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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 5 — Mar. 1, 2010
  • pp: 4829–4837
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Beaming thermal emission from hot metallic bull’s eyes

S. E. Han and D. J. Norris  »View Author Affiliations


Optics Express, Vol. 18, Issue 5, pp. 4829-4837 (2010)
http://dx.doi.org/10.1364/OE.18.004829


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Abstract

We theoretically examine thermal emission from metallic films with surfaces that are patterned with a series of circular concentric grooves (a bull’s eye pattern). Due to thermal excitation of surface plasmons, theory predicts that a single beam of light can be emitted from these films in the normal direction that is narrow, both in terms of its spectrum and its angular divergence. Thus, we show that metallic films can generate monochromatic directional beams of light by a simple thermal process.

© 2010 Optical Society of America

1. Introduction

Electromagnetic waves known as surface plasmons (SPs) can be excited at the surface of a metal [1

1. H. Raether, Surface Plasmons (Springer-Verlag, Berlin, 1988).

]. These waves propagate at the interface with an intensity that decays both in the bulk metal and the surrounding medium. Because they can occur at optical frequencies, they allow interesting interactions with light. In particular, when light is coupled to SPs, the optical field can be concentrated in extremely small volumes. This has implications for applications from sensors to solar energy [2

2. A. Polman, “Plasmonics applied,” Science 322, 868–869 (2008). [CrossRef] [PubMed]

]. Earlier, this effect was mostly applied in spectroscopic techniques such as surface enhanced Raman spectroscopy, where intense local fields can provide high sensitivity. More recently, the ability to couple light to SPs has been examined for reducing optical circuit elements (waveguides, switches, etc.) to sizes smaller than the optical wavelength [3

3. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003). [CrossRef] [PubMed]

, 4

4. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006). [CrossRef] [PubMed]

]. As a result, the field of plasmonics has arisen to study how man-made metallic structures can control the generation and manipulation of SPs.

This was encouraged in part by the unexpected observation of extraordinary optical transmission through thin metal films perforated with an array of sub-wavelength holes [5

5. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998). [CrossRef]

]. Because the holes are periodically spaced, diffraction can excite SPs when one side of the film is illuminated. The SPs can then transmit energy through the holes and re-radiate on the opposite side of the film. This can lead to transmission much higher than would be expected for sub-wavelength holes. Later, the same effect was observed when a single hole in a metal film was surrounded by circular concentric grooves patterned on both sides of the film [6

6. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. García-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002). [CrossRef] [PubMed]

]. In this structure, referred to as a bull’s eye, the SPs interact with the grooves to couple in and out of the film. More importantly, even with a sub-wavelength hole, the bull’s eye produces an output beam that is directional [6

6. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. García-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002). [CrossRef] [PubMed]

, 7

7. L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90, 167401 (2003). [CrossRef] [PubMed]

], a useful property.

However, in these experiments light was utilized to create the SPs. Indeed, most of plasmonics uses an optical source. An alternative that has only begun to be explored in these devices is thermal excitation of SPs [8

8. Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P. A. Lemoine, K. Joulain, J. P. Mulet, Y. Chen, and J. J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature 444, 740–743 (2006). [CrossRef] [PubMed]

]. For example, SPs could be produced electrically by heating either the entire structure or a specific location within it. Because plasmonic devices can be small, very low power would be required. Through proper design, the resulting SPs could be launched with specific properties. Additional flexibility and new phenomena could result.

As one example, here we theoretically study thermal emission from metallic bull’s eyes. Our structures are similar to those in Ref. [6

6. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. García-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002). [CrossRef] [PubMed]

] except we treat a tungsten film that is patterned on only one side and without the central hole. We find that when such a bull’s eye is heated a directional monochromatic beam should be emitted in the normal direction. Thus, a simple thermal process should provide a novel source of infrared radiation.

Prior work has considered the ability of patterned surfaces [9–13

9. P. J. Hesketh, J. N. Zemel, and B. Gebhart, “Organ pipe radiant modes of periodic micromachined silicon surfaces,” Nature (London) 324, 549–551 (1986). [CrossRef]

] or periodically structured solids [14–19

14. C. M. Cornelius and J. P. Dowling, “Modification of Planck blackbody radiation by photonic band-gap structures,” Phys. Rev. A 59, 4736–4746 (1999). [CrossRef]

] to modify the emissivity of a material. Most efforts have aimed to obtain emission features that are sharper than the broad black-body-like spectrum expected for unstructured solids. Indeed, narrow spectra have been reported, particularly for simple surface gratings. For example, one-dimensionally (1D) periodic grooves in a metal film can lead to a sharp emission peak. However, its wavelength depends on the propagation direction of the emission [11

11. J. J. Greffet, R. Carminati, K. Joulain, J. P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature (London) 416, 61–64 (2002). [CrossRef] [PubMed]

]. Consequently, highly directional monochromatic beams of thermal emission have not been obtained.

2. Origin of Nearly Monochromatic Beaming of Thermal Emission

To consider if a bull’s eye can provide such a beam, we examined these structures theoretically. This can be helpful since, as seen in simple gratings [9

9. P. J. Hesketh, J. N. Zemel, and B. Gebhart, “Organ pipe radiant modes of periodic micromachined silicon surfaces,” Nature (London) 324, 549–551 (1986). [CrossRef]

, 10

10. A. Heinzel, V. Boerner, A. Gombert, B. Blasi, V. Wittwer, and J. Luther, “Radiation filters and emitters for the NIR based on periodically structured metal surfaces,” J. Mod. Opt. 47, 2399–2419 (2000).

], the exact structure of the grooves can strongly influence the emission properties. However, because the bull’s eye lacks translational symmetry, standard techniques for electromagnetic simulations of periodic structures are inefficient. Thus, such techniques have only been applied to bull’s eyes with a small number of circular grooves [20

20. H. Caglayan, I. Bulu, and E. Ozbay, “Extraordinary grating-coupled microwave transmission through a subwavelength annular aperture,” Opt. Express 13, 1666–1671 (2005). [CrossRef] [PubMed]

]. For a more thorough study, we assume the opposite limit of very large bull’s eyes with a constant shape and size of the grooves. For circularly symmetric structures, it is convenient to expand the electric field E as

Ejrϕz=nEj,nrzeinϕ,
(1)

where j = r, ϕ, z are cylindrical coordinates and n is the angular momentum quantum number. By defining the electric fields E±,n as

E±,n=Eϕ,n±iEr,n,
(2)

we can write the Helmholtz equation as [21

21. E. Popov, M. Nevière, A.-L. Fehrembach, and N. Bonod, “Optimization of plasmon excitation at structured apertures,” Appl. Opt. 44, 6141–6154 (2005). [CrossRef] [PubMed]

]

[2r2+1rr(n±1)2r2+k2+2z2]E±,nrz=0,
(3)

where k is the wavevector. When kr ≫ ∣n ± 1∣, Eq. (3) becomes

[2r2+2z2+k2]E±,nrz0.
(4)

If we replace r by x in Eq. (4), it becomes equivalent to the Helmholtz equation in Cartesian coordinates with 2/∂y2 = 0. This implies that bull’s eye structures can be approximated by linear gratings when kr ≫ ∣n ± 1∣. Note that this condition is easier to satisfy for a smaller angle of incidence because n = ±1 for normal incidence and ∣n∣ increases as the incidence angle increases. Throughout the following, we consider the limit kr ≫ ∣n ± 1∣. In addition, we assume that the size of our bull’s eyes is much larger than the SP coherence length. In this case, we can neglect contributions from the boundary and center of the bull’s eye and approximate the structure with a series of linear gratings oriented around the center. The optical properties can then be determined by averaging the response of linear gratings evenly distributed in all directions in the plane. Specifically, we will average the absorptivity, as it is equivalent to emissivity according to Kirchhoff’s law [22

22. S. E. Han, “Theory of thermal emission from periodic structures,” Phys. Rev. B 80, 155108 (2009). [CrossRef]

].

Fig. 1. Calculated absorptivity versus in-plane wavevector for unpolarized light at λ of (a) 4.078, (b) 3.502, and (c) 3.069μm for a tungsten 1D grating ruled along y and periodic in x with period a = 3.5μm. The circular plot boundary is k = 2π/λ. The cross-section of the grooves is rectangular with a depth of 165 nm and a width of 2.625 μm.

We can visualize this with Fig. 1, which displays a contour plot of the absorptivity versus in-plane wavevector for a 1D tungsten grating [23

23. J. B. Pendry and A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett.69, 2772–2775 (1992). A unit cell is discretized by a 80×80 mesh. For 25°C, we used the dielectric function for tungsten from D. W. Lynch and W. R. Hunter, in Handbook of Optical Constants of Solids, edited by E. D. Palik (Academic Press, Orlando, 1985). [CrossRef] [PubMed]

]. The grating is ruled along y and periodic in x with period a = 3.5μm. Strong absorption will occur when the incident light satisfies the momentum matching condition for coupling to SPs [24

24. F. Marquier, C. Arnold, M. Laroche, J. J. Greffet, and Y. Chen, “Degree of polarization of thermal light emitted by gratings supporting surface waves,” Opt. Express 16, 5305–5313 (2008). [CrossRef] [PubMed]

], that is, when the incident light gains just enough parallel momentum from the grating to propagate parallel to the surface. The curves of high intensities in Fig. 1 show when this condition is satisfied. When shifted by an integer multiple of the reciprocal lattice vector of the grating (2π/a), these curves lie on the circular boundary of the plot. The boundary represents light with k = 2π/λ where k is the wavevector parallel to the xy-plane and λ is the optical wavelength. Three cases are shown in Fig. 1: (a) λ > a, (b) λ ~ a, and (c) λ < a. When λ is close to a, the plot has two dots at kx = ±2π/a and two arcs meeting at the origin.

Using such absorptivity vs. wavevector plots for 1D gratings, we can determine the absorptivity due to SPs for large bull’s eyes by averaging around a circle with radius k = (k2x + k2y)1/2. Since k can be converted to an angle θ from the surface normal through k = 2πsinθ/λ, we can extract the angular dependence of the absorptivity by considering a series of k circles that increase in radius from zero to the plot boundary. From Fig. 1, we see that for λ > a and λ < a the absorptivity will be negligible when the k circle is very near the origin as no absorption curves exist there. As k increases, the circle will eventually intersect the curves that lie closest to the origin. This will lead to a feature in the absorptivity at a given θ. However, because we average around the entire circumference of the k circle, such a feature will be weak. Further, it should become weaker as the k circle increases in radius. Similarly, because the first intersection between the circle and the absorption curve moves away from the origin as λ diverges from a, the contribution to the absorptivity will also decrease away from the λa resonance condition. For the special case of resonance [Fig. 1(b)], the two absorption curves cross at the origin. Thus, averaging over a very small k circle near the origin will lead to a large contribution to the absorptivity. (Note that Fig. 1(b) has a very small dot of high intensity at the center of the k circle.) Consequently, on resonance the bull’s eye should exhibit an intense peak near θ = 0 and will be small otherwise. This qualitative analysis suggests that the absorptivity due to SPs in the bull’s eye will decrease as: (i) θ increases and (ii) λ diverges from a. Moreover, a sharp absorption peak should be observed when the surface is illuminated near normal incidence at λa. Due to Kirchhoff’s law, this implies a highly directional monochromatic beam will be emitted in the normal direction when the bull’s eye is heated. The maximum intensity of this beam will be approximately half that of a black body at the same temperature because SPs couple only to one polarization. We also note that this beaming behavior arises from the circular structure of the bull’s eye and does not occur in linear gratings.

Fig. 2. Calculated (a) emissivity spectra at various angles θ from the surface normal and (b) angular dependence of emissivity at the peak maximum λ = 3.502 μm for a tungsten bull’s eye structure with groove parameters as in Fig. 1. The dielectric function for 25 °C was used. The spectrum at θ = 0 is enlarged in the inset of (a).

3. Shallow Grooves

For a quantitative treatment we need to set the depth of the grooves. We started by considering shallow grooves, i.e. those which have a depth much smaller than the wavelength but still sufficiently deep to couple to the incident light strongly. Following the approach described above, we calculated the expected emissivity for large tungsten bull’s eyes with a = 3.5 μm. We varied the width and depth of the rectangular-shaped grooves to minimize the spectral width of the emission feature while keeping the emissivity above 0.5. When the groove depth is less than 250 nm, the structural parameters listed in Fig. 1 lead to our best results. By averaging over k circles we obtained spectra for this structure for various angles, as shown in Fig. 2(a). As predicted, the calculated emissivity for θ ≠ 0 exhibits weak features both above and below λ = a that increase in separation with increasing angle. More importantly, for θ = 0 a sharp peak appears for λa with a spectral width at half maximum of less than 1 nm [Fig. 2(a) inset], which represents a quality factor, Q = λλ, of 5130. Figure 2(b) shows the calculated angular dependence of the emissivity at the peak wavelength λ = 3.502 μm. The angular width at half maximum is only 0.021°, which is about seven times narrower than that reported for a 1D tungsten grating emitting near 4μm [12

12. M. Laroche, C. Arnold, F. Marquier, R. Carminati, J. J. Greffet, S. Collin, N. Bardou, and J. L. Pelouard, “Highly directional radiation generated by a tungsten thermal source,” Opt. Lett. 30, 2623–2625 (2005). [CrossRef] [PubMed]

]. Previously, it was noted that this 1D result is already comparable to the angular width of a typical He-Ne laser of 0.12° at λ = 3.39 μm [12

12. M. Laroche, C. Arnold, F. Marquier, R. Carminati, J. J. Greffet, S. Collin, N. Bardou, and J. L. Pelouard, “Highly directional radiation generated by a tungsten thermal source,” Opt. Lett. 30, 2623–2625 (2005). [CrossRef] [PubMed]

].

The small angular width of our beam implies a long spatial coherence on a source plane [25

25. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

]. This coherence is caused by the SPs and their coupling to the propagating beam [11

11. J. J. Greffet, R. Carminati, K. Joulain, J. P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature (London) 416, 61–64 (2002). [CrossRef] [PubMed]

, 12

12. M. Laroche, C. Arnold, F. Marquier, R. Carminati, J. J. Greffet, S. Collin, N. Bardou, and J. L. Pelouard, “Highly directional radiation generated by a tungsten thermal source,” Opt. Lett. 30, 2623–2625 (2005). [CrossRef] [PubMed]

]. The coherence length LSP of the SPs for our structure is estimated to be λθ = 2692λ which would be the largest SP coherence length reported for this wavelength region. Indeed, this is longer than LSP for a flat surface, 1/Im(k) = 312λ. Recently, it was discussed how LSP of a grating can be larger than that for a flat surface [26

26. N. Dahan, A. Niv, G. Biener, Y. Gorodetski, V. Kleiner, and E. Hasman, “Enhanced coherency of thermal emission: Beyond the limitation imposed by delocalized surface waves,” Phys. Rev. B 76, 045427 (2007). [CrossRef]

, 27

27. G. Biener, N. Dahan, A. Niv, V. Kleiner, and E. Hasman, “Highly coherent thermal emission obtained by plasmonic bandgap structures,” Appl. Phys. Lett. 92, 081913 (2008). [CrossRef]

]. We also caution that other radiating modes unrelated to the SPs can contribute and decrease the coherence length of the overall field [22

22. S. E. Han, “Theory of thermal emission from periodic structures,” Phys. Rev. B 80, 155108 (2009). [CrossRef]

].

The above calculation utilized the dielectric function of tungsten at 25°C. Because we are interested in thermal emission of bull’s eyes that are at elevated temperatures, we also considered the thermal expansion of the structures and changes in the dielectric function of the metal. For the optimized structure, Fig. 3 shows the expected peak wavelength, quality factor Q, and angular width versus temperature. The peak wavelength λmax increases with temperature due to thermal expansion. The total shift is about 40 nm between 25 and 2000°C, which indicates that fine temperature tuning of λmax should be possible. The quality factor of the beam decreases with increasing temperature due to changes in the dielectric function, which shortens the SP life time. However, even at 2000°C, Q should still be ~ 1000. The angular width Δθ of the beam shows the opposite behavior for the same reason. Figure 3c predicts that Δθ should remain comparable to that of a conventional He-Ne laser (0.12°) even at 1500°C.

Fig. 3. Calculated temperature dependence of (a) the peak wavelength λmax, (b) the quality factor Q, and (c) the angular width Δθ for the tungsten bull’s eye structure with groove parameters as in Fig. 1. (a) and (b) are for the surface normal direction and (c) is at λmax.

4. Deep Grooves

The ability of metallic bull’s eyes to produce thermal beams is not limited to structures with shallow grooves. We also considered 1D metallic gratings with grooves of depth ~ λ/2. In this case, cavity modes are excited inside the grooves and couple coherently to each other by the delocalized SPs. At the frequency of the cavity resonance, the coherence length can be larger than on a flat surface [26

26. N. Dahan, A. Niv, G. Biener, Y. Gorodetski, V. Kleiner, and E. Hasman, “Enhanced coherency of thermal emission: Beyond the limitation imposed by delocalized surface waves,” Phys. Rev. B 76, 045427 (2007). [CrossRef]

]. This leads to a narrow angular width of emission in the plane (xz) perpendicular to the grating (y). However, at other frequencies, the cavity modes are not excited. Consequently, thermal emission of a narrow frequency band can be emitted in a narrow angle in the xz-plane. When these gratings have a bull’s eye pattern, the background emission can be smoother than the bull’s eyes with shallow grooves.

Fig. 4. Calculated (a) emissivity spectra at various angles θ from the surface normal and (b) angular dependence of emissivity at the peak maximum λ = 3.502 μm for a tungsten bull’s eye structure supporting a coupled cavity resonance. The cross-section of the groove is rectangular and the period, depth, and width of the groove are 3.5, 1.825, and 1.925 μm, respectively.

This is demonstrated in Fig. 4, which shows the calculated emissivity for a large tungsten bull’s eye with grooves having a period, depth, and width of 3.5, 1.825, and 1.925 μm, respectively. Due to the cavity resonances, the absorptivity versus in-plane wavevector plot in Fig. 1 will change in several ways. First, the intensities along the arcs will be mostly smaller for the coupled cavities because, in these cases, resonances do not occur at every point along the arcs except at the center of the k circle when λa. Second, absorptivity may be larger at wavevectors not along the arcs due to the increased surface area of the gratings. However, because of the cavity mode, the absorptivity at λa will be strong at the center of the k circles. In Fig. 4, the groove parameters were optimized as before to minimize the spectral width while keeping the emissivity above 0.5. In this case, compared to Fig. 2, both the peak at λ = 3.502 μm for θ = 0 (Q = 2270) and the angular width at half maximum (Δθ =0.036°) are slightly broader. However, the small features observed at off-normal directions in Fig. 2 have disappeared. Even though a small peak at λ = 3.85 μm is growing as the angle increases, the background emission is in general smoother than for the shallow grooves. This is the consequence of using the cavity mode that couples to the normal direction.

Fig. 5. Calculated (a) emissivity spectra at various angles θ from the surface normal and (b) angular dependence of the emissivity at the peak maximum (λ = 557 nm) for a silver bull’s eye structure supporting a coupled cavity resonance. The cross-section of the groove is rectangular and the period, depth, and width of the groove are 550, 280, and 358 nm, respectively.

5. Beaming at Shorter Wavelengths

6. Comparison to Other Structures

7. Conclusion

We have shown that thermal excitation of SPs on bull’s eyes should lead to spectrally narrow beams of light with small angular divergence. These beams arise due to the interaction of SPs with the circular symmetry of the structure. Although the emission intensity is less than conventional lasers due to the limitation imposed by Planck’s radiation law, such beams can still provide useful sources as they have other laser-like qualities. Tungsten structures with shallow grooves should allow very high quality factors (Q ~ 1000) and angular widths less than 0.2° when heated up to 2000°C. Moreover, using deep groove structures, smaller unwanted emission features that do not contribute to the beam can be reduced. As here we restricted our treatment to rectangular grooves, additional optimization may be possible by allowing the cross-section of the grooves to vary. Finally, we found that, although challenging, beaming of thermal emission at visible wavelengths should also be possible by careful choice of metal and design parameters. More broadly, by understanding the physics of heated plasmonic structures, a variety of interesting optical phenomena should be possible.

Acknowledgments

This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-FG02-06ER46438 and used resources at the University of Minnesota Supercomputing Institute. SEH thanks the Samsung Foundation for financial support.

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1.

H. Raether, Surface Plasmons (Springer-Verlag, Berlin, 1988).

2.

A. Polman, “Plasmonics applied,” Science 322, 868–869 (2008). [CrossRef] [PubMed]

3.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003). [CrossRef] [PubMed]

4.

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006). [CrossRef] [PubMed]

5.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998). [CrossRef]

6.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. García-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002). [CrossRef] [PubMed]

7.

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90, 167401 (2003). [CrossRef] [PubMed]

8.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P. A. Lemoine, K. Joulain, J. P. Mulet, Y. Chen, and J. J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature 444, 740–743 (2006). [CrossRef] [PubMed]

9.

P. J. Hesketh, J. N. Zemel, and B. Gebhart, “Organ pipe radiant modes of periodic micromachined silicon surfaces,” Nature (London) 324, 549–551 (1986). [CrossRef]

10.

A. Heinzel, V. Boerner, A. Gombert, B. Blasi, V. Wittwer, and J. Luther, “Radiation filters and emitters for the NIR based on periodically structured metal surfaces,” J. Mod. Opt. 47, 2399–2419 (2000).

11.

J. J. Greffet, R. Carminati, K. Joulain, J. P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature (London) 416, 61–64 (2002). [CrossRef] [PubMed]

12.

M. Laroche, C. Arnold, F. Marquier, R. Carminati, J. J. Greffet, S. Collin, N. Bardou, and J. L. Pelouard, “Highly directional radiation generated by a tungsten thermal source,” Opt. Lett. 30, 2623–2625 (2005). [CrossRef] [PubMed]

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H. Caglayan, I. Bulu, and E. Ozbay, “Extraordinary grating-coupled microwave transmission through a subwavelength annular aperture,” Opt. Express 13, 1666–1671 (2005). [CrossRef] [PubMed]

21.

E. Popov, M. Nevière, A.-L. Fehrembach, and N. Bonod, “Optimization of plasmon excitation at structured apertures,” Appl. Opt. 44, 6141–6154 (2005). [CrossRef] [PubMed]

22.

S. E. Han, “Theory of thermal emission from periodic structures,” Phys. Rev. B 80, 155108 (2009). [CrossRef]

23.

J. B. Pendry and A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett.69, 2772–2775 (1992). A unit cell is discretized by a 80×80 mesh. For 25°C, we used the dielectric function for tungsten from D. W. Lynch and W. R. Hunter, in Handbook of Optical Constants of Solids, edited by E. D. Palik (Academic Press, Orlando, 1985). [CrossRef] [PubMed]

24.

F. Marquier, C. Arnold, M. Laroche, J. J. Greffet, and Y. Chen, “Degree of polarization of thermal light emitted by gratings supporting surface waves,” Opt. Express 16, 5305–5313 (2008). [CrossRef] [PubMed]

25.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

26.

N. Dahan, A. Niv, G. Biener, Y. Gorodetski, V. Kleiner, and E. Hasman, “Enhanced coherency of thermal emission: Beyond the limitation imposed by delocalized surface waves,” Phys. Rev. B 76, 045427 (2007). [CrossRef]

27.

G. Biener, N. Dahan, A. Niv, V. Kleiner, and E. Hasman, “Highly coherent thermal emission obtained by plasmonic bandgap structures,” Appl. Phys. Lett. 92, 081913 (2008). [CrossRef]

28.

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990). [CrossRef]

29.

R. H. Jordan and D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution,” Opt. Lett. 19, 427–429 (1994). [CrossRef] [PubMed]

30.

J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987). [CrossRef] [PubMed]

OCIS Codes
(230.1950) Optical devices : Diffraction gratings
(230.6080) Optical devices : Sources
(240.6680) Optics at surfaces : Surface plasmons

ToC Category:
Optics at Surfaces

History
Original Manuscript: January 4, 2010
Revised Manuscript: February 16, 2010
Manuscript Accepted: February 16, 2010
Published: February 23, 2010

Citation
S. E. Han and D. J. Norris, "Beaming thermal emission from hot metallic bull’s eyes," Opt. Express 18, 4829-4837 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-5-4829


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