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Optical circulation and isolation based on indirect photonic transitions of guided resonance modes Yu Shi, Seunghoon Han, and Shanhui Fan ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.7b00420 • Publication Date (Web): 28 Jun 2017 Downloaded from http://pubs.acs.org on June 28, 2017
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ACS Photonics
Optical circulation and isolation based on indirect photonic transitions of guided resonance modes
Yu Shi1, Seunghoon Han2, and Shanhui Fan1, a) 1
Department of Electrical Engineering, Ginzton Laboratory, Stanford University, Stanford, California 94305, USA
2
Samsung Advanced Institute of Technology, Samsung Electronics, Suwon 443-803, Korea a)
Author to whom correspondence should be addressed. Email:
[email protected].
Abstract: While there has been enormous progress in meta-surface designs, most meta-surfaces are constrained by the Lorentz reciprocity. Breaking reciprocity, however, enables additional functionalities and greatly expands the applications of meta-surfaces. Here, we introduce a realistic non-reciprocal meta-surface that can achieve optical circulation and isolation. This device consists of a photonic crystal slab that supports two bands of guided resonances, and upon a temporal modulation in each unit cell with a spatially-varying phase, an indirect photonic transition can be induced between the guided resonances, which breaks Lorentz reciprocity without the use of magneto-optic materials. We provide direct first-principle numerical simulations, using the multi-frequency finite-difference frequency-domain (FDFD) method, to demonstrate that this device can achieve optical circulation and isolation with no back reflection under realistic modulation frequency and modulation strength. Key words: Metamaterials. Optical isolation. Dynamic modulation.
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In recent years, there have been significant advancements in the manipulation of light using two dimensional meta-materials, or meta-surfaces1–3. Meta-surfaces have been used in performing refraction and diffraction4–6, manipulating the polarization of light7,8, as well as realizing special optical physics on a two-dimensional platform, such as the generation of optical spin Hall effects9 and optical vortices10. Moreover, there are now significant recent efforts in developing non-reciprocal meta-surfaces11–19. Breaking Lorentz reciprocity enables meta-surfaces to be used for important functionalities such as optical isolation and circulation, which are crucial in signal processing and for laser feedback protection, and which cannot be realized in any reciprocal structures20–22. There are two major approaches towards creating non-reciprocity that can be used to achieve complete optical isolation and circulation. Both approaches have been considered in meta-surfaces. The first method is to use magneto-optic materials11,12,23,24. Under a static magnetic field bias, magneto-optic materials exhibit a non-symmetric permittivity tensor, which can be used in meta-surfaces to break the reciprocity between forward and backward propagating modes, resulting in optical isolation11,12. This approach can be purely passive, which is attractive. However, standard optoelectronic materials typically do not have magneto-optic effects, and consequently it is difficult to apply this approach in most standard optoelectronic platforms. The second method is to apply a dynamic modulation to the dielectric constant of the device. Such modulation also breaks reciprocity and can lead to complete optical isolation13,14,16,19,25–34. The dynamic modulation approach is an active approach that requires external energy input, but it is compatible with most standard optoelectronic materials. Therefore, there have been several recent proposals that use the dynamic modulation approach for constructing non-reciprocal metasurfaces13,14,16,18,19.
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In this paper, we focus on the dynamic modulation approach towards the creation of nonreciprocal meta-surfaces. In spite of the significant advancements as summarized above, there still remain significant challenges in realizing such dynamic non-reciprocal meta-surfaces in the optical frequency range. For example, one approach for achieving non-reciprocal response in dynamically modulated structures is based on the concept of photonic transition26–28,31–35. In this approach, one considers a photonic system supporting two optical modes at frequencies and
. Modulation of the system at a frequency = − can induce a photonic transition between these two optical modes. In such a transition, the upward and downward transition in frequency acquires opposite phases, which can be used to create non-reciprocal responses. There have been several attempts seeking to implement this concept in meta-surfaces13,14,16,18,19. However, there is a very large frequency difference between the modulation frequency and the optical frequency. The maximum modulation frequency typically used in electro-optic modulation is usually on the order of 10 GHz36, whereas a typically optical frequency is around 200 THz. A realistic design of a dynamic non-reciprocal meta-surface must take into account such a large frequency difference. In the present work, we introduce a design of non-reciprocal meta-surface that operates in optical frequency while being driven by a refractive index modulation at tens of gigahertz based on the concept of photonic transition. This design incorporates two essential considerations. First, the device needs to have independent modes that are separated by an attainable modulation frequency of a few gigahertz. Second, these modes must have high enough quality factors so that they can be spectrally resolved. To meet these considerations, we exploit guided resonances in a photonic crystal slab. First, with a proper choice of the thickness of the slab, the slab supports guided resonances forming multiple photonic bands, and as a result, for a given modulation 3 ACS Paragon Plus Environment
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frequency , it is always possible to find two optical modes in the structure with their
frequencies differing by . Secondly, with a proper choice of slab parameters, these guided
resonances can achieve high quality factors. Thus, the photonic crystal slab system naturally satisfies the considerations as outlined above. By using a pair of such high-quality-factor guided resonances and by applying a dynamic modulation with its modulation frequency and modulation wave-vector matching the frequency and momentum differences of these two resonances, we demonstrate theoretically and numerically that this device can break reciprocity in the optical range. In the lossless limit, one can achieve perfect optical circulation. Even with material loss that is realistic in the free-carrier-based modulation schemes in silicon, this device is still capable of attaining complete optical isolation with no back reflection. As a concrete implementation, we consider in two dimensions a one-dimensional photonic crystal slab as shown in Fig. 1(a), which consists of a perfect electric conductor (PEC) substrate, an air superstrate, and a dielectric slab waveguide with periodic grooves along the -
direction spaced at a periodicity of = 550 nm. The dielectric slab has a thickness of 625 nm, and it has a relative permittivity of 11.56, which is similar to that of silicon in the optical range. The periodic grooves are 100 nm in width and 75 nm in height, and they have a relative permittivity of 12.25. Without loss of generality, we study the transverse magnetic (TM) polarization, where the nonzero field components are , , and . We first focus our analysis
by assuming no material loss is present, which enables this device to act as a perfect optical circulator. Then, we introduce material loss that may be associated with modulation and show that this device can still achieve complete optical isolation.
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We start by analyzing the properties of this photonic crystal slab with no time-dependent
permittivity modulation. Because of the periodicity along , according to Bloch’s theorem, such a structure supports modes with the following form37–39:
, , = , ,
(1)
where is the Bloch wave vector in the direction, which is real, , is the complex Bloch field profile that is periodic in , such that , = + , , and is the complex
frequency of the guided mode. For a photonic crystal slab made with lossless materials, the imaginary component of captures the rate at which the guided resonance mode radiates its power into free space. With a standard frequency-domain eigenmode-solver technique37, we can numerically
compute this photonic crystal slab’s band diagram. We choose a spatial resolution of Δ = Δ = 25 nm, and the band diagram is plotted on the left side of Fig. 1(b). This plot shows that around
a frequency of 200 THz, the structure supports many different guided modes. For the purpose of subsequent discussions, we highlight three independent modes, which are labeled on Fig. 1(b).
The mode labeled |1〉 has a frequency = 2( × 201.1461 THz and a wave-vector = 0, the
mode labeled |2〉 has a frequency = 2( × 201.1225 THz and a wave-vector = 0.4(/ ,
and the mode labeled |3〉 has the same frequency as mode |2〉 and a wave-vector / =
− = −0.4(/ . From the same eigenmode solver, we deduce that due to radiative loss, mode
|1〉 has a quality factor of 0 = 1.90 × 102 and resonance linewidth 23 = 1.1 GHz, and modes
|2〉 and |3〉 have quality factors of 0 = 0/ = 1.05 × 104 and resonance linewidth 23 = 23/ =
0.2 GHz. On the right side of Fig. 1(b), we show the Bloch field profiles , of these guided
resonances. With respect to the center-plane of the slab in the x-z plane, Mode |1〉 is approximately odd, and mode |2〉 and |3〉 are approximately even. Furthermore, modes |2〉 and 5 ACS Paragon Plus Environment
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|3〉 have the same field profile due to mirror symmetry. Mode |1〉 couples to normally incident
and out-going plane waves, whereas mode |2〉 (|3〉) couples to obliquely incident and out-going
plane waves with positive (negative) horizontal wave-vectors. Following the concepts of Ref. 28, the reciprocity in this system can be broken by
creating an indirect photonic transition between modes |1〉 and |2〉, which can be realized by applying a dynamic modulation to the permittivity in the regions as indicated by the dashed lines in Fig. 1(a). These regions are chosen such that there is a significant overlap of the two modes integrated over the modulation region, as shown in Fig. 1(c). Mathematically, the permittivity of the modulated structure is described by27,28
5, , = 56 , + 7, cos; +
