## Extremely sub-wavelength THz metal-dielectric wire microcavities |

Optics Express, Vol. 20, Issue 27, pp. 29121-29130 (2012)

http://dx.doi.org/10.1364/OE.20.029121

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### Abstract

We demonstrate minimal volume wire THz metal-dielectric micro-cavities, in which all but one dimension have been reduced to highly sub-wavelength values. The smallest cavity features an effective volume of 0.4 µm^{3}, which is ~5.10^{−7} times the volume defined by the resonant vacuum wavelength (λ = 94 µm) to the cube. When combined with a doped multi-quantum well structure, such micro-cavities enter the ultra-strong light matter coupling regime, even if the total number of electrons participating to the coupling is only in the order of 10^{4}, thus much less than in previous studies.

© 2012 OSA

## 1. Introduction

1. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from Conductors, and Enhanced Non-Linear Phenomena,” IEEE Trans. Microw. Theory Tech. **47**(11), 2075–2084 (1999). [CrossRef]

3. E. Strupiechonski, G. Xu, N. Isac, M. Brekenfeld, Y. Todorov, A. M. Andrews, C. Sirtori, G. Strasser, A. Degiron, and R. Colombelli, “Sub-diffraction-limit semiconductor resonators operating on the fundamental magnetic resonance,” Appl. Phys. Lett. **100**(13), 131113 (2012). [CrossRef]

4. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics **1**(1), 41–48 (2007). [CrossRef]

*LC*circuit [2

2. M. W. Klein, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Single-slit split-ring resonators at optical frequencies: limits of size scaling,” Opt. Lett. **31**(9), 1259–1261 (2006). [CrossRef] [PubMed]

5. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking Surface Plasmons with Structured Surfaces,” Science **305**(5685), 847–848 (2004). [CrossRef] [PubMed]

7. P. Biagioni, J.-S. Huang, and B. Hecht, “Nanoantennas for visible and infrared radiation,” Rep. Prog. Phys. **75**(2), 024402 (2012). [CrossRef] [PubMed]

9. A. P. Hibbins, J. R. Sambles, C. R. Lawrence, and J. R. Brown, “Squeezing Millimeter Waves into Microns,” Phys. Rev. Lett. **92**(14), 143904 (2004). [CrossRef] [PubMed]

10. Y. Todorov, L. Tosetto, J. Teissier, A. M. Andrews, P. Klang, R. Colombelli, I. Sagnes, G. Strasser, and C. Sirtori, “Optical properties of metal-dielectric-metal microcavities in the THz frequency range,” Opt. Express **18**(13), 13886–13907 (2010). [CrossRef] [PubMed]

11. P. Jouy, Y. Todorov, A. Vasanelli, R. Colombelli, I. Sagnes, and C. Sirtori, “Coupling of a surface plasmon with localized subwavelength microcavity modes,” Appl. Phys. Lett. **98**(2), 021105 (2011). [CrossRef]

10. Y. Todorov, L. Tosetto, J. Teissier, A. M. Andrews, P. Klang, R. Colombelli, I. Sagnes, G. Strasser, and C. Sirtori, “Optical properties of metal-dielectric-metal microcavities in the THz frequency range,” Opt. Express **18**(13), 13886–13907 (2010). [CrossRef] [PubMed]

_{nm0}modes. The fundamental resonance for this system corresponds to two degenerate modes, the TM

_{010}and TM

_{100}. The resonance condition is imposed by the side

*s*of the square, which is approximately equal to λ/2

*n*, where λ is the resonant wavelength, and

*n*is the refractive index of the dielectric core. The typical thickness

*L*of the core is, however, much smaller than the wavelength,

*L*<< λ.

12. C. Ciuti, G. Bastard, and I. Carusotto, “Quantum vacuum properties of the intersubband cavity polariton field,” Phys. Rev. B **72**(11), 115303 (2005). [CrossRef]

13. Y. Todorov, A. M. Andrews, R. Colombelli, S. De Liberato, C. Ciuti, P. Klang, G. Strasser, and C. Sirtori, “Ultrastrong Light-Matter Coupling Regime with Polariton Dots,” Phys. Rev. Lett. **105**(19), 196402 (2010). [CrossRef] [PubMed]

15. M. Geiser, F. Castellano, G. Scalari, M. Beck, L. Nevou, and J. Faist, “Ultrastrong Coupling Regime and Plasmon Polaritons in Parabolic Semiconductor Quantum Wells,” Phys. Rev. Lett. **108**(10), 106402 (2012). [CrossRef] [PubMed]

*V*, as the coupling constant 2Ω

_{R}, the Rabi splitting, increases as 2Ω

_{R}~

*N*

_{e}is the number of electrons in the cavity. Electro-optic devices based on the strong coupling regime are expected to provide an increased radiative efficiency as the characteristic time of the Rabi oscillations,

*T*

_{R}= 2π/Ω

_{R}is much faster than the typical spontaneous emission time [16

16. C. Ciuti and I. Carusotto, “Input-output theory of cavities in the ultrastrong coupling regime: The case of time-independent cavity parameters,” Phys. Rev. A **74**(3), 033811 (2006). [CrossRef]

17. S. De Liberato and C. Ciuti, “Quantum theory of electron tunneling into intersubband cavity polariton states,” Phys. Rev. B **79**(7), 075317 (2009). [CrossRef]

*N*

_{e}in the device, while there are only two bright states corresponding to the polaritons. The injection probability for each bright state is therefore on the order of 1/

*N*

_{e}, which is a very low number since in a typical microcavity

*N*

_{e}~10

^{5}. A possible approach to circumvent this problem is to use the smallest possible cavity volume

*V*, so that the ratio

*N*

_{e}/

*V*is preserved but with a reduced number of electrons

*N*

_{e}.

10. Y. Todorov, L. Tosetto, J. Teissier, A. M. Andrews, P. Klang, R. Colombelli, I. Sagnes, G. Strasser, and C. Sirtori, “Optical properties of metal-dielectric-metal microcavities in the THz frequency range,” Opt. Express **18**(13), 13886–13907 (2010). [CrossRef] [PubMed]

*s*is kept equal to λ/2

*n*. For the smallest of these devices the ratio

*V/λ*is ~5.10

^{3}^{−7}, with the resonant vacuum wavelength λ = 95 µm. Furthermore, we have used these wire cavities to study the light-matter strong coupling regime. The comparison with square patch cavities shows that the Rabi splitting is not reduced in the wire resonators, while the total number of electrons in the system is reduced by more than an order of magnitude.

## 2. Wire microcavities

*L*, wafer-bonded on a gold-coated surface. A thin metallic stripe with length

*s*and width

*w*is deposited on the dielectric, with

*s*>>

*w*. The length

*s*sets the resonant frequency of the fundamental mode TM

_{100}through the formula:In this formula,

*c*is the speed of light, and

*n*is the effective index of the mode, that depends both on the modal guided index of the wire-semiconductor-metal waveguide, and the phase of the effective reflectivity at the open ends of the resonator [10

_{eff}**18**(13), 13886–13907 (2010). [CrossRef] [PubMed]

*n*are close to the bulk GaAs index in the THz domain (λ ~100 µm),

_{eff}*n*= 3.55. In order to obtain resonances in the THz range, the typical strip length is

*s*~10 µm, while the other dimensions are kept very sub-wavelength:

*w*,

*L*< 1 µm. Figure 1(b) shows a dense periodic array of such structures. Nanofabrication has been be carried out using an electron beam lithography facility which includes a laser interferometer controlled stage for ensuring tenth of nanometer precision over millimeter length scales. The overall sample surface is 2x2 mm

^{2}. Figures 1(c)-1(f) show the vertical component of the electric field (

*E*

_{z}) of the fundamental TM

_{100}cavity resonance, for a thickness

*L*= 1 µm (Figs. 1(c,d)) and

*L*= 300 nm (Figs. 1(e,f)) in the (

*x,z*) and (

*y,z*) plane. These simulations were performed with the COMSOL finite element commercial software which resolves for the electromagnetic eigenmodes of the structure. The metallic layers are described with a frequency independent complex refractive index 150 + 250

*i*and the semiconductor layers with an index 3.55. As expected, the field is strongly confined between the two metals, and displays a standing wave-like pattern along the strip,

*E*

_{z}~cos(

*x*π/

*s*). In the case of the

*L*= 1 µm structure, there is a partial leakage in the air and a slight spreading in the

*y*-direction (Figs. 1(c,d)). In the case of the

*L*= 300 nm structure the vertical electric field

*E*

_{z}is quasi homogeneous in the two directions perpendicular to the stripe (

*y*,

*z*). In this case the expression for the effective volume is the geometrical volume of the structure,

*V*=

*swL*= λ

*wL*/(2

*n*). We will experimentally demonstrate that this formula is a very good approximation also for thicker structures, despite the partial leak in the air and the fringing fields around the resonator corners.

_{eff}*d*= 5 µm and

_{x}*d*= 2 µm. The unit cell of the array is

_{y}*Σ*= (

*d*

_{x}+

*s*)x(

*d*

_{y}+

*w*) = 51 μm

^{2}and remains very sub-wavelength. The only propagating diffracted order is therefore the 0th order, corresponding to the specular reflection. Such dense arrays allow increasing the contrast of the reflectivity dips that correspond to the resonant excitation of the cavity modes [10

**18**(13), 13886–13907 (2010). [CrossRef] [PubMed]

*s*= 12 µm, and two sets of samples with different thickness

*L*were used,

*L*= 1 µm and

*L*= 300 nm. In Fig. 2(a) we report the reflectivity results with a lateral dimension of

*w*= 1 µm, while in Fig. 2(b) we have

*w*= 100 nm. In Fig. 2(b), the higher incident angle (45°) for the

*w*= 100 nm wires allowed for a further improvement of the contrast of the reflectivity features, as the contrast increases with the incident angle [10

**18**(13), 13886–13907 (2010). [CrossRef] [PubMed]

_{100}cavity mode as described in Fig. 1. The resonance frequencies for the thinner (

*L*= 300 nm) structures is around 3.5 THz, and equal to 4 THz for the

*L*= 1 µm structures. These values agree very well with the eigenvalues provided by the finite element method: 3.7 THz and 3.9 THz respectively, that correspond to the contour plots in Figs. 1(c-f). Considerable broadening of the reflectivity dips is observed for the micro-cavites with a width of

*w*= 100 nm. Nevertheless, the clear reflectivity dip indicates that the mode is laterally confined by a strip width

*w*which is 3 orders of magnitude smaller than the resonant wavelength. For the smallest structure (

*w*= 100 nm and

*L*= 300 nm) the incoming free-space photon with a wavelength λ = 100µm is squeezed in a volume

*V =*0.4 µm

^{3}that is 5.10

^{−7}times the wavelength cubed λ

^{3}.

*y*-lateral dimension of the stripe leads to spreading effects and an increase of the fringing fields. In order to investigate these effects, we have realized two sets of arrays, with thickness

*L*= 300 nm and

*L*= 1 μm, in which the lateral gap

*d*

_{y}between the wires is varied (see Fig. 1(b)). The gold stripe dimensions are kept constant:

*w*= 1 µm and

*s*= 12 µm. The separation

*d*

_{x}was also fixed to

*d*

_{x}= 3 µm. In Fig. 3(a) , we report the measured resonant frequencies as a function of

*d*

_{y}and in Fig. 3(b) we plot the corresponding effective index

*n*

_{eff}from Eq. (1). We observe that the frequency increases up to

*d*

_{y}= 2μm and then remains constant. This behavior can be simply explained assuming the evanescent coupling between adjacent resonators in a tight binding model [18]. Similar trend has been observed with metallic nanoparticles in the visible, for separations larger than the conductive contact [19

19. T. Atay, J.-H. Song, and A. V. Nurmikko, “Strongly interacting plasmon nanoparticle pairs: from dipole−dipole interaction to conductively coupled regime,” Nano Lett. **4**(9), 1627–1631 (2004). [CrossRef]

*ν*and

_{∞}*ν*correspond to the limit for

_{0}*d*

_{y}going towards ∞ and 0 respectively. The fit provides a typical length scale for the extension of the exponential tails that is

*d*= 1.9 µm and

_{0}*d*= 1.3 µm for the

_{0}*L*= 1 µm and

*L*= 300 nm respectively. This result indicates that the fringing fields decrease as the thickness is reduced, as qualitatively observed in Fig. 1. Note that the asymptotic values of

*ν*(2.7 THz for

_{0}*L*= 300 nm, 3.2 THz for

*L*= 1 μm) correspond to the resonant frequencies that would have been obtained for an infinite waveguide. Indeed in the limit

*d*

_{y}= 0 all wires merge together into an infinite stripe of a width

*s*.

*d*is revealed by the numerical simulations of the

_{0}*L*= 1 µm structure, presented in Figs. 3(c) and 3(d). Figure 3(c) is an enlarged version of the

*E*

_{z}plot of Fig. 1(d), where we have saturated the color map in order to render visible the spreading fields. This figure reveals very important field spreading in the plane of the metallic strip. In Fig. 3(d) we plot the variations of the field

*E*

_{z}as a function of

*y*along to the cuts A an B defined in Fig. 3(c). The cut A correspond to the horizontal plane of the metallic strip, and the cut B corresponds to the region of strong field confinement, between the two metal layers. This plot shows that the field remains strongly confined under the stripe (cut B) and decays very fast away from the double-metal region. On the contrary, along the cut A, the field features a slow decay away from the metallic edges. The variation of

*E*

_{z}in this region is very well fitted by an exponential decay law with

*d*

_{0}= 1.5 µm, in fairly good agreement with the experimental data. Note from Fig. 3(d) that the amplitude of these exponential wings is almost an order of magnitude less than the amplitude of the field confined between the two metals, we therefore consider its contribution to the total energy density to be negligible. This was also confirmed from the numerical computation of the integral

*L*= 1 µm structure and for 97% for the

*L*= 0.3 µm structure.

*n*

_{eff}of the structures, deduced from Eq. (1), is plotted in Fig. 3(b), together with the bulk GaAs index (

*n*= 3.55). This plot indicates that when the cavities are uncoupled, the effective index remains close, but inferior to the bulk index. The difference is stronger for the thicker structure: Δ

*n*= −0.5. This effect comes from a partial leak of the electric field in the air as the thickness is increased, as visible in the simulations in Fig. 1(c), 1(d). When the thickness of the devices is reduced the effective index is increased, reaching values close to the refractive index of the bulk core material due to a reduced leakage in the air, as clearly visible from the simulations in Fig. 1(e), 1(f). Therefore this study indicates that the electric field is well confined under stripe for structures with transverse aspect ratios

*w*/

*L > 1*.

## 3. Intersubband polaritons as a near field probe

13. Y. Todorov, A. M. Andrews, R. Colombelli, S. De Liberato, C. Ciuti, P. Klang, G. Strasser, and C. Sirtori, “Ultrastrong Light-Matter Coupling Regime with Polariton Dots,” Phys. Rev. Lett. **105**(19), 196402 (2010). [CrossRef] [PubMed]

15. M. Geiser, F. Castellano, G. Scalari, M. Beck, L. Nevou, and J. Faist, “Ultrastrong Coupling Regime and Plasmon Polaritons in Parabolic Semiconductor Quantum Wells,” Phys. Rev. Lett. **108**(10), 106402 (2012). [CrossRef] [PubMed]

*z*-axis [20]. Therefore the ISB polarization is an effective tool to probe the dominant

*E*

_{z}component of the mode. The vacuum Rabi splitting is provided by the formula:where

*N*is the number of quantum wells inserted in the microcavity,

_{QW}*N*is the population difference between the fundamental and excited subband of the QW expressed as sheet electron density,

_{2D}*e*is electron charge,

*m**is the effective mass of the electrons,

*ε*is the dielectric constant of the medium, and

*f*is the oscillator strength for the transition. Note that in Eq. (2) the bare Rabi frequency is renormalized by the quantity Ψ

_{12}^{2}, which is a geometrical factor, describing the spread of the electric field [21

21. S. Zanotto, R. Degl’Innocenti, L. Sorba, A. Tredicucci, and G. Biasiol, “Analysis of line shapes and strong coupling with intersubband transitions in one-dimensional metallodielectric photonic crystal slabs,” Phys. Rev. B **85**(3), 035307 (2012). [CrossRef]

*z*-component of the field and the total electromagnetic energy. For a perfect TM

_{0}mode of a double metal waveguide we have Ψ

^{2}= 1, whereas for more complex waveguide structures, that combine dielectric and metallic confinement, the typical value observed is Ψ

^{2}= 0.5 [21

21. S. Zanotto, R. Degl’Innocenti, L. Sorba, A. Tredicucci, and G. Biasiol, “Analysis of line shapes and strong coupling with intersubband transitions in one-dimensional metallodielectric photonic crystal slabs,” Phys. Rev. B **85**(3), 035307 (2012). [CrossRef]

**18**(13), 13886–13907 (2010). [CrossRef] [PubMed]

13. Y. Todorov, A. M. Andrews, R. Colombelli, S. De Liberato, C. Ciuti, P. Klang, G. Strasser, and C. Sirtori, “Ultrastrong Light-Matter Coupling Regime with Polariton Dots,” Phys. Rev. Lett. **105**(19), 196402 (2010). [CrossRef] [PubMed]

^{2}= 1. We then compare the polariton splittings obtained from an identical QW medium, but processed once in square patch cavities, and once in the wire cavities to determine the spread factor Ψ

^{2}of the latter.

_{0.15}Ga

_{0.85}As hetero-structure similar to the design reported in Ref [13

**105**(19), 196402 (2010). [CrossRef] [PubMed]

*N*

_{QW}= 25 quantum wells, with an overall thickness

*L*= 1.3 µm. The heterostructure was probed in multi-pass transmission measurements, that displayed a single 3.86 THz peak at

*T*= 4 K. The peak corresponds to the transition between the two confined subbands of the well, with the electrons occupying dominantly the fundamental subband, with an estimated areal density of

*N*

_{2D}= 1.7 × 10

^{11}cm

^{−2}per quantum well.

*s*between 6 µm and 20 µm were fabricated and measured. The width of the wires was

*w*= 1µm. The results are summarized in Fig. 4 .

*T*= 300

*K*where the effect of the electronic absorption can be neglected. This measurements allow us to determine the minimal polariton splitting that corresponds to the vacuum Rabi splitting provided by Eq. (2) (Fig. 4(c)). From this data we deduce 2Ω

_{R}= 1.38 THz for the square patch cavities and 2Ω

_{R}= 1.29 THz for the wire microcavities. The slightly smaller (−6%) splitting observed for the wire type resonators, indicates more field leakage, with a spread factor of Ψ

^{2}= 0.87. This measurement confirms the strong confinement of the wire microcavities, despite the strongly reduced lateral size. Furthermore, the total number of electrons per well for these resonators was 1.1x10

^{4}– 3.9x10

^{4}electrons per well, which is an order of magnitude less than the typical values of 2x10

^{5}required for square-patch resonators. These results indicate the potential of the concept of wire resonators for exploring the ultra-strong light-matter coupling regime with strongly reduced number of electronic dark states.

## 4. Summary and conclusion

*V*Ψ

_{eff}= V_{0}/^{2}is the effective volume of the mode with

*V*the geometrical volume. We refer to Eq. (4) as a “squeezing factor”, since it describes how a free propagating photon can be squeezed into a microcavity volume. In our wire-cavities only one dimension of the mode is commensurable with the wavelength in the material, therefore Eq. (4) becomes:

_{0}= swL^{2}= 1) of electric field in a TM

_{100}mode with

*n*

_{eff}=

*n*. The dots correspond to the real structures. The values of

*L*,

*w*and Ψ

^{2}used in this plot have been reported in the right upper part of Fig. 5, next to each curve. To take into account the spreading fields we have used Ψ

^{2}= 0.87 for structures with

*L*≥ 1 µm as measured from the polariton splitting (see previous section) . When

*L*= 0.3 µm we estimate Ψ

^{2}= 1.0 due to the higher cavity effective index,

*n*

_{eff}, which increases the confinement. We have also taken into account the variations of the refractive index owing to the dispersion of the semiconductor. The highest value attained for our structures is

*R*= 6100, well above the diffraction limit

*R*= 1 that can be obtained in a dielectric cavity [22

22. E. Peter, P. Senellart, D. Martrou, A. Lemaître, J. Hours, J. M. Gérard, and J. Bloch, “Exciton-Photon Strong-Coupling Regime for a Single Quantum Dot Embedded in a Microcavity,” Phys. Rev. Lett. **95**(6), 067401 (2005). [CrossRef] [PubMed]

23. H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. **96**(9), 097401 (2006). [CrossRef] [PubMed]

*R =*93, and squeezing factors of

*R*= 225 have been recently observed in the THz spectral range [3

3. E. Strupiechonski, G. Xu, N. Isac, M. Brekenfeld, Y. Todorov, A. M. Andrews, C. Sirtori, G. Strasser, A. Degiron, and R. Colombelli, “Sub-diffraction-limit semiconductor resonators operating on the fundamental magnetic resonance,” Appl. Phys. Lett. **100**(13), 131113 (2012). [CrossRef]

24. E. M. Purcell, H. Torrey, and R. Pound, “Resonance Absorption by Nuclear Magnetic Moments in a Solid,” Phys. Rev. **69**(1-2), 674 (1946). [CrossRef]

25. C. Walther, G. Scalari, M. Beck, and J. Faist, “Purcell effect in the inductor-capacitor laser,” Opt. Lett. **36**(14), 2623–2625 (2011). [CrossRef] [PubMed]

26. Y. Todorov, I. Sagnes, I. Abram, and C. Minot, “Purcell Enhancement of Spontaneous Emission from Quantum Cascades inside Mirror-Grating Metal Cavities at THz Frequencies,” Phys. Rev. Lett. **99**(22), 223603 (2007). [CrossRef] [PubMed]

*F*= (6/π

_{p}^{2})

*RQ*, with

*Q*the quality factor of the resonator. For the cavities with the highest

*R*we have

*Q*~5.0, and a very high Purcell factor can be estimated

*F*

_{p}~2x10

^{4}. However, from the practical point of view, we need rather to evaluate the rate of photons collected outside the cavity, which can be estimated as

*F*/

_{p}Q*Q*

_{rad}, where

*Q*

_{rad}represents the radiative quality factor of the wire microcavity. Regarding the structure as a microwave patch antenna, we can evaluate that

*Q*

_{rad}~λ

^{2}/

*wL*[27], and therefore the increased confinement compensates exactly for the decreased radiative losses and there is no net benefit in terms of radiative efficiency. Indeed, in this case

*Q*is dominated by the metal loss, and

*Q*

_{rad}>>

*Q*. The benefit of the reduced cavity volume is more pertinent when regarding the structure as receiving antenna, rather than emitting one. In that case the Purcell factor measures the enhancement of the electromagnetic field energy density in the cavity with respect to the amplitude of the impinging wave. Such structures can therefore be useful for far-infrared detectors, with an increased ratio between photocurrent and dark current. This point will be developed elsewhere.

## Acknowledgments

## References and links

1. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from Conductors, and Enhanced Non-Linear Phenomena,” IEEE Trans. Microw. Theory Tech. |

2. | M. W. Klein, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Single-slit split-ring resonators at optical frequencies: limits of size scaling,” Opt. Lett. |

3. | E. Strupiechonski, G. Xu, N. Isac, M. Brekenfeld, Y. Todorov, A. M. Andrews, C. Sirtori, G. Strasser, A. Degiron, and R. Colombelli, “Sub-diffraction-limit semiconductor resonators operating on the fundamental magnetic resonance,” Appl. Phys. Lett. |

4. | V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics |

5. | J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking Surface Plasmons with Structured Surfaces,” Science |

6. | S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express |

7. | P. Biagioni, J.-S. Huang, and B. Hecht, “Nanoantennas for visible and infrared radiation,” Rep. Prog. Phys. |

8. | M. J. Adams, |

9. | A. P. Hibbins, J. R. Sambles, C. R. Lawrence, and J. R. Brown, “Squeezing Millimeter Waves into Microns,” Phys. Rev. Lett. |

10. | Y. Todorov, L. Tosetto, J. Teissier, A. M. Andrews, P. Klang, R. Colombelli, I. Sagnes, G. Strasser, and C. Sirtori, “Optical properties of metal-dielectric-metal microcavities in the THz frequency range,” Opt. Express |

11. | P. Jouy, Y. Todorov, A. Vasanelli, R. Colombelli, I. Sagnes, and C. Sirtori, “Coupling of a surface plasmon with localized subwavelength microcavity modes,” Appl. Phys. Lett. |

12. | C. Ciuti, G. Bastard, and I. Carusotto, “Quantum vacuum properties of the intersubband cavity polariton field,” Phys. Rev. B |

13. | Y. Todorov, A. M. Andrews, R. Colombelli, S. De Liberato, C. Ciuti, P. Klang, G. Strasser, and C. Sirtori, “Ultrastrong Light-Matter Coupling Regime with Polariton Dots,” Phys. Rev. Lett. |

14. | P. Jouy, A. Vasanelli, Y. Todorov, A. Delteil, G. Biasiol, L. Sorba, and C. Sirtori, “Transition from strong to ultrastrong coupling regime in mid-infrared metal-dielectric-metal cavities,” Appl. Phys. Lett. |

15. | M. Geiser, F. Castellano, G. Scalari, M. Beck, L. Nevou, and J. Faist, “Ultrastrong Coupling Regime and Plasmon Polaritons in Parabolic Semiconductor Quantum Wells,” Phys. Rev. Lett. |

16. | C. Ciuti and I. Carusotto, “Input-output theory of cavities in the ultrastrong coupling regime: The case of time-independent cavity parameters,” Phys. Rev. A |

17. | S. De Liberato and C. Ciuti, “Quantum theory of electron tunneling into intersubband cavity polariton states,” Phys. Rev. B |

18. | U. Rössler, |

19. | T. Atay, J.-H. Song, and A. V. Nurmikko, “Strongly interacting plasmon nanoparticle pairs: from dipole−dipole interaction to conductively coupled regime,” Nano Lett. |

20. | M. Helm, in |

21. | S. Zanotto, R. Degl’Innocenti, L. Sorba, A. Tredicucci, and G. Biasiol, “Analysis of line shapes and strong coupling with intersubband transitions in one-dimensional metallodielectric photonic crystal slabs,” Phys. Rev. B |

22. | E. Peter, P. Senellart, D. Martrou, A. Lemaître, J. Hours, J. M. Gérard, and J. Bloch, “Exciton-Photon Strong-Coupling Regime for a Single Quantum Dot Embedded in a Microcavity,” Phys. Rev. Lett. |

23. | H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. |

24. | E. M. Purcell, H. Torrey, and R. Pound, “Resonance Absorption by Nuclear Magnetic Moments in a Solid,” Phys. Rev. |

25. | C. Walther, G. Scalari, M. Beck, and J. Faist, “Purcell effect in the inductor-capacitor laser,” Opt. Lett. |

26. | Y. Todorov, I. Sagnes, I. Abram, and C. Minot, “Purcell Enhancement of Spontaneous Emission from Quantum Cascades inside Mirror-Grating Metal Cavities at THz Frequencies,” Phys. Rev. Lett. |

27. | P. W. C. Hon, A. A. Tavallaee, Q.-S. Chen, B. S. Williams, and T. Itoh, “Radiation Model for Terahertz Transmission-Line Metamaterial Quantum-Cascade Lasers,” IEEE Trans. THz Sci. Technol. |

**OCIS Codes**

(130.5990) Integrated optics : Semiconductors

(260.3090) Physical optics : Infrared, far

(270.5580) Quantum optics : Quantum electrodynamics

(140.3945) Lasers and laser optics : Microcavities

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: October 23, 2012

Revised Manuscript: December 5, 2012

Manuscript Accepted: December 5, 2012

Published: December 14, 2012

**Citation**

Cheryl Feuillet-Palma, Yanko Todorov, Robert Steed, Angela Vasanelli, Giorgio Biasiol, Lucia Sorba, and Carlo Sirtori, "Extremely sub-wavelength THz metal-dielectric wire microcavities," Opt. Express **20**, 29121-29130 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-27-29121

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### References

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