Observation of coherent perfect absorption at an exceptional point

Model and setup

• The experiment studies absorbing EPs in a coupled system of optical microcavities described by temporal coupled-mode theory.The setup uses coupled whispering-gallery-mode microcavities with intrinsic losses, intercavity coupling, and two taper waveguides.Source: Theoretical model and experimental setup · pp. 4–5
• The two-by-two scattering matrix describes reflection and transmission between four physical ports reduced to two relevant channels.Negligible backscattering means only the clockwise mode in one cavity and counterclockwise mode in the other are excited for the chosen inputs.Source: Theoretical model and experimental setup · p. 5

Exceptional-point conditions

• Without parameter tuning, the model has distinct resonance and zero eigenfrequency pairs and therefore no EPs.At resonances one scattering eigenvalue diverges, whereas at zeros one eigenvalue vanishes while the other remains finite.Source: Theoretical model and experimental setup · p. 5
• With absorption, resonant and absorbing EP conditions generally differ because poles and zeros are no longer complex conjugates.As coupling increases, the two zeros can meet first at an EP and separate before the two poles merge at a higher coupling.Source: Theoretical model and experimental setup · p. 6

CPA EP

• A CPA EP occurs when two zeros meet on the real-frequency axis, producing steady-state perfect absorption.Within the model, this requires total intrinsic loss γ1 + γ2 to balance total radiative coupling γc1 + γc2.Source: Theoretical model and experimental setup · p. 6
• A generic CPA EP has an anomalous quartic absorption lineshape, while symmetric coupling can make resonant and absorbing EPs coincide in a non-generic CPA EP.The non-generic case also produces singular scattering behavior and coincides with an EP of the scattering matrix.Source: Theoretical model and experimental setup · pp. 6–7

Experimental distinction

• The experiment realizes and compares resonant and absorbing EPs by varying the intercavity coupling strength κ.The zeros meet at κ = κ_th, while the poles meet later at approximately 1.32κ_th.Source: Resonant EPs and absorbing EPs · p. 7
• The measured pole and zero transitions confirm that resonant and absorbing EPs generally occur at different coupling strengths.After the transition, zeros and poles share approximately the same imaginary frequency component rather than the same real component.Source: Resonant EPs and absorbing EPs · p. 7
• The ordering of absorbing and resonant EPs can reverse when the system parameters follow the alternative differential-loss scenario.In that scenario, the absorbing EP appears after the resonant EP as κ increases.Source: Resonant EPs and absorbing EPs · p. 7

Lineshape signatures

• EPs in systems with arbitrary loss need not produce the commonly assumed single dip or peak in transmission or reflection spectra.Both the absorbing EP and resonant EP spectra in this setup exhibit an absorbing doublet.Source: Resonant EPs and absorbing EPs · p. 7
• At zero detuning, the optical field tends to localize in μR2, while detuning redistributes energy toward the more lossy μR1 and reduces output power.The reduction occurs because the narrow linewidth of μR2 sharply decreases its intracavity energy away from zero detuning.Source: Resonant EPs and absorbing EPs · p. 8
• The detuning-dependent output reduction disappears for the CPA EP because the relevant zero reaches the real axis.The preceding behavior occurs when neither EP lies on the real axis.Source: Resonant EPs and absorbing EPs · p. 8

One-channel CPA EP

• For one-channel CPA EPs, critical coupling is reached by balancing cavity-waveguide coupling against intrinsic loss and tuning the intercavity coupling.The measured spectrum is well fit by a quartic function, unlike a Lorentzian fit based on the measured coupling.Source: One-channel and two-channel CPA EPs · p. 8

Two-channel CPA EP

• Two-channel CPA requires simultaneous coherent illumination of ports 1 and 3 in the absorbing eigenchannel of the scattering matrix.The inputs require balanced amplitudes and a π/2 relative phase for the specified eigenstate [1, − i].Source: One-channel and two-channel CPA EPs · pp. 8–9
• The non-generic CPA EP is realized with symmetric waveguide-cavity coupling and occurs simultaneously with the resonant EP.Its signature is two degenerate zeros at a real frequency.Source: One-channel and two-channel CPA EPs · pp. 8–9
• At the non-generic CPA EP, output amplitudes from the two channels have the characteristic equal-amplitude scattering structure.Experimentally, |r1|^2 and |t1|^2 are nearly equal at zero detuning, while |r2|^2 and |t2|^2 approach equality with a small discrepancy.Source: One-channel and two-channel CPA EPs · p. 9
• The measured two-port output spectra have similar lineshapes and show strong, though non-ideal, absorption at the non-generic CPA EP.The non-ideal absorption is attributed to fluctuations in optical phases and coupling parameters.Source: One-channel and two-channel CPA EPs · p. 9
• The total absorption lineshape is quadratic rather than quartic for eigenchannel input at the non-generic CPA EP.This behavior is attributed to the non-analytic behavior of the scattering-matrix eigenvalues around zero detuning.Source: One-channel and two-channel CPA EPs · p. 9

• The two output signals oscillate in phase as the relative input phase changes and reach their minimum at ϕ = −π/2.Source: Phase response at the two-channel CPA EP · pp. 9–10
• The total output shows a large oscillation against the relative phase, with modulation depth defined from the maximum and minimum output power.Source: Phase response at the two-channel CPA EP · p. 10
• Output from port 2 exhibits a single peak, whereas output from port 4 shows a doublet.Source: Phase response at the two-channel CPA EP · p. 10
• The total-output modulation depth exceeds 0.8 over a frequency range above 500 MHz, indicating broadband large phase sensitivity at the CPA EP.Source: Phase response at the two-channel CPA EP · p. 10
• Transmission magnitude alone is difficult to use for absorbing-EP identification because it is influenced by both poles and zeros, whereas another transmission quantity is pole-sensitive and can search for resonant EPs.Source: Discussion and outlook · p. 10
• A generic CPA EP produces perfect absorption with a quartic lineshape for the S-matrix eigenstate input, while a non-generic CPA EP produces quadratic output spectra.Source: Discussion and outlook · p. 10

CPA-EP excitation

• Balanced optical power in the two input channels is required to obtain the non-generic CPA-EP input eigenvector used experimentally.Loss differences in fiber-taper paths can create unequal powers, so thermo-optic calibration is used to restore equal amplitudes.Source: Materials and Methods · p. 20
• The experiment measures scattered output signals versus the relative phase between the two inputs.Outputs are monitored from the two output ports while the relative phase is controlled through the optical paths.Source: Materials and Methods · p. 20

Phase control

• Unequal optical-path lengths create frequency differences, beat notes, and relative-phase oscillations in the scattered-output spectra.Keeping Δl = 0 fixes the relative phase during frequency scans for CPA-EP measurements.Source: Measurement of scattered output… · pp. 20–21
• For sufficiently large Δl, rapid phase oscillations allow the output response at approximately fixed detuning to be measured as a function of relative phase.The method works well for Δl ≫ 10 m under the stated scan-rate and linewidth conditions.Source: Measurement of scattered output… · p. 21

Measurement limitations

• Sinusoidal EOM phase modulation can be inaccurate because the phase-voltage calibration and finite data-collection speed limit precise extrema extraction.The voltage must be set to Vπ for a π phase shift, while limited frames and measurement speed hinder capturing maximum and minimum output powers.Source: Measurement of scattered output… · p. 21

Theoretical framework

• The coupled microcavity system is analyzed using temporal coupled-mode theory (TCMT).The model describes two directly coupled microcavities with resonant frequencies and intrinsic loss rates.Source: S1. Theoretical framework · p. 22
• The Hamiltonian and diagonal coupling matrix encode inter-cavity coupling and coupling to waveguide channels.The inter-cavity coupling strength is κ, while the channels contribute through the coupling matrix.Source: S1. Theoretical framework · p. 22

Scattering formulation

• The TCMT scattering matrix incorporates optical dissipation through the effective Hamiltonian H0 − iD†D/2.This framework supports calculation of scattering behavior, poles, and zeros.Source: S1. Theoretical framework · p. 22

Model assumptions

• The principal analytic case assumes equal cavity resonance frequencies, while other detuned cases may support EP studies in schemes such as anti-PT symmetry.The matched-resonance restriction is adopted for simplicity and does not exhaust possible parameter solutions.Source: S1. Theoretical framework · p. 22

Resonant EP analysis

• Resonances are obtained from the eigenvalues of the effective Hamiltonian and correspond to the poles of the scattering matrix.For matched cavity resonances, ω1 = ω2 = ω0, the resonance expressions simplify.Source: S2. Resonant EP · pp. 22–23

Zero degeneracy

• Absorbing EPs are derived by calculating degeneracies of the zeros of the scattering matrix.The analysis defines Ω1,2 and obtains the zero-degeneracy condition from the scattering formulation.Source: S3. Absorbing EP · pp. 23–24

Scattering formulation

• The scattering matrix is constructed from the coupled-cavity model and has symmetric form, indicating reciprocal wave transport.Its eigenvalues are then used to obtain the scattering zeros.Source: S3. Absorbing EP · p. 23

EP comparison

• The critical κ for an absorbing EP can be larger, smaller, or identical to the critical κ for a resonant EP.Thus, the two EP conditions represent distinct degeneracies with independently varying coupling thresholds.Source: S3. Absorbing EP · p. 24
• When κ increases, the absorbing EP occurs before the resonant EP if γ1 > γ2, whereas the resonant EP occurs first if γ1 < γ2.The ordering is supported by the reported experimental and simulation results.Source: S3. Absorbing EP · p. 24

CPA-EP conditions

• A coherent perfect absorption EP requires an absorbing EP whose zero occurs at a real frequency.The additional requirement is imposed through the imaginary part of the zero-frequency condition.Source: S4. CPA EP · pp. 24–25

CPA-EP eigenvectors

• Under the CPA-EP conditions, the scattering matrix and its eigenvectors are evaluated to identify the zero-output input state.The eigenvector expressions depend on the coupling and intrinsic-loss parameters.Source: S4. CPA EP · p. 25

Input conditions

• For generic CPA EPs with γc1 ≠ γc2, the two input amplitudes are unequal, whereas γc1 = γc2 gives balanced input amplitudes.The balanced case is used experimentally because its input eigenvector is easier to find.Source: S4. CPA EP · p. 25
• When γc1 = γc2, both scattering eigenvalues vanish and their eigenvectors coalesce, producing a scattering-matrix EP alongside the wave-operator EP.This coincidence is exact only within the TCMT approximation and is not expected for the exact physical-system scattering matrix because of other resonances.Source: S4. CPA EP · p. 25

• A scattering EP occurs where the S-matrix eigenvalues degenerate and their eigenvectors coalesce.The degenerate eigenvalue need not be zero.Source: S5. Scattering EP · p. 25
• Within TCMT, the CPA EP requires symmetric coupling strengths in the two channels.For γc1 = γc2 = γc, the S-matrix takes a defective form.Source: S5. Scattering EP · p. 26
• Near a CPA EP, the eigenvector remains unchanged under arbitrary detuning, so the system remains at an EP but generally loses zero eigenvalue.This persistence is a TCMT prediction and does not hold exactly beyond the approximation.Source: S5. Scattering EP · pp. 26–27
• Symmetric coupling is necessary for absorbing and resonant EPs to coincide within TCMT, whereas exact scattering calculations generally place them only near each other.The simultaneous condition does not generally survive outside the TCMT approximation.Source: S5. Scattering EP · p. 27

• Reflection spectra are influenced by both poles and zeros, so dips need not occur at their real-frequency positions.Without PT symmetry, poles and zeros are not generally complex-conjugate pairs.Source: S6. The reflection and transmission… · p. 27
• Reflection minima and maxima occur near, but not exactly at, the real parts of zeros and poles, respectively.The spectrum can be interpreted through geometric distances from the real-frequency axis to complex zeros and poles.Source: S6. The reflection and transmission… · p. 27
• When κ > κ_th for the resonant EP, separated pole real parts cannot split the central transparency window.The poles lie farther from the real axis than the zeros, making zero-related influence stronger.Source: S6. The reflection and transmission… · p. 28
• Transmission peaks occur when the laser frequency equals the real part of the poles, enabling inference of resonant EPs.Experiments located the resonant EP by decreasing intercavity coupling until two transmission peaks evolved.Source: S6. The reflection and transmission… · p. 28
• Real-frequency eigenvector probing cannot directly reveal zero and pole locations because poles affect the output and shift local minima.The experiment retrieves zero locations by fitting all system parameters.Source: S6. The reflection and transmission… · p. 28

• At an absorbing EP, the scattered-output spectrum displays a doublet even though the zeros coalesce.Two poles with different imaginary parts account for this lineshape.Source: S7. The lineshape of the spectrum at… · p. 28
• Resonant and absorbing EPs coincide when coupling channels are symmetric or when two optical modes have identical resonant frequencies and loss rates.These are sufficient and necessary conditions in the generalized TCMT treatment.Source: S7. The lineshape of the spectrum at… · p. 29
• With symmetric coupling and properly tuned intercavity coupling, zeros and poles can become degenerate simultaneously regardless of other conditions.This case can also realize a scattering EP and, with equal mode frequencies, a three-way EP coalescence.Source: S7. The lineshape of the spectrum at… · pp. 29–30
• Engineering identical modes while maintaining asymmetric coupling is experimentally difficult.Asymmetric coupling breaks gauged PT symmetry, preventing simultaneous scattering-EP occurrence in that case.Source: S7. The lineshape of the spectrum at… · p. 30
• When complete symmetry satisfies both coincidence conditions, an EP requires one intermode coupling direction to vanish.A coupled clockwise-counterclockwise microcavity is given as an example.Source: S7. The lineshape of the spectrum at… · p. 30
• EP presence is inferred from parameter curve fitting rather than a uniquely identifying reflection or transmission lineshape.Spectral peaks and dips may not mark exact resonance or zero positions.Source: S7. The lineshape of the spectrum at… · p. 30

• For a non-generic CPA EP, the total output power scales quadratically near zero detuning: |vout|2~δ2.The quadratic behavior follows from the output spectrum derived for the non-generic configuration.Source: S8. Lineshape of the spectrum at a CPA… · pp. 30, 32
• At a one-channel CPA EP, the reflection spectrum has a quartic lineshape, with R1~δ4 as δ→0.Source: S8. Lineshape of the spectrum at a CPA… · pp. 30–31
• For a generic two-channel CPA EP with asymmetric waveguide-cavity couplings, one S-matrix eigenvalue approaches zero at zero detuning.This eigenvalue defines the perfect-absorption input channel.Source: S8. Lineshape of the spectrum at a CPA… · p. 31
• |σ1|2~δ4 as δ→0 in the generic two-channel case, producing a quartic output spectrum under eigenvector excitation.Source: S8. Lineshape of the spectrum at a CPA… · p. 31
• At a non-generic CPA EP, |t1|2 and |t2|2 show single peaks, while |r2|2 retains two dips because both poles and zeros influence the spectrum.Experimental spectra at the CPA EP show single absorption dips, whereas moving away from the EP produces a doublet in one output spectrum.Source: S8. Lineshape of the spectrum at a CPA… · p. 32
• Changing the relative phase between equal-amplitude inputs can split a single output dip near the non-generic CPA EP.The experiment and simulations use phase-controlled two-field excitation to probe the output spectra.Source: S8. Lineshape of the spectrum at a CPA… · p. 32, Figure S6 · p. 38, Figure S7 · p. 39