Chiral and degenerate perfect absorption on exceptional surfaces

Engineering light-matter interactions using non-Hermiticity, particularly through spectral degeneracies known as exceptional points (EPs), is an emerging field with potential applications in areas such as cavity quantum electrodynamics, spectral filtering, sensing, and thermal imaging. However, tuning and stabilizing a system to a discrete EP in parameter space is a challenging task. Here, we circumvent this challenge by operating a waveguide-coupled resonator on a surface of EPs, known as an exceptional surface (ES). We achieve this by terminating only one end of the waveguide with a tuneable symmetric reflector to induce a nonreciprocal coupling between the frequency-degenerate clockwise and counterclockwise resonator modes. By operating the system at critical coupling on the ES, we demonstrate chiral and degenerate perfect absorption with squared-Lorentzian lineshape. We expect our approach to be useful for studying quantum processes at EPs and to serve as a bridge between non-Hermitian physics and other fields that rely on radiation engineering.


I. Experimental setup and theoretical model 14
Here we introduce the experimental and the theoretical model in detail and provide information 15 on the processing of the experimentally collected data. We derive the expressions for 16 transmission, reflection, and absorption spectra of our experimental system for the inputs in the 17 clockwise (CW) and counterclockwise (CCW) directions and provide simulation results to clarify 18 the effects of various experimental parameters. 19 20 a) Experimental setup 21 Schematics of the setup is given in Fig. S1. Light from a tunable laser (1440nm band) is used to 22 probe the transmission and reflection spectra of a waveguide-coupled resonator. Our resonator 23 is an on-chip whispering-gallery-mode (WGM) microsphere resonator, which is fabricated 24 through CO2 laser reflow of a silica microdisk resonator. The waveguide used to couple light in 25 and out of the resonator is a sub-micron tapered fiber, which is fabricated using 26 S2 Figure S1 | Experimental setup used in this study to investigate chiral perfect absorption on an 28 exceptional surface (ES heat-and-pull method. The detector 1 is used to monitor reflection (transmission) spectra 34 whereas the detector 2 is used to monitor the transmission (reflection) spectra for light fields 35 input in the CW (CCW) direction (i.e., left incidence: CW and right incidence: CCW). The fiber-36 loop with a polarization controller simulates a symmetric tuneable end-mirror. Reflectivity of the 37 end-mirror is the same for light input in the CW and CCW directions. Similarly, the transmittivity 38 for CW and CCW inputs is the same. The reflection magnitude and phase of this end-mirror are 39 tuned using a polarization controller and a phase shifter (PS). In this way, the coupling from the 40 CW input to the CCW direction is controlled by tuning the reflectivity of the end-mirror in the 41 range 0 to 1.0. Polarization controllers (PC) are used at each end of the tapered fiber to set the 42 proper polarization and to correct if any polarization rotation takes place. VOA placed after the 43 laser is used to control the input power. The isolator in front of the laser diode is placed to 44 prevent or minimize reflections from the optical components into the laser. Optical switch placed 45 after the VOA is used to select the input direction (i.e., CW or CCW). We note this platform allows 46 us to independently control and tune the waveguide-resonator coupling and the strength of 47 unidirectional coupling between the CW and CCW modes of the resonator.

b) Theoretical Model and numerical simulations
Similarly, we can describe the field at 2 as 111 Then the transmission spectrum of the system is found as 115 Note the extra term of 1 2 ( ∆+Γ) 2 in the expression for compared to already implies that the 119 lineshape of will be significantly different than that of . Using the relation + + 120 = 1, we can write the absorption for the input in the CW direction as: 121 123 124 which reveals that the absorption spectrum is a superposition of a Lorentzian (i.e., spectrum) 125 and a squared Lorentzian function (i.e., spectrum). This implies that one can set the 126 lineshape of the absorption spectrum by finely tuning the system parameters. For example, when 127 the system is set at critical coupling ( 0 = 1 , that is Γ = 0 = 1 ), absorption is written as 128 which with the choice of completely reflecting end-mirror ( = 0, = 1) reduces to , implying a squared-Lorentzian lineshape, and with the choice of completely 133 transmitting end-mirror ( = 1, = 0 ) reduces to Lorentzian lineshape. Thus, provided that 0 / 1 is kept constant, one can tune the absorption 135 lineshape from a Lorentzian to a squared-Lorentzian form by varying reflectivity of the end-mirror 136 (Fig.S2). Similarly, one can tune the lineshape by tuning the waveguide-resonator coupling 137 strength (varying 0 / 1 ) if and are kept constant (Fig. S3). 138 139 ii.

Input in the CCW direction (left incidence, backward direction) 140
We now consider the case that we have input only in the backward direction (CCW input). The equals to 0 (blue), 1/4 (green), 1/2 (red), 3/4 (purple), and 1 (black). These settings simulate 152 end-mirrors with zero-reflection (blue), 25% reflection (green), 50% reflection (red), 75% reflection 153 (purple), and 100% reflection (black), respectively. c, Enlarged view of the top part of the spectra 154 shown in b. Transition of the spectra from Lorentzian lineshape to a flat-top quartic lineshape is 155 clearly seen as the reflectivity of the end-mirror is increased. S7 to the detector 1 (i.e., no back-reflection into the CW mode). The field input to the waveguide 157 after the fiber-loop reflector is , . There is no input in the CW direction, thus we have 158 , = 0 and = 0 (i.e., no coupling between the CW and CCW modes and input only in the 159 CCW direction). Thus, it is enough to consider only the modified rate equation: 160 The field detected at 1 for the input in CCW direction gives the transmission spectrum as: 188 For the CCW input, we calculate the reflection as 194 which is constant for all frequencies and is significantly different from which exhibits a 198 squared Lorentzian lineshape. We can then use the relation + + = 1 to write 199 absorption for the input in the CCW direction as 200 which reveals an absorption spectrum with Lorentzian lineshape. At critical coupling ( 0 = 1 , 204 that is Γ = 0 = 1 ), absorption spectra for a CCW input becomes 205 iii. Intracavity field intensity at exceptional surfaces of different coupling regimes 210 S9 us to observe the intracavity field for CW and CCW inputs at different coupling regimes and 212 hence on different exceptional surfaces (Figs. S4 and S5). We performed the simulations for two 213 specific cases, that is for a perfectly reflecting end-mirror (Fig. S4) and for an end-mirror with half 214 reflecting and half transmitting, | | 2 = | | 2 = 1/2 (Fig.S5). Simulations show the formation of a 215 standing-wave like pattern inside the resonator only for the CW input. This is because, the 216 reflector is placed at only one of the waveguide ends (the end in the CW direction) and thus the 217 light transmitted through the waveguide-coupled resonator in the CW direction is back reflected 218 in the CCW direction. As a result, there are two fields propagating in the CW and CCW directions 219 in the resonator even if the input to the system is in the CW direction. The response of the system 220 for these end-mirrors (i.e., the fully reflective and the partially reflective end-mirrors) differ 221 significantly for the CCW input: In Fig. S4 there is no field in the resonator (thus no absorption for 222 CCW input) because the CCW input is fully reflected and does not reach the resonator. In  in the CCW direction (thus there is absorption for CCW input). Since there is no end-mirror in the 244 CCW input direction, there is only CCW traveling wave in the resonator for the CCW input (Fig.  245   S5). This asymmetric response for the CCW input is observed at all exceptional surfaces 246 associated with waveguide-resonator coupling regimes. Simulations also show that the ES 247 emerging at the critical coupling leads to the highest intracavity-field intensity for CW input. On 248 a given ES associated with a coupling strength, the intracavity field intensity is stronger for the 249 CW input than for the CCW input. 250 251 c) Normalization procedure to assess absorption in experimentally obtained spectra

252
To correctly assess the absorption on exceptional surfaces, it is important that all losses in the 253 system (i.e., including the insertion and return losses of various components and the propagation 254 S11 losses in the connecting fibers but not the resonator-related losses) incurred during light 255 propagation in the experimental setup are measured and are considered in the normalization 256 process. We have measured these spurious losses in the path of the fields for input both in the 257 CW and CCW directions from the setup input-point until the detection at the detectors 1 and 258 2 . Since these paths are different for different input directions, the losses are different, and they 259 should be measured individually and included in the normalization of the associated 260 experimentally obtained spectra. We measured these losses by recording off-resonant (i.e., 261 without the resonator) transmission (reflection) and reflection (transmission) for the input in the 262 CW (CCW) direction at detectors 1 and 2 , respectively. 263 264 i.

Input in the Clockwise (CW) Direction 265
If there were no losses at all in the system, off-resonant transmitted and reflected field intensities 266 for an input with intensity / in the CW direction would be given as The theoretical model predicts that when the system is at critical coupling, reflection spectra on 319 the ES exhibits squared Lorentzian lineshape with perfect absorption occurring at the ES 320 frequency. The experimentally obtained reflection spectra reveals the expected squared 321 Lorentzian lineshape (i.e., flat bottom) at the critical coupling at all end-mirror reflectivity values 322 (Fig. S6). As seen in the normalized reflection spectra, we have = 0 at the ES-frequency. 323

Moreover,
at the ES-frequency is zero because the system is at critical coupling. Then, using 324 where Γ = ( 0 + 1 )/2 is modified as the coupling regime is changed by varying the resonator-337 waveguide gap (i.e., varying 1 ). We have obtained exceptional surfaces at the critical coupling If all parameters of the system are kept constant but only the resonator-waveguide coupling 381 strength is modified, a new ES will emerge at each coupling regime. However, only the ES at the 382 (Fig. S9). We experimentally obtained the reflection spectra and calculated the absorption 384 spectra as = 1 − (Note that in this case = 0). As the system moves away from 385 the critical coupling, the ES emerging at the new coupling regimes does not satisfy the perfect 386 absorption condition. We also observe that while the ES at the critical coupling clearly exhibits a 387 quartic lineshape in the reflection and absorption spectra, this feature is not clear as the system 388 moves away from critical coupling.