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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 27 — Dec. 17, 2012
  • pp: 29121–29130
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Extremely sub-wavelength THz metal-dielectric wire microcavities

Cheryl Feuillet-Palma, Yanko Todorov, Robert Steed, Angela Vasanelli, Giorgio Biasiol, Lucia Sorba, and Carlo Sirtori  »View Author Affiliations


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 µm3, 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 104, thus much less than in previous studies.

© 2012 OSA

1. Introduction

Recently, there have been numerous demonstrations of photonic resonators able to confine the light in volumes that are much smaller than the free space wavelength λ. All these resonators are commonly based on metals, but fall into two big categories. The prototype of the first category is the split-ring resonator [1

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

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]

], used as unit cell for meta-materials [4

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

]. Such resonator basically acts as an electronic 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]

], in which there is no wavelength cut- off, as the displacement current arising from the time varying electric field is negligible in this case. The second type of resonator relies on propagation effects. In this case the resonant condition is set by the round trip of a photon bouncing between two boundaries, while in one or both directions perpendicular to light propagation metallic (or plasmonic) boundary conditions permit to shrink the extension of the electromagnetic field in a very sub-wavelength scale [5

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

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

].

In particular, double-metal microcavities are able to support modes that are highly sub-wavelength with respect to the distance between the metal layers [8

8. M. J. Adams, An Introduction to Optical Waveguides, (John Wiley & Sons, Chichester, 1981).

]. This is often exploited in the microwave frequency range to fabricate waveguides and patch antennas [9

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]

]. Similar structures have been used at terahertz frequencies and in the mid-infrared [10

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

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]

]. The reflectivity spectra from arrays of such microcavities display deep reflection minima at resonance frequencies, indicating their ability to strongly absorb an incoming electromagnetic fields regardless its incident direction [10

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]

]. For these experiments, typically square shaped micro-cavities were used, consisting of a metallic back-plane, a dielectric layer and a top metal layer which is patterned into square patches. Each cavity support separate confined TMnm0 modes. The fundamental resonance for this system corresponds to two degenerate modes, the TM010 and TM100. The resonance condition is imposed by the side s of the square, which is approximately equal to λ/2n, 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.

The sub-wavelength confinement of the light field between the metal layers was exploited to reach the light-matter ultra-strong coupling regime [12

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]

] in the THz and MIR frequency range [13

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

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]

]. For these studies, the gap between the metal is filled with doped quantum wells (QWs) displaying a strong intersubband absorption. In this case, the light-matter interaction is enhanced by the strongly subwavelength volume of the resonator V, as the coupling constant 2ΩR, the Rabi splitting, increases as 2ΩR ~Ne/V, where Ne 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, TR = 2π/ΩR 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]

]. However, the obstacle for achieving a device with an increased electro-optic conversion are the so called “dark” states [17

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]

], which have a quantum degeneracy equal to the number of electrons Ne 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/Ne, which is a very low number since in a typical microcavity Ne ~105. A possible approach to circumvent this problem is to use the smallest possible cavity volume V, so that the ratio Ne/V is preserved but with a reduced number of electrons Ne.

In this paper we report on a minimal volume wire cavity resonators in the THz spectral region. These structures are similar to the patch cavities reported in Ref [10

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]

], however now two dimensions of the resonators have been shrunk to highly sub-wavelength values, and only the length of the resonator s is kept equal to λ/2n. For the smallest of these devices the ratio V/λ3 is ~5.10−7, 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

To probe experimentally the microcavity resonances, we have performed reflectivity measurements on dense arrays of structures as those shown in Fig. 1(b). In our setup a polarized beam from a Globar lamp of a Bruker interferometer (IFS66) is focused on the sample and the reflected intensity is measured with a cooled Bolometer detector. The incident angle was either 10° or 45°. The absolute reflectivity is obtained by dividing the spectra on a reference obtained from unpatterned gold mirror. The parameters of the arrays, as defined in Fig. 1(b) were dx = 5 µm and dy = 2 µm. The unit cell of the array is Σ = (dx + s)x(dy + w) = 51 μm2 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

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]

]. The results of the measurements, performed with four different microcavities, are reported in Fig. 2
Fig. 2 Reflectivity spectra for cavities with s = 12 µm and various strip widths w, for core thicknesses L = 1 µm and L = 300 nm. a) w = 1 µm b) w = 100 nm. The insets in (a) and (b) show SEM pictures of the microcavity arrays employed in the experiments.
. The polarization of the incident light was chosen so that the electric field of the incident beam is parallel to the wires, and the incident angle was 10° for Fig. 2(a) and 45° for Fig. 2(b). The length of the stripe was fixed to 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

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]

].

All the spectra clearly display the resonant reflectivity dip that corresponds to the excitation of the fundamental TM100 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 µm3 that is 5.10−7 times the wavelength cubed λ3.

As already seen in Fig. 1(d), the strong reduction of the 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 dy 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 dx was also fixed to dx = 3 µm. In Fig. 3(a)
Fig. 3 (a) Measured resonant frequencies of micro-resonators with s = 12 µm and w = 1 µm, as a function of the lateral wire-wire gap dy (logarithmic scale). The dashed lines are an exponential fit according to lawν=ν(νν0)exp(dy/d0). Circles: L = 1 µm, triangles: L = 300 nm. (b) The corresponding effective index as deduced from Eq. (1). The dashed line corresponds to the bulk GaAs index, n = 3.55. (c) Vertical electric field Ez for the L = 1 µm resonator (identical to Fig. 1(d)) with a color map saturated below the maximum field amplitude, in order to render visible the spreading fields in the y-direction. (d) Logarithmic plots of Ez (normalized to its maximal value E0z) as a function of y along the two cuts A and B indicated in Fig. 3(c).
, we report the measured resonant frequencies as a function of dy and in Fig. 3(b) we plot the corresponding effective index neff from Eq. (1). We observe that the frequency increases up to dy = 2μm and then remains constant. This behavior can be simply explained assuming the evanescent coupling between adjacent resonators in a tight binding model [18

18. U. Rössler, Solid State Theory, (Springer, 2009).

]. 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]

]. The frequency shift Δν is then proportional to the coupling constant that depends on the overlap between the evanescent tails of two adjacent cavities. This picture leads to an exponential law Δv=ν(νν0)exp(dy/d0) where the values of ν and ν0 correspond to the limit for dy going towards ∞ and 0 respectively. The fit provides a typical length scale for the extension of the exponential tails that is d0 = 1.9 µm and d0 = 1.3 µm for the 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 ν0 (2.7 THz for 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 dy = 0 all wires merge together into an infinite stripe of a width s.

The physical meaning of the quantity d0 is revealed by the numerical simulations of the L = 1 µm structure, presented in Figs. 3(c) and 3(d). Figure 3(c) is an enlarged version of the Ez 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 Ez 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 Ez in this region is very well fitted by an exponential decay law with d0 = 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 ε|Ez|2dydzfrom the 2D plots in Figs. 1(d) and 1(f). We have found that the contribution of the double region in this integral accounts for 85% for the L = 1 µm structure and for 97% for the L = 0.3 µm structure.

The effective index neff 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: Δn = −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

When a microcavity mode of the wire-resonators described above is tuned into resonance with the intersubband (ISB) transition of quantum wells inserted inside the microcavity, the electronic polarization and the electromagnetic field enter the strong coupling regime, giving rise to two intersubband polariton states [13

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

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]

]. The energy separation between the polaritons, the Rabi splitting, is proportional to the square root of the spatial overlap between the electronic polarization and the electric field of the cavity mode. The polariton splitting can therefore be used as a precise near field probe of the electromagnetic confinement of the structure. To understand this point one has to recall that the electronic absorption between ISB levels is strongly anisotropic, and it is sensitive only to the electric field component parallel to the growth axis of the wells, the z-axis [20

20. M. Helm, in Intersubband Transitions in Quantum Wells: Physics and Device Applications I, H. C. Liu, F. Capasso, eds. (Academic Press, 2000).

]. Therefore the ISB polarization is an effective tool to probe the dominant Ez component of the mode. The vacuum Rabi splitting is provided by the formula:
ΩR2=Ω02Ψ2        with         Ω02=e2f12N2DNQW4m*εε0L
(2)
where NQW is the number of quantum wells inserted in the microcavity, N2D is the population difference between the fundamental and excited subband of the QW expressed as sheet electron density, e is electron charge, m* is the effective mass of the electrons, ε is the dielectric constant of the medium, and f12 is the oscillator strength for the transition. Note that in Eq. (2) the bare Rabi frequency is renormalized by the quantity Ψ2, 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]

]:
Ψ2=VEz2d3xV(Ex2+Ey2+Ez2)d3x
(3)
This quantity is the ratio between the electromagnetic energy contained in the z-component of the field and the total electromagnetic energy. For a perfect TM0 mode of a double metal waveguide we have Ψ2 = 1, whereas for more complex waveguide structures, that combine dielectric and metallic confinement, the typical value observed is Ψ2 = 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]

].

The structures that are closest to the ideal situation of almost zero field leakage are the stripe or square patch double-metal cavities [10

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]

,13

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]

]. We consider the latter as a reference where Ψ2 = 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 Ψ2 of the latter.

The core material for the cavities was GaAs/Al0.15Ga0.85As hetero-structure similar to the design reported in Ref [13

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]

], with 32 nm wide quantum wells and 20 nm barriers. The structures were modulation doped in the barriers (5nm from the well). They contained a repeat of NQW = 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 N2D = 1.7 × 1011 cm−2 per quantum well.

The minimal energy splitting between the ISB polaritons can be inferred by measuring the polariton frequencies in cavities with variable detuning from the ISB resonance at 3.86 THz. For this purpose, both square patch and wire cavities with different length 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
Fig. 4 (a) Typical polariton resonances obtained in low temperature reflectivity measurements for nearly resonant square (dotted curve) and wire cavity (solid curve) arrays. (b) Polariton frequencies measured for both type of structures as a function of the microcavity resonance. (c) Polariton splitting as a function of the cavity frequency. The minimal polariton splitting corresponds to the vacuum Rabi splitting 2ΩR.
.

Figure 4(a) reports on typical reflectivity spectra measured at low temperature. Instead of a single reflectivity dip, these spectra display two polariton dips corresponding to the cavity-ISB coupled polarization modes. In Fig. 4(b) we report on systematic studies of the polariton frequencies as a function of the frequency of the bare cavity resonance, measured at 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 2ΩR = 1.38 THz for the square patch cavities and 2ΩR = 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 Ψ2 = 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 1.1x104 – 3.9x104 electrons per well, which is an order of magnitude less than the typical values of 2x105 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

In order to summarize the achievement of highly subwavelength volumes for the wire-resonator reported in this work, we use a dimensionless geometrical figure of merit that describes the light confinement:

R=(λ/2n)3Veff
(4)

where Veff = V0/Ψ2 is the effective volume of the mode with V0 = swL 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:

R=neffψ24wLn3λ2
(5)

The large light confinement also means an increased Purcell factor [24

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]

] for the spontaneous emission of sources inserted inside such microcavities. It is therefore interesting to comment on the possibility of such structures to being used for radiative enhancement of THz semiconductor sources [25

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

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]

]. The Purcell factor can be expressed as Fp = (6/π2)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 Fp ~2x104. 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 FpQ/Qrad, where Qrad represents the radiative quality factor of the wire microcavity. Regarding the structure as a microwave patch antenna, we can evaluate that Qrad2/wL [27

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. 2(3), 323–332 (2012).

], 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 Qrad >> 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

We gratefully acknowledge support from the French National Research Agency in the frame of its Nanotechnology and Nanosystems program P2N, Project No. ANR-09-NANO-007 and financial support from the ERC grant “ADEQUATE”.

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. 47(11), 2075–2084 (1999). [CrossRef]

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]

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]

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]

6.

S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express 14(5), 1957–1964 (2006). [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]

8.

M. J. Adams, An Introduction to Optical Waveguides, (John Wiley & Sons, Chichester, 1981).

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]

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]

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. 98(23), 231114 (2011). [CrossRef]

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]

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]

18.

U. Rössler, Solid State Theory, (Springer, 2009).

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]

20.

M. Helm, in Intersubband Transitions in Quantum Wells: Physics and Device Applications I, H. C. Liu, F. Capasso, eds. (Academic Press, 2000).

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]

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]

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]

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. 2(3), 323–332 (2012).

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

  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]
  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]
  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. Photonics1(1), 41–48 (2007). [CrossRef]
  5. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking Surface Plasmons with Structured Surfaces,” Science305(5685), 847–848 (2004). [CrossRef] [PubMed]
  6. S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express14(5), 1957–1964 (2006). [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]
  8. M. J. Adams, An Introduction to Optical Waveguides, (John Wiley & Sons, Chichester, 1981).
  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. Express18(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]
  12. C. Ciuti, G. Bastard, and I. Carusotto, “Quantum vacuum properties of the intersubband cavity polariton field,” Phys. Rev. B72(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]
  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.98(23), 231114 (2011). [CrossRef]
  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]
  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. A74(3), 033811 (2006). [CrossRef]
  17. S. De Liberato and C. Ciuti, “Quantum theory of electron tunneling into intersubband cavity polariton states,” Phys. Rev. B79(7), 075317 (2009). [CrossRef]
  18. U. Rössler, Solid State Theory, (Springer, 2009).
  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]
  20. M. Helm, in Intersubband Transitions in Quantum Wells: Physics and Device Applications I, H. C. Liu, F. Capasso, eds. (Academic Press, 2000).
  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. B85(3), 035307 (2012). [CrossRef]
  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]
  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]
  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.2(3), 323–332 (2012).

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