: March 1, 2011
Revised Manuscript: April 14, 2011
Manuscript Accepted: April 30, 2011
Published: May 9, 2011

Citation
L. Zhao and Wenhui Duan, "Slow-light-enhanced codirectional couplers with negative index materials," Opt. Express 19, 10088-10101 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-11-10088


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References

  1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
  2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [PubMed]
  3. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005). [PubMed]
  4. N. Fang, Z. Liu, T.-J. Yen, and X. Zhang, “Regenerating evanescent waves from a silver superlens,” Opt. Express 11, 682–687 (2003), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-7-682 . [PubMed]
  5. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [PubMed]
  6. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). [PubMed]
  7. H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics 2, 295–298 (2008).
  8. S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature (London) 466, 735–738 (2010).
  9. S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105, 127401 (2010). [PubMed]
  10. A. Fang, Th. Koschny, and C. M. Soukoulis, “Self-consistent calculations of loss-compensated fishnet metamaterials,” Phys. Rev. B 82, 121102(R) (2010).
  11. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).
  12. A. C. Peacock and N. G. R. Broderick, “Guided modes in channel waveguides with a negative index of refraction,” Opt. Express 11, 2502–2510 (2003), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-20-2502 . [PubMed]
  13. K. Halterman, J. Elson, and P. Overfelt, “Characteristics of bound modes in coupled dielectric waveguides containing negative index media,” Opt. Express 11, 521–529 (2003), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-6-521 . [PubMed]
  14. K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,” Phys. Rev. B 73, 085104 (2006).
  15. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature (London) 450, 397–401 (2007).
  16. K. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, “Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures,” Appl. Phys. Lett. 89, 201103 (2006).
  17. X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95, 071111 (2009).
  18. V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Experimental observation of the trapped rainbow,” Appl. Phys. Lett. 96, 211121 (2010).
  19. Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. USA 108, 5169–5173 (2011). [PubMed]
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