## Slow-light-enhanced codirectional couplers with negative index materials |

Optics Express, Vol. 19, Issue 11, pp. 10088-10101 (2011)

http://dx.doi.org/10.1364/OE.19.010088

Acrobat PDF (808 KB)

### 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

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]

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]

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]

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. 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]

## 2. Theoretical model

*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, 21], 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.

*isolated*waveguide

*A*, the spatial distributions of its relative permittivity and permeability are given by Similarly, for the

*isolated*waveguide

*B*, we have The propagation of the guided waves in the two isolated waveguides is thus described by Maxwell’s equations, 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**

*) has been removed.*

_{m}*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, 21], 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., 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 where

*ε*=

_{r}*ε*+

_{rA}*ε*–

_{rB}*ε*

_{r}_{3}and

*μ*=

_{r}*μ*+

_{rA}*μ*–

_{rB}*μ*

_{r}_{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, 28], one can obtain where

*ẑ*represents the unit vector in the

*z*direction (see Fig. 1). We can take the scalar product of both sides of Eq. (6) with

**E**

*,*

_{A}**H**

*) and (*

_{A}**E**

*,*

_{B}**H**

*)] 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*

_{B}*x*in the transverse direction of the codirectional coupler (see Fig. 1). The parameters

*κ*and

_{AB}*κ*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 (

_{BA}**E**

*,*

_{A}**H**

*) and (*

_{A}**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, 12, 15, 16], 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 κ*

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]

_{AB}and κ_{BA}between the two waveguides.*k*along the waveguides (i.e.,

_{z}*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 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 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

*L*, the signal will be equally split into the two waveguides.

_{c}## 3. Numerical simulations and results

*A*and

*B*are the TE modes with the electric fields

**E**

*and*

_{A}**E**

*polarized in the*

_{B}*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

*x*

_{1}=

*w*is the width of waveguide

*A*,

*ŷ*is the unit vector in the

*y*direction,

*k*is the transverse wave vector perpendicular to the waveguide in the core region 0 ≤

_{x}*x*≤

*x*

_{1},

*k*

_{0}is the wave vector of the incident optical field in vacuum,

*x*< 0,

*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

*x*

_{2}=

*w*+

*s*,

*x*

_{3}= 2

*w*+

*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**

*and*

_{A}**H**

*. For simplicity, we here omit the expressions. Additionally, to determine the eigenmodes in waveguide*

_{B}*A*, we can use the continuity conditions at

*x*= 0 and

*x*=

*w*, and find which determines the transverse wave vectors

*k*for the discrete eigenmodes guided in waveguide

_{x}*A*. Likewise, for waveguide

*B*, we can obtain the

*same*expression.

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

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]

*ω*/2

_{p}*π*= 10 GHz is the electronic plasma frequency,

*ω*

_{0}/2

*π*= 4 GHz is the magnetic resonance frequency, and

*F*= 0.56 is a factor depending on the structures of NIMs.

*In our following numerical discussions, for simplicity, we fix the refractive index of the NIM used in our scheme to be n*= −1. Using Eq. (27), one can find that the frequency of the incident microwave field is

^{NIM}*ω/*2

*π*= 5.2 GHz, the wavelength in the vacuum is

*λ*= 5.77 cm, the relative permittivity is

**292**, 77–79 (2001). [PubMed]

*ε*

_{r}_{2}= 1.96 and

*μ*

_{r}_{2}= 1). But, the claddings can be made of either PIMs (e.g., air) or NIMs. In this way, one can construct different configurations to elicit a more in-depth understanding regarding how the slow-light effect significantly influences the coupling strength between the two waveguides.

*A*and

*B*have the same width

*w*= 5.5 cm and the separation distance between the two waveguides is

*s*= 5 cm.

- For the configuration shown in Fig. 2(a), we have
*n*_{1}=*n*_{3}= −1 with*ε*_{r}_{1}=*ε*_{r}_{3}= −2.70 and*μ*_{r}_{1}=*μ*_{r}_{3}= −0.37, and*n*_{2}= 1.4 with*ε*_{r}_{2}= 1.96 and*μ*_{r}_{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. - For the configuration shown in Fig. 2(b), we have
*n*_{1}= −1 with*ε*_{r}_{1}= −2.70 and*μ*_{r}_{1}= −0.37,*n*_{2}= 1.4 with*ε*_{r}_{2}= 1.96 and*μ*_{r}_{2}= 1, and*n*_{3}= 1 with*ε*_{r}_{3}= 1 and*μ*_{r}_{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. - For the configuration shown in Fig. 2(c), we have
*n*_{1}= 1 with*ε*_{r}_{1}= 1 and*μ*_{r}_{1}= 1,*n*_{2}= 1.4 with*ε*_{r}_{2}= 1.96 and*μ*_{r}_{2}= 1, and*n*_{3}= −1 with*ε*_{r}_{3}= −2.70 and*μ*_{r}_{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]. - 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}*ε*_{r}_{3}= 1 and*μ*_{r}_{1}=*μ*_{r}_{3}= 1, and*n*_{2}= 1.4 with*ε*_{r}_{2}= 1.96 and*μ*_{r}_{2}= 1.

*κ*| = |

*κ*| = |

_{AB}*κ*| given by Eq. (19) or (21) and the shortest coupling length

_{BA}*L*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

_{c}*P*of the guided mode in each isolated waveguide [11, 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]

*P*of the guided mode in isolated waveguide

_{A}*A*, which is defined as [11, 12

*z*component of the Poynting vector along waveguide

*A*,

**E**

*is given by Eq. (24), and*

_{A}**H**

*can be derived from the Maxwell’s Eqs. (3) and Eq. (24). Due to the symmetry of the configurations, for isolated waveguide*

_{A}*B*, one can obtain

*P*=

_{B}*P*=

_{A}*P*. Moreover, we also numerically verify the weak-coupling parameters

*R*

_{1}−

*R*

_{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

*μ*−

_{rA}*μ*

_{r}_{3}=

*μ*−

_{rB}*μ*

_{r}_{3}= 0. Thus,

*R*

_{2}and

*R*

_{6}are not applicable (N/A) in this case, and the expressions of

*κ*and

_{AB}*κ*given by Eqs. (19) and (21) can be reduced to those of the conventional codirectional couplers given by Refs. [20, 21].

_{BA}*Additionally, note that, for simplicity, we only consider the coupling between the eigenmodes in the individual waveguides having the minimum transverse wave vectors k*

_{x}*given by Eq. (26).*

*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**

*) and (*

_{A}**E**

*,*

_{B}**H**

*)]. Therefore, for the same energy flux in the denominators of Eqs. (19) and (21), we should further examine the spatial overlap of (*

_{B}**E**

*,*

_{A}**H**

*) and (*

_{A}**E**

*,*

_{B}**H**

*). 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*

_{B}**E**

*[i.e.,*

_{A}*E*(

_{Ay}*x*)] given by Eq. (24) and

**E**

*[i.e.,*

_{B}*E*(

_{By}*x*)] given by Eq. (25) for both cases (ii) and (iii) in Fig. 3, where the common

*z*-dependent propagation term exp(

*ik*) in the expressions is ignored. As can clearly be seen in this figure, the refractive-index profile in case (ii) causes that

_{z}z**E**

*and*

_{A}**E**

*are located close to each other [Fig. 3(a)], whereas the refractive-index profile in case (iii) causes that*

_{B}**E**

*and*

_{A}**E**

*are located far away from each other [Fig. 3(b)]. As a result, the spatial overlap between*

_{B}**E**

*and*

_{A}**E**

*in case (ii) is larger than that in case (iii). Hence, case (ii) has a larger coupling coefficient |*

_{B}*κ*| than case (iii) although both cases have the same energy flux in each isolated waveguide.

*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], 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*

_{1}−

*R*

_{8}are much smaller than 1, which verifies the validity of the weak-coupling condition in our scheme.

*ω/*2

*π*= 4.88 GHz, the wavelength in the vacuum is

*λ*= 6.14 cm, the relative permittivity is

*ε*

_{r}_{2}= −3.20, the relative permeability is

*μ*

_{r}_{2}= −0.70, and thus

*n*

_{1}=

*n*

_{3}= 1. For comparison, we also construct a conventional codirectional coupler with two PIM cores (

*n*

_{2}= 1.5,

*ε*

_{r}_{2}= 2.55, and

*μ*

_{r}_{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.*

*L*and the frequency

_{c}*ω/*2

*π*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.

## 4. Discussion

*L*. In order to correctly investigate this situation, the improved coupled-mode formulations could be employed to describe the strong coupling regime [28].

_{c}*ω/*2

*π*= 5.03 GHz, the wavelength in the vacuum is

*λ*= 5.97 cm, the relative permittivity of the NIM core is

*ε*

_{r}_{2}= −2.96, the relative permeability of the NIM core is

*μ*

_{r}_{2}= −0.53, and thus

*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.

**E**

*,*

_{A}**H**

*) and (*

_{A}**E**

*,*

_{B}**H**

*) have different propagation constants*

_{B}*k*(i.e.,

_{z}*δ*=

*k*−

_{z,A}*k*where

_{z,B}*δ*is the phase-mismatch factor), based on the theoretical derivations in Refs. [20, 21], the maximum power-coupling efficiency in the phase-mismatched cases can be equivalently defined by 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.

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. 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]

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]

## 5. Conclusion

## Acknowledgments

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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 |

**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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