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Using Subwavelength Diffraction Gratings to Design Open Electromagnetic Cavities Matthieu Dupré, Mathias Fink, and Geoffroy Lerosey* Institut Langevin, ESPCI ParisTech and CNRS UMR 7587, 1 rue Jussieu, 75005 Paris, France (Received 17 July 2013; published 29 January 2014) In this Letter we propose to use subwavelength diffraction gratings as very good semitransparent mirrors for electromagnetic waves to design open cavities. To do so, we replace part of the walls of a cavity by such a grating. We numerically and analytically link the grating characteristics to the spectral properties of the realized open cavity. Then we demonstrate that the eigenmodes of the cavity can be transmitted perfectly through the grating to the exterior, thereby turning a point source inside the cavity into a very directive source. We investigate the effect of disorder, which leads to isotropic radiation patterns, and perform experiments in the microwave domain in order to support our claims. Finally, we present an example of application of the concept in fundamental physics, by measuring from outside the eigenmodes of a disordered microwave cavity. DOI: 10.1103/PhysRevLett.112.043902
PACS numbers: 41.20.-q, 84.40.-x
Wave propagation in microwave cavities is a major topic of interest in various fields of research ranging from fundamental physics with the notions of quantum fidelity, level repulsion of states, or atomic emission rates [1–7], to electrical engineering, which embraces electromagnetic compatibility and antenna measurements. The behavior of waves in such complex media can be analytically calculated for the case of regular cavities and many statistical approaches have been proposed to predict it in chaotic ones [8–12]. To date, most works devoted to these propagation media concentrated on the study of the wave fields inside the cavities, since the latter are usually 2D or 3D shapes enclosed by metallic walls which conceal the electromagnetic fields perfectly. The usual way to couple such cavities to the exterior consists of carving a hole which supports propagating modes in one of its metallic walls, hence, realizing so-called open microwave cavities in fundamental physics, or cavity backed slot antennas in electrical engineering [13,14]. In the optical domain, open cavities have been studied for a long time, since metals are no longer good conductors. Indeed, at those frequencies, a silver film 100 nm thick constitutes a semitransparent mirror. Similarly high contrast gratings or distributed Bragg filters, both being variants of photonics crystals, have been used recently in vertical-cavity surface-emitting lasers [15–17]. In this Letter, we propose to use subwavelength metallic binary diffraction gratings as semitransparent mirrors to design open cavities. We focus our study on microwave cavities, yet our approach remains valid up to the visible. We show that a subwavelength diffraction grating that closes a microwave cavity becomes transparent at the cavity resonances, and transmits all the energy coming from a source placed inside it, analogous to Wood anomalies or extraordinary optical transmission [18–20]. We first link the spectral properties of such an open cavity to the dimensions of the grating. Then, we numerically and analytically demonstrate that a regular cavity coupled to a subwavelength 0031-9007=14=112(4)=043902(5)
grating can turn a point source into a highly directive wide aperture antenna, while introducing disorder leads to isotropic radiation patterns. We validate our approach with experiments in the microwave range. Finally, we show an example of application of the concept in fundamental physics. Spectral properties.—Coupling a microwave cavity to a subwavelength metallic grating involves various characteristic lengths. The ratio of the typical size of the cavity to the wavelength with an exponent related to the dimensionality of the cavity governs the density of modes of the cavity. The grating period and filling of metal, which we will refer to here as the fill factor, are related to its transmittance through the wavelength as well. We first quantify the effects of the grating dimensions on the spectral properties of the open cavity. At long distances from apertures, it is known that the transmission of a single 2D hole of a radius which is small compared to the wavelength in a thin screen behaves as T ∝ ða=λÞ4 [21,22]. Since a binary metallic grating is an array of such holes, this underlines two noticeable facts: the transmission of the grating is frequency dependent and can be precisely tuned over a given spectral range through its dimensions. It is clear that the Q factor of the cavity, which represents the amount of stored energy divided by the amount of dissipated energy per cycle, can be tuned using the properties of the grating as well. In fact, we can show (see Supplemental Material [23]) that for a 3D cavity and 2D grating, if there are only radiative losses, Q is linked to the transmission of a slot T as QðωÞ ¼
2 ω V cavity ; c NTðωÞ Sgrating
(1)
where V cavity and Sgrating are, respectively, the volume of the cavity and the surface of the grating. Hence, one can
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tune the bandwidth or linewidth of the modes of the open cavity by playing with, for instance, the fill factor of the grating. We verify this fact with a set of simulations realized using a finite element method code (FEM). For the sake of simplicity we study a 2D rectangular cavity of infinite dimension in the vertical direction, made out of perfectly electric conductor, with a point source placed inside. Only one wall is replaced by a grating [Fig. 1(a)]. The electric field is polarized along the grating slots, the wavelength λ varies between 3 and 30 cm for frequencies from 10 to 1 GHz. We note Lx and Ly the dimensions of the cavity, Λ the period of the grating, N and a the number and width of the grating slots. These define the fill factor r of the binary metallic grating: a ¼ ð1 − rÞΛ. We first study the transmission in free space of the grating whose dimensions are given in Fig. 1. The results clearly indicate that the ratio a=λ defines the transmission of the grating: transmissions from 10−5 to 10−1 are achievable [Fig. 1(b)]. The exponent found is 2 instead of 4 due to the dimensionality of the holes, which is 1 rather than 2 in the Bethe formula [22,24]. We next study the influence of the fill factor on the Q factor of the fundamental mode of the cavity (around 530 MHz), whose dimensions are given in Fig. 1. FEM simulation results are presented which display the energy spectrum of the cavity near its fundamental resonance for different fill factors [Fig. 1(c)]. From those simulations, the Q factor is extracted and plotted on Fig. 1(d) and ranges from 400 to 6000, while r varies only from 1% to 30%.
As r is increased, Q is increased and the fundamental resonance is slightly shifted closer to its zero loss theoretical value. Overall, this figure shows that metallic subwavelength gratings can make very good and simple semitransparent mirrors. Using this property, one can study the statistics of designed open chaotic cavities as in [25–29], or design antennas or detectors of given bandwidth. Far field emission of a regular or disordered cavity.—If a point source is placed inside the 2D cavity, it is able to excite any mode of the cavity indexed by the integers nx , ny that is not zero at the source position. At resonance, the energy builds up in the cavity and, due to energy conservation, 100% of the energy stored in the cavity is transmitted through the grating if we neglect Ohmic losses. In other words, the impedance of the point source placed in the open cavity is adapted through the resonance as a consequence of the Purcell effect [30]. Excited modes impose a certain incident field on the grating, thus radiating energy only for the associated resonance frequencies and in a certain bandwidth given by the linewidth of the mode. In a rectangular cavity, the incident field Ei can be decomposed in terms of two propagative waves of wave vector k ¼ ðknx ; kny Þ, where ki ¼ πni =Li . With an infinite grating, the diffracted field can be written as a discrete sum of an infinite set of plane waves, that can be indexed by an integer n. In the case of a subwavelength grating, all orders are evanescent, except the order n ¼ 0. Hence, an incident plane wave of wave vector k, which forms an angle θ normal to the grating, radiates to the far field a plane wave with the same wave vector. For an infinitely extended cavity, this gives directly the angle of emission for a given mode: tan θ ¼
ny Lx : nx Ly
(2)
Fraunhofer theory gives a convenient way to formulate the radiation pattern of our finite size system for small angles. Noting tðyÞ as the grating transmission function, Ei the wave incident on the grating, and F the Fourier transform, we can write the far field EðyÞ at a distance x from the grating (see details in the Supplemental Material [23]): 2πy E ¼ F ½tF ½Ei ; (3) λx FIG. 1 (color online). (a) The cavity (Lx ¼ 30 cm, Ly ¼ 71 cm. Grating: N ¼ 30 slots, period Λ ¼ 1.5 cm) is composed of metallic walls, one of which is replaced by a subwavelength binary grating, and a small antenna is placed inside. (b) Simulated transmission of an infinite 1D subwavelength grating (Λ ¼ 1 cm, r ¼ 0.5) in free space as a function of a=λ. (c) Simulated energy spectrum near the fundamental resonance of the cavity depicted in (a) as a function of the fill factor r of the grating. (d) Q factor of the fundamental mode of the cavity as a function of r.
which leads to the direction of emission (see Supplemental Material [23]) for a rectangular cavity: ny Ly
tan θ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2ffi : n nx þ Lyy Lx
(4)
Equation (4) is consistent with Eq. (2) only at small angles, or, equivalently, for nx ≪ ny, which is consistent with our
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assumptions. Figure 2(a) shows a FEM simulation of the emitted energy (in dB) of a cavity, as a function of the frequency and direction of radiation. Numerous modes (nx , ny ) are clearly visible and form curves. Each curve regroups modes of a given nx and different ny , and the next curve regroups the modes indexed by nx þ 1. Here, Q is around 100 and decreases with frequency since a=λ increases. We can see a slight asymmetry that is due to the fact that the antenna is located at coordinates that are incommensurable with the lengths of the cavity.
FIG. 2 (color online). Normalized simulated emitted energy (in dB) for: Lx ¼ 8.5 cm, Ly ¼ 51 cm, N ¼ 45, Λ ¼ 1 cm, r ¼ 0.1. (a) Regular cavity and regular grating. Inset: example of the corresponding modes inside the cavity. (b) to (g) top to bottom: disordered grating, disordered cavity, and disordered cavity with a disordered grating. Left: one realization, and right: average over 40 realizations.
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Equation (2) fits the angles of the simulation at 5°. This proves that a cavity opened by a grating transforms a point source into a very large and directive aperture, which can even be the whole cavity. Since a cavity can support many modes, each one associated to an emission angle, it is clear that cleverly placing a set of sources inside a designed cavity, one can generate any radiation pattern. We now study the effect of disorder on the system by placing scatterers inside the cavity or randomizing the period and fill factor of the grating. Equation (3) remains valid but the field inside the cavity Ei or the transmittance t are now random. Figure 2(b) shows FEM simulation results obtained for a cavity with a random grating. We can see that the symmetry of the radiation pattern is broken, and that the modes widths are broader. However, this only acts as as small perturbation: the subwavelength grating is seen as a semireflecting mirror by the wave, and its details are not relevant. Simulation of a cavity disordered with 20 scatterers of radius 1 cm is presented in Fig. 2(d). The far field is the convolution of the transmittance by a speckle like field Ei ; it is random and isotropic, a signature of the eigenmode of the random cavity. Figure 2(f) shows a realization of a disordered cavity coupled to a random grating and expectedly combines the effects of both precedent cases. All those effects are naturally more visible when averaging over the realizations of disorder [Figs. 3(e) and 3(f)]. It means that introducing disorder in the system, especially in the cavity, permits us to evolve from a directive source or sensor to an isotropic one. Experimental validation.—The experimental setup is shown on Fig. 3(a) : a network analyzer measures the transmission coefficient S12 between a small vertically polarized monopole antenna placed inside the cavity, and a vertically polarized horn antenna placed in its far field (2 m). Hence, we consider only transverse electric modes, making this setup comparable to our simulation one. The cavity dimensions are given in Fig. 3, with an array of 1D slots of finite size as shown in Fig. 3(a). The results are shown on Fig. 3(b) for a regular cavity with a periodic grating, and they are consistent with the simulation of Fig. 2(a), but with a lower dynamics due to Ohmic losses in the cavity. We do see distinct modes, each one emitting energy in a discrete direction and forming curves according to Eq. (2). Around 6 GHz, we can see unexpected modes that are due to the third dimension of the cavity. Figure 3(c) shows the spectrum of emitted energy for different directions, i.e., three horizontal lines of Fig. 3(b); it shows that an excited mode emits energy in a specific direction only. Figure 3(d) shows a realization of a random grating, and it confirms the results obtained numerically. Figure 3(e) shows a realization of a disordered cavity randomly filled with 7 metallic scatterers of dimensions from 1 to 3 cm. Lower frequency modes are lost because the field must vanish on the metallic scatterers. However, higher modes are still visible which do not form curves anymore: the
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FIG. 4 (color online). (a) Picture of the 2D open disordered cavity (30 × 18 × 1 cm) without the upper grating. (b) Measurement setup. (c) Measured jS12 j spectrum of the cavity, averaged over all the positions. (d) Normalized maps of jS12 j for various frequencies, displaying the eigenmodes of the disordered cavity.
FIG. 3 (color online). (a) Schematic view of the experimental set-up with the picture of the cavity. A network analyzer records the S12 transmission matrix between a remote horn antenna and an small monopole placed inside the cavity. A motor can rotate the cavity to study the directivity. (b), (d) and (e) measured jS12 j (in dB) of the far field emission of respectively: a regular cavity, a cavity with a random grating, and a disordered cavity. (c) Spectrum of the jS12 j for 3 directions of emission, corresponding to three horizontal lines of (b).
radiation pattern looks like a speckle. Overall these measurements do validate our FEM simulations: the energy is emitted for discrete angles and discrete frequencies for a regular cavity, but the radiation pattern tends to become statistically isotropic when disorder is introduced. An application in fundamental physics.—Finally, we want to prove that our approach has many applications in fundamental physics. The experimental investigation of chaotic or disordered cavities is generally quite complicated due to the difficulty to access inside information without deeply modifying the system or its boundaries. In [25], for instance, a solution is proposed to measure the eigenmodes of 2D regular or chaotic cavities, yet it is limited to empty ones. In [28], a clever yet complicated and long process is used to probe the field inside a disordered 2D cavity as noninvasively as possible. Our idea can replace both techniques since one now simply has to close the upper part of the 2D cavity, whether empty or filled with scatterers, by a subwavelength diffraction grating of tuned transmission. This grating can be used as a window to measure noninvasively the field inside the cavity. Furthermore, it does not change the boundary conditions
of the cavity since the wave mostly feels the metal due to the subwavelength nature of the grating. This is what we show in Fig. 4: a 2D cavity of dimensions 30 × 18 × 1 cm3 is filled randomly with 16 cylindrical aluminum scatterers of radius 0.75 cm [Fig. 4(a)]. The top closing grating has a period of 2 mm with a fill factor of 75% (area of metal over the total area of the grating). We place a very electrically short (1 mm, noninvasive) monopole inside the cavity, while another one slightly longer (5 mm) is mounted on a 2D translation stage and placed above the top closing grating. We use a network analyzer to measure the S12 between the two probes [Fig. 4(b)]. This gives us access to the modes of the cavity and its spectrum in various locations [Fig. 4(c)]. Examples of modes of the measured empty cavity are proposed in the Supplemental Material [23], while here we map the modes of the disordered one [Fig. 4(d)]. Clearly the modes are nicely obtained from outside the cavity, in a nonpertubative way, which is ideal for the study of chaotic systems, for instance. In conclusion, we have proposed to design open microwave cavities using subwavelength metallic diffraction gratings. We have analyzed numerically, theoretically, and experimentally the spectral properties and far field emission patterns of such ordered or disordered open cavities. We have underlined applications and given clues which could lead to the design, throughout the spectrum, of smart antennas, sensors, or detectors. Finally, we have proposed a simple yet sound demonstration of the interest of the concept to study the physics of open chaotic or random cavities [25–29]. Finally, we add that these open cavities will also have a great potential when associated with the concept of time reversal [31–33]. M. D. acknowledges funding by the French “Ministère de la Défense, Direction Générale de l’Armement”. This
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work is supported by LABEX WIFI (Laboratory of Excellence within the French Program “Investments for the Future”) under references ANR-10-LABX-24 and ANR-10-IDEX-0001-02 PSL*.
*
[email protected] [1] S. Haroche and D. Kleppner, Phys. Today 42, No. 1, 24 (1989). [2] P. So, S. M. Anlage, E. Ott, and R. N. Oerter, Phys. Rev. Lett. 74, 2662 (1995). [3] R. A. Méndez-Sánchez, U. Kuhl, M. Barth, C. H. Lewenkopf, and H.-J. Stöckmann, Phys. Rev. Lett. 91, 174102 (2003). [4] U. Kuhl, M. Martínez-Mares, R. A. Méndez-Sánchez, and H.-J. Stöckmann, Phys. Rev. Lett. 94, 144101 (2005). [5] S. Hemmady, X. Zheng, E. Ott, T. M. Antonsen, and S. M. Anlage, Phys. Rev. Lett. 94, 014102 (2005). [6] J. Barthélemy, O. Legrand, and F. Mortessagne, Europhys. Lett. 70, 162 (2005). [7] B. T. Taddese, G. Gradoni, F. Moglie, T. M. Antonsen, E. Ott, and S. M. Anlage, New J. Phys. 15, 023025 (2013). [8] M. Gutzwiller, J. Math. Phys. (N.Y.) 12, 343 (1971). [9] S. W. McDonald and A. N. Kaufman, Phys. Rev. A 37, 3067 (1988). [10] H. Alt, H. D. Gräf, H. L. Harney, R. Hofferbert, H. Lengeler, A. Richter, P. Schardt, and H. A. Weidenmüller, Phys. Rev. Lett. 74, 62 (1995). [11] H.-J. Sommers, Y. V. Fyodorov, and M. Titov, J. Phys. A 32, L77 (1999). [12] U. Kuhl, R. Höhmann, J. Main, and H.-J. Stöckmann, Phys. Rev. Lett. 100, 254101 (2008). [13] A. F. Stevenson, J. Appl. Phys. 19, 24 (1948). [14] W. Watson, The Physical Principles of Wave Guides and Transmissions (Oxford University Press, New York, 1947). [15] C. F. R. Mateus, M. Huang, Y. Deng, A. R. Neureuther, and C. Chang-Hasnain, IEEE Photonics Technol. Lett. 16, 518 (2004).
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[16] C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. ChangHasnain, and Y. Suzuki, IEEE Photonics Technol. Lett. 16, 1676 (2004). [17] M. C. Huang, Y. Zhou, and C. J. Chang-Hasnain, Nat. Photonics 1, 119 (2007). [18] R. W. Wood, Philos. Mag. 4, 396 (1902). [19] T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, Nature (London) 391, 667 (1998). [20] J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, Phys. Rev. Lett. 83, 2845 (1999). [21] H. A. Bethe, Phys. Rev. 66, 163 (1944). [22] C. J. Bouwkamp, Rep. Prog. Phys. 17, 35 (1954). [23] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.112.043902 for theoretical derivations of the Q factor of the open cavity as well as of its radiation pattern, and experimental and numerical studies of the 2D cavity studied in the last part of the paper. [24] K. A. Milton, J. Schwinger, L. L. DeRaad, Jr., and W.-Y. Tsai, Classical Electrodynamics (Perseus Books, Westview Press, New York, 1998). [25] J. Stein and H.-J. Stöckmann, Phys. Rev. Lett. 68, 2867 (1992). [26] J. Stein, H.-J. Stöckmann, and U. Stoffregen, Phys. Rev. Lett. 75, 53 (1995). [27] S. Rotter, F. Libisch, J. Burgdörfer, U. Kuhl, and H.-J. Stöckmann, Phys. Rev. E 69, 046208 (2004). [28] D. Laurent, O. Legrand, P. Sebbah, C. Vanneste, and F. Mortessagne, Phys. Rev. Lett. 99, 253902 (2007). [29] O. Xeridat, C. Poli, O. Legrand, F. Mortessagne, and P. Sebbah, Phys. Rev. E 80, 035201 (2009). [30] E. M. Purcell, Phys. Rev. 69, 674 (1946). [31] G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, Phys. Rev. Lett. 92, 193904 (2004). [32] M. Davy, J. de Rosny, J.-C. Joly, and M. Fink, C.R. Phys. 11, 37 (2010). [33] S. K. Hong, B. T. Taddese, Z. B. Drikas, S. M. Anlage, and T. D. Andreadis, J. Electromagn. Waves Appl. 27, 1262 (2013).
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Using Subwavelength Diffraction Gratings to Design Open Electromagnetic Cavities Matthieu Dupré, Mathias Fink, and Geoffroy Lerosey* Institut Langevin, ESPCI ParisTech and CNRS UMR 7587, 1 rue Jussieu, 75005 Paris, France (Received 17 July 2013; published 29 January 2014) In this Letter we propose to use subwavelength diffraction gratings as very good semitransparent mirrors for electromagnetic waves to design open cavities. To do so, we replace part of the walls of a cavity by such a grating. We numerically and analytically link the grating characteristics to the spectral properties of the realized open cavity. Then we demonstrate that the eigenmodes of the cavity can be transmitted perfectly through the grating to the exterior, thereby turning a point source inside the cavity into a very directive source. We investigate the effect of disorder, which leads to isotropic radiation patterns, and perform experiments in the microwave domain in order to support our claims. Finally, we present an example of application of the concept in fundamental physics, by measuring from outside the eigenmodes of a disordered microwave cavity. DOI: 10.1103/PhysRevLett.112.043902
PACS numbers: 41.20.-q, 84.40.-x
Wave propagation in microwave cavities is a major topic of interest in various fields of research ranging from fundamental physics with the notions of quantum fidelity, level repulsion of states, or atomic emission rates [1–7], to electrical engineering, which embraces electromagnetic compatibility and antenna measurements. The behavior of waves in such complex media can be analytically calculated for the case of regular cavities and many statistical approaches have been proposed to predict it in chaotic ones [8–12]. To date, most works devoted to these propagation media concentrated on the study of the wave fields inside the cavities, since the latter are usually 2D or 3D shapes enclosed by metallic walls which conceal the electromagnetic fields perfectly. The usual way to couple such cavities to the exterior consists of carving a hole which supports propagating modes in one of its metallic walls, hence, realizing so-called open microwave cavities in fundamental physics, or cavity backed slot antennas in electrical engineering [13,14]. In the optical domain, open cavities have been studied for a long time, since metals are no longer good conductors. Indeed, at those frequencies, a silver film 100 nm thick constitutes a semitransparent mirror. Similarly high contrast gratings or distributed Bragg filters, both being variants of photonics crystals, have been used recently in vertical-cavity surface-emitting lasers [15–17]. In this Letter, we propose to use subwavelength metallic binary diffraction gratings as semitransparent mirrors to design open cavities. We focus our study on microwave cavities, yet our approach remains valid up to the visible. We show that a subwavelength diffraction grating that closes a microwave cavity becomes transparent at the cavity resonances, and transmits all the energy coming from a source placed inside it, analogous to Wood anomalies or extraordinary optical transmission [18–20]. We first link the spectral properties of such an open cavity to the dimensions of the grating. Then, we numerically and analytically demonstrate that a regular cavity coupled to a subwavelength 0031-9007=14=112(4)=043902(5)
grating can turn a point source into a highly directive wide aperture antenna, while introducing disorder leads to isotropic radiation patterns. We validate our approach with experiments in the microwave range. Finally, we show an example of application of the concept in fundamental physics. Spectral properties.—Coupling a microwave cavity to a subwavelength metallic grating involves various characteristic lengths. The ratio of the typical size of the cavity to the wavelength with an exponent related to the dimensionality of the cavity governs the density of modes of the cavity. The grating period and filling of metal, which we will refer to here as the fill factor, are related to its transmittance through the wavelength as well. We first quantify the effects of the grating dimensions on the spectral properties of the open cavity. At long distances from apertures, it is known that the transmission of a single 2D hole of a radius which is small compared to the wavelength in a thin screen behaves as T ∝ ða=λÞ4 [21,22]. Since a binary metallic grating is an array of such holes, this underlines two noticeable facts: the transmission of the grating is frequency dependent and can be precisely tuned over a given spectral range through its dimensions. It is clear that the Q factor of the cavity, which represents the amount of stored energy divided by the amount of dissipated energy per cycle, can be tuned using the properties of the grating as well. In fact, we can show (see Supplemental Material [23]) that for a 3D cavity and 2D grating, if there are only radiative losses, Q is linked to the transmission of a slot T as QðωÞ ¼
2 ω V cavity ; c NTðωÞ Sgrating
(1)
where V cavity and Sgrating are, respectively, the volume of the cavity and the surface of the grating. Hence, one can
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tune the bandwidth or linewidth of the modes of the open cavity by playing with, for instance, the fill factor of the grating. We verify this fact with a set of simulations realized using a finite element method code (FEM). For the sake of simplicity we study a 2D rectangular cavity of infinite dimension in the vertical direction, made out of perfectly electric conductor, with a point source placed inside. Only one wall is replaced by a grating [Fig. 1(a)]. The electric field is polarized along the grating slots, the wavelength λ varies between 3 and 30 cm for frequencies from 10 to 1 GHz. We note Lx and Ly the dimensions of the cavity, Λ the period of the grating, N and a the number and width of the grating slots. These define the fill factor r of the binary metallic grating: a ¼ ð1 − rÞΛ. We first study the transmission in free space of the grating whose dimensions are given in Fig. 1. The results clearly indicate that the ratio a=λ defines the transmission of the grating: transmissions from 10−5 to 10−1 are achievable [Fig. 1(b)]. The exponent found is 2 instead of 4 due to the dimensionality of the holes, which is 1 rather than 2 in the Bethe formula [22,24]. We next study the influence of the fill factor on the Q factor of the fundamental mode of the cavity (around 530 MHz), whose dimensions are given in Fig. 1. FEM simulation results are presented which display the energy spectrum of the cavity near its fundamental resonance for different fill factors [Fig. 1(c)]. From those simulations, the Q factor is extracted and plotted on Fig. 1(d) and ranges from 400 to 6000, while r varies only from 1% to 30%.
As r is increased, Q is increased and the fundamental resonance is slightly shifted closer to its zero loss theoretical value. Overall, this figure shows that metallic subwavelength gratings can make very good and simple semitransparent mirrors. Using this property, one can study the statistics of designed open chaotic cavities as in [25–29], or design antennas or detectors of given bandwidth. Far field emission of a regular or disordered cavity.—If a point source is placed inside the 2D cavity, it is able to excite any mode of the cavity indexed by the integers nx , ny that is not zero at the source position. At resonance, the energy builds up in the cavity and, due to energy conservation, 100% of the energy stored in the cavity is transmitted through the grating if we neglect Ohmic losses. In other words, the impedance of the point source placed in the open cavity is adapted through the resonance as a consequence of the Purcell effect [30]. Excited modes impose a certain incident field on the grating, thus radiating energy only for the associated resonance frequencies and in a certain bandwidth given by the linewidth of the mode. In a rectangular cavity, the incident field Ei can be decomposed in terms of two propagative waves of wave vector k ¼ ðknx ; kny Þ, where ki ¼ πni =Li . With an infinite grating, the diffracted field can be written as a discrete sum of an infinite set of plane waves, that can be indexed by an integer n. In the case of a subwavelength grating, all orders are evanescent, except the order n ¼ 0. Hence, an incident plane wave of wave vector k, which forms an angle θ normal to the grating, radiates to the far field a plane wave with the same wave vector. For an infinitely extended cavity, this gives directly the angle of emission for a given mode: tan θ ¼
ny Lx : nx Ly
(2)
Fraunhofer theory gives a convenient way to formulate the radiation pattern of our finite size system for small angles. Noting tðyÞ as the grating transmission function, Ei the wave incident on the grating, and F the Fourier transform, we can write the far field EðyÞ at a distance x from the grating (see details in the Supplemental Material [23]): 2πy E ¼ F ½tF ½Ei ; (3) λx FIG. 1 (color online). (a) The cavity (Lx ¼ 30 cm, Ly ¼ 71 cm. Grating: N ¼ 30 slots, period Λ ¼ 1.5 cm) is composed of metallic walls, one of which is replaced by a subwavelength binary grating, and a small antenna is placed inside. (b) Simulated transmission of an infinite 1D subwavelength grating (Λ ¼ 1 cm, r ¼ 0.5) in free space as a function of a=λ. (c) Simulated energy spectrum near the fundamental resonance of the cavity depicted in (a) as a function of the fill factor r of the grating. (d) Q factor of the fundamental mode of the cavity as a function of r.
which leads to the direction of emission (see Supplemental Material [23]) for a rectangular cavity: ny Ly
tan θ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2ffi : n nx þ Lyy Lx
(4)
Equation (4) is consistent with Eq. (2) only at small angles, or, equivalently, for nx ≪ ny, which is consistent with our
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assumptions. Figure 2(a) shows a FEM simulation of the emitted energy (in dB) of a cavity, as a function of the frequency and direction of radiation. Numerous modes (nx , ny ) are clearly visible and form curves. Each curve regroups modes of a given nx and different ny , and the next curve regroups the modes indexed by nx þ 1. Here, Q is around 100 and decreases with frequency since a=λ increases. We can see a slight asymmetry that is due to the fact that the antenna is located at coordinates that are incommensurable with the lengths of the cavity.
FIG. 2 (color online). Normalized simulated emitted energy (in dB) for: Lx ¼ 8.5 cm, Ly ¼ 51 cm, N ¼ 45, Λ ¼ 1 cm, r ¼ 0.1. (a) Regular cavity and regular grating. Inset: example of the corresponding modes inside the cavity. (b) to (g) top to bottom: disordered grating, disordered cavity, and disordered cavity with a disordered grating. Left: one realization, and right: average over 40 realizations.
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Equation (2) fits the angles of the simulation at 5°. This proves that a cavity opened by a grating transforms a point source into a very large and directive aperture, which can even be the whole cavity. Since a cavity can support many modes, each one associated to an emission angle, it is clear that cleverly placing a set of sources inside a designed cavity, one can generate any radiation pattern. We now study the effect of disorder on the system by placing scatterers inside the cavity or randomizing the period and fill factor of the grating. Equation (3) remains valid but the field inside the cavity Ei or the transmittance t are now random. Figure 2(b) shows FEM simulation results obtained for a cavity with a random grating. We can see that the symmetry of the radiation pattern is broken, and that the modes widths are broader. However, this only acts as as small perturbation: the subwavelength grating is seen as a semireflecting mirror by the wave, and its details are not relevant. Simulation of a cavity disordered with 20 scatterers of radius 1 cm is presented in Fig. 2(d). The far field is the convolution of the transmittance by a speckle like field Ei ; it is random and isotropic, a signature of the eigenmode of the random cavity. Figure 2(f) shows a realization of a disordered cavity coupled to a random grating and expectedly combines the effects of both precedent cases. All those effects are naturally more visible when averaging over the realizations of disorder [Figs. 3(e) and 3(f)]. It means that introducing disorder in the system, especially in the cavity, permits us to evolve from a directive source or sensor to an isotropic one. Experimental validation.—The experimental setup is shown on Fig. 3(a) : a network analyzer measures the transmission coefficient S12 between a small vertically polarized monopole antenna placed inside the cavity, and a vertically polarized horn antenna placed in its far field (2 m). Hence, we consider only transverse electric modes, making this setup comparable to our simulation one. The cavity dimensions are given in Fig. 3, with an array of 1D slots of finite size as shown in Fig. 3(a). The results are shown on Fig. 3(b) for a regular cavity with a periodic grating, and they are consistent with the simulation of Fig. 2(a), but with a lower dynamics due to Ohmic losses in the cavity. We do see distinct modes, each one emitting energy in a discrete direction and forming curves according to Eq. (2). Around 6 GHz, we can see unexpected modes that are due to the third dimension of the cavity. Figure 3(c) shows the spectrum of emitted energy for different directions, i.e., three horizontal lines of Fig. 3(b); it shows that an excited mode emits energy in a specific direction only. Figure 3(d) shows a realization of a random grating, and it confirms the results obtained numerically. Figure 3(e) shows a realization of a disordered cavity randomly filled with 7 metallic scatterers of dimensions from 1 to 3 cm. Lower frequency modes are lost because the field must vanish on the metallic scatterers. However, higher modes are still visible which do not form curves anymore: the
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FIG. 4 (color online). (a) Picture of the 2D open disordered cavity (30 × 18 × 1 cm) without the upper grating. (b) Measurement setup. (c) Measured jS12 j spectrum of the cavity, averaged over all the positions. (d) Normalized maps of jS12 j for various frequencies, displaying the eigenmodes of the disordered cavity.
FIG. 3 (color online). (a) Schematic view of the experimental set-up with the picture of the cavity. A network analyzer records the S12 transmission matrix between a remote horn antenna and an small monopole placed inside the cavity. A motor can rotate the cavity to study the directivity. (b), (d) and (e) measured jS12 j (in dB) of the far field emission of respectively: a regular cavity, a cavity with a random grating, and a disordered cavity. (c) Spectrum of the jS12 j for 3 directions of emission, corresponding to three horizontal lines of (b).
radiation pattern looks like a speckle. Overall these measurements do validate our FEM simulations: the energy is emitted for discrete angles and discrete frequencies for a regular cavity, but the radiation pattern tends to become statistically isotropic when disorder is introduced. An application in fundamental physics.—Finally, we want to prove that our approach has many applications in fundamental physics. The experimental investigation of chaotic or disordered cavities is generally quite complicated due to the difficulty to access inside information without deeply modifying the system or its boundaries. In [25], for instance, a solution is proposed to measure the eigenmodes of 2D regular or chaotic cavities, yet it is limited to empty ones. In [28], a clever yet complicated and long process is used to probe the field inside a disordered 2D cavity as noninvasively as possible. Our idea can replace both techniques since one now simply has to close the upper part of the 2D cavity, whether empty or filled with scatterers, by a subwavelength diffraction grating of tuned transmission. This grating can be used as a window to measure noninvasively the field inside the cavity. Furthermore, it does not change the boundary conditions
of the cavity since the wave mostly feels the metal due to the subwavelength nature of the grating. This is what we show in Fig. 4: a 2D cavity of dimensions 30 × 18 × 1 cm3 is filled randomly with 16 cylindrical aluminum scatterers of radius 0.75 cm [Fig. 4(a)]. The top closing grating has a period of 2 mm with a fill factor of 75% (area of metal over the total area of the grating). We place a very electrically short (1 mm, noninvasive) monopole inside the cavity, while another one slightly longer (5 mm) is mounted on a 2D translation stage and placed above the top closing grating. We use a network analyzer to measure the S12 between the two probes [Fig. 4(b)]. This gives us access to the modes of the cavity and its spectrum in various locations [Fig. 4(c)]. Examples of modes of the measured empty cavity are proposed in the Supplemental Material [23], while here we map the modes of the disordered one [Fig. 4(d)]. Clearly the modes are nicely obtained from outside the cavity, in a nonpertubative way, which is ideal for the study of chaotic systems, for instance. In conclusion, we have proposed to design open microwave cavities using subwavelength metallic diffraction gratings. We have analyzed numerically, theoretically, and experimentally the spectral properties and far field emission patterns of such ordered or disordered open cavities. We have underlined applications and given clues which could lead to the design, throughout the spectrum, of smart antennas, sensors, or detectors. Finally, we have proposed a simple yet sound demonstration of the interest of the concept to study the physics of open chaotic or random cavities [25–29]. Finally, we add that these open cavities will also have a great potential when associated with the concept of time reversal [31–33]. M. D. acknowledges funding by the French “Ministère de la Défense, Direction Générale de l’Armement”. This
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work is supported by LABEX WIFI (Laboratory of Excellence within the French Program “Investments for the Future”) under references ANR-10-LABX-24 and ANR-10-IDEX-0001-02 PSL*.
*
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