Abstract
Recently, synthetic optical materials represented via nonHermitian Hamiltonians have attracted significant attention because of their nonorthogonal eigensystems, enabling unidirectionality, nonreciprocity and unconventional beam dynamics. Such systems demand carefully configured complex optical potentials to create skewed vector spaces with a desired metric distortion. In this paper, we report optically generated nonHermitian photonic lattices with versatile control of real and imaginary sublattices. In the proposed method, such lattices are generated by vectorfield holographic interference of two elliptically polarized pump beams on azobenzenedoped polymer thin films. We experimentally observe violation of Friedel’s law of diffraction, indicating the onset of complex lattice formation. We further create an exact paritytime symmetric lattice to demonstrate totally asymmetric diffraction at the spontaneous symmetrybreaking threshold, referred to as an exceptional point. On this basis, we provide the experimental demonstration of reconfigurable nonHermitian photonic lattices in the optical domain and observe the purest exceptional point ever reported to date.
Introduction
Engineered optical materials have been extensively studied in the pursuit of new materials with exotic properties unavailable from natural substances, especially within the context of photonic crystals^{1} and metamaterials^{2}. In these artificial systems, new functionalities have been found, generally by manipulating the real part of permittivity (or permeability) to achieve a desired spatial distribution. Synchronizing such systems with carefully configured imaginary permittivities, a new approach inspired by an abstract theoretical concept based on nonHermitian Hamiltonians, is now of great interest in both theory and experiment. This emerging strategy is motivated in part by the experimental feasibility of optical systems for mimicking paritytime (PT) symmetric quantum systems without any conflict with the Diracvon Neumann’s Hermiticity axiom for physical observables. In seminal papers on PTsymmetric Hamiltonians^{3,4,5}, Bender and coworkers showed that such Hamiltonians display completely realvalued energy spectra below certain phasetransition thresholds, referred to as exceptional points. This property leads to skewed eigenvector spaces where the system’s time evolution and spectral features are remarkably distinct from standard Hermitian systems^{6,7,8,9}.
In optics, one can produce a system having a PTsymmetric Hamiltonian by engineering a complex dielectric distribution ɛ(r) that is invariant under spatial inversion and has simultaneous loss–gain interchange, that is, ɛ*(–r)=ɛ(r), where r is a position vector. Therefore, the notion of PT symmetry and its associated phenomena in optics are accessible experimentally, revealing abstract nonHermitian dynamics and further providing technologically relevant underlying physics. Introducing several representative reports, among a wide variety of theoretical and experimental works along this line, Regensburger et al.^{10} proposed a coupled fibreoptic network structure for PTsymmetric timedomain lattices as an experimental testbed. They experimentally showed intriguing optical effects such as exact symmetry breaking of the system’s eigenmodes at the exceptional point, Bloch oscillations and nonreciprocal reflection, which are characteristic features of skewed eigenvector spaces. Moreover, purely spatial, complex photonic lattices have shown unconventional beam dynamics associated with spectrally singular Bragg scattering^{6}, asymmetric or solitary optical propagation^{11,12,13,14} and counterintuitive uniformintensity wave solutions in nonuniform media^{15}. Exploiting these phenomena for device applications, nonreciprocal transmission^{16} and unidirectionalreflection elements^{17,18,19} based on photonic integrated circuits were experimentally demonstrated without introducing any nonlinear or gyrotropic material to break timereversal symmetry. Of fundamental importance in this context is the development of versatile platforms where arbitrary complex optical potentials can be precisely configured and nonHermitian optical effects experimentally explored.
Although some proofofconcept experiments have been reported, synthesizing nonHermitian optical structures generally demands challenging highprecision fabrication involving multiple etching and deposition steps with deepsubwavelength interstep alignment tolerance^{16,17,19,20,21,22}. For example, a PTsymmetric dielectric function demands Re(ɛ) and Im(ɛ) profiles of opposite spatial parity. Creating such a profile is clearly nontrivial in the context of conventional nanophotonic architectures. Consequently, further realizing such structures in reconfigurable platforms is presently a formidable task in the optical domain. Taking a completely different approach in this paper, we report comprehensive nonHermitian photonic lattice generation using vectorholographic interference in azodyedoped polymer (azopolymer) thin films. Under the influence of polarized periodic optical fields, azo dyes in polymer matrices simultaneously induce surface relief, birefringence and dichroic subgratings due to molecular migration and reorientation triggered by photoisomerization and relaxation processes^{23,24}. These subgrating components induced by a single holographic vector field are precisely deployed to synthesize a desired nonHermitian system. Therefore, robust and reconfigurable nonHermitian photonic lattices can be generated using the proposed method.
Results
Formation of complex photonic lattices
The essence of our proposed method is illustrated schematically in Fig. 1a. Two coherent electric fields E_{1} and E_{2} of different polarization states are incident to form a periodic pump field E_{P}(x)≡E_{1}+E_{2}=E_{P}(x±Λ) in an azopolymer thin film. E_{P} leads to the simultaneous formation of a surfacerelief grating and of a dichroic absorption grating, each contributing a modulation to the real and imaginary dielectric function, respectively. These two subgrating components originate from distinct mechanisms: the former is generated by optical gradient forces and consequent migration of azobenzene–polymer complexes^{22}, while the latter is formed by polarizationinduced reorientation of azodye molecules^{23}. Considering these two mechanisms for a given periodic pump field E_{P}, the dielectric function ɛ(x) in the azopolymer film is written ɛ(x)=ɛ_{avg}+Δɛ_{R}(x)+iΔɛ_{I}(x) with the real (Δɛ_{R}) and imaginary (Δɛ_{I}) modulations determined by
Here, χ is the electric susceptibility of the host polymer, 〈···〉_{t} implies timeaveraging of the argument, and α=tan^{−1}(E_{Py}/E_{Px}) and ϕ_{xy}=arg(E_{Px})−arg(E_{Py}) are local polarization parameters. The empirical constants C_{R} and C_{I} are fixed for a given dye doping concentration and film thickness. See Supplementary Note 1 and Supplementary Figs 1 and 2 for additional details on these relations. We note in equations (1) and (2) that the dominant contributions result from the second derivative ^{2}E_{Px}^{2}/x^{2} of the intensity for Δɛ_{R}(x) and the polarization contrast E_{Px}^{2}−E_{Py}^{2} for Δɛ_{I}(x). Therefore, one can judiciously control relative magnitudes and phases of Δɛ_{R}(x) and Δɛ_{I}(x) by tuning the polarization state of E_{1} and E_{2} to form a desired distribution in the single vectorholographic pump field E_{P}.
We confirm experimentally the proposed concept using 100nmthick PMMA thin films doped with Disperse Red 1 (SigmaAldrich) azodye at a 15% molecular concentration. A Nd:YAG laser operating at a wavelength of 532 nm and supplying a continuouswave power of 30 mW is used as a coherent pump source. We show two representative combinations of E_{1} and E_{2} along with the consequent E_{P}, and dielectric function modulations Δɛ_{R}(x) and Δɛ_{I}(x) over one period (1.356 μm) in Fig. 1b,c. The experimental profiles measured by phaseshifting interferometry agree quantitatively with theoretical predictions using equations (1) and (2). See Supplementary Notes 2 and 3 for the description of the measurement method and Supplementary Figs 3 and 4 for details of the experimental configuration. The generated complex photonic lattices can be conveniently expressed by firstharmonic sinusoidal modulations in the real and imaginary dielectric functions:
where K=2π/Λ with Λ being the period of the modulation. Here we define the balance factor ξ∈[0, 1] and relative phase difference δ∈[−π, π] as the primary parameters determining the nonHermitian properties of the lattice.
Optical beam dynamics in the complex lattices
The evolution of a photonic state through the lattice is described by:
Considering configurations allowing only the p=−1, 0 and +1 diffraction orders as propagating through the lattice, we express the state vector such that A(z)〉=[A_{−1}(z) A_{0}(z) A_{+1}(z)]^{T} with A_{p}(z) being the amplitude of the pth diffraction order. For a ypolarized plane wave, the 3 × 3 Hamiltonian matrix H is given by
where the constants denote the Fourier coefficients of Δɛ(x) at the ±1 harmonic orders, respectively, and k_{0} is the vacuum wavenumber. (See Supplementary Note 2 for details.) In this matrix representation, the parity operation P implies a matrix transpose such that PH=H^{T} while the time reversal operation T is defined as a complexconjugate transpose such that TH=(H*)^{T}. Therefore, the PT operation in our case yields PTH=H*. Consequently, the given Hamiltonian H is PT symmetric for η_{±1}*=η_{±1} at δ=π/2 in this formulation.
Solving the eigenvalue problem Hu_{v}〉=α_{v}u_{v}〉 yields a set of eigenvectors {u_{v}〉} and corresponding eigenvalues {α_{v}} representing stationary Floquet–Bloch modes in the lattice and grouptransport momenta, respectively. We define the skewness parameters of the vector space defined by {u_{v}〉} as c_{vμ}=〈u_{v}u_{μ}〉, producing values between 0 for orthogonal eigenvectors and 1 for eigenvectors merging at an exceptional point (EP). In Supplementary Note 2 and Supplementary Table 1, we provide closedform expressions for α_{v}, u_{v}〉, and c_{vμ}. The character of the eigenvectors for the case of Δɛ(x) given by equation (3) is shown in Fig. 2a, which identifies the dominant amplitudes at the three allowed diffraction channels. The first eigenstate u_{1}〉 is an antisymmetric combination of the p=+1 and −1 diffraction orders while u_{2}〉 is a symmetric combination of the p=+1 and −1 diffraction orders. These two eigenvectors form a merging pair at the EPs (ξ, δ)=(1/2, ±π/2) and thus generate a skewed vector space. This property is clearly visualized in Fig. 2b where we present the cosine angle θ_{12}≡cos^{−1}(c_{12}) between u_{1}〉 and u_{2}〉 as a measure of the geometrical distance between these two states in the canonical Hilbert space. Intriguing nonHermitian optical effects occur near the EPs characterized by θ_{12}=0. At these points, u_{1}〉 and u_{2}〉 are identical, so the vector space displays extreme skewness leading to interesting properties such as unidirectional or nonreciprocal energy transport^{16,17,18,19,21,25} and a wormholelike effect on state evolution in the Hilbert space^{7,8,9}.
To experimentally confirm the skewed subspace formed by u_{1}〉 and u_{2}〉, we observe violation of Friedel’s law of diffraction^{6,26,27} and totally asymmetric diffraction at exact EPs as immediate consequences of nonorthogonal eigenvectors. In detail, the contrast ratio Γ≡I_{+1}/I_{−1} for the two firstorder diffraction intensities produced under incidence by a single ypolarized plane wave as an optical probe, is directly connected to the skewness parameter c_{12}:
for 0≤δ<π. For −π<δ<0, the same relation applies for the inverse ratio I_{−1}/I_{+1} (see Supplementary Note 3 for the derivation). Clearly, Γ=1 for Hermitian configurations having c_{12}=0, that is, for θ_{12}=π/2. As EPs are approached, that is, (ξ, δ)→(1/2, ±π/2), then c_{12}→1 and consequently Γ diverges, implying that firstorder diffraction becomes totally asymmetric. Although details are not presented in this letter, the EPs for a xpolarized probe are identical to those for a ypolarized probe and the same consequence in the contrast ratio applies once one switches δ to δ+π because the imaginary subgrating profiles Δɛ_{I}(x) for orthogonal probe polarizations take opposite signs. Interestingly, this implies that polarization flipping between the x and ypolarizations can induce a highextinction switching of Γ between 0 and ∞. In addition, a continuous change in Γ is obtainable by rotating the probe polarization angle with respect to the periodicity axis of the lattice.
Insitu measurement
An experimental setup for realtime contrastratio measurements of complex lattice formations is shown in Fig. 3. Two coherent pump beams from one Nd:YAG laser (operating at a wavelength of 532 nm and an output power of 30 mW) are prepared to form E_{1} and E_{2} with the desired polarization states using quarterwaveplate halfwaveplate pairs to generate a vectorholographic pump field E_{P} in the azopolymer film. A ypolarized probe beam from an Ar^{+} laser, operating at a wavelength of 488 nm and an output power of 3 mW, is incident on the azopolymer film at a surfacenormal angle. The probe laser wavelength is selected to match the absorption maximum of the azodye polymer. The I_{+1} and I_{−1} intensities diffracted from the probe beam are monitored to acquire Γ in the time window over which the complex lattice is formed.
We perform contrastratio measurements in the time domain with different polarization settings as outlined in the left inset of Fig. 4a. The fixed polarization parameters are the longaxis angle ψ_{2} of E_{2} and the ellipticity e_{1} and e_{2} of E_{1} and E_{2}, respectively, while the longaxis angle ψ_{1} of E_{1} is varied from −54.8° to −40°. From the results shown in Fig. 4a, we clearly observe maxima over the recording time that ranges from 6.5 to 8.5 min (highlighted as the orange band) for several cases of ψ_{1}. The time origin corresponds to the time at which the pump beam was turned on. The time response of the measurement is understood from the different timescales required for the formation of the real and imaginary subgratings. The imaginary subgrating is formed from molecular reorientation processes which take a few seconds to buildup, while the real subgrating forms from the comparatively slower migration of azobenzene–polymer complexes which needs ∼10^{3} s for complete buildup. Assuming a typical exponential relaxation for this process, the balance factor ξ over the recording timespan t follows:
where ξ_{∞}=ξ(t→∞) and τ_{R} is relaxation time of the molecular migration process for a given film thickness and pumpfield intensity. Clearly, ξ monotonically decreases from 1 at t=0 to ξ_{∞} at t>>τ_{R}. Using phaseshifting interferometry, we estimate the time dependence of ξ for ψ_{1}=−54.0° as plotted along the top of Fig. 4a with schematic illustrations of the corresponding Δɛ_{R}(x) and Δɛ_{I}(x) profiles. Thus, Fig. 4a clearly confirms that the contrast ratio Γ is maximized for ξ=1/2, where the eigenvector skewness parameter c_{12} is highest for a given phasedifference parameter δ. Moreover, the contrast ratio near ξ=1/2 displays a diverging behaviour for ψ_{1}=−54.0° as this condition further satisfies the phasedifference requirement of δ=π/2 for the EPs, as indicated in Fig. 4b which plots δ calculated as a function of ψ_{1} using equations (1)–(3), , . In Fig. 4c, we compare the measured Γ versus ψ_{1} for ξ=1/2 with the model based on equation (1)–(4), , , . The quantitative agreement between the experiment and model again confirms that our method is efficient for accessing exact EPs associated with PTsymmetry breaking. We note that the experimental Γ value, well in excess of 200, is remarkably higher than the previously reported values of ∼5.4 in [19] and ∼14 in [21], where the complex lattices were generated by lithographic and angledeposition methods, respectively. It is also higher than the reported value of Γ∼7±1 generated by a complex photonic lattice in an azopolymer film by Birabassov et al.^{27} Their approach is based on spatially modulated spectral hole burning that permanently changes the material properties in an irreversible manner. To the best of our knowledge, the observed EP is purest among those ever obtained previously. Therefore, our proposal is very promising for generating precise nonHermitian photonic lattices solely using finely controlled optical instruments.
Discussion
Although the results summarized in Fig. 4 are taken from temporally varying complex lattices, stationary lattices with ξ=ξ_{∞} are obtained when the pump fields are continuously applied over a recording time t>>τ_{R}, which is of the order of 10 min in our experimental configuration. Here we have selected a sufficiently low ξ_{∞} (≈0.28) to produce a temporally tuned parametric scan over a sufficiently broad parameter range to clearly observe exceptional point behaviour. The ξ_{∞} value is in principle adjustable to higher values (up to 0.5) by tuning the pumppolarization parameters. Alternatively, incorporating a dielectric layer on top of an azopolymer film provides another way to tune the ξ_{∞} value, following the elastic modulus and thickness of the dielectric layer. Turning off the pump fields leads to complete annihilation of the imaginary sublattice in a few seconds due to thermal relaxation at room temperature as the photoisomerization and consequent molecular reorientation processes cease to occur. In contrast, the generated real sublattice persists in the form of a surfacerelief grating until another pump or erasing fields drive new periodic gradient forces or planarizing potentials, respectively.
Importantly, the obtained results demonstrate the successful experimental realization of reconfigurable nonHermitian photonic lattices in the optical domain. We note that reconfigurable nonHermitian systems are of great importance for exploring chiral EP dynamics^{28,29} and the associated asymmetric state interchange between orthogonal eigenmodes^{30,31}. Although effects associated with EPs are observable in temporally static, spacevariant systems^{31}, experimental realization of reconfigurable nonHermitian systems provides rigorous means to study true timedomain dynamics and tunable devices taking advantage of temporally varying system parameters. Previously, a mechanically tunable microwave cavity was introduced to show a quasiadiabatic state flip during parametric encircling around an EP^{29}. More recently, a reconfigurable excitonpolaritonic nonHermitian microcavity was generated using a pumpfield imaging configuration^{32}. However, migration of these concepts into photonic systems in the optical domain is formidable. A mechanically tunable single or dual mode optical cavity should involve intricate micro or nano electromechanical system architectures with deep subwavelength fabrication tolerance. The pumpfield imaging method for excitonpolaritonic tunable cavities is also experimentally futile in photonic systems because of extremely weak nonlinear interaction between pure photons. In this context, the proposed nonHermitian holographic lattice platform alleviates such difficulties and thereby can be used to further investigate the chiral EP dynamics with remarkably improved experimental feasibility and parametric precision.
Previously, holographic lattice generation in azopolymers has been widely studied to realize optical memory devices permitting stable reconstruction of data with multiple read–write–erase cycles below a certain photobleaching threshold intensity^{33}. In our case, we have confirmed that significant photo bleaching is observed at a pump intensity of 1.7 W cm^{−2} for a recording time of over 2 h. In Supplementary Note 4 and Supplementary Fig. 5, we provide an experimental result on the complex lattice reconfiguration that shows stable formation of reconfigured complex lattices on a single sample spot with multiple write–erase–rewrite cycles. We confirm a stable reconfiguration time of ∼1 min for a pump intensity of 1.7 W cm^{−2} for a total recording time of 40 min and a robust contrast ratio tuning in response to the varying pumppolarization parameters. In the experiment that leads to the data presented in Fig. 4, we have used again a single sample spot for multiple write–erase–rewrite cycles at a lower pump intensity of 30 mW cm^{−2}. In this case, no photobleaching effects were identified and reconfiguring a complex lattice into another form takes a fairly longer time—about 10 min, as required for the stable formation of the real sublattice. In fact, a much shorter reconfiguration time is possible with a slight modification of the sample geometry. We note that creation and annihilation of the imaginary sublattice takes a few seconds, following the relaxation time of azodyes into the stable transphase state aligned in a preferred orientation for a given pumpfield condition. Therefore, if a real sublattice is predefined, for example using a standard lithographic technique or other available means, one can readily obtain stable reconfiguration within a few seconds using this strategy.
In conclusion, we propose vectorholographic generation of nonHermitian photonic lattices in azopolymer thin films. We demonstrate, in both theory and experiment, versatile control of skewed eigenvector spaces in complex photonic lattices synthesized using the proposed method. We observe an extremely high asymmetry in the diffraction intensities at exact EPs, clearly confirming precise complex lattice formations. Notably, the proposed method generates reconfigurable nonHermitian photonic lattices on demand. Moreover, replacing the azopolymer with photorefractive media may lead to more efficient and dynamic complex lattices with greater controllability by means of electrooptic interaction or nonlinear optical properties^{34}. Our approach is of strong interest for further investigation. For instance, introducing optical gain by doping azopolymer films with laser dyes not only compensates average losses but also may enable observation of much broader classes of nonHermitian optical effects including saturatedgaininduced nonreciprocal light transmission^{35,36}, spectral singularities and associated unconventional laser oscillation effects^{6}, optical solitons^{11,13} and uniformintensity Bloch wave beams^{15}. On this basis, we envisage further experimental work on nonHermitian spectral band engineering with subwavelength periodicity, the optical realization of encircling EPs with geometric phase effects, and attendant device applications such as switchable unidirectional couplers.
Data availability
The data that support the findings of this study are available from the corresponding authors on request.
Additional information
How to cite this article: Hahn, C. et al. Observation of exceptional points in reconfigurable nonHermitian vectorfield holographic lattices. Nat. Commun. 7:12201 doi: 10.1038/ncomms12201 (2016).
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Acknowledgements
This research was supported in part by the Basic Science Research Program (NRF2015R1A2A2A01007553) and by the Global Frontier Program through the National Research Foundation (NRF) of Korea funded by the Ministry of Science, ICT & Future Planning (NRF2014M3A6B3063708).
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Affiliations
Department of Physics, Hanyang University, Seoul 133791, Korea
 Choloong Hahn
 , Youngsun Choi
 , Jae Woong Yoon
 , Seok Ho Song
 & Cha Hwan Oh
School of Electrical Engineering and Computer Science, University of Ottawa, 800 King Edward Avenue, Ottawa, Ontario, Canada K1N 6N5
 Pierre Berini
Department of Physics, University of Ottawa, 150 Louis Pasteur, Ottawa, Ontario, Canada K1N 6N5
 Pierre Berini
Centre for Research in Photonics, University of Ottawa, 25 Templeton Street, Ottawa, Ontario, Canada K1N 6N5
 Pierre Berini
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Contributions
C.H. provided the original idea. C.H., S.H.S and C.H.O. initiated the work. C.H. and Y.C. performed the experiments. C.H., Y.C. and J.W.Y. developed the theory and models. All authors discussed and interpreted the results. J.W.Y., C.H., S.H.S. and P.B. wrote the manuscript. C.H. and Y.C. contributed equally to the work.
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The authors declare no competing financial interests.
Corresponding authors
Correspondence to Jae Woong Yoon or Seok Ho Song.
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