Asymmetric Light Propagation in Active Photonic Structures

Transcription

1 Wesleyan University Asymmetric Light Propagation in Active Photonic Structures by Samuel Kalish Class of 2013 A thesis submitted to the faculty of Wesleyan University in partial fulfillment of the requirements for the Degree of Bachelor of Arts with Departmental Honors in Physics Middletown, Connecticut April, 2012

2 Abstract This thesis will present a theoretical and numerical analysis of optical systems exhibiting non-reciprocity, with potential applications in the field of integrated opto-electronic circuits. We will carefully engineer structures with highly-asymmetric transport properties, where traits such as reflectance and transmittance are dependent upon the incident direction of the interrogating waves. This will be achieved through appropriately engineered amplifying and attenuating mechanisms in spatial arrangements obeying paritytime symmetries, as well as the implementation of magneto-optical elements. It will be demonstrated that such structures are good candidates for the development of novel devices serving the purposes of optical isolators and unidirectional lasers in integrated photonic technologies.

3 Acknowledgments First, I would like to thank Professor Tsampikos Kottos, who has been exceptional in his role as my teacher and mentor over the past several years. You have not only taught me a great deal of physics, but have also pushed and motivated me to do the best work that I could do. Second, I would like to thank Hamidreza Ramezani and Zin Lin for being both terrific colleagues and great friends. To the members of the WTICS group, both past and future, I am glad to count myself among you and wish you great success as a part of the group and beyond. I extend thanks to the faculty, staff, and students that make up the Wesleyan Physics Department, who have all been an indispensable part of my time here. Last, but certainly not least, I would like to thank my family and friends both at Wesleyan and beyond who have played an important role in my life. Thank you. In addition, financial support from an NSF ECCS grant and Wesleyan University scholarship is greatly appreciated.

4 Contents 1 Introduction 1 2 Scattering in PT -Symmetric Media PT Symmetry One-Dimensional Wave Scattering Passive Conservation Relations PT -Symmetric Conservation Relations Light Transport in PT -Symmetric Random Media Mathematical Model Transmittance Reflectance Summary PT -Symmetric Magneto-Optical Isolator Model and Symmetries Scattering Formalism Asymmetric Transport Summary Magneto-Optical Unidirectional Laser Scattering Formalism iii

5 5.2 Asymmetry in Dispersion Relation Unidirectional Lasing Conclusion 45

6 List of Figures 2.1 Diagram of transfer matrix Diagram of scattering matrix One-dimensional PT -symmetric multi-layered random medium We report lnt against the system size for: a system with only gain and loss and n R = n 0 constant (upper red); a random layer medium with n I = 0 (middle black); and a PT -symmetric disorder medium (lower green) Numerically extracted localization length ξ γ (W ) for various gain and loss parameters γ (not indicated in the figure), rescaled with ξ 0 (W ) versus the scaling parameter ξ 0 (W )/ξ γ (0) Typical reflectances R L and R R versus system size (number of layers, L), for various values of disorder, W, and gain/loss, γ The asymptotic value of lnr versus the scaled parameter Λ Scattering setup of a micro-cavity with PT -symmetric gain/loss distribution and asymmetric transport The magnitudes of the eigenvalues of the trimer system (a) The left/right reflectances vs. ω. (b) The difference T l T r between the left and right transmittances vs. ω. (c) The quality factor Q T vs. ω 35 v

7 4.4 (Upper) Dispersion relation ω(k) of the infinite periodic anisotropic Bragg reflector with permittivity contrasts ɛ = 3 and ɛ = 12 and δ = 1, φ = 0. (Lower) Quality factor Q T (red dashed line) of a PT -symmetric nonreciprocal magneto-optical cavity when it is embedded in the anisotropic Bragg reflector associated with the set-up of the upper sub-figure Schematic of a periodic system of trimer units Left: The dispersion relation of the passive, infinite system. Right: ω k, the group velocity within the system (a) The dispersion relation of the passive, infinite system. (b) Tracing the zeros of M 22 of the single-trimer system as, γ, the gain, is increased. (c) A good measure of the asymmetry, Q = T R+R L T L +R R vs. ω for the 20-unit system Transmission and reflection coefficient sums on the left and right sides of the system, as well as the ratio of the sums Q, shown for systems of 10-, 20-, and 30-trimers

8 Chapter 1 Introduction Achieving non-reciprocal wave propagation is of fundamental importance for many engineering devices, and it is at the heart of exciting concepts in physics and applied mathematics. The engineering challenge consists in realizing on-chip integrated nonreciprocal devices like isolators and circulators, capable of controlling and directing the energy flow at will. These are indispensable components of optical networks aimed at preventing adverse reflections, interference, and feedback, and constitute the building blocks of a variety of transport-based devices, like rectifiers, pumps, switches and transistors. Currently, such unidirectional elements rely almost exclusively on naturally weak magneto-optical effects, which require magnetic bias over large volumes, incompatible with integrated optoelectronic technology. Attempts to address this problem using resonant enhancement of magneto-optical interactions usually results in prohibitively strong losses, which preclude the use of many magneto-optical materials with otherwise excellent physical properties. In this thesis, we revolutionize the way in which nonreciprocal phenomena are realized and integrated in complex optoelectronic systems. Our efforts to achieve asymmetric transport will combine the use of active and passive gyrotropic structures with specific mathematical symmetries like parity-time (PT ). We will also explore nanophotonic metamaterials with carefully engineered frequency 1

9 2 dispersion, leading to highly enhanced non-reciprocal response, unattainable in conventional materials. At the same time, our architectures, guided by precise physical models and advanced mathematical concepts, will ensure that losses and other unwanted effects are significantly reduced, or even eliminated. This thesis is based on a research activity which culminated in two publications [1, 2] with another under preparation. A focus point of these results was in proposing nonreciprocal photonic structures which guide the flow of light in a spatially asymmetric way. More specifically, the structure of the thesis is as follows. In chapter 2, we will introduce the theoretical groundwork for general scattering formalism through the use of the transfer and scattering matrices. Specifically, we will develop generalized conservation relations for parity-time symmetric systems and indicate their consequences in transmittances and reflectances. A brief comparison with the well-known case of passive systems will also be presented. Motivated by the fact that in most photonic structures layer-imperfections and impurities are a reality, we will investigate, in chapter 3, we will investigate the effects of disorder in PT -symmetric slab geometries. We will develop a scaling theory for the localization length of these systems and identify the effects from the interplay between destructive interferences (Anderson localization) and gain and loss elements. Furthermore, we will show that although left/right transmittance is the same, there is an asymmetric reflection which depends on the side that the incident wave enters the sample. In chapter 4, we discuss the effects of magnetic elements in parity-time symmetric optical structures. The previous generalized conservation relations are expanded in order to include the effects of magnetic non-reciprocity. We demonstrate our theoretical analysis by using a basic 3-layer photonic microcavity. The asymmetric transmission is furthermore dramatically enhanced via embedding in an anisotropic Bragg grating. Remarkably, such asymmetry persists regardless of the incident light polarization. This photonic architecture may be used as the building block for chip-scale non-reciprocal

10 3 devices such as optical isolators and circulators. In chapter 5, we step outside the bounds of PT -symmetry and discuss the possibility of creating a mirror-less unidirectional lasing cavity. A Mirror-less Unidirectional Laser (MUL) propagates all the lasing power into a single direction as opposed to an arbitrary splitting of the light power into two directions. The cavity can be either a finite open (mirror-less) active structure or a ring cavity that amplifies a light beam that propagates in a desirable direction. This structure is based on an engineered asymmetric dispersion relation, which can be achieved by appropriately arranging magneto-optical layers together with anisotropic dielectric layers. The inclusion of gain enforces asymmetric lasing action by selecting modes with non-reciprocal propagation velocity. Potential applications of such a device are as an element to gyroscopes for measuring angular acceleration. Finally, we will give our conclusions in the last chapter of the thesis.

11 Chapter 2 Scattering in PT -Symmetric Media Parity (P) and time-reversal (T ) symmetries are basic notions within physics. Of recent interest have been systems which do not independently obey either (P) or (T ) symmetries, but do obey a combined PT symmetry. In the specific framework of optics, PT symmetry can be realized in a medium with the property that the complex refractive index satisfies n( z) = n (z). This implies that gain and loss elements are incorporated in an antisymmetric way, so that the creation and absorption of photons occur in a balanced manner and that the net gain or loss is zero. The numerous intriguing features that have been demonstrated by PT -symmetric optical structures include power oscillations, non-reciprocity of light propagation [3 6], non-reciprocal Bloch oscillations [7, 8], and unidirectional invisibility [9]. In the nonlinear domain, such pseudo-hermitian non-reciprocal effects can be used to realize a new generation of on-chip isolators and circulators [10]. In this chapter, we will discuss the basic principles and the associated transport properties of PT -symmetric optical structures by developing a transfer matrix formalism. 4

12 Section 2.1. PT Symmetry 5 Because of the active elements of the systems under investigation, we are no longer working within the traditional realm of flux conservation, but will develop new conservation relations that emerge due to underlying symmetries. These weaker constraints will allow us to explore the anomalous transport properties of PT -symmetric scattering systems, such as super-unitary transmittance, asymmetric reflectance, and optical isolation. 2.1 PT Symmetry Parity (P) and time-reversal (T ) transformations are fundamental symmetry operations in physics. They are mathematically defined by their actions on the position operator ˆx and the momentum operator ˆp, in addition to the time parameter t [11]. P is a linear operator which acts as a spacial reflection, inverting both position and momentum: P : ˆx ˆx; ˆp ˆp (2.1) T is an anti-linear operator which acts as a complex conjugation, inverting time: T : i i; ˆx ˆx; ˆp ˆp (2.2) In the context of quantum mechanics, PT -symmetric systems are described by a Hamiltonian, H, which commutes with the combined PT operator, that is [PT, H] = 0. While such Hamiltonians are in general non-hermitian, there remains a possibility that they will exhibit entirely real spectra. As γ, the gain/loss parameter (that controls the degree of non-hermiticity), is increased and reaches a critical value γ P T, one can observe a spontaneous breaking of the PT -symmetry. Here the eigenfunctions of H cease to be eigenfunctions of the PT operator, thought the H and PT operators still commute. For values of γ > γ P T, know as the broken phase of the system, the spectrum becomes partially or entirely complex, in contrast to the case of γ < γ P T, known as the exact phase of the system, where the spectrum is entirely real.

13 Section 2.2. One-Dimensional Wave Scattering One-Dimensional Wave Scattering In this thesis, we consider the scattering and transport properties of one- and twodimensional optical systems. Therefore, here we will develop the theoretical groundwork in the simple case of general one-dimensional scattering. As we consider linear media, the Fourier decomposition of the fields makes evolution in time trivial once we know the spatial modulation of the Fourier components. We thus consider the time-independent scattering problem, the solution of which is contained in both the transfer matrix, which describes the wave propagation in mode space from one end of the scattering domain to the other, and the scattering matrix which relates incoming waves with outgoing waves. Consider a one-dimensional optical medium of infinite length, where the z-axis is the propagation axis of the wave. We are primarily concerned with propagation of electromagnetic waves through a finite segment of length L of inhomogeneous scattering element, located within z L/2. Without loss of generality, we assume that the electric field E is linearly polarized in the y-direction, perpendicular to the direction of propagation z. If the field is traveling in a purely dielectric material, the vectorial Maxwell s equations reduce to the one-dimensional Helmholtz equation [12] 2 E y (z, t) z 2 = n2 (z) c 2 2 E y (z, t) t 2 (2.3) where n(z) is the index of refraction of the optical medium, c is the speed of light in a vacuum, E y is the y-component of E, and E x = E z = 0. Outside of the scattering element, for z > L/2, the optical refractive index takes a constant value n 0 (homogeneous leads). Without loss of generality, we can take n 0 = 1, as in air or free space. Within the scattering element, for z L/2, n(z) is an inhomogeneous function of position, which has potentially complex values. An entirely real refractive index corresponds to energy or flux-conserving media, while a complex refractive index represents amplifying or dissipating elements. Assuming a wavepacket, a Fourier decomposition of E y (z, t)

14 Section 2.2. One-Dimensional Wave Scattering 7 gives E y (z, t) = dk 2π φ(k)e(z; k)e iω kt (2.4) where E(z; k) is a stationary scattering state of the time-independent Helmholtz equation d 2 E(z; k) dz 2 + n(z)2 ω 2 k c 2 E(z; k) = 0 (2.5) with the corresponding dispersion relation given by ω k = ck, with k identified as the free-space wave vector associated with the frequency ω k. Furthermore, we decompose E(z; k) into forward and backward-traveling components in the leads. E(z; k) = E f eikz + E b e ikz for z L/2 = C 1 f(k; z) + C 2 g(k; z) for z L/2 (2.6) = E + f eikz + E + b e ikz for z L/2 where f(z; k) and g(z; k) are two independent solutions to Eq. (2.5), whose linear combination establishes the stationary field within the inhomogeneous scattering region at the frequency ω k (or equivalently, the free-space wave vector k). The actual field is determined by the specific nature of n(z) and appropriate boundary conditions at z = L/2. In principle, if n(z) and the boundary conditions are known, E(z; k) can be completely solved using numerical or analytical methods. In the interest of investigating the transmission and reflection properties of the scattering region we introduce the transfer matrix M(k), which connects the field amplitudes on the left of the sample (E f,e b ) to the amplitudes on the right of the sample (E+ f,e+ b ). The continuity of E(z; k) and its first derivative E (z; k) at z = ±L/2 gives us a system of linear equations E f e ikl/2 + E b eikl/2 = C 1 f( L/2; k) + C 2 g( L/2; k) ike f e ikl/2 ike b eikl/2 = C 1 f ( L/2; k) + C 2 g ( L/2; k) E + f eikl/2 + E + b e ikl/2 = C 1 f(l/2; k) + C 2 g(l/2; k) ike + f eikl/2 ike + b e ikl/2 = C 1 f (L/2; k) + C 2 g (L/2; k)

15 Section 2.2. One-Dimensional Wave Scattering 8 Figure 2.1: Diagram of transfer matrix This allows us to eliminate the constants C 1 and C 2, and to rewrite the left and right field amplitude in matrix form as where L M(k) = eik 2 e ik L 2 ike ik L 2 ike ik L 2 1 E+ f E + b = M(k) E f E b (2.7) f( L ; k) 2 g( L ; k) 2 f( L ; k) 2 g( L ; k) 2 f ( L ; k) 2 g ( L ; k) 2 f ( L ; k) 2 g ( L ; k) 2 1 L e ik 2 e ik L 2 ike ik L 2 ike ik L 2 Here the basis vectors are 1 = e ikz and 0 = e ikz, so that the field E f e ikz E b e ikz is represented by the amplitude vector E f. We can now express the transmission and reflection of the system under consideration in terms of the elements of M(k). For a monochromatic wave incident on the left side of the system (left incidence), E + b E b (2.8) = 0 and so the complex transmission amplitude is t L = E+ f, while the complex reflection amplitude is r E L = E b. The real-valued f E f transmittance and reflectance are then the absolute squares of theses complex values, T L = t L 2 and R L = r L 2 respectively. Then, from Eq. (2.7), taking E + b = 0, we can

16 Section 2.2. One-Dimensional Wave Scattering 9 obtain [13, 14] t L = det(m) M 22 (2.9) Similarly, we take the case of right-incidence, that is E f amplitudes t R = E b E + b and r R = E+ f, solving to obtain E + b r L = M 21 M 22 (2.10) = 0, and define the complex t R = 1 M 22 (2.11) r R = M 12 M 22 (2.12) As noted earlier, another complete expression of the solution to the problem at hand is the scattering matrix S(k), which connects the incoming components of the electric field to the outgoing components. That is E b E + f = S(k) E f E + b (2.13) For the case of left incidence, we can take E + b = 0, E + f = t L E f, and E b = r L E f, which demonstrates that S 11 = r L and S 21 = t L. Likewise, the right incidence case yields S 12 = t R and S 22 = r R. Thus the elements of the scattering matrix are in fact the complex amplitudes, S(k) = r L t L t R r R (2.14) The scattering matrix provides a more intuitive understanding of the transport of a system than the transfer matrix; however, the transfer matrix can be preferable in some situations. For instance, when calculating the transfer matrix for two adjacent systems with transfer matrices M 1 (k) and M 2 (k), the combined transfer matrix is simply M 2 (k)m 1 (k), since the effective field to the left of the right subsystem is exactly equal to the calculated field to the right of the left subsystem.

17 Section 2.3. Passive Conservation Relations 10 Figure 2.2: Diagram of scattering matrix 2.3 Passive Conservation Relations An optical medium with an entirely real refractive index, that is with no amplification or dissipation, observes conservation of energy. In the steady state, the electromagnetic flux going into the scattering region must be either reflected or transmitted, so that the transmittance and reflectance obey T + R = 1 (2.15) Since n(z) is real, clearly n (z) = n(z) and so the existence of any solution of the Helmholtz equation E(z; k) implies the existence of another solution E (z; k) with the same wavevector k. That is, E (z; k) = E b e ikz + E f e ikz for z L/2 = E + b e ikz + E + f e ikz for z L/2 And so since we are working with linear media, the same transfer matrix applies, so that E+ b E + f = M(k) E b E f (2.16)

18 Section 2.3. Passive Conservation Relations 11 We take the complex conjugate of this equation and multiply with 0 1 to ob- 1 0 tain E+ f E + b = 0 1 M (k) E f E b (2.17) so comparing with Eq. (2.7), we see that M(k) = 0 1 M (k) 0 1 (2.18) particularly M 11 = M 22 (2.19) M 12 = M 21 (2.20) Additionally, from Eq. (2.8) we know that the determinant of the transfer matrix is det(m(k)) = f( L 2 ; k)g ( L 2 ; k) g( L 2 ; k)f ( L 2 ; k) f( L 2 ; k)g ( L 2 ; k) g( L 2 ; k)f ( L 2 ; k) (2.21) The stationary Helmholtz equation ensures that the Wronskian (fg gf ) is a constant function of position. Because f and g are defined as solutions of the Helmholtz equation, d 2 f(z; k) dz 2 d 2 g(z; k) dz 2 + n2 (z)ω 2 k c 2 f(z; k) = 0 (2.22) + n2 (z)ω 2 k c 2 g(z; k) = 0 (2.23) Multiplying Eq. (2.22) by g and Eq. (2.23) by f and taking the difference, we have Therefore f(z; k) d2 g(z; k) dz 2 g(z; k) d2 f(z; k) dz 2 = 0 d dz (f(z; k)g (z; k) g(z; k)f (z; k)) = 0 f(z; k)g (z; k) g(z; k)f (z; k) = const. (2.24) det(m(k)) = 1 (2.25)

19 Section 2.4. PT -Symmetric Conservation Relations 12 That is, M 11 M 22 M 12 M 21 = 1, which together with Eqs. (2.19), (2.20) gives 1 M 21 2 M 22 2 = 1 M R L = T L (2.26) Additionally, det(m(k)) = 1 with Eqs. (2.9), (2.11) tells us that t L = t R, so the above derivation is valid for both left and right incident waves, so we indeed conclude that T + R = 1 in general for such systems. 2.4 PT -Symmetric Conservation Relations When considering optical media with a complex refractive index, with amplifying or dissipative elements, energy is no longer conserved. Therefore, the passive conservation relation no longer holds and T + R 1 in general. However, the scattering modes of systems that obey PT -symmetry satisfy a generalized conservation relation which reveals the underlying symmetries of the scattering target. Recalling the general solution to the Helmholtz equation and the definition of the scattering matrix, E b E + f = S(k) E f E + b (2.27) we note that since the system is PT -symmetric, we also have a solution E+ b E f = S(k) Comparing these two solutions, it is clear that E+ f E b (2.28) S (k) = S 1 (k) (2.29) Because the terms of the S-matrix are exactly the complex transmission and reflection amplitudes, it is straightforward to show from Eq. (2.29) that [15] r L r R = 1 t 2 (2.30)

20 Section 2.4. PT -Symmetric Conservation Relations 13 r L t = r Lt (2.31) r R t = r Rt (2.32) Above we have used the fact that since det(m) = 1 [13, 14] for linear non-magnetic system, this means that t L = t R = t, but r L r R in general. The transmittance T = t 2 is thus the same for left and right incidence. On the other hand, the reflectances for left and right incidence, R L = r L 2 and R R = r R 2, are not necessarily the same. In particular, Eq. (2.30) implies that RL R R = 1 T (2.33) where R L and R R are, in general, not equal. Eq. (2.33) is an interesting generalization of the more familiar conservation relation T + R = 1, which applies in the absence of active elements. In the PT -symmetric case, the geometric mean of the two reflectances, RL R R replaces the passive reflectance R.

21 Chapter 3 Light Transport in PT -Symmetric Random Media The propagation of waves through complex media, whether manufactured or naturallyoccurring is an interdisciplinary field of research that addresses systems as diverse as classical, quantum, and atomic-matter waves. However, beneath this diversity, such systems share many commonalities when it comes to the study of their properties of wave transport. One such characteristic is wave-interference phenomena, which result in a halting of the propagation of waves in the system, the degree of which is linked to the randomness of the medium. First described by Anderson in the framework of quantum (electronic) waves [16] and experimentally confirmed in recent years in experiments with matter [17, 18] and classical waves [19 27], such localization occurs in the presence of a disordered potential, realized in optical media through a random index of refraction. Due to wave interference in such structures, the field within becomes strongly localized around some point within the system, exponentially decaying like e x/ξ, where ξ is the localization length of the system. In the case of optics, a system of a length scale larger than the localization length will express the localization as an exponential decay of the transmission. While this localization of classical waves has been well-understood for 14

22 15 some time [28], the propagation of light in random media containing active elements has only been scrutinized relatively recently [29 36]. The lack of a conservation law for photons allows for the coherent absorption and amplification of light in such optical systems. The combination of localization and active elements has produced results such as the dual symmetry of absorption and amplification for the average transmittance and the localization length [34, 35], the sharpness of the back-scattering coherent peak and the statistics of super-reflectance and transmittance [29 33]. In addition, there has been work done on disordered PT systems, focusing on the spectral properties of the corresponding PT -Hamiltonians [37 40], but with little investigation of their transport properties. In this chapter, we will examine the transmittance and reflectance through one-dimensional (1D) PT -symmetric systems with disorder in the index of refraction (see Fig. 3.1). We show that the exponential decay rate of transmittance, which defines the inverse localization length ξ 1, is associated with the harmonic sum of the localization length ξ 0 (W ) of a passive system with the same degree of randomness and the amplification length ξ γ (0) of a periodic PT -symmetric system with the same degree of gain and loss. Furthermore, we find that the asymptotic value of the reflectance follows a singleparameter scaling law which is dictated by the ratio Λ = ξ 0 (W )/ξ γ (0). Finally, we show that while the transmission processes are reciprocal to left- and right-incident waves, the reflection is enhanced from one side and is inversely suppressed from the other, thus allowing such PT -symmetric random media to act as unidirectional coherent absorbers (see Fig. 3.1). First, we introduce the system under investigation and the mathematical modeling used in this chapter in section 3.1. In section 3.2, we examine the scaling of the transmittance of the system. Finally, in section 3.3 we present the scaling law and asymptotic behavior of the reflectance.

23 Section 3.1. Mathematical Model 16 Figure 3.1: The gain and loss refraction index profile is uniform [see Eq. (1)] with the loss side on the left (light green color) and the gain side (dark red color) on the right of the structure. The real part of the refraction index contrast n R is random, uniformly distributed around n 0, with the symmetry n R ( z) = n R (z). For large enough system sizes (or strong disorder and/or large gain and loss) the system acts as a highperformance absorber if the incident wave is entering from the lossy side of the structure (light green arrows), while it super-reflects if the incident wave enters the structure from the gain side (dark red arrows). 3.1 Mathematical Model We consider a one-dimensional (1D) active, disordered sample with refractive index n(z) = n 0 + n R (z) + in I (z) in the interval z < L/2. The sample is embedded in a homogeneous medium with uniform refractive index n 0. Without loss of generality, for this investigation we assume n 0 = 1. Here, n R is the real index contrast and n I is the gain and loss spatial profile. In order for our system to obey PT -symmetry, that is n( z) = n (z), we enforce that n R ( z) = n R (z) and n I ( z) = n I (z). For this to be experimentally realizable in optics [3 5], these amplitudes must be small (i.e. n R, n I n 0 ). We simplify the sample under consideration by assuming that it is composed of an even number of layers of uniform width d, each with a constant real refractive index contrast n R (z j d/2 z z j + d/2) = n R j given by a random variable with a uniform distribution from ( W, +W ), satisfying the PT -symmetric constraint n R (z j ) = n R ( z j ). For layers to the left of the sample center we impose n I = γ for

24 Section 3.1. Mathematical Model 17 L/2 < z < 0 and for layers to the right of center we impose n I = γ for 0 < z < L/2, where γ 0 is the fixed gain and loss parameter. It can then be observed that we have n 0 + n R (z) + iγ for L/2 < z < 0 n(z) = n 0 + n R (z) iγ for 0 < z < L/2 (3.1) which obeys the constraints of PT -symmetry. Although the majority of our simulations below have been done for n I (±z) = γ, we have also checked that our results apply for the case that n I (±z) = γ + δn I where δn I is a random variable given by a uniform distribution centered at zero. Since the qualitative features remain the same, we will not distinguish between these two cases. As has been discussed in chapter 2, in active PT -symmetric structures we have developed a generalized conservation relation RL R R = 1 T (3.2) In considering Hermitian systems, where R L = R R = R and T 1, Eq. (3.2) reduces to R = 1 T, or the conventional T + R = 1. However, the systems under investigation in this chapter are non-hermitian and exhibit non-identical reflectances for left and right incident waves, connected as described in Eq. (3.2). In the following sections, we will describe the scaling properties of transmittance T and left and right reflectances R L and R R from such optical structures with respect to the disorder strength W, the gain and loss parameter γ, and the wave vector of the incoming wave. We have used various random refraction index contrasts W (0, 0.5), and gain and loss parameters γ (0, 0.01) which is typical for optical media. In our numerical simulations we use slabs with L = 10 to L = 10 4 of layers, each having width d = 1. The logarithmic averages lnt and lnr are performed over 10 4 disorder realizations.

25 Section 3.2. Transmittance 18 Figure 3.2: We report lnt against the system size for: a system with only gain and loss and n R = n 0 constant (upper red); a random layer medium with n I = 0 (middle black); and a PT -symmetric disorder medium (lower green). 3.2 Transmittance The transport properties of passive (γ = 0) disordered optical systems have been extensively studied. It has been shown that systems of a large enough length scale exhibit exponential decay of transmittance with respect to system size (middle black line in Fig. 3.2). This decay is characterized by the localization length ξ 0 (W ), which is related to the degree of randomness in the system and is defined by 1/ξ 0 (W ) lim lnt /L (3.3) L For 1D media, the subject of this investigation, ξ 0 (W ) 1/W 2. We consider the other limiting case, the non-disordered (W = 0) active PT -symmetric system. In this case, n R = 0 and so the system ( 3.1) is reduced to two components with

26 Section 3.2. Transmittance 19 Figure 3.3: The symbols and colors indicate different wavelengths k of the incident wave, and disorder strengths W. The black line indicates the theoretical prediction of Eq. (3.11). The meshed symbols correspond to some typical ξ γ (W ) for the scenario where the imaginary part of the refraction index is n I = γ + δn I where δn I is a random variable uniformly distributed around zero. uniform real refractive index n 0 and imaginary refractive index n I = γ for L/2 < z < 0 and n I = γ for 0 < z < L/2, where each component has length Ld/2. The electric field in this system, with scattering boundary conditions, can be solved analytically to obtain an expression for the transmittance: T = 8(1 + γ 2 ) 2 γ 2 (4 + γ 2 )[γ 2 cosh(2klγ)cos(2kl)] γ 4 γ 6 + 4γ 2 [1 + cos(kl)cosh(klγ) γ(2 + γ 2 )sin(kl)sinh(klγ)] (3.4) As in the passive disordered case, we are concerned with the domain where L is very large, and so the cosh(2klγ) term dominates and T decays exponentially (upper red line in Fig. 3.2). Additionally, enforcing the experimentally relevant condition γ 1

27 Section 3.2. Transmittance 20 causes Eq. (3.4) to reduce to T 16e 2kγL γ 4 (4 + γ 2 ) and so we consider a similar quantity to the passive localization length, where (3.5) 1/ξ γ (0) lim lnt/l 2kγ (3.6) L defines the attenuation and amplification length ξ γ (0). In considering the small system length domain, we see that the transmittance is approximately constant, with T 1. However, as L approaches a critical length L c, large oscillation in T emerge (upper red line in Fig. 3.2) before the exponential decay described above. Defining an approximation of this critical length by the condition T (L c ) = 1, we find that L c 1 ( ) 2kγ ln 16 γ 4 (4 + γ 2 ) (3.7) The existence of such a critical length scale is characteristic of optical media containing gain and is related to the lasing threshold for which T diverges. Below this length, stimulated emission enhances transmittance in the gain medium, while at larger length scales transmittance is reduced. The slope on either side of this maximum of lnt is approximately symmetric. In PT -symmetric media, as considered here, the enhancement of transmittance for L < L c due to the gain is balanced by the attenuating affects of the loss in the system, so T 1 as it would be in a passive system of uniform refractive index. Despite this, near the lasing threshold, the balanced gain and loss does not entirely compensate for the divergent behavior of T and so significant oscillations away from T = 1 are observed. We now consider a system combining both the PT -symmetric active elements and disorder. The behavior of the transmittance of such a system is shown by the lower green line in Fig In order to understand the exponential decay of this system, we consider the simplified layer geometry with index of refraction described by Eq. (3.1). Because of the general multiplicative nature of transfer matrices, the transfer matrix of the total PT -symmetric system is the product of the transfer matrix M l associated with the

28 Section 3.2. Transmittance 21 lossy subsystem and the transfer matrix M g associated with the gain subsystem. The transmittance through the combined system is then given by T = T lt g 1 r l r g 2 (3.8) Thus we need only know the scaling behavior of the terms on the right hand side of Eq. (3.8) to develop a scaling for T. In Ref. [35], these terms were studied and it was shown that both absorption and amplification lead to the same exponential decay of the transmittance, which in both cases is enhanced with respect to a passive disordered medium by the strength of the gain or loss rate. That is, lnt l = lnt g = [2kγ + ξ 0 (W ) 1 ]L/2]. In the case of gain, this somewhat surprising statement can be explained by enhanced internal reflections at the boundaries. Also due to [35], we have the duality relation r l r g = 1 for the reflections of a lossy medium (l) and a gain medium (g) with equivalent amount of gain and loss. For such systems of length L/2 > ξ 0 (W ), we can show that lnt = lnt l + lnt g 2 ln 1 r l r g = lnt l + lnt g 2 ln 2[1 cos(2θ)] (3.9) where θ is the phase of the reflection amplitude r L. Assuming that θ is a random variable uniformly distributed on the interval [0, 2π] [36], the last term after performing the average over the random variable θ is finite, and so lim lnt/l = [2kγ + ξ 0(W ) 1 ] (3.10) L Through the above argument, we conclude that the localization length of the PT - symmetric disordered system is given by ξ γ (W ) 1 = ξ γ (0) 1 + ξ 0 (W ) 1 (3.11) The above definition is equivalent to the equation ξ 0 (W )/ξ γ (W ) = 1 + ξ 0 (W )/ξ 0 (W ), and so in Fig. 3.3 we present the results of our numerical simulations for a PT -symmetric disordered sample, and demonstrate that the extracted localization length fits nicely with the scaling behavior predicted by our theoretical argument.

29 Section 3.3. Reflectance Reflectance In this section, we proceed to analyze the reflectances of the PT -symmetric disordered system in Fig In the case of random media with only gain or only loss of gain/loss strength γ, it has been shown in Ref. [35] that the reflectances of these two distinct systems obey the reciprocal relation R l R g = 1 (3.12) Specifically, it was found that in the material with gain R g > 1, while for a lossy material R l = R 1 g < 1. Here it should be noted that in each of these systems, the value of the reflectances does not distinguish between left- and right-incidence, that is R L = R R. This lack of directionality no longer holds when we move to PT -symmetric systems, where R L R R in general, a phenomenon observed for periodic PT -symmetric systems in Ref. [9]. In addition, in this chapter we have shown that for a PT -symmetric system with random index of refraction, transmittance decays exponentially with system size at a rate of 1/ξ γ (W ) as defined in Eq. (3.11). So for large system sizes, T 1 and so the relation in Eq. (3.2) describes the relationship between the reflectances: R L R R 1 (3.13) The similarity of this relation to Eq. (3.12) encapsulates that the PT -symmetric system simultaneously reflects as a gain medium in the case of left incidence, with R L < 1, and as a lossy medium in the case of right incidence, with R R > 1. This asymmetric and reciprocal reflective behavior is demonstrated through numerical simulations in Fig We define RL and R R as the geometric means of the asymptotic reflectance, so that, RL = exp( lnr L ) and RR = exp( lnr R ) when L. Further, we turn to investigating the scaling behavior of RL and R R as a function of the disorder W and the gain and loss parameter γ. We speculate that the following one-parameter scaling law

30 Section 3.3. Reflectance 23 Figure 3.4: Typical reflectances R L and R R versus system size (number of layers, L), for various values of disorder, W, and gain/loss, γ. describes this asymptotic value: RL (γ, W ) = RR (γ, W ) = f(λ), where Λ = ξ 0 (W )/ξ γ (0) (3.14) We test this through numerical simulation, and extract the asymptotic reflectances for various values of γ and W and plot them against the scaling variable Λ in Fig For realistic values of the gain and loss parameter γ 10 3, we find that the data nicely follow the one-parameter scaling hypothesis [36]. As the scaling parameter increases (due to decreasing W or increasing γ), RL decreases and R R increases. This behavior would make the proposed structure useful as a unidirectional coherent absorber with absorption tuned by adjustments of the scaling parameter Λ. As a final note, it should be recognized that the structure discussed in this chapter is different from the one suggested in Ref. [41], where it is shown that a disordered system with a single absorbing element causes coherent enhanced absorption if the phases of the

31 Section 3.4. Summary 24 Figure 3.5: For large values of Λ, lnr increases, indicating that R L (i.e., reflectance for an incident wave entering the sample from the lossy side) diminishes. In this domain, the sample acts as a unidirectional absorber. We use the same symbol and color coding for our data as in Fig input field are appropriately manipulated. The problem addressed here is fundamentally different in that we have broadband absorption in one direction without a need for phase manipulation of the incoming wave. 3.4 Summary In this chapter, we have discussed the transport properties of a one-dimensional PT - symmetric disordered layered system. We have demonstrated that the localization length ξ γ (W ), defined as the inverse decay rate of the transmittance with respect to system size, is smaller than both the localization length of the passive disordered sys-

32 Section 3.4. Summary 25 tem, ξ 0 (W ), and the attenuation and amplification length of the non-disordered PT - symmetric system, ξ γ (0), and is in fact described by the harmonic sum of the two. In addition, the system under investigation exhibits reflectance dependent on the direction of the incident wave, enhanced when incident on the gain side and suppressed when incident on the lossy side. Finally, this suppression or enhancement of the reflectance is dictated by a one-parameter scaling Λ = ξ 0 (W )/ξ γ (0) and allows us to use such structures as unidirectional quasiperfect coherent absorbers.

33 Chapter 4 PT -Symmetric Magneto-Optical Isolator Global communication and computer science are subjects of great interest to the modern scientific community, and thus so is the development of technology related to the advancement of data transmission and processing. Photonic integrated circuits (PICs) are one such technology that shows great promise for many applications, including nextgeneration optical networks, optical interconnects, tunable photonic oscillators, wavelength division multiplexed systems, coherent transceivers, and optical buffers [42 44]. The many advantages of PICs for these applications include higher performance, reduced device footprint, lower component-to-component coupling losses and lower power consumption. However, in order for PICs to be built on a useful scale for such things, there are challenges that must be overcome, such as the realization of novel classes of non-reciprocal integrated photonic devices that allow one-directional flow of information, that is, optical isolators and circulators [45]. Current realizations of such unidirectional elements rely almost exclusively on magneto-optical effects, such as the magnetic Faraday rotation caused by non-reciprocal circular birefringence. 26

34 27 However, at optical frequencies, all such non-reciprocal effects (NRE) are very weak. This requires most effective non-reciprocal devices to be so large as to be unsuitable for most desired applications. A natural method of overcoming this limitation is the implementation of the magneto-optical material into an optical resonator, which can be a photonic structure with feature sizes comparable to the light wavelength. Despite the effectiveness of these resonators, typical of the structures are undesirable effects, like enhanced absorption, linear birefringence, and nonlinear effects. The most damaging of these effects is absorption which, under resonance conditions, can dramatically affect the functionality of the optical devices, causing deviation of the transmitted light polarization from linear to elliptic (non-reciprocal circular dichroism), which significantly compromises the performance of the optical isolator. Enhanced absorption also can result in a significant power loss and even moderate absorption can lower the quality factor of the optical resonator by several orders of magnitude, significantly compromising its performance as a Faraday rotation enhancer. Some other solutions that have been presented to overcome the absorption problem include the replacement of the uniform magnetic material with a slow-wave magneto-photonic structure [46]. This type of structure can enhance asymmetry of transmittance and reduce absorption, but is ineffective at frequencies higher than those in the microwave range. An additional alternative involves the incorporation of gain and loss with non-linearity [10], but inherits the typical limitations of non-linear materials, i.e. restrictions to specific power ranges. In this chapter, we demonstrate that active magneto-photonic structures in which the spatial destribution of gain and loss displays a special, anti-linear symmetry (see Fig. 4.1) can be utilized to achieve strongly-asymmetric transport at infrared and optical frequencies, with a magnetic element of minimal size, ideal for on-chip isolator technologies. In classical optics, this involves the inclusion of regions of balanced gain and loss, as well as the careful construction of the index of refraction [3], making use of the framework of PT -symmetry discussed above.

35 Section 4.1. Model and Symmetries 28 Figure 4.1: The left/right (green/red) slab is a lossy/gain non magnetic layer with inplane anisotropy. The middle (arsenic) slab is a passive ferromagnetic material with magnetization M 0 (indicated with the arrows inside the layer). (b) A generalized PT - symmetric micro-cavity embedded in an anisotropic Bragg reflector. First, the model and appropriate symmetries will be introduced in section 4.1. Next, the scattering formalism will be discussed in section 4.2. Finally, the results of our investigation will be presented in section 4.3, and our conclusions will be given in section Model and Symmetries The system that we investigate, shown in Fig. 4.1(a), is a microcavity consisting of three components: a central magnetic layer sandwiched between two active (one with gain and the other with loss) anisotropic layers distributed in such a way that the whole structure exhibits an antilinear symmetry. The magnetic layer provides a nonreciprocal circular birefringence (magnetic Faraday effect) in order to break the Lorentz reciprocity, which would otherwise prevent the development of any asymmetry in the transmission properties of the structure. The magnetic circular birefringence breaks the Lorentz reciprocity, however, it is not sufficient to provide asymmetry in forward and backward transmission, which requires broken space inversion symmetry. This is achieved through the use of misaligned birefringent layers as suggested in [47]. Such

36 Section 4.1. Model and Symmetries 29 a system, with no incorporated active gain or loss, may exhibit asymmetric transport for a single input polarization, but active elements are absolutely necessary to achieve asymmetry in forward and backward transmission averaged over polarizations. With all of our elements: a magnetic component and active but balanced gain and loss in anisotropic layers, we achieve polarization-independent asymmetry and maintain generalized unitary relations for the energy flux conservation. Enhancement of this asymmetry is achieved via embedding of the above-described system in an anisotropic Bragg grating, shown in Fig. 4.1(b). The anisotropic grating forms broad frequency domains at its pseudo-gaps which creates an effective high-q optical resonator, enhancing the influence of the magnetic component. In order to model this structure, we define the non-hermitian permittivity tensor for each of our components. For our lossy anisotropic layer, the permittivity tensor is given by ɛ + iγ + δcos(2φ 1 ) δsin(2φ 1 ) 0 ˆɛ Al = δsin(2φ 1 ) ɛ + iγ δcos(2φ 1 ) ɛ zz (4.1) while the permeability tensor is unity. Similarly, for the gain anisotropic layer, the permittivity tensor is ɛ iγ + δcos(2φ 2 ) δsin(2φ 2 ) 0 ˆɛ Ag = δsin(2φ 2 ) ɛ iγ δcos(2φ 2 ) 0 (4.2) 0 0 ɛ zz with a permeability tensor of unity as well. Above, δ describes the magnitude of in-plane anisotropy, γ is the gain/loss parameter, and the angles φ 1 and φ 2 define the orientation of the principle axes in the xy-plane. The magnetic component has as a permittivity tensor ɛ iα 0 ˆɛ F = iα ɛ ɛ zz (4.3)

37 Section 4.2. Scattering Formalism 30 and permeability tensor µ iβ 0 ˆµ F = iβ µ µ zz (4.4) Here α and β are the gyrotropic parameters responsible for the Faraday rotation, dependent upon the static components of the magnetic field H 0. Outside of our scattering region, we assume that the permittivity takes the constant form ɛ ˆɛ = 0 ɛ ɛ 0 Taking the center of our magnetic component to be the axis of symmetry, and noting that changing H 0 H 0 and M 0 M 0 implies [47] α α and β β, we can define an antilinear generalized PT symmetry, Π-symmetry, under which our system is invariant. The operator Π = PT is composed of the traditional antilinear timereversal operator T which performs transpose complex conjugation, and a modified linear operator P. The operator P = PΘ is a combination of the parity operator P, representing spatial inversion r r, and the exchange operator Θ which changes φ 1 φ 2. Notice that in the case that φ 1 = φ 2, the exchange operator Θ effects no change, and the symmetry of our system reduces to traditional PT -symmetry. 4.2 Scattering Formalism In order to develop our scattering formalism, we begin with the time-harmonic Maxwell equations: E( r) = i ω c ˆµ H( r), H( r) = i ω c ˆε E( r) (4.5) with the solution E( r) = e i(kxx+kyy) E(z), H( r) = e i(k xx+k yy) H(z) (4.6)

38 Section 4.2. Scattering Formalism 31 We assume normal propagation, i.e. k x = k y = 0. the solutions of Eq. (4.6) for E(z) in the left (l) and right (r) side of the scattering region, are written in terms of the forward and backward traveling waves: where E l,r (α, β, φ 1, φ 2, z) = A l,r e ikz + B l,r e ikz (4.7) A l,r = [A l,r x (α, β, φ 1, φ 2 ) A l,r y (α, β, φ 1, φ 2 )] T (4.8) B l,r = [B l,r x (α, β, φ 1, φ 2 ) B l,r y (α, β, φ 1, φ 2 )] T (4.9) The 4 4 transfer matrix M(α, β, φ 1, φ 2 ) describes the relation between the electric field on the left and right sides of the scattering region (taking units c = 1): Ar B r = M Al B l (4.10) M = M 11 M 12 (4.11) M 21 M 22 Because our system is invariant under the Π operation, we can obtain another solution of the Maxwell equations by performing the operation on our solution in Eq. (4.7): E l,r (α, β, φ 1, φ 2 )Π ( E l,r ( α, β, φ 2, φ 1 )) (4.12) Now applying to the right hand side of Eq. (4.12) the transfer matrix M(α, β, φ 1, φ 2 ) we obtain Al ( α, β, φ 2, φ 1 ) B l ( α, β, φ 2, φ 1 ) = M(α, β, φ 1, φ 2 ) Ar ( α, β, φ 2, φ 1 ) B r ( α, β, φ 2, φ 1 ) Performing complex conjugation and taking α α, β β, and φ 1 Eq. (4.10) we arrive at Ar ( α, β, φ 2, φ 1 ) B r ( α, β, φ 2, φ 1 ) = M ( α, β, φ 2, φ 1 ) Al ( α, β, φ 2, φ 1 ) B l ( α, β, φ 2, φ 1 ) (4.13) φ 2 on (4.14)

39 Section 4.2. Scattering Formalism 32 Together with Eq. (4.13), it is clear that M(α, β, φ 1, φ 2 )M ( α, β, φ 2, φ 1 ) = ˆ1 (4.15) We now define the scattering matrix, Bl A r S = S rl t l Al t r r r B r (4.16) (4.17) which connects the incoming components of the electric field to the outgoing components. We can deduce the elements of the scattering matrix by considering the cases of left incidence and right incidence. In the first case, we suppose that the only incoming electric field is on the left side of the system (and scaled to 1 for convenience). That is, A l = 1 and B r = 0. From Eqs. (4.16) and (4.17), we now have Bl A r = rl t l t r r r 1 (4.18) 0 and so B l = r l and A r = t l. Substituting into Eq. (4.10) we obtain tl = M 11 M 12 1 (4.19) 0 M 21 M 22 r l. So t l = M 11 + M 12 r l and 0 = M 21 + M 22 r l, which can be solved to obtain r l = M 1 22 M 21 (4.20) t l = M 11 M 12 M 1 22 M 21 (4.21) our complex transmission and reflection coefficients for left-incident waves. In the case of right incidence, where there is no incoming electric field on the left side of the system, we similarly take A l = 0 and B r = 1. From Eqs. (4.16) and (4.17), here we have Bl A r = rl t l t r r r 0 (4.22) 1

40 Section 4.2. Scattering Formalism 33 and so B l = t r and A r = r r. Substituting into Eq. (4.10) we obtain rr = M 11 M 12 1 M 21 M 22 So r r = M 12 t r and 1 = M 22 t r, which can be solved to obtain 0 t r (4.23) t r = M 1 22 (4.24) r r = M 12 M 1 22 (4.25) Thus, Eqs. (4.20), (4.21), (4.24), and (4.25) express the elements of the scattering matrix in terms of the transfer matrix, and so we can extend the relation of Eq. (4.15) to the scattering matrix to obtain the conservation relation: PS ( α, β, φ 2, φ 1 )PS(α, β, φ 1, φ 2 ) = ˆ1 (4.26) where P = 0 1 is the parity operator in the channel space. This is a weaker con- 1 0 straint than unitary of the scattering matrix S S = ˆ1, which does not hold for an active system, such as the one in this investigation. In a passive system respecting the typical unitary relation, each eigenvalue e n of S is unimodular, that is, e n = 1, and so the corresponding eigenvector e n undergoes no amplification or dissipation. In an active magneto-optical system, this is not true in general. An analogous condition on the eigenvalues of such a system can be obtained by rewriting Eq. (4.26) as PS ( α, β, φ 2, φ 1 )P = S 1 (α, β, φ 1, φ 2 ) and evaluating both sides of the equation operating on an eigenvector of the scattering matrix e n, so that we have PS ( α, β, φ 2, φ 1 )P e n = S 1 (α, β, φ 1, φ 2 ) e n = e 1 n e n (4.27) Moreover, since T ΘS(α, β, φ 1, φ 2 )T Θ = S ( α, β, φ 2, φ 1 ), the left side of Eq. (4.27) is exactly Π 1 S(α, β, φ 1, φ 2 )Π e n and so S(α, β, φ 1, φ 2 )Π e n = e 1 n ( α, β, φ 1, φ 2 ) e n (4.28)

41 Section 4.3. Asymmetric Transport 34 Figure 4.2: Notably, in the frequency range ω the system is in the exact phase, with all eigenvalues unimodular, while for all other displayed frequencies the system is in the broken phase. This tells us that we can make conditional statements about the eigenvalue e n. If e n is, in addition to being an eigenvector of S, an eigenvector of Π, then from Eq. (4.28) we have that e n( α, β, φ 2, φ 1 )e n (α, β, φ 1, φ 2 ) = 1, and so e n is unimodular. If this holds for all four eigenvectors of the scattering matrix, we say that the system is in the exact phase. On the other hand, if e n is not an eigenvector of Π, we have that e m( α, β, φ 2, φ 1 )e n (α, β, φ 1, φ 2 ) = 1, where e m is the eigenvalue corresponding to some other eigenvector e m of S. The eigenvectors e n and e m then are not unimodular, but have the property that the two eigenvectors are paired, one experiencing dissipation and the other experiencing an equivalent amount of amplification, as indicated in Fig Here we say that the system is in the broken phase.

42 Section 4.3. Asymmetric Transport 35 Figure 4.3: All layers have the same thickness d, the phase misalignment is φ = π, the anisotropy is δ = 1, while the gain/loss parameter is γ = Furthermore, the real part of the permittivity for the birefringent layers is ɛ = 9, for the magneto-optical layer is ɛ = while we assume that ɛ 0 = 1. The Faraday gyrotropic parameters are α = 0.925, β = 0, and µ = Asymmetric Transport As is clear in the expressions shown in Eqs. (4.20), (4.21), (4.24), and (4.25), the transmission and reflection coefficients for left and right incidence are, in general, not the same, so it is reasonable to expect asymmetric transport. We now discuss the asymmetric transport exhibited by the trimer system shown in Fig. 4.1(a). From [47], choosing a misalignment angle φ = φ 2 φ 1 not equal to 0 or π/2 causes asymmetry between forward and backward wave propagation. If the system is passive, γ = 0, the sum rule resulting from the flux conservation and unitarity of the scattering matrix prohibit a polarization-averaged asymmetric transmission [48]. However, the introduction of gain and loss into the system frees us from this restriction. Defining the reflectances and transmittances R r 2 and T t 2, in Fig. 4.3(a), we present the results of our numerical simulations for the left and right reflectances R l,r averaged over all possible polarizations of the incoming wave. The difference between our polarization-averaged transmittances T l T r is presented in Fig. 4.3(b). These results demonstrate that the reflectances and transmittances for left and right incident waves are different

43 Section 4.4. Summary 36 from one another. Before this work, asymmetry in reflectances has been achieved with other systems with antilinear symmetry [9], but such asymmetry in transmittances is a new phenomenon which requires both magneto-optical materials as well as active gain and/or loss materials. We seek to quantify the asymmetric transmittance of the system in Fig. 4.3(c), which displays the asymmetric quality factor Q T, defined as Q T = T l T r T l + T r (4.29) The asymmetric transport of the magneto-optical micro-cavity can be further amplified by the embedding of the trimer unit between two anisotropic Bragg mirrors as shown in Fig. 4.1(b). In comparison to an isotropic Bragg grating, which exhibits clearly defined bands and gaps in the corresponding transmission spectrum, the introduction of anisotropy creates pseudo-gaps in the transmission spectrum, as indicated in the highlighted regions of Fig The lower subfigure of Fig. 4.4 demonstrates that we observe greatly-enhanced nonreciprocity, again represented by Q T, within these frequency windows for a system with Bragg gratings composed of only 45 layers. The Bragg and trimer system exhibits a dramatically increased degree of asymmetry because the magneto-optical cavity, when within the Bragg structure, acts as a high-quality optical resonator containing magneto-optical material. The light passing through the system thus resides within the magnetic material for a much longer period of time. Regardless of the direction of travel within the cavity, the Faraday rotation maintains one sign, and so the amount of Faraday rotation can be understood to be directly proportional to the amount of time a photon resides within the material. 4.4 Summary It has previously been demonstrated that non-reciprocal effects, such as Faraday rotation, can be enhanced through resonance conditions (see [49 51] and references therein), however, these conditions also enhance the absorption inherent in the very materials which produce these effects, compromising the performance of such non-reciprocal

44 Section 4.4. Summary 37 Figure 4.4: The micro-cavity has the same parameters as in Fig. 2. The transmittance spectrum of the periodic Bragg is also shown in order to identify the pseudo-gap frequency windows (green highlighted areas) where the QT -factor takes large values. devices. In this chapter, it has been shown that a unit also containing balanced implementation of gain and loss, as shown in Fig. 4.1(a), can overcome this challenge. This both allows for an enhancement of non-reciprocal effects without the energy-loss issue, as well as producing a strong polarization-independent transmission asymmetry, the defining property of an optical isolator. Because of the relative simplicity and potentially small size of the components necessary for this design, this may be used as the essential component for chip-scale non-reciprocal devices such as optical isolators and circulators.

45 Chapter 5 Magneto-Optical Unidirectional Laser In this chapter, we show that we can achieve highly-unidirectional lasing behavior in an active magneto-photonic system. In order to achieve this target, we will have to resolve a two-fold problem: we must design a system that exhibits lasing, as well as highly asymmetric scattering behavior at the lasing threshold. In chapter 4 of this thesis we have demonstrated that an active, anisotropic magneto-optical system can be used as an effective polarization-independent isolator, so this provides an adequate basis upon which to conduct our investigation. The basic unit of our system is a three-layer cavity (trimer), composed of a central magnetic layer sandwiched between two anisotropic layers. In chapter 4, we have shown that asymmetric transmission can be achieved by a combination of the non-reciprocal circular birefringence imposed by the magnetic component and the broken space-inversion symmetry, brought about by the misaligned birefringent layers. In producing the dispersion relation for the closed periodic system with no gain, we see highly asymmetric characteristics. Particularly, we find frequencies for which only 38

46 Section 5.1. Scattering Formalism 39 Figure 5.1: Schematic of a periodic system of trimer units. forward-moving (positive k) or backward-moving (negative k) waves can pass through the system. An alternative possible asymmetric characteristic of the dispersion relation, more relevant to the work at hand, is the possibility of frozen modes with dω dk = 0 for only one direction of traveling waves [47]. We then make use of a scattering formalism to identify frequencies at which the active, gain-pumped system will exhibit lasing. In combination with the asymmetric properties of the dispersion relation for the closed system, we will demonstrate the possibility of high-quality unidirectional lasing in open cavities composed of periodic arrangements of the trimer unit, as indicated in Fig Finally we will investigate the scaling of the unidirectional laser activity of our system as it grows larger. 5.1 Scattering Formalism As in the previous chapter, we approach the scattering and transport properties of our system through the use of the transfer matrix M and the scattering matrix S. The transfer matrix connects the electric field on the left side of the system to the field on