Generalized Fano lineshapes reveal exceptional points in photonic molecules

The optical behavior of coupled systems, in which the breaking of parity and time-reversal symmetry occurs, is drawing increasing attention to address the physics of the exceptional point singularity, i.e., when the real and imaginary parts of the normal-mode eigenfrequencies coincide. At this stage, fascinating phenomena are predicted, including electromagnetic-induced transparency and phase transitions. To experimentally observe the exceptional points, the near-field coupling to waveguide proposed so far was proved to work only in peculiar cases. Here, we extend the interference detection scheme, which lies at the heart of the Fano lineshape, by introducing generalized Fano lineshapes as a signature of the exceptional point occurrence in resonant-scattering experiments. We investigate photonic molecules and necklace states in disordered media by means of a near-field hyperspectral mapping. Generalized Fano profiles in material science could extend the characterization of composite nanoresonators, semiconductor nanostructures, and plasmonic and metamaterial devices.

where 0 and 0 are the electric and magnetic fields of the single cavity; describes the spatial separation between the cavities; Δ is the dielectric constant difference between the coupled and the isolated system. Supplementary Equation (1) highlights how the photon tunneling is governed by the spatial overlap of the electric fields, thus depending on the system geometry and on the dielectric permittivity.
Supplementary Figure 1: Schematics of a photonic molecule. Two optical cavities are coupled by the overlapping coupling parameter , given by Supplementary Equation (1). For each single cavity i is the resonant field, i is the resonance and i is the loss (i =1,2).
In order to find the normal modes of the coupled system we solve the system: The solutions of Supplementary Equation (2) are normal modes with eigenvalues A , B and linewidths A , B . For sake of simplicity, we deal with the case where the frequency detuning is zero 1 = 2 ≡ o and we define 2 = 1 + 2 . Therefore the key parameter becomes the loss difference = 1 − 2 , in fact the normal modes are given by two classes of solutions, distinguished by the condition 2 < | | (weak coupling, WC), and 2 > | | (strong coupling, SC) [2]. The two cases are summarized by: It is worth noting that within this definition SC does not mean that Rabi oscillation will show up, this requires a more stringent condition > . So, the two normal modes differ either by broadening (WC) or by frequency (SC). However, exactly at the transition, when 2 = | 1 − 2 |, the system manifests a singularity that exhibits A = B and A = B , known as exceptional point (EP) [2,3]. By solving the eigenvalues and eigenmodes problem for the coupling matrix in Supplementary Equation (2), we find that it is diagonalized by applying the matrix and its inverse −1 : So that Supplementary Equation (2) becomes: The amplitudes of the normal modes A and , with the respect to the uncoupled ones, 1 and 2 , are given by: Note that the amplitudes of A and B are not-normalized and not-orthogonal, since we are dealing with an open system: The last term of Supplementary Equation (8) holds in close proximity of the EP if Ω ≪ , and then for 2~| | (at the first order in Ω). In this case emerges the existence of the EP singularity, where the two normal modes A and B tend to become the same mode ( 1 + 2 ) [3]. The exceptional point is related to anomalous effects in many physical fields from photonics to quantum mechanics [4][5][6]. Here, we focus on the experimental detection of modes in close proximity to the EP, therefore with almost identical frequency and loss. This occurs when the relation 2 = | | ≠ 0 is satisfied. The EP singularity does not correspond to the trivial case of two modes with zero-coupling that can be realized by identical cavities largely separated, which do not show interesting physics. We experimentally achieve the condition 2 ~| |≠ 0 for modes exhibiting a large spatial overlap. In close proximity of the EP, in principle we could find either SC regime (2 > | |) with normal modes with same losses and a small frequency splitting Ω or WC regime (2 < | |) with normal modes with same frequency and a small splitting of the losses Γ. Often in the SC case also the condition > holds, giving rise to Rabi oscillations as a function of time between the two symmetric and antisymmetric eigenstates. However, in the case where 2 > | | but < , two different eigenvalues are present, but no Rabi oscillations occur. In photonics no distinction is made between these two regimes [2]. In the field of cavity polariton this circumstance was tentatively defined as critical coupling, where the eigenvalues (frequencies) are split and the modes are mixed, but without Rabi oscillations occurrence [7]; still this definition has not been used by the scientific community and the condition 2 > | | is always defined as strong coupling. In all the presented cases around the transition (2 ~| |) the solutions are two different modes with almost the same spectral properties, that represent the best achievable approximation of EP. In real coupled systems (e.g. dielectric and plasmonic structures) the mathematical requirements for being exactly at the EP singularity can be achieved only within the fabrication tolerance. Still, they have been attracted great interest, not only in relation to Electromagnetic Induced Transparency (EIT) and Autler Townes splitting (ATS) effects, but also for studying the transition from coupled lasing modes to a single-amplifying laser [4], a reversal of the pump dependence in coupled quantum cascade lasers [8] and the non-reciprocal wave propagation in a coupled two-channel system [9]. In order to detect almost-degenerate modes in close proximity to the EP, we investigate coupled modes found to be in the intermediate coupling regime where both conditions 2 > | | and < 1 , 2 hold.

Supplementary Note 2: Analytical derivation of standard Fano lineshape with 1 resonance and 1 mode
Supplementary Figure 2: Schematics of the interaction between a single resonant mode and a continuum. The single mode is given by a decaying induced dipole. The continuum is given by an instantaneous induced dipole. They are represented both in time and in frequency domains.
Here, we derive the relationship between the semiclassical approach and the standard Fano lineshape. We consider the optical material response as it is due to induced dielectric dipoles, which can be split in two contributions: a nonresonant (instantaneous) response and a resonant (delayed and damped) response. The model schematics is sketched in Supplementary Figure 2 both in the time and frequency domains. Assuming a delta-like temporal excitation (instantaneous scattering) at =0 and a phase delay of the resonant part (single mode), we have: where NR is the non-resonant amplitude, R is the resonant amplitude, ( ) is the Heavyside step function, which accounts for causal response. In the frequency domain, it can be expressed as: Here ⁄ is a Lorentzian with resonant frequency , spectral broadening and phase of the resonant dipole equal to 1 ( ) = { ( − ) + } (the amplitude R is real and positive). From Supplementary Equation (10) we derive the predicted lineshapes for absorption, resonant scattering (RS) and photoluminescence (PL) experiments. First, following [10] we consider absorption measurements, where the total absorption abs ( ) is proportional to the imaginary part of ( ). Then we have: The lineshape of the resonant absorption therefore depends on the phase . By using a different approach we consider the overall Fano lineshape intensity ( ) given by [11]: where q is the Fano parameter; ( + 0 ) is the background non-resonant signal and therefore the resonant contribution is equal to [ ( ) − ( + 0 )]. Thus, by mapping the resonant contribution in Supplementary Equation (11) with the resonant contribution in Supplementary Equation (12) we obtain: that provides the relations = −cotan{ /2} ; 0 = R /(1 + 2 ). Note that we subtracted a flat background with respect to [11]. Supplementary Equation (13) is one main result of [10]. In elastic scattering experiments the signal is proportional to | ( )| 2 and in most experiments the non-resonant (instantaneous) scattering is much larger that the resonant (delayed) contribution. Similarly to the absorption, resonant scattering measurements (RS) give Fano profiles, but the relation between and is slightly different. Assuming no absorption from the non-resonant contribution (dielectric material with NR real) the total elastic scattered signal can be written as: The scattering signal near 0 comes from the interference between the emission from the resonant and non-resonant dipoles. It lies on top of a non-resonant scattering (background) signal due to the instantaneous response of the material. Since elastic scattering is a coherent signal, it brings information on the phase of the optical response. Indeed, it can be either a positive or a negative signal with respect to the background. Thus, mapping the RS lineshape 1 ( ) after background subtraction with Fano lineshape, we find: Then in resonant scattering measurements the relation between and is: The expressions 1 ( , ), 1 ( ) and ( ) evaluated in Supplementary Equations (15) and (16) give the lineshapes reported in Supplementary Figure 3. Note that for '= + the resonant optical response changes sign. Also the Fano parameters and 0 change according to ′ = −1/ ; 0 ′ = 2 0 both for absorption or scattering. To describe the inversion of the optical signals in the Fano approach, we can use the property that the transformation (̃→ −1/ ′ ; 0 → − 2 0 ′) does not change the Fano lineshape (i.e. the parameters ′ , 0 ′ are not unequivocally defined for a given Fano lineshape). It follows that for '= + , the resulting Fano lineshape can be described by both the parameter pairs ( ′ = −1/ ; 0 ′ = 2 0 ) and (̃= ; 0 = − 0 ). The latter description has the advantage to highlight the inversion of the Fano lineshape in a straightforward approach. Finally, in PL experiments the signal is proportional to the squared modulus of the resonant part of the dipole: PL ( ) is a Lorentzian lineshape, which does not depend on the phase since it is an incoherent signal. In Supplementary Figure 3 a) the phase 1 ( ) and the amplitude | 1 ( )| are reported by red and black curves, respectively. Supplementary Figure 3 b) shows the intensity | 1 ( )| 2 as would be obtained by performing an incoherent measurement, such as PL. Supplementary Figures 3 c) (17) that represents a Lorentzian lineshape. c)-d) Interference term { 1 ( ) + + 2 } for two notable cases = and = (3/2) , respectively. e)-f) Scattering amplitude 1 ( ) given by Supplementary Equation (15) for the two cases of c) and d), respectively. They correspond to Fano profiles with = 1 and ≫ 1, respectively.

Supplementary Note 3: Analytical derivation of generalized Fano lineshape with 1 resonance and 2 modes
Supplementary Figure 4. Schematics of the interaction between two degenerate and coherent modes with the continuum. The two modes show a phase difference of π between them. The continuum is given by an instantaneous induced dipole. They are represented both in time and in frequency domains.
We derive the RS ( ) lineshape for the case of two degenerate modes A and B, which are the two almost degenerate normal modes in proximity to the EP. We can follow two routes (dipole derivation, Fano derivation) leading to complementary understanding of the physics of resonant scattering in case of one spectral resonance and two modes.
In case of two modes, as the ones reported in the schematics of Supplementary Figure 4, the dipole global response in the time domain is: Where we use the same parameters of the previous derivation. In the frequency domain Supplementary Equation (18) becomes: Note, first of all, that by using an incoherent response the signal is given, for any choice of the parameters, by the sum of the modulus square of the two dipoles: which is a sum of two Lorentzian profiles. An interesting behaviour emerges when the spectral properties of the two modes are extremely close. In this case the PL signal gives an overall Lorentzian lineshape from which no information on the relative phase between the two modes can be retrieved. Phase information, which can be obtained in RS experiments, may help in separating two contributions well below the spectral Rayleigh limit. By using Fano derivation we find for the total elastic scattering signal: where the two single Fano resonances 1, A ( ) and 1,B ( ) are given by Supplementary Equation (15) and related to mode A and B, respectively and A and B are linked to A and B by Supplementary Equation (16), respectively. The resonant scattering results in the sum of two different Fano profiles, which are determined by the interference of mode A and mode B with the non-resonant background, respectively. The point is that, due to the coherent nature of RS, the resulting spectral lineshape can be very different from standard Fano formula, and we define this lineshape as generalized Fano profile. Obviously, in both derivations RS signal for perfect degeneracy ( A = B ; A = B ; A = experimental near-field approach needed for addressing the sufficient spatial resolution, always gives tip induced frequency shifts and additional mode losses, which are unavoidably different for the two modes. Therefore, in photonics resonators a slightly variation from the perfect degeneracy condition always holds and in particular: Here, since the two modes correspond to the normal modes introduced in Supplementary Note 1 we have that = 2Ω. The cases where ( A ≫ ; A ≫ ) can be practically consider as a degeneracy, since in PL measurements Lorentzian lineshapes of mode A and mode B cannot be distinguished. Let us focus on these two cases. We also want to link the signal given in Supplementary Equation (21) with the resonant response as deriving from a single individual effective dipole: with 2 ( ), 2 ( ), 2 quantities describing the resonant response, to be derived. We are going to show that this approach is seminal for understanding the link between standard Fano and generalized Fano profile.
Case i): 2 ( ) Let us consider ( = 0; = A = B ≫ ) ; in this case (corresponding to close proximity of the EP on the SC side) analytical expressions for the resonant scattering can be derived. We define A = ; B = + > for a direct comparison with the case of one resonance and one mode. Then in the Fano derivation, we have at the zeroth order in .
This can be written as a standard Fano profiles in some cases; this is obvious for A = B and A = B . The interesting case is when A = − B = 0 and A = B = , where obviously the resonant part of the zero-th order in vanishes. Therefore, in this case we have to evaluate the first order in : Where 2 is the resonant part of RS 1 ( ), with the definition: The coherent response of the system in close proximity to the EP on the SC side, or equivalently of two almost degenerate modes with the same losses, is a peculiar case of generalized Fano lineshapes and can be expressed in a simple analytical formula. In the dipole approach, this particular case can be addressed by defining: A = B = R ; B = A + = + and then we have: Assuming that the optical response is given by an effective resonant dipole: Thus, the functions | 2 ( )| and 2 ( ) are defined by: In Supplementary Figure 5 is shown a summary of these formula. Note that | 2 ( )| is a Lorentzian (and not the square root of a Lorentzian, as | 1 ( )| is for a single mode) while the resonant phase 2 ( ) jumps by 2 across , as highlighted by Supplementary Figure 5 a). Following the previous derivation, the scattered signal, subtracted the flat non-resonant contribution, can be written as: Then the 2 jump of 2 ( ) across the resonance leads to anomalous lineshapes for RS. In particular, in Supplementary , respectively. e) and f) Scattering amplitude 2 ( ) for the two cases of c) and d), respectively. g)-h) Standard Fano profiles of the single modes 1, A ( ) and 1,B ( ), whose subtraction gives the generalized profiles of e) and f), respectively. Here, for a fast comparison we imposed a larger detuning = /4.

Case ii): 3 ( )
Here we consider the case where ( = 0; A = B = ; B = A − = − < ), thus corresponding to close proximity of the EP on the WC side. In this case analytical expressions for the resonant scattering can be derived from the Fano analysis. At the zero-th order in we find: which can be written as a standard Fano profiles in most cases. As analysed in case i), the interesting case to focus on is A = − B = 0 and A = B = , where the zero-th order in vanishes. At the first order in we find: This is a second peculiar case of generalized Fano lineshape. Note that the shape does not depend on the variation , but only its amplitude. In addition, the resonant part of RS 1 ( ), 3 , can be related to the resonant part of RS 1 ( ), 2 , as: In the dipole approach, case ii) is given by choosing: A = B = R ; B = A + = + . Then we have: Assuming that the optical response is given by an effective resonant dipole, we have: The right part of this equation defines the functions | 3 ( )| and 3 ( ) by: Where ( ) is the Heaviside step function. Note that | 3 ( )| is a non-Lorentzian lineshape and the resonant phase 3 ( ) jumps by 3 across the resonance. Indeed a 2 jump is due to the phase of 2 ( ) and an additional discontinuous jump at = is associated to the change of sign of the detuning ( − ). Following the previous derivation, the scattered signal can be written as:

Supplementary Note 4: Probing a coupled system with a waveguide or with a near-field probe
Coupled photonic systems in close proximity to the strong to weak coupling transition have been largely investigated by means of the detection of light transmission through an adjacent waveguide [12][13][14]. In order to compare our approach with the literature, we compare the spectral response of the resonant scattering through a waveguide (WG) and a near-field probe.

Case i): Single cavity
Here we analyze the case of one single cavity probed by a WG. The system is sketched in Supplementary Figure 7  We model the system by following the approach of [2]. In the scalar approximation, the coupled mode theory describes the electric field amplitude that propagates in the WG and that is confined in the cavity by the parameters W and 1 , respectively [15]. They are defined such that the input power propagating towards the cavity is given by W = | W | 2 , the output power propagating in the WG after the cavity results Out = | Out | 2 and that the energy stored in the cavity is C = | 1 | 2 . Note that W and 1 are both proportional to the electric fields, but they do not have the same units: [ W ]/[ 1 ] = Hz 1/2 . For a stationary pumping we have: Where is the cavity resonance, 1 is the cavity loss rate outside the WG, is the cavity loss rate in the WG, is the WG-cavity coupling, defined as a real quantity with unit Hz 1/2 . The loss rate W is proportional to | | 2 ; for sake of comparison with EIT models [2] we assume W ≪ 1 . This means that the Q factor of the cavity is not significantly changed by the WG coupling. So 1 is given by: Where we have defined the frequency-dependent cavity response 1 ( ) . By evaluating the field at the end of the WG we get: Finally, the transmission ( ) normalized to the transmission of the bare WG is: where we used the hypothesis of a small WG-cavity coupling: 2 1 ( ) ≪ 1 ( is a real quantity). Supplementary Equation (45) coincides with the model used for describing EIT and ATS [3]. In Supplementary Figure 7 b) we report the observable transmission change: Δ = − 1 = 2 { 2 1 ( )}. The transmission at the end of the WG shows a Lorentzian dip, whose symmetry does not depend on the WG to cavity coupling , being only an amplitude factor, and, as reported in Supplementary Equation (46), it measures the field in the cavity normalized to the field in the WG. In the SNOM experiment, sketched in Supplementary Figure 8 a), the single cavity is both pumped and observed in the near-field at the apex of the tip. This approach is denominated resonant back-scattering and it gives Fano lineshapes as the results of the interference between the non-resonant field scattered back by the sample surface and the resonant field in the cavity [16]. Then the resonant scattering intensity is given by the superposition of two fields as in the case of the WG. Indeed, we have shown in [16] that a resonant forward scattering experiment (i.e. a near-field transmission) would give similar results. The equations that describe this model are given in [16] and can be recast in the formalism of the coupled mode theory used in Supplementary Equation (41) and (42). So, w here represents the field inside the tip and the WG-cavity coupling is replaced by a tip-cavity position-dependent coupling ̃( ), since the near-field tip can move on the sample surface. Some observed lineshapes are Lorentzian dips but in general they belong to the wider class of standard Fano profiles [16]. This means that ̃ is a complex quantity, with a phase that is related to the mode phase-distribution. This phase defines the shape of the Fano profiles and the value or the dipole-phase by Supplementary Equations (16). These lineshapes are reported on Supplementary  Figure 8 and they clearly correspond to the standard Fano profiles obtained in Supplementary Note 2. In summary, with respect to the waveguide approach the near-field tip case has the advantage of mapping of the mode phase. The resonant scattering through near-field tip, RS ( ), is given by: For sake of simplicity, we do not explicit the dependence of ̃( ) on the tip position. If we assume RS~2 {̃2 1 ( )}, it would imply a phase shift by 4 ⁄ of the phase defined in Supplementary Equation (47). Since we are not investigating the mode-phase retrieval, the conclusion of our analysis would always be that nearfield experiments show a variety of Fano profiles as a function of the tip position, in agreement with recent experiments [16].

Case ii): Two coupled cavities
For two coupled cavities the EIT or ATS experiments reported to date are performed by means of a WG coupled only to one single cavity. On the contrary, near-field tip can be positioned either on a single cavity or on the mode overlap region. Supplementary Figure 9 a) shows the sketch of an experiment using WG, which tests, for instance, cavity #1. The system is described by Supplementary Equation (2) with the insertion of the WG field w and WG-cavity coupling 1 . As in Supplementary Equation (43) we find: where is the intercavity coupling and 1 the coupling between the WG and cavity #1. Here we define the spectral response of the system as: then, in the limit of small coupling 1 2 , the normalized transmission change can be written as: Figure 9: Photonic molecule tested by a waveguide coupled to a single cavity. a) Schematic of the photonic molecule where only cavity #1 is coupled to the waveguide (WG) by the coefficient . W and Out describe the fields in the waveguide before and after the interaction with the photonic molecule. 1 ( 2 ) and 1 ( 2 ) are the resonant field and the losses of cavity #1 (cavity #2), respectively. is the intercavity coupling. b) Transmission Δ , as given by Supplementary Equation (50), for the WG coupled to cavity #1 ( 1 = 10 2 ) for different value of the intercavity coupling . At the exceptional point, for = 0 = | 1 − 2 |/2, and for ~0 the spectrum shows a clear EIT peak. c) Δ for the WG coupled to cavity #2. Even at the exceptional point there is no evidence of the EIT signature.
We consider the typical case in which EIT and ATS are reported, that is: 1 = 2 = , and 1 ≫ 2 [2]. In Supplementary Figure 9 b)-c) we investigate as a function of the coupling the normalized transmission change Δ (for 1 / 2 = 10) as given by Supplementary Equation (50), when the WG is coupled to the cavity #1 and #2, respectively. Note the striking difference in Δ when probing the two cavities: the transparency window emerges only by probing the high loss cavity (cavity #1). The coupling is varied with respect to the value 0 = | |/2 that corresponds to the exceptional point (when the normal modes coalesce and A = B , A = B ). In the strong coupling regime, when ≫ 0 , a clear ATS profile with mode frequency splitting with two Lorentzian dips is observed. If the WG is coupled to cavity#1 around the EP transition (~0), the normal modes tend to coalesce and similar lineshapes, with a sharp transparency window, are observed, as reported in Supplementary Figure 10 a). Still the effect is denominated ATS when it occurs in strong coupling ( > 0 ) and EIT when in weak coupling ( < 0 ) regime. For the WG coupled to cavity #1 is we also analyze the lineshapes as a function of the ratio 1 / 2 , as shown in Supplementary Figure 10. In each row we used a different coupling strength: = 1.2 0 (ATS), = 0 (EP), = 0.8 0 (EIT), respectively. Clearly for large values of 1 / 2 (around 10) a deep transparency window is always observed, with slight variations between ATS and EIT, that nevertheless can be used to discriminate between the two effects [2]. Moreover, by performing further analysis, in the range 0.9 0 < < 1.1 0 the lineshapes tend to be identical, thus meaning that the transition across the EP is smooth. For moderate loss mismatch, 1 / 2 < 2, the transparency window is missing, even at the EP. In summary, the waveguide scheme shows a signature of the EP transition only for large loss mismatch and only if the WG is coupled to the low-Q cavity.
Supplementary Figure 10: Normalized transmission change through a waveguide coupled only to the higher loss cavity of the photonic molecule. Δ , as given by Supplementary Equation (50) for the waveguide coupled only to cavity #1, as a function of the intercavity coupling with respect to 0 = | 1 − 2 |/2, and as a function of the single cavity loss ratio 1 / 2 .
To highlight the meaning of these results, we analytically solve the problem on the normal mode basis recalling Supplementary Note 1, where the transformation matrix is defined by Supplementary Equation (5): This leads to: Supplementary Equation (52) applies both to SC regime, where 2Ω = √4 2 − 2 is a real quantity, and to WC regime where 2Ω = − Γ = − √ 2 − 4 2 is a pure imaginary quantity. Then the normal eigenvectors result: By pumping cavity #1 we excite both normal modes with different couplings given by: and each normal mode has a Lorentzian response, A ( ) and B ( ), with a single pole. Then, since the WG coupled to cavity #1, to get the WG transmission we evaluate 1 : or: Then, given Supplementary Equation (50) Δ = 2 [ 1 1 / W ], both EIT and ATS arise from the sum of the response of the two normal modes. In ATS for a large frequency splitting (Ω ≅ 2 ≫ δγ) the field 1 is given by the in phase contribution response of the two normal modes: Approaching the degeneration, that is close to the exceptional point (2 ≅ δγ , Ω ≅ 0), but still in strong coupling, we have: In weak coupling 1 is the same as Supplementary Equation (58) with the substitution Ω=-Γ. The term ( B − A ) in Supplementary Equation (58) comes from the destructive interference between the responses of the normal modes and it corresponds to a generalized Fano lineshape. We conclude that near the EP the transparency window is a generalized Fano response, which is the signature of proximity to degeneration. This is highlighted in Supplementary  Figure 11 where In near-field experiments the tip can be coupled, with a complex term, to both cavities. The corresponding equations (still imposing 1 = 2 = ) are: Where ̃2 gives the coupling between the tip and cavity #2. By changing the tip position we can vary the ratio ̃1/̃2. Then by defining: the detected resonant change is a sum of two Fano profiles, each given by Supplementary Equation (46), thus the transmission through the near-field probe RS ( ) results: where ̃1 = |̃1|exp ( 1 /2), ̃2 = |̃2|exp ( 2 /2). In order to make a comparison with the case where the WG probes cavity #1, we assume ̃2 = 0. The transmission change through a near-field tip in different positions on a photonic molecule as a function of the coupling strength are shown in Supplementary Figure 12, for the same parameters used in Supplementary Figure 9. When 1 = 0 the SNOM lineshape coincides with the WG transmission, that is with the imaginary part of the system response 1 . However, by selecting 1 = /2 the lineshapes become dispersive, and near the EP (both in EIT or ATS) the SNOM approach offers the possibility to measure also the real part of the response 1 , which in the WG approach is related to the slow light effect [12,17]. In summary, the same results of the WG setup, both in EIT and in ATS, can be retrieved by the SNOM approach, which also gives access to the real part of 1 . The main novelty of the near-field approach is the detection of generalized Fano lineshapes near the EP for any values of 1 / 2 . By exploiting the collection from both cavities, we can cancel the two standard Fano contributions in Supplementary Equation (58), by balancing the collection (̃1 2 = −̃2 2 ). In order to compare the results to the WG approach, in Supplementary Figure 13 we show the calculation performed using Supplementary Equation (61). The generalized Fano lineshape 2 , fingerprint of the proximity to the EP, is always observed. Approaching the EP, the lineshape does not change, but its amplitude tends to zero. Exactly at the EP we get zero signal due to complete destructive interference. Still, for any small deviation from the EP (or even for ̃1 2 ≠ −̃2 2 ) the SNOM approach furnishes generalized Fano lineshapes, as reported on Supplementary Figure 14. In conclusion, our method is a generalization of the WG coupling commonly used for probing EIT or ATS transmission peaks. The near-field approach retrieves the same results when probing only one single cavity but possess the noteworthy advantage to explore also the real part of the system response, related to the group velocity effect.
Moreover, the transition between ATS and EIT can be distinguishable by a sudden change of the generalized Fano lineshapes observed in resonant scattering as a function of the inter-cavity coupling, thus extending the possibilities of the WG approach.
Supplementary and/or Fano parameter A = B + , in addition to and/or . In these cases, the problem must be handles numerically. In the main text the near-field experimental data of the photonic molecule reported Fig. 4 are fitted by Supplementary Equation (21). In Supplementary Table 1 we show the fitting output parameters. Note that the central frequency and broadening in the different fits slightly depends on the detection points. This is due to the tip perturbation effect as discussed in [16].  Fig. 4 b) of the main text.
Many other generalized Fano profiles were predicted and observed. In Supplementary Figure 16

Supplementary Note 7: Detecting small detuning
Here we discuss the possibility to detect small spectral detuning between two modes (or two arbitrary optical signals), whenever the other parameters are well characterized. Let us consider, for example, the transmission signals from two independent resonant modes with broadening B = A + and frequency B = A , that result in the profile given by Supplementary Figure 3

Supplementary Note 8: Laser assisted oxidation
In order to compensate the fabrication-induced detuning between the two single cavity modes we performed a local nano-oxidation of the GaAs membrane induced by Ar-laser light (514 nm). The laser illuminates the sample through the near-field dielectric probe with a power of 2.5 mW. The non-thermal process induces a gentle blue-shift of the photonic modes [19]. Before oxidizing the sample (t=0) we mapped the mode distributions, as reported in Supplementary  . This demonstrates that a disorder induced detuning overcomes the coupling between the two cavities. Then, we locally oxidized the right part of the cavity where P1 is localized [see insets of Fig. 4 a)]. Both resonant wavelengths of P1 and P2 performed a blue-shift, as highlighted in Supplementary  Figure 9.0 d). The splitting variation for sequential oxidation processes gradually decreases. In fact, by exposing the Ar-laser light on the right side of the cavity system, we tune the wavelength of P1 more towards the blue than P2. Finally, at t=29 min we obtain almost degenerate modes with PL spatial distribution delocalized over the entire photonic molecule, as reported in Supplementary Figure 9.0 e)-f). By continuing the oxidation process we found that the wavelength splitting P1-P2 increases, as reported in Fig.3 a), and that the mode distributions are localized on opposite cavities; that is the larger (lower) wavelength mode P1 (P2) is found on the left (right) cavity. These features demonstrate that we performed a variation of the sign of the detuning , thus approaching the zero-value. In order to reproduce the wavelength splitting of the coupled modes, Ω =P1-P2, as a function of the oxidation time (reported in the main manuscript) we used the coupled mode theory: Where the detuning is given by = [ √ − 0 − ] as reported in [20]. The value = | 1 − 2 | = 0.05 nm is measured at the beginning of the oxidation process when the two modes are uncoupled because of the large detuning ( 1 = 0.71 nm and 1 = 0.76 nm). The fitted curve does not cross the zero value, indicating that the modes perform an anticrossing as a function of detuning and therefore they are effectively coupled. In fact, Ω is larger than zero at its minimum, where the detuning is almost zero. The fit output parameter 0 = 3 min accounts for the thin oxide layer already present when starting the oxidation process. The parameter , which accounts for the oxidation strength, is consistent with similar oxidations reported in [20]. Finally, the coupling strength is = 0.03 nm, thus resulting in a strong-coupling interaction (2 > ) between the two resonators, even if Rabi oscillation and an effective photon hopping cannot be achieved, since < 1 , 2 .