Observation of exceptional points in reconfigurable non-Hermitian vector-field holographic lattices

Recently, synthetic optical materials represented via non-Hermitian 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 non-Hermitian photonic lattices with versatile control of real and imaginary sub-lattices. In the proposed method, such lattices are generated by vector-field holographic interference of two elliptically polarized pump beams on azobenzene-doped 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 parity-time symmetric lattice to demonstrate totally asymmetric diffraction at the spontaneous symmetry-breaking threshold, referred to as an exceptional point. On this basis, we provide the experimental demonstration of reconfigurable non-Hermitian photonic lattices in the optical domain and observe the purest exceptional point ever reported to date.


Supplementary Note 1: Expressions for real and imaginary sub-grating profiles
The azobenzene dyes have two isomer phases referred to as 'trans' and 'cis' molecules. 1,2 The trans phase is more stable than the cis phase and the trans molecules preferentially absorb photons with polarization parallel to the molecular long-axis. 3,4 Consequently, the isomerization processes under a polarized pump field results in net stationary population of aligned trans molecules with their long axis perpendicular to the pump polarization. Anisotropic absorption due to this process is called angular hole burning, [5][6][7] and is the dominant mechanism responsible for the formation of the imaginary sub-grating in our method. Considering angular hole burning in our excitation configuration (see Fig. 1a in the main text), we model the modulation of the imaginary dielectric function as: where P is the pump intensity, CI is an empirical constant,  is the angle of the major axis of the polarization ellipse of EP from the probe polarization that experiences the modulated absorption, and a is the ratio of the minor to major axis of the polarization ellipse of EP. Note that a = 1 for circular polarization and a = 0 for linear polarization. For a TE-polarized probe having an electric field oscillating along the y axis, equation (1) is written as: ( For a TM-polarized probe having an electric field oscillating along the x axis, the same relation is valid with the last term on the right-hand side replaced with |EPy| 2 -|EPx| 2 . Introducing new polarization state parameters  = tan -1 (|EPy/EPx|) and xy = arg(EPx)-arg(EPy), equation (2) can be alternatively expressed by equation (2) Substituting equation (3) into equation (2), we obtain the periodic profile of I(x) for given pump fields. Now, we describe the real dielectric function modulation due to migration of azobenzene-polymer complexes 7,8 due to an optical gradient force acting along the axis of the periodicity. The time-averaged optical gradient force is written as: where P = 0EP is the induced dipole moment density, …t implies time-averaging of the argument, 0 and  denote the permittivity of vacuum and the susceptibility of the azo polymer, respectively. Molecular migration due to the periodic gradient force induces a surface-relief grating (SRG) that represents a modulation in the real dielectric function. In more detail, the azo molecules near positive or negative local maxima migrate in the x directions. The result of this process is modulation of the real dielectric function modeled as: where CR is an empirical constant. Substituting equation (4) into equation (5) yields equation (1) of the main text.
Applying the expression for the pump field in equation (3) to equations (2) and (5), we calculate the phase parameter  in equation (3) of the main text as a function of pump polarization angles 1 and 2. Supplementary Fig. 2 shows a 1-2 map of  for one exemplary combination of a1 = 0 and a2 = 0.23. We clearly confirm that an arbitrary  value in the full range of [0, 2] is accessible by adjusting 1 and 2. The contours indicated by black dashed curves represent  = /2 or 3/2 for the exact PT-symmetry condition. Finding optimal pump polarization conditions for experiments employing different polarization control schemes, one can generate a 1-2 map of  for other combinations of a1 and a2.

Hamiltonian formulation
Since R and I in our experimental configuration are of the order of 10 -2 or 10 -1 , we take the slowly varying envelope approximation to mathematically describe the evolution of the probe electric field E in the generated complex lattices.
Assuming E(x,z) = eyẼ(x,z)exp(ikz) at the angular frequency  with ey being the unit vector along the y axis, (x) = avg+(x), and k = avg 1/2 /c, the frequency-domain Maxwell's equations yield the following wave equation: Here, we assume the coordinate system illustrated in Fig. 1a of the main text. Using Fourier decomposition and the Floquet-Bloch theorem, we express (x) and Ẽ(x,z) as: where K = 2/ is the grating wavenumber. By substituting these two expressions into equation (6), we obtain a coupled-mode equation Introducing a column vector |A(z)  [… Am(z) …] T , equation (9) is expressed as a Hamiltonian matrix equation: The Hamiltonian matrix elements are given by (11)

Contrast ratio in terms of skewness parameters
For normal incidence, the initial state is given by |A(0) = [0 E0 0] T , and the diffracted wave at z = d in the short-propagation limit k0d << 1 is determined by evaluating equation (10)

Supplementary Note 3: Phase-shifting interferometry for complex dielectric function measurement
Our measurement method is based on the approach used in Refs. 9 and 10. Briefly explaining the method with reference to the schematic illustration in Supplementary Fig. 3, two coherent beams from a frequency doubled Nd:YAG laser are incident on the azo polymer film at an angle of incidence . E1 is elliptically polarized and E2 is linearly polarized. They form an interference pattern in azo polymer which generates a complex refractive index grating. The in-plane wavevector component of the incident field satisfies the Bragg condition kx = K/2 with the grating wavenumber being K = 2/. Therefore, the electric field in equation (8) By substituting equation (20) into equation (6), we obtain a coupled-mode equation for the m-th harmonic component as The Hamiltonian matrix elements in this case are given by: . Here, we assume  << 1 for an optically thin azo polymer film. Shifting the interference pattern by moving a mirror on a piezoelectric translation stage in one branch of the pump fields, changes the initial phase . The -dependent output field is then obtained as