Coherent virtual absorption for discretized light
Stefano Longhi


[ 









Dipartimento di Fisica, Politecnico di Milano and Istituto di Fotonica e Nanotecnologie del Consiglio Nazionale delle Ricerche, Piazza L. da Vinci 32, I-20133 Milano, Italy (stefano.longhi@polimi.it)



Coherent virtual absorption (CVA) is a recently-introduced phenomenon for which exponentially growing waves incident onto a conservative optical medium are neither reflected nor transmitted, at least transiently. CVA has been  associated to complex zeros of the scattering matrix and can be regarded as the time reversal of the decay process of a quasi-mode sustained by the optical medium.  Here we consider CVA for discretized light transport in coupled resonator optical waveguides or waveguide arrays and show that a distinct kind of CVA, which is not related to complex zero excitation of quasi-modes, can be observed. This result suggests that scattering matrix analysis can not fully capture CVA phenomena. 


130.3120, 290.5839, 000.1600
 ] 



The ability of tailoring light absorption in a medium exploiting interference effects has sparked much interest in the past recent years with a wealth of potential applications ranging from  optical sensing to optical modulation, optical switching, polarization control, etc. <cit.>; for a recent review see <cit.>. Coherent perfect absorption (CPA) refers to the possibility for a dissipative medium to fully absorb suitable patterns of incoming waves; it can be regarded as the time reversal of  lasing <cit.>. In a multiport linear system, incoming and outgoing fields are related by the scattering matrix S. CPA is realized when an eigenvalue of S vanishes at some real frequency. In a recent work <cit.>, Baranov and collaborators introduced the concept of coherent virtual absorption (CVA) in lossless reciprocal media, which is based on the excitation of the medium with suitable incoming waves with complex frequency. While CPA modes are associated to zeros of the scattering matrix on the real axis, CVA corresponds to complex  zeros of the scattering matrix, i.e. existence of quasi-bound optical modes. Unlike conventional CPA, in CVA the excitation fields have an amplitude that increases exponentially in time and space, similar to Gamow^'s states introduced in earlier theories of quantum mechanical decay <cit.>. CVA can be viewed as the time reversal process <cit.> of the light decay process in a high-quality passive resonator <cit.>.



In this Letter we reveal a distinct form of coherent virtual absorption, which is neither related to complex zero eigenvalues nor does it strictly require existence of quasi-bound optical eigenmodes. Specifically, we consider
transport  of discretized light in coupled resonator optical waveguides (CROW) <cit.> or in evanescently-coupled optical waveguide arrays <cit.>, and show that a kind of virtual absorption can be observed even in structures that do not sustain any resonance mode, i.e. beyond the conditions considered in previous work <cit.>. Such a kind of CVA stems from the property that specially-designed  waveforms, synthesized using time reversal symmetry, can show half-space  discrete self-focusing on the lattice.

We consider light transport in a linear chain of CROW <cit.> [Fig.1(a)] or, similarly, spatial propagation of light in arrays of evanescently coupled optical waveguides <cit.> [Fig.1(b)]. For the sake of definiteness, we will refer mostly to the former physical system.  The CROW is composed by a chain resonators with the same resonance frequency ω_0, which are evanescently-coupled with a coupling constant κ. A defective resonator, with detuned resonance frequency ω_0+δω_0, is placed at site n=0, and coupled to its neighboring resonators with a modified coupling constant x κ, with x ≤ 1; see Fig.1(a). The defective resonator could be lossy, with a loss rate γ (γ=0 in the conservative limit). Light transport in the CROW is described by coupled-mode equations <cit.>

    i dc_n/dt    =    ω_0 c_n -κ (c_n+1+c_n-1)     |n| ≥ 2 
    
    i dc_±1/dt    =    ω_0 c_± 1 -κ c_±2-x κ c_0 
    
    i dc_0/dt    =     (ω_0+δω_0-i γ) c_0-x κ (c_1+c_-1)

where c_n(t) is the field amplitude of the resonator mode at site n. The scattering properties of the defective resonator are embodied in the 2 ×2 scattering matrix 

    S= (
    [ r t; t r ])

where r=r(q) and t=t(q) are the spectral reflection and transmission amplitudes, respectively. These are determined from the solutions to Eq.(1) of the form

    c_n(t)= {[ [ A exp(iqn) +B exp(-iqn) ] exp(-i Ω t)                                  n ≤ -1;                           E exp(-i Ω t)                                     n=0;  [ C exp(-iqn)+D exp(iqn) ] exp(-i Ω t)                                   n ≥ 1 ].

where (A,C) and (B,D) are the amplitudes of incoming and outgoing waves, respectively [Fig.1(a)], q is the Bloch wave number with 0 ≤ Re(q) ≤π, and 

    Ω(q)=ω_0-2 κcos q

is the lattice dispersion relation of the homogeneous lattice <cit.>. The scattering matrix relates the amplitudes of outgoing and incoming waves according to (B,D)^T=S (A,C)^T.
The explicit form of r(q) and t(q) read

    r(q)=ρ / κ - 2(x^2-1) cos q/2 x^2exp(iq)- 2 cos q - ρ / κ
    
     t(q)=2i x^2 sin q /2 x^2exp(iq)- 2 cos q - ρ / κ

where we have set ρ≡δω_0-i γ. Note that, for a real Bloch wave number q, the frequency Ω(q) is real and the scattered states (3) are stationary and propagative waves.

Like for light scattering in continuous media, CPA and CVA based on real and complex zeros of the scattering matrix  can be found for discretized light on a lattice. 
The condition det (S)=0, corresponding to a zero eigenvalue of S, implies r=-t or r=t. Only the former condition r=-t, corresponding to symmetric mode excitation A=C, is relevant and yields the following  equation for the Bloch wave number q

    (x^2-1) cos q-i x^2 sin q= (δω_0-i γ)/ ( 2 κ)

For CPA, we require q to be real. Like in other CPA systems  <cit.>, this implies some dissipation γ in the medium and a constraint between frequency detuning δω_0 and dissipation γ.





In a dissipationless structure, i.e. for γ=0, the condition det(S)=0 can be satisfied for a complex Bloch wave number q=q_0, corresponding to incoming (Gamow^'s-like) waves with an exponentially increasing amplitude in time. For the sake of simplicity, let us assume δω_0=0 and x ≤ 1, so that the defect does not sustain any bound state. A resonance (quasi-bound) state, with lifetime determined by the imaginary part of Ω(q_0), is found as the complex root of Eq.(7) with γ=δω_0=0


    q_0=π /2 - (i/2)  ln(1-2x^2).

 Note that for x ≪ 1 one has q_0 ≃π/2+i x^2, and the complex energy of the resonance mode is given by Ω≃ω_0 + 2 i κ  x^2. In this limit, two incoming waves, exponential growing in time t and space n according to
 
    c_n(t) ≃exp [-i(π/2)|n|+x^2 |n|+2 κ x^2 t-i ω_0 t]

(n ≠ 0),  realize CVA, in which the energy of incoming waves is fully stored in the defective resonator. Figure 2 shows an example of CVA based on complex zero excitation, in which the exponential waves at initial time are  truncated at |n|=200 [Fig.2(b)]. This is precisely the scenario considered in Ref.<cit.>, and therefore it will not be further analyzed.

Here we wish to suggest a different route toward virtual absorption, which is not associated to the existence of resonance modes.  To this aim, let us consider the behavior of the Bloch wave number q_0 [Eq.(8)] in complex plane,  which is shown in Fig.1(c) together with the corresponding behavior of the frequency Ω=ω_0-2 κcos q_0. Clearly, a phase transition is found at x=1/ √(2). 
For x< 1 / √(2), Re(q_0)=π/2 is independent of x while Im(q_0) increases as x increases, corresponding to lowering of the Q-factor (lifetime) of the quasi-mode. At x=1/ √(2), q_0 goes to infinity, i.e. det(S) does not vanish nowhere in complex q plane and the quasi-mode disappears.  For 1/ √(2) < x ≤ 1, q_0 is purely imaginary and the frequency Ω becomes real, indicating that the solution (3) with B=D=0 is stationary in time but exponentially growing in space (anti-evanescent wave), and thus unphysical.

 A natural question then arises: while for x< 1 / √(2) the structure sustains a quasi-bound mode and CVA can be observed by complex zero eigenvalue excitation, what happens when crossing the phase transition point at x=1/ √(2)? Can virtual absorption be found for x> 1 / √(2)? The main result of this Letter is that virtual absorption does not strictly require the existence of quasi bound modes. Exploiting the very general concept of time-reversal symmetry, one can synthesize half-space  self-focusing wave packet distributions  such that, for a finite (yet arbitrarily long) time interval, an initially broad wave, localized at n<0 sites, shrinks  and stores energy in few lattice sites around n=0, with negligible transmission in the |n|>1 half space.
 
 To gain some insights into the synthesis method, let us indicate by U_n,m(t) the propagator of Eq.(1) with δω_0=γ=0, so that 
 
    c_n(t)=∑_m U_n,m(t) c_m(0) exp(-i ω_0 t).

  The propagator U_n,m(t) can be calculated in a closed form for special values of x, namely for x=1 (homogeneous lattice) and for x= 1 / √(2), i.e. at the phase transition point. For x=1, one has U_n,m(t)=i^n-m J_n-m(2 κ t) <cit.>, while for x= 1 /√(2) the expression of U_n,m(t) is more involved and not given here for the sake of brevity (see <cit.>). For other values of x, U_n,m(t) can be computed numerically.  CVA based on time reversal can be exemplified by considering two simple cases.

  (i) As a first example, let us assume c_n(0)=δ_n,0. After some time t=T, a broad symmetric waveform is found as a result of wave spreading in the lattice, which is given by c_n(T)=U_n,0(T) with |c_-n(T)|^2=|c_n(T)|^2.  Its time reversal corresponds to a symmetric self-focusing wave. In fact, assuming as an initial condition c_n(0)=U_n,0^*(T), complex conjugation corresponds to time reversal and the spatially-broad wave distribution shrinks and completely stores energy at the n=0 lattice site at the focusing time t=T, i.e. |c_n(T)|=δ_n,0. Note that this phenomenon of self-focusing as time reversal of discrete spreading in the lattice is a very general one and is not related to complex zero eigenmodes.
 
  (ii) As a second example, let us consider excitation of the site n=0 and of a few other sites around n=0 with tailored phases and amplitudes. In this case, wave spreading in the lattice becomes asymmetric, and its time reversal thus corresponds to asymmetric CVA, i.e. CVA with unbalanced wave amplitudes from the two sides of the n=0 site. For example, let us consider the simplest case x=1 and let us assume c_n(0)=δ_n,0+iΔ ( δ_n,1+δ_n,-1); one then readily obtains 

    c_n(T)=i^n J_n( 2 κ t) (1 + n Δ/κ T).

Note that the waveform broadens as κ T increases and the symmetry |c_-n|^2=|c_n|^2 is broken for a non-vanishing value of Δ. Since the Bessel J_n function has its maxima at n ≃± 2 κ T for κ T ≫ 1, by choosing Δ=-1/2 the distribution |c_n(T)|^2 turns out to be strongly asymmetric around n=0, with the major excitation in the n<0 half lattice. Time reversal thus corresponds to a self-focusing wave with dominant excitation in the left half-space of the lattice. If more than three waveguides with properly-tailored amplitudes and phases are excited at t=0, the pattern |c_n(T)|^2 can be entirely localized in the half space n ≤ 0, so that its time reversal would correspond to  CVA with unidirectional excitation of the lattice. This is a distinctive feature as compared to CVA based on complex zeros of the scattering matrix eigenvalues, where bidirectional excitation is required [see Eq.(9)].
To realize CVA with one-side excitation, let us assume the initial condition 

    c_n(0)=U_n,0^*(T)H_n

 with envelope H_n defined by

    H_n= {[ exp( β n)    n ≤ -1;         0     n ≥ 0 ].

and 0<β≪ 1. The waveform (12) is localized in the n<0 half-space and is obtained from the deformation and truncation of the time reversal of the impulse response U_n,0 of the lattice. Extended numerical simulations indicate that the waveform (12) undergoes half-space transient self focusing in the time interval (0,T), with c_n(T) being non vanishing for few sites around n=0 (focusing condition). The instantaneous transmittance 𝒯(t), defined by 

    𝒯(t) ≡∑_n=n_0^∞ |c_n(t)|^2/∑_n=-∞^∞ |c_n(0)|^2

  (n_0>0 is some small integer), turns out to be negligible for t<T, showing an abrupt transition to a non-negligible value at t>T. This result is a rather general one and turns out to be rather independent of the small parameter β, which can be used to flatten the wave excitation amplitudes over the lattice sites at initial time. Examples of self-focusing  waveforms, corresponding to virtual absorption for unidirectional wave excitation, are shown in Fig.3 for a few values of x. For times smaller than the focusing time t=T, the initially broadened wave packet shrinks toward the n=0 site but is not able to cross this site, as if virtual absorption would occur (actually energy is increasingly stored in a few sites around n=0). However, at times larger than T wave scattering (either transmission or reflection) is clearly visible. Interestingly, such a method to synthesize discrete wave packets  can be applied for x<1 / √(2) as well, i.e. below the phase transition point [Fig.3(c)]. In this case virtual absorption is clearly not related to the complex zero of the scattering matrix eigenvalue: in fact, the wave packet excitation greatly deviates from the exponential Gamow^'s state, and excitation is unidirectional rather than bidirectional [compare Fig.2 and Fig.3(c)].Physical observation of virtual absorption without complex zero excitation.  An experimentally accessible system for the observation of CVA of discretized light is provide by spatial light propagation in waveguide arrays <cit.>, where the propagation distance z along the array plays the same tole as time t in a CROW. A suggested experimental setup is shown in Fig.4(a). A waveguide array with homogeneous waveguide spacing d [corresponding to x=1 and δω_0=0 as in Fig.3(b)] comprises two sections, of length L_1 and L_2 ≥ L_1. The first section is used to synthesize the waveform (12), which then propagates in the second section to realize CVA. The array is excited at the n=0 waveguide and discrete diffraction occurs from z=0 up to z=L_1. At z=L_1 half waveguides of the array, at sites n ≤ 0, are interrupted for a short length s, preventing further propagation of light in the second section of the array.  s is chosen of the order (or smaller) than the coupling length ∼ 1 / κ, yet sufficiently long to prevent light re-coupling after the interruption. In the short distance s, the other half waveguides of the array are alternately segmented to realize π phase slips between adjacent guides <cit.>. The phase slips realize the analog of time reversal <cit.>. The optical pattern that propagates in the second section of the array is thus the waveform (12) with β=0. Figure 4(b) shows an example of simulated discrete light propagation in a 8-cm-long array comprising 80 waveguides, fabricated by fs-laser-writing in fused silica <cit.>, for a lattice period  d=24  μm, corresponding to a coupling constant κ≃ 0.4   mm^-1  <cit.>, and segmentation/interruption length ∼ 2.6 mm <cit.>. Clearly, in the second array section the discrete beam undergoes half self-focusing, realizing CVA without complex zero eigenmodes. Fluorescence imaging techniques <cit.> can be used to experimentally reproduce propagation maps of Fig.4(b).
 
  In this work the phenomenon of coherent virtual absorption, recently introduced in <cit.>, has been investigated for discretized light transport. The main conclusion is that virtual absorption does not necessarily require complex zero excitation of quasi-modes. This result suggests that scattering matrix analysis does not fully capture the entire physics of the phenomenon and motivates further theoretical and experimental research,  opening up some major questions. For example, can virtual absorption without complex zero eigenvalue excitation be observed for light waves in continuous media? Can transient virtual absorption be controlled by exploiting material nonlinearity?  


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