## Resonant coupling from a new angle: coherent control through geometry |

Optics Express, Vol. 21, Issue 14, pp. 16504-16513 (2013)

http://dx.doi.org/10.1364/OE.21.016504

Acrobat PDF (1283 KB)

### Abstract

We demonstrate that interference of absorption pathways can be used to control resonant coupling of light to guided modes in a manner analogous to quantum coherent control or electronically induced transparency. We illustrate the control of resonant coupling that interference affords using a plasmonic test system where tuning the phase of a grating is sufficient to vary the transfer of energy into the surface plasmon polariton by a factor of over 10^{6}. We show that such a structure could function as a one-way coupler, and present a simple explanation for the underlying physics.

© 2013 OSA

## 1. Introduction

1. J. M. Fraser, A. I. Shkrebtii, J. E. Sipe, and H. M. van Driel, “Quantum interference in electron-hole generation in noncentrosymmetric semiconductors,” Phys. Rev. Lett. **83**, 4192–4195 (1999) [CrossRef] .

2. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today **50**, 36–42 (1997) [CrossRef] .

3. W. Wan, Y. Chong, L. Ge, H. Noh, A. Douglas Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science **311**, 889–892 (2011) [CrossRef] .

4. B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Norlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Materials **9**, 707–715 (2010) [CrossRef] .

6. L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nature Photon. **4**, 182–187 (2010) [CrossRef] .

## 2. Single component coupling

*k*parallel to the surface, to a final mode, associated with

_{i}*k*, that occurs when an integer multiple of the grating wavevector,

_{f}*G*, spans their difference, where

*m*is an integer and

*G*= 2

*π*/Λ for a grating with period Λ. We sketch out this process in Fig. 1(a), showing two pathways for the coupling: for the top pathway two

*G*transitions are required to couple to the guided mode [

*m*= 2 in Eq. (1)], while for the bottom pathway only one 2

*G*transition is needed (

*m*= 1). Physically, the difference between the two cases would be the period of the grating, which would have to double for the situation shown on top, relative to the bottom. We explicitly note that if either, or both, of

*k*and

_{i}*k*represent a guided mode such as a SPP then the coupling is resonant; in this case, even for an arbitrary grating profile, only the component that fulfills Eq. (1) with the smallest possible

_{f}*m*need be considered to accurately model the coupling, even for relatively large amplitude gratings with coupling efficiencies in excess of 80% [8

8. N. Rotenberg and J. E. Sipe, “Analytic model of plasmonic coupling: Surface relief gratings,” Phys. Rev. B **83**, 045416 (2011) [CrossRef] .

9. J. Chandezon, M. T. Dupuis, G. Cornet, and D. J. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” Opt. Soc. Am. **72**, 839–846 (1982) [CrossRef] .

12. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A **12**, 1077–1086 (1995) [CrossRef] .

*k*

_{0}= 2

*π*/

*λ*is the wavevector of light in a vacuum and

*ε*is the complex dielectric function of gold. We first consider the case where only one coupling pathway is available, where

_{m}*λ*= 1500 nm and Λ = 1200 nm or 2400 nm, corresponding to the lower and upper pathways of Fig. 1(a), respectively. Note that when Λ = 2400 nm we expect several diffracted orders, as well as the SPP to result from the grating diffraction as depicted in Fig. 1(b).

*R*

^{(n)}associated with the different reflected orders, using a coordinate transformation method [9

9. J. Chandezon, M. T. Dupuis, G. Cornet, and D. J. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” Opt. Soc. Am. **72**, 839–846 (1982) [CrossRef] .

10. L. Li, J. Chandezon, G. Granet, and J.-P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. **38**, 304–313 (1999) [CrossRef] .

*R*

^{(0)}, (c) and top panel of (d)] and in the 1-order diffraction [

*R*

^{(1)}, bottom of (d)] but, surprisingly, it appears in a different manner the different reflectivities. In the example of the shorter period grating (Λ = 1200 nm) where there are no diffracted orders, energy that would otherwise reflect is coupled to SPPs, and we observe a dip in the reflectivity for angles near −16.7 degrees. A similar transfer of energy to SPPs occurs for the longer period grating (Λ = 2400 nm), where a dip in

*R*

^{(1)}indicates that energy that would otherwise be diffracted is coupled into the SPP. But for the longer period grating the specular reflection is enhanced by the presence of the SPP, as energy flows from the SPP to this order and we observe a peak in

*R*

^{(0)}. This suggests that in the example of the longer period grating we can view both the dip in

*R*

^{(1)}and the peak in

*R*

^{(0)}as a flow of energy between the SPP and the reflected orders, but with a different phase, such that the interference is constructive for

*R*

^{(0)}and destructive for

*R*

^{(1)}.

*h*(

*y*) as and hence, in

*k*-space, the amplitude of a grating component with wavevector

*pG*will be represented by Fourier coefficients which we write as

*h*. To ‘see’ the plasmonic field, the diffracted field must first excite the plasmonic field through interactions with the grating in a +

_{±pG}*G*transition, and then ‘return’ through a −

*G*transition. And, as we have previously shown, the effect of the SPP on the diffracted order will be proportional to

*h*

_{G}h_{−G}, which results in a

*π*phase shift [8

8. N. Rotenberg and J. E. Sipe, “Analytic model of plasmonic coupling: Surface relief gratings,” Phys. Rev. B **83**, 045416 (2011) [CrossRef] .

*G*and hence the effect will be proportional to

*π*phase shift.

*c*is a constant that depends on the geometry and material properties of the system, and likewise for the 1200 nm grating the coupling amplitude is

8. N. Rotenberg and J. E. Sipe, “Analytic model of plasmonic coupling: Surface relief gratings,” Phys. Rev. B **83**, 045416 (2011) [CrossRef] .

*energy*to the SPP is proportional to

## 3. Double harmonic gratings

*a*= 90 nm,

_{G}*a*

_{2G}= 5 nm, and

*φ*= 0 degrees, where the amplitudes have been carefully selected to match the coupling strength of the components. In this case, both transitions from the incident radiation to the SPP shown in Fig. 1(a) are available simultaneously. We emphasize that the two coupling channels – the

*G*and the 2

*G*pathways – excite the same plasmonic mode, albeit with different phases, and hence we expect that they interfere. Indeed, we observe a dispersive line-shape in the reflectivity, which is reminiscent of a Fano resonance, and which clearly demonstrates that the phases of the excited SPPs play an important role in the overall plasmonic coupling. The source of the interference is particularly obvious if we again consider the energy flow into the SPP. When the two pathways are available, then the coupling is proportional to

*n*denotes the different diffracted orders. In this manner, the absorption includes both the direct ohmic losses as well as the energy which is first coupled into the SPPs and therefore does not re-radiate into free-space. Figure 2(b) shows this absorption and allows for the clear separation of the two mechanisms: the direct ohmic losses appear as a relatively flat ambient value of ∼ 0.019, while the plasmonic coupling appears as a sharp spectral bump near −16.7 degrees which peaks at a value ∼ 0.040. The difference between the two values, Δ

*A*, represents the portion of energy which couples to the SPPs. While the interference between the coupling channels can manifest as either an amplitude increase or decrease of the different diffracted modes, the net absorption is always increased due to the plasmonic coupling.

*φ*, as shown in Fig. 3. Even with this simple geometry, it is possible to suppress the energy that flows into SPPs by a factor of over 10

^{6}. For example, for negative angles of incidence near −16.7 degrees [Fig. 3(a)], when the 2

*G*grating component lags behind that of the

*G*grating by almost a quarter period, we find that the SPPs coupled by the two grating components are in phase and Δ

*A*peaks above 0.04. Conversely, it is possible to minimize Δ

*A*to 5 × 10

^{−8}when the 2

*G*component leads by a quarter period and destructive interference quenches the plasmonic coupling. Remarkably, the suppression of the coupling occurs for all angles of incidence, peaking at values of Δ

*A*< 6 × 10

^{−8}as shown in the inset of this figure.

*φ*at 88° and determining the absorption spectrum for both positive and negative angles of incidence. As Fig. 3(b) shows, for this relative phase light that is incident at +16.7 degrees is maximally coupled to SPPs, while almost no coupling occurs for light incident from the negative direction. In essence, we break the coupling symmetry of this structure merely by properly setting the relative phase of the two grating components: when light is incident from the positive direction the coupling

*G*and 2

*G*coupling channels constructively interfere, while for negative incidence the two channels interfere destructively and no coupling occurs. This symmetry breaking is shown in Fig. 3(c), where we map the phase-dependence of Δ

*A*. As expected the structure, and hence the coupling, is symmetric for

*φ*= 0, ±180 degrees. The extreme cases, denoted by

*1*through

*4*, are shown schematically in Fig. 3(d), and are found near

*φ*= ±90 degrees; here we explicitly show when plasmonic coupling occurs (

*1*,

*3*, dark state) and when it does not (

*2*,

*4*, perfect-diffractor), appearing as a sort of plasmonic diode. We attribute the 2 degree offsets with respect to ±90 degrees to the inclusion of higher-order coupling processes in our calculations.

*3*and

*4*of Fig. 3(d), whose corresponding absorption spectra are shown in part (b) of this figure. Their absorption spectra differ greatly, due to SPP excitation for light incidence from the left and not from the right. Thus one might naively expect that the coefficients of reflection in the two situations would also differ greatly and, since the two situations correspond to a swap of source and detector, that this would violate reciprocity.

13. S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on Nonreciprocal light propagation in a silicon photonic circuit,” Science **335**, 38 (2012) [CrossRef] [PubMed] .

*3*) than for negative angles (case

*4*), the specular reflection spectra of the two cases are identical, respecting reciprocity. The ‘missing’ energy can be located in the diffracted orders. In Fig. 4(b), which shows the 1-order reflection, we observe a plasmonic feature for incidence near 16.7 degrees but not near −16.7 degrees. Remarkably, the interference in the coupling processes essentially acts to suppress energy transfer from the specular reflection to the SPP, while maximizing that from the diffracted orders. Not only is it possible to tune the magnitude of the plasmonic coupling using the relative phase (

*φ*) of the two grating components, but varying this parameter also allows us to change which of the diffracted orders provides the energy for the SPP.

*a*and

_{G}*a*

_{2G}, of Eq. (3). Generally, increasing the amplitude of a grating component leads to a corresponding increase of the plasmonic coupling. Consequently, if we wish to effectively interfere the different coupling channels, an increase or decrease of one component must be accounted for with a similar change of the other component. However, since the 2

*G*channel is a first-order coupling mechanism, while the

*G*coupling is a second-order event, there is no linear correspondence to the relative changes of the components. Figure 5 shows the phase-dependent plasmonic absorption for different grating heights As expected, by increasing the

*a*

_{2G}amplitude and suitably matching

*a*we are able to fix the minimum coupling near zero while increasing the maximal coupling. In fact, by increasing

_{G}*a*

_{2G}from 5 to 30 nm (and

*a*from 90 to 208 nm) we triple the plasmonic coupling, from 0.042 to 0.12. However, as this figure shows, the achievable gain in plasmonic coupling saturates with increasing amplitudes. This is perhaps unexpected since single component gratings can couple in excess of 0.9 of the energy to plasmonic modes. However, we note that at the peak, the energy couples from the diffracted modes to the SPP and hence the effectiveness of this coupler is limited by its diffraction efficiency. Further, we hypothesize that for the large grating amplitudes, and in particular for a large

_{G}*a*component, higher order processes can no longer be neglected. That is, with these large amplitudes it might not be possible to optimize both the minimum and maximum achievable coupling with only two grating components, and third or fourth grating harmonics might be required.

_{G}## 4. Historical context

**83**, 045416 (2011) [CrossRef] .

9. J. Chandezon, M. T. Dupuis, G. Cornet, and D. J. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” Opt. Soc. Am. **72**, 839–846 (1982) [CrossRef] .

11. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A **12**, 1068–1076 (1995) [CrossRef] .

*G*component is much larger than that of the 2

*G*component, for example in the cases shown in Figs. 2 and 3

*a*= 90 nm and

_{G}*a*

_{2G}= 5 nm. Hence, in our case we have two coupling pathways of equal amplitude interfering, which affords us the extreme degree of control over the plasmonic coupling that has been demonstrated above. In contrast, for blazed diffraction gratings, where there there is no resonant coupling, the amplitudes of the first couple higher harmonics tend to be comparable to the fundamental component. Here, the higher grating harmonics fine-tune the diffraction. Alternatively, a very asymmetric profile can be used to, for example, maximize the

*h*

_{−G}Fourier component with respect to the

*h*component and in doing so diffract, or even couple [15

_{G}15. N. Bonod, E. Popov, L. Li, and B. Chernov, “Unidirectional excitation of surface plasmons by slanted gratings,” Opt. Express **15**, 11427–11432 (2007) [CrossRef] [PubMed] .

16. W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B **54**, 6227–6244 (1996) [CrossRef] .

*G*) grating component couples directly to the SPP and hence the higher grating harmonics (2

*G*, 3

*G*, etc.), which have smaller amplitudes, are the ones that provide higher order coupling pathways. Again, as with the blazed gratings, the higher harmonics provide a way to

*fine-tune*the plasmonic coupling, in contrast to our work where the coupling pathways of the two harmonics interfere with equal amplitudes. In essence, we introduce the concept of using the phase between the grating components to completely change the way a grating couples light to SPPs, rather than just controlling the fine details.

18. G. Maisons, M. Carras, M. Garcia, O. Parillaud, B. Simozrag, X. Marcadet, and A. De Rossi, “Substrate emitiing index coupled quantum cascade lasers using biperiodic top metal gratings,” Appl. Phys. Lett. **94**, 151104 (2009) [CrossRef] .

16. W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B **54**, 6227–6244 (1996) [CrossRef] .

## 5. Conclusion and outlook

15. N. Bonod, E. Popov, L. Li, and B. Chernov, “Unidirectional excitation of surface plasmons by slanted gratings,” Opt. Express **15**, 11427–11432 (2007) [CrossRef] [PubMed] .

19. I. Dolev, M. Volodarsky, G. Porat, and A. Arie, “Multiple coupling of surface plasmons in quasiperiodic gratings,” Opt. Lett. **36**, 1584–1586 (2011) [CrossRef] [PubMed] .

20. B. le Feber, J. Cesario, H. Zeijlemaker, N. Rotenberg, and L. Kuipers, “Exploiting long-ranged order in quasiperiodic structures for broadband plasmonic excitation,” Appl. Phys. Lett. **98**, 201108 (2011) [CrossRef] .

^{6}is the interference between the different channels. In principle, our design is only limited by the diffraction efficiency of the grating structure, which routinely approaches 0.8 in commercial gratings. This suggests that with optimization, grating couplers based on our paradigm could essentially act like photonic diodes, where energy is only coupled to a specific channel for one direction of incidence. Further, although for our demonstration we used a single bi-harmonic grating, similar control over the plasmonic coupling should be achievable using more easily fabricated structures such as one where the grating harmonics are layered. This control could be made even more powerful if one could actively control the phase of one of the grating components, perhaps in a linear [21

21. N. Rotenberg, M. Betz, and H. M. van Driel, “Ultrafast all-optical coupling of light to surface plasmon polaritons on plain metal surfaces,” Phys. Rev. Lett. **105**, 017402 (2010) [CrossRef] [PubMed] .

22. J. Renger, R. Quidant, N. van Hulst, S. Palomba, and L. Novotny, “Free-space excitation of propagating surface plasmon polaritons by nonlinear four-wave mixing,” Phys. Rev. Lett. **103**, 266802 (2009) [CrossRef] .

18. G. Maisons, M. Carras, M. Garcia, O. Parillaud, B. Simozrag, X. Marcadet, and A. De Rossi, “Substrate emitiing index coupled quantum cascade lasers using biperiodic top metal gratings,” Appl. Phys. Lett. **94**, 151104 (2009) [CrossRef] .

23. C. Ruppert, J. Neumann, J. B. Kinzel, H. J. Krenner, A. Wixforth, and M. Betz, “Surface acoustic wave mediated coupling of free-space radiation into surface plasmon polaritons on plain metal films,” Phys. Rev. B **82**, 081416(R) (2010) [CrossRef] .

24. T. Utikal, M. I. Stockman, A. P. Heberle, M. Lippitz, and H. Giessen, “All-optical control of the ultrafast dynamics of a hybrid plasmonic system,” Phys. Rev. Lett. **104**, 113903 (2010) [CrossRef] [PubMed] .

## Acknowledgment

## References and links

1. | J. M. Fraser, A. I. Shkrebtii, J. E. Sipe, and H. M. van Driel, “Quantum interference in electron-hole generation in noncentrosymmetric semiconductors,” Phys. Rev. Lett. |

2. | S. E. Harris, “Electromagnetically induced transparency,” Phys. Today |

3. | W. Wan, Y. Chong, L. Ge, H. Noh, A. Douglas Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science |

4. | B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Norlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Materials |

5. | M. Born and E. Wolf, |

6. | L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nature Photon. |

7. | H. Raether, |

8. | N. Rotenberg and J. E. Sipe, “Analytic model of plasmonic coupling: Surface relief gratings,” Phys. Rev. B |

9. | J. Chandezon, M. T. Dupuis, G. Cornet, and D. J. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” Opt. Soc. Am. |

10. | L. Li, J. Chandezon, G. Granet, and J.-P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. |

11. | M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A |

12. | M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A |

13. | S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on Nonreciprocal light propagation in a silicon photonic circuit,” Science |

14. | M. C. Hutley, |

15. | N. Bonod, E. Popov, L. Li, and B. Chernov, “Unidirectional excitation of surface plasmons by slanted gratings,” Opt. Express |

16. | W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B |

17. | R. A. Watts, A. P. Hibbins, and J. R. Sambles, “The influence of grating profile on surface plasmon polariton resonances recorded in different diffracted orders,” J. Mod. Opt. |

18. | G. Maisons, M. Carras, M. Garcia, O. Parillaud, B. Simozrag, X. Marcadet, and A. De Rossi, “Substrate emitiing index coupled quantum cascade lasers using biperiodic top metal gratings,” Appl. Phys. Lett. |

19. | I. Dolev, M. Volodarsky, G. Porat, and A. Arie, “Multiple coupling of surface plasmons in quasiperiodic gratings,” Opt. Lett. |

20. | B. le Feber, J. Cesario, H. Zeijlemaker, N. Rotenberg, and L. Kuipers, “Exploiting long-ranged order in quasiperiodic structures for broadband plasmonic excitation,” Appl. Phys. Lett. |

21. | N. Rotenberg, M. Betz, and H. M. van Driel, “Ultrafast all-optical coupling of light to surface plasmon polaritons on plain metal surfaces,” Phys. Rev. Lett. |

22. | J. Renger, R. Quidant, N. van Hulst, S. Palomba, and L. Novotny, “Free-space excitation of propagating surface plasmon polaritons by nonlinear four-wave mixing,” Phys. Rev. Lett. |

23. | C. Ruppert, J. Neumann, J. B. Kinzel, H. J. Krenner, A. Wixforth, and M. Betz, “Surface acoustic wave mediated coupling of free-space radiation into surface plasmon polaritons on plain metal films,” Phys. Rev. B |

24. | T. Utikal, M. I. Stockman, A. P. Heberle, M. Lippitz, and H. Giessen, “All-optical control of the ultrafast dynamics of a hybrid plasmonic system,” Phys. Rev. Lett. |

**OCIS Codes**

(050.1960) Diffraction and gratings : Diffraction theory

(240.6690) Optics at surfaces : Surface waves

(260.5740) Physical optics : Resonance

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: April 24, 2013

Revised Manuscript: June 1, 2013

Manuscript Accepted: June 4, 2013

Published: July 2, 2013

**Citation**

N. Rotenberg, D. M. Beggs, J. E. Sipe, and L. Kuipers, "Resonant coupling from a new angle: coherent control through geometry," Opt. Express **21**, 16504-16513 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-14-16504

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### References

- J. M. Fraser, A. I. Shkrebtii, J. E. Sipe, and H. M. van Driel, “Quantum interference in electron-hole generation in noncentrosymmetric semiconductors,” Phys. Rev. Lett.83, 4192–4195 (1999). [CrossRef]
- S. E. Harris, “Electromagnetically induced transparency,” Phys. Today50, 36–42 (1997). [CrossRef]
- W. Wan, Y. Chong, L. Ge, H. Noh, A. Douglas Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science311, 889–892 (2011). [CrossRef]
- B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Norlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Materials9, 707–715 (2010). [CrossRef]
- M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
- L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nature Photon.4, 182–187 (2010). [CrossRef]
- H. Raether, Surface Plasmons, edited by G. Hohler, (Springer, Berlin, 1988).
- N. Rotenberg and J. E. Sipe, “Analytic model of plasmonic coupling: Surface relief gratings,” Phys. Rev. B83, 045416 (2011). [CrossRef]
- J. Chandezon, M. T. Dupuis, G. Cornet, and D. J. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” Opt. Soc. Am.72, 839–846 (1982). [CrossRef]
- L. Li, J. Chandezon, G. Granet, and J.-P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt.38, 304–313 (1999). [CrossRef]
- M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A12, 1068–1076 (1995). [CrossRef]
- M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A12, 1077–1086 (1995). [CrossRef]
- S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on Nonreciprocal light propagation in a silicon photonic circuit,” Science335, 38 (2012). [CrossRef] [PubMed]
- M. C. Hutley, Diffraction Gratings (Academic Press, New York, 1982).
- N. Bonod, E. Popov, L. Li, and B. Chernov, “Unidirectional excitation of surface plasmons by slanted gratings,” Opt. Express15, 11427–11432 (2007). [CrossRef] [PubMed]
- W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B54, 6227–6244 (1996). [CrossRef]
- R. A. Watts, A. P. Hibbins, and J. R. Sambles, “The influence of grating profile on surface plasmon polariton resonances recorded in different diffracted orders,” J. Mod. Opt.46, 2157–2186 (1999).
- G. Maisons, M. Carras, M. Garcia, O. Parillaud, B. Simozrag, X. Marcadet, and A. De Rossi, “Substrate emitiing index coupled quantum cascade lasers using biperiodic top metal gratings,” Appl. Phys. Lett.94, 151104 (2009). [CrossRef]
- I. Dolev, M. Volodarsky, G. Porat, and A. Arie, “Multiple coupling of surface plasmons in quasiperiodic gratings,” Opt. Lett.36, 1584–1586 (2011). [CrossRef] [PubMed]
- B. le Feber, J. Cesario, H. Zeijlemaker, N. Rotenberg, and L. Kuipers, “Exploiting long-ranged order in quasiperiodic structures for broadband plasmonic excitation,” Appl. Phys. Lett.98, 201108 (2011). [CrossRef]
- N. Rotenberg, M. Betz, and H. M. van Driel, “Ultrafast all-optical coupling of light to surface plasmon polaritons on plain metal surfaces,” Phys. Rev. Lett.105, 017402 (2010). [CrossRef] [PubMed]
- J. Renger, R. Quidant, N. van Hulst, S. Palomba, and L. Novotny, “Free-space excitation of propagating surface plasmon polaritons by nonlinear four-wave mixing,” Phys. Rev. Lett.103, 266802 (2009). [CrossRef]
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