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University of Ljubljana Faculty of Mathematics and Physics Seminar Ib PT symmetry in optics Author: Jakob Frontini Mentor: doc. dr. Miha Ravnik Abstract In this seminar the notion of non-Hermitian parity-time (PT) symmetry is presented in the context of advanced optics and photonics. Recently, PT symmetry was explored as exciting possible mechanism for designing and compensating losses in optical metamaterials. The selected realisations of PT symmetric systems in optics are presented, including two coupled PT symmetric waveguides, a PT symmetric Bragg scatterer and a PT microring laser. June 10, 2016 Contents 1 Introduction 1 2 Theoretical background 2 3 Coupled waveguides 3 4 Unidirectional invisibility 4 5 Single-mode laser 7 6 Conclusion 10 References 10 1 Introduction The development of new artificial structures and materials is one of the major research challenges in optics today. In most research so far, the design has been based on manipulating the refractive index profile. There is no doubt as to how useful this is, since this has led to the design and fabrication of photonic crystals [1] and photonic crystal fibres and to the exploration of nanoplasmonics [2] and negative-index metamaterials [3]. Recently, a new approach has been proposed: simultaneously using gain and loss as a way of achieving optical behaviour that is at present unattainable with standard arrangements. This is done by carefully designing the profile of not only the real, but also the imaginary part of refractive index of the structure. Positive imaginary part causes gain and negative part causes loss. Loss is abundant in physical systems and is typically considered to be a problem. Gain, however, is important in optoelectronics because it provides a mean to induce lasing or to overcome losses. The question that arises is whether new artificially made optical structures with balanced gain and loss can be produced. Quantum mechanics is based on Hermitian operators that have real eigenvalues. But a Hamiltonian does not need to be Hermitian in order to have real eigenvalues. There is an alternative formulation of quantum mechanics, where the requirement of hermiticity is replaced by a more general requirement of space-time reflection symmetry (PT symmetry). Certain optical systems are governed by equations that are similar to Schrödinger equation, so the concept of PT symmetry can be extended to such systems. This means that PT symmetry is important for developing new materials. This seminar is organised as follows. In Chapter 2 the concept of parity-time (PT) symmetric Hamiltonians in quantum mechanics is introduced and a corresponding concept in optics is established. Then a few realisations of PT symmetric systems in optics are presented: in Chapter 3 a PT optical coupled system, in Chapter 4 a PT symmetric Bragg scatterer and in Chapter 5 a PT microring laser. Conclusions are given in Chapter 6. 1 2 Theoretical background New materials were recently developed by using the notion of “parity-time symmetry” in optical systems. In 1998, Bender and Boettcher [4] showed that wide class of nonHermitian Hamiltonians can actually possess entirely real spectra as long as they respect parity-time symmetry. In quantum mechanics, the action of parity operator P̂ is defined by the relation (i, x, p) → (i, −x, −p), and that of time reversal operator T̂ by (i, x, p) → (−i, x, −p), Where x and p represent position and momentum operators, respectively and i is the imaginary unit. The P̂ T̂ symmetry of the Hamiltonian Ĥ means that it commutes with the P̂ and T̂ operators: Ĥ P̂ T̂ = P̂ T̂ Ĥ (2.1) Thus Ĥ and P̂ T̂ may share a common set of eigenfunctions. Let us consider a Hamiltonian Ĥ = p2 /2 + V (x) with a complex potential V (x). It can be shown that the condition for PT symmetry reduces to the requirement that the complex potential involved satisfies [5] V (x) = V ∗ (−x) (2.2) In other words, the real part of the potential must be an even function of position and that of the imaginary part must be odd. In this configuration, the eigenfunctions are no longer orthogonal and hence the eigenmodes vector space is skewed. Even more fascinating is the existence of a sharp, symmetry-breaking transition. In the broken parity-time symmetry the eigenvalues of the system cease to be all real. The notions of parity-time symmetry can be experimentally explored and used in optics. Because such photonic systems are typically considered classical, they can be realised without introducing any conflict with the hermiticity of quantum mechanics. What allows this duality between quantum mechanics and optics is the formal equivalence of the wave equation to the Schrödinger equation ∂ψ h̄2 ∂ 2 ψ =− + V (x)ψ. ih̄ ∂t 2m ∂x2 (2.3) For example, under paraxial propagation conditions the electric field envelope E of the optical beam is governed by the equation: i ∂E 1 ∂ 2E + + k0 [nR (x) + inI (x)]E = 0, ∂z 2k ∂x2 (2.4) where k0 = 2π/λ, k = k0 n0 , λ is the wavelength of light in vacuum and n0 represents the refractive index of the substrate. nR (x) is the real part of the index modulation, and nI (x) denotes the imaginary component leading to gain or loss. We can see the similarities between 2.3 and 2.4. The complex refractive index plays the role of the complex potential and PT symmetry in the context of optics demands: n(x) = n∗ (−x). (2.5) By solving Schrödinger equation we get energy eigenvalues, which are propagation constants in the case of paraxial optics. 2 3 Coupled waveguides In integrated optics, a single PT element can be realised in the form of a coupled optical system. Rüter et al. [6] studied two PT symmetric waveguides fabricated from iron-doped LiNbO3 . If the distance between the two waveguides is small enough that the evanescent tails of the guided modes overlap, then they become coupled. One of the waveguides provides gain for the guided light while the other experiences the equal amount of loss. Under these conditions, the optical field dynamics in the two coupled waveguides are described by: dE1 γ i − i E1 + κE2 = 0, (3.1) dz 2 dE2 γ i + i E2 + κE1 = 0, (3.2) dz 2 where E1,2 represent field amplitudes in channels 1 and 2, κ is the coupling constant and γ is the effective gain coefficient. We can rewrite the system of equations into a matrix differential equation ! ! ! d E1 i γ2 −κ E1 · =i . (3.3) −κ −i γ2 E2 dz E2 q The eigenvalues of the matrix in 3.3 are λ± = ±κ 1 − (γ/2κ)2 . The general solution can be written as (E1 , E2 ) = c1 exp(iλ+ z)u+ + c2 exp(iλ− z)u− , where u± are the eigenvectors and c1 and c2 are constants, which can be calculated from initial conditions. The behaviour of this non-Hermitian system can be explained by considering the structure of its eigenvectors, above and below the point γ/(2κ) = 1. Below this threshold the eigenvectors are given by u± = (1, ± exp(±iθ)), with corresponding eigenvalues being ± cos θ, where sin θ = γ/2κ. At phase transition, the modes coalesce to u± = (1, i), where amplitudes in both channels have the same magnitude. Above threshold (γ > 2κ), u± = (1, i exp(∓θ)), where in this range cosh θ = γ/2κ and the eigenvalues (propagation constants in the language of optics) are ∓ sinh θ. Unlike Hermitian systems, these eigenmodes are no longer orthogonal. This leads to important implications for beam dynamics including a non-reciprocal response and power oscillations. For a conventional Hermitian system (γ = 0), any superposition of the eigenmodes leads to reciprocal wave propagation: the light obeys left-right symmetry. The situation changes when gain and loss are involved in the coupled system (Fig. 1). If gain is below threshold, the relative phase difference between the two fields increases with increasing gain from their initial values at 0 and π, and finally, at threshold they coalesce at θ = π/2. Remarkably, light propagation is now non-reciprocal: the pattern of beam propagation differs depending on whether the initial excitation is on the left or right waveguide. This behaviour is even more drastic above threshold. In this regime, light always leaves the sample from channel 1, regardless of the input. This can be explained by noting that the two complex eigenvalues occur in complex conjugate pairs, with the corresponding amplitudes either exponentially increasing or decaying. Thus only one eigenmode effectively survives. The experimental set-up is shown in Fig. 2. The coupled waveguide system in the experiment [6] was based on Fe-doped LiNbO3 . Optical gain was provided through twowave mixing using the material’s photorefractive nonlinearity. A mask on top of the 3 Figure 1: Light propagation in a PT material. Left (right) panels correspond to initial excitation in the left (right) waveguide in each case. Waveguides are coloured according to the gain/loss parameter (γ): gray for passive (γ = 0), red for gain and green for loss. (a) In a passive system, the total power (orange line) remains constant. (b) System below the threshold γ < 2κ. Variation of total power can be seen. (c) γ > 2κ. Beam power grows exponentially and the beam propagation is again non-reciprocal with respect to the mirror axis of the two waveguides. [7] sample was used to partially block the pump beam, to provide input in only one channel. The output intensity and the phase relation between the channels (using interference with a reference plane wave) were monitored using a CCD camera. Losses in the system arise from the optical excitations of electrons from Fe2+ centres to the conduction band. The optical two-wave mixing gain has a finite response time, so the evolution of the intensity at the output was monitored as a function of time t. 4 Unidirectional invisibility Lin et al [8] studied the possibility of synthesizing PT symmetric objects which can act as unidirectional invisibility media near the spontaneous PT symmetry breaking points (exceptional points). The subject of cloaking physics has attracted considerable interest in the recent years. Lin proposed a new approach to achieve invisibility. Rather than surrounding a scatterer with a cloak medium, in their case the invisibility arises because of spontaneous PT symmetry breaking. This can be accomplished via designing a PT synthetic Bragg structure. They considered scattering from such a structure (Fig. 3) and investigated the consequences of PT symmetry in the scattering process. 4 Figure 2: Experimental set-up. An argon-ion laser beam (wavelength 514.5 nm) is coupled into the arms of the structure fabricated on a photorefractive LiNbO3 substrate. An amplitude mask blocks the pump beam from entering channel 2, thus enabling two-wave mixing gain only in channel 1. The output intensity and phases were monitored with a CCD camera. Figure 3: PT symmetric Bragg scatterer. The wave entering from the left (upper figure) goes through the structure unaffected, while a wave entering the same structure from the right (lower figure) experiences enhanced reflection. Passive gratings (involving no gain or loss) can act as high efficiency reflectors around the Bragg wavelength. Instead, a PT symmetric object at the PT symmetric breaking point is reflectionless over all frequencies around the Bragg resonance when light is incident from one side of the structure while from the other side its reflectivity is enhanced. Additionally, in this same regime the transmission phase vanished, which is necessary condition for avoiding detectability. Surprisingly, these effects persist even in the presence of Kerr nonlinearities. These effects were demonstrated on a PT symmetric periodic structure or grating with a refractive index profile n(z) = n0 + n1 cos(2βz) + in2 sin(2βz) for |z| < L/2. This grating is surrounded by a homogeneous medium with a refractive index n0 for |z| > L/2. In practice, refractive index modulations are small, e.g. n1 , n2 n0 . The grating wave number β is related to its spacial periodicity Λ via β = π/Λ and in the absence of gain (n2 = 0) the periodic index modulation leads to a Bragg reflection close to the Bragg angular frequency ωB = cβ/n0 where c is the speed of light in vacuum. In this arrangement, a time-harmonic electric field obeys the Helmholtz equation: ∂ 2 E(z) ω 2 2 + 2 n (z)E(z) = 0, ∂z 2 c 5 (4.1) which formally coincides with stationary 1D Schrödinger equation: ∂2 2m(V (z) − Ek ) ψk (z) = 0. ψk (z) − 2 ∂z h̄ (4.2) We would like to obtain the transmission T and reflection R coefficients for left (L) and right (R) incidence waves. It turns out that for k ≈ β close to the Bragg point, they can be written as |λ|2 T = (4.3) |λ|2 cos2 (λL) + δ 2 sin2 (λL) RL = (n1 − n2 )2 k 2 /4n20 ; δ 2 + |λ cot(λL)|2 RR = (n1 + n2 )2 k 2 /4n20 , δ 2 + |λ cot(λL)|2 (4.4) q where λ = δ 2 − (n1 − n2 )2 k 2 /4n20 and δ = β − k is the detuning. The transmission is the same for left and right incidence. For n2 = 0 one gets the standard scattering features of periodic Bragg structures (RL = RR ). Close to the Bragg point δ = 0 the reflection in the large L limit becomes unity (transmission becomes zero), see Fig 4a. Instead, if n2 6= 0, an asymmetry between left and right reflection starts to develop. Figure 4: (a) Numerical evaluation from Eq. 4.1 of transmission T and reflection R for a Bragg grating. The following parameters were used: n0 = 1, n1 = 10−3 , L = 12.5π and β = 100. In case of a PT grating, the system is at the exceptional point when n2 = n1 . In this case, RL is diminished for a broad frequency range, while RR is enhanced, in agreement with the theoretical predictions. (b) Transmission phase φt as a function of the detuning δ for the PT symmetric system of Fig. 3. The results of the exceptional point (n1 = n2 ) are compared with the passive structure n2 = 0. (c) The corresponding transmission delay times τt as a function of δ. This asymmetry becomes most pronounced at n2 = n1 . At the Bragg point δ = 0. the transmission is identical to unity, T = 1, while the reflection for the left incident waves is RL = 0 (see Fig. 4a). This is a manifestation of the PT nature of this periodic structure. At the same time, the reflection for right incident grows with the size L of the grating as 2 RR = L k 2 n1 n0 2 sin(Lδ) Lδ 6 2 δ→0 2 −−→ L k 2 n1 n0 2 . (4.5) This was confirmed by using numerical simulations. This phenomenon is referred to as unidirectional reflectivity. Furthermore, equations 4.4 indicate that a transformation n2 → −n2 reverts the reflectivity of the system, allowing for reflectionless behaviour for the right incident waves, i.e. RR = 0, while left incident waves now follow Eq. 4.5. In other words, the phase delay between the real and imaginary part refractive index dictates the unidirectional reflectivity of the system. Reflectionless potentials in one-dimensional scattering configuration are in general not invisible. This because the transmitted wave might depend on energy, thus leading to distortion after passing through the sample. It is therefore necessary to examine the phase φt of the transmission t = |t| exp(iφt ) and compare it to the phase acquired by a wave propagating in a grating-free environment (φt = 0). It turns out that the phase φt close to the Bragg point is φt = arctan λ − tan(λL) + Lδ. δ (4.6) At n1 = n2 , which means that λ = δ, the resulting transmission phase is φt = 0. Thus interference measurements will fail to detect this periodic structure. Although the above analytical result is performed near the Bragg point δ ≈ 0, numerical results shown in Fig. 4b indicate that these effects are valid over a wide range of frequencies. The results are compared to a passive (n2 = 0) Bragg grating in Fig. 4b. The dependence of the transmission delay time τt ≡ dφt /dk on the detuning δ was also analysed. This quantity provides information about the time shift experienced by a transmitted wave packet when its average position is compared to a corresponding one in the absence of the scattering medium. Using Eq. 4.6, it shown that at the PT symmetry breaking point the transmission delay time is τt = 0. The results for a structure at n1 = n2 are shown in Fig. 4c together with those for a passive case. 5 Single-mode laser Laser resonators support large number of closely spaced modes. As a result, the outputs from such laser are subject to fluctuations and instabilities. During recent decades, different strategies have been explored to achieve single mode operation, which is needed for enhanced laser performance with higher monochromaticity, less mode competition and better beam quality. Obtaining single mode operation depends on sufficient gain and loss modulations, but factors such as inhomogeneous gain saturation can impede such modulation. A general design concept with flexible design of cavity modes is desired. Feng et al. [10] used the PT symmetry breaking concept to delicately manipulate gain and loss of a microring resonator and observed single-mode laser oscillation of a whispering-gallery mode. Whispering-gallery modes (WGMs) are a type of wave that can travel around a concave surface. In a ring cavity, two counter-propagating modes interfere with each other and form two orthogonal standing waves (WGMs) in the ring resonator. These two WGMs are the eigenmodes of the microring resonator. They take place no matter whether or not there is additional modulation in the ring resonator. Without the coupling between the clockwise and counter-clockwise modes by the additional gain and loss modulation, two WGMs would share the same eigen wavenumber; while with 7 modulation, the coupling is introduced and the wavenumbers of two WGMs become different. In a ring cavity without any modulation, the clockwise mode and counter-clockwise mode do not cross-talk. The mode equations can be written as dA = iβ0 A, R dφ dB = iβ0 B, R dφ (5.1) where A and B are the clockwise and the counter-clockwise travelling waves in a harmonic form of A = A0 exp(iβRφ) and B = B0 exp(iβRφ), R is the radius of the resonator and φ is the azimuthal angle. The wavenumbers associated with two standing modes are the same as the original wavenumbers of the clockwise and counter-clockwise modes: β = β0 . If the ring cavity is pumped, the created optical amplification makes β0 complex. If gain is sufficient to compensate for the loss, lasing occurs. Since there is no limitation to the azimuthal order of the WGMs, multi WGMs corresponding to different azimuthal orders typically lase simultaneously, showing the multi-mode lasing spectrum similar to Fig. 7A. When the PT modulation is added, the equations 5.1 have to be modified. Assuming a small variation of index change and negligible scattering loss in the PT microring resonator, the coupled mode equations can be written as dA = iβ0 A + in00 κB, R dφ dB = iβ0 B − in00 κA, R dφ (5.2) where n00 denotes the index modulation in only the imaginary part. The wave numbers of the WGMs can be obtained, β = β0 ± iκn00 . manifesting non-threshold PT symmetry breaking in PT microring lasers (Fig. 5B,C). This thresholdless condition is robust against the nonlinearity during the pump process. For example, the real part index change by the optical nonlinear effects only affects β0 in Eq. 5.2. However, with any arbitrary values for β0 , the bifurcation in the imaginary part of β takes place immediately and the resulted PT symmetry breaking is thresholdless with respect to n00 . The PT microring laser can in principle support all WGM orders. However, the continuous rotational symmetry is only valid for the desired WGM order, because the introduced PT gain-loss modulation was specifically designed for the desired WGM order and it does not affect the non-desired ones. Consequently, the coupled mode equations for the non-desired WGMs can be written as in Eq. 5.1, meaning that they experience balanced gain/loss modulation and thus remain below the lasing threshold. The PT-synthetic microring resonator was designed with 500 nm thick InGaAsP multiple quantum wells, which have a high gain coefficient, on an InP substrate (Fig. 5A). Periodically formed Cr-Ge structures on top of the InGaAsP introduce loss in such fashion that PT symmetry is satisfied for the desired WGM order. In the lasing mode, electric fields are confined mainly in the amplification sections, exhibiting a net gain coefficient 6. For other WGMs, electric fields in both energy-degenerate WGMs are uniformly distributed in gain and loss regimes and so gain and loss average out. 8 Figure 5: (A) Schematic of the PT microring laser which, consists of Cr/Ge structures on top of InGaAsP microring to mimic a gain/loss modulation. The diameter and width of the microring resonator are 8.9 µm and 900 nm, respectively. Here, the designed azimuthal order is m = 53. (B and C) The same eigenfrequency (197.6 THz) and complex conjugate imaginary eigenspectra for the two modes at m = 53. Figure 6: WGMs of different azimuthal orders in the PT microring laser. (A and B) Electric field intensity distributions for lasing and absorption modes at m = 53. The two modes have conjugate gain/loss coefficients. (C and D) Electric field intensity distributions at m = 54. The two modes share the same eigenfrequency and a similar loss coefficient. The PT microring laser with Cr/Ge modulations was fabricated using overlay electron beam lithography and plasma etching. It was optically pumped with a femtosecond laser. At pumping intensities above the lasing threshold, the single mode lasing peak is seen at the wavelength of 1513 nm, confirming the theoretical predictions. The lasing linewidth is about 1.7 nm, corresponding to a quality factor of about 890, that is limited by the surface roughness of the sample. The lasing threshold at peak pump power density is about 600 MW cm−2 . For comparison, a control sample of a same-sized microring resonator without the additional index modulation was fabricated. As expected, a typical multimode lasing spectrum with different azimuthal orders m was observed (Fig. 7). Relative to the PT microring laser, it can be seen that the resonance peaks for the same azimuthal order of 9 m = 53 are well matched. The power efficiency and the lasing threshold are also similar, because the introduced loss minimally affects the desired lasing mode. An additional PT microring laser with azimuthal order m = 55 was also fabricated. Its lasing emission also agrees with the conventional WGM laser. By changing the desired azimuthal order of the structured PT modulation, the single mode lasing frequency can be efficiently selected. Figure 7: Comparison between PT laser and typical microring laser. (A) Multimode lasing spectrum from a typical microring WGM laser, showing a series of lasing modes corresponding to different azimuthal orders m. (B) Single-mode lasing spectra of the PT microring lasers operating at the m = 53 and m = 55, consistent with the lasing wavelength for the same m in (A). This shows that PT microring lasers efficiently select the desired lasing mode and do not alter the original WGMs. 6 Conclusion It was shown that the initial proposal of PT symmetric systems in quantum mechanics can be extended to optics. Therefore optics has become an interesting platform for studying the fundamentals of PT symmetry. The strategic modulation of gain and loss in the PT symmetry breaking condition can broaden optical science at both semiclassical and quantum levels. Throughout the seminar we demonstrated some intriguing features of PT synthetic materials, such as PT phase transition, unidirectional invisibility and single-mode lasing. This could lead to a new generation of optical devices, materials and networks, including, for example, unidirectional on-chip devices and high power laser system. Finally, ideas from PT symmetry may provide a viable route to overcoming losses that have so far hindered progress in other areas of applied physics such as plasmonics and metamaterials. References [1] J. Joannopoulos, P. Villeneuve and S. Fan, ”Photonic crystals: Putting a new twist on light”, Nature 386, 143 (1997). [2] W. Barnes, A. Dereux and T. Ebbesen, ”Surface plasmon subwavelength optics”, Nature 424, 824 (2003). 10 [3] J. Pendry, ”Negative refraction makes a perfect lens”, Phys. Rev. Lett. 85, 3966 (2000). [4] C. Bender and S. Boettcher, ”Real spectra in non-Hermitian Hamiltonians having PT symmetry”, Phys. Rev. Lett. 80, 5243 (1998). [5] A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides and U. Peschel, ”Parity-time synthetic photonic lattices”, Nature 488, 167 (2012). [6] C. E. Rueter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev and D. Kip, ”Observation of parity-time symmetry in optics”, Nat. Phys. 6, 192 (2010). [7] T. Kottos, ”Broken symmetry makes light work”, Nat. Phys. 6, 166 (2010). [8] Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao and D. N. Christodoulides, ”Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures”, Phys. Rev. Lett. 106 (2011). [9] Y. D. Chong, L. Ge and A. D. Stone, ”PT -Symmetry Breaking and Laser-Absorber Modes in Optical Scattering Systems”, Phys. Rev. Lett. 106, 093902 (2011). [10] L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang and X. Zhang, ”Single-mode laser by parity-time symmetry breaking”, Science 346, 972 (2014). 11