Main

Before we introduce the concept of spacetime reflection in optics, we first briefly outline some of the basic aspects of this symmetry within the context of quantum mechanics. In general, a Hamiltonian (where and are position and momentum operators respectively, m is mass and V is the potential) is considered to be P T symmetric, , provided that it shares common eigenfunctions with the P T operator1,16,17,18,19,20,21. This condition corresponds to an exact or unbroken P T symmetry, as opposed to that of broken P T symmetry, where, even though is still valid, and P T (or any other antilinear operator) possess different eigenvectors22. For the case considered here, given that the action of the parity P and time T operators is defined as and , respectively, it then follows that a necessary (but not sufficient) condition for a Hamiltonian to be P T symmetric is . In other words, P T symmetry requires that the real part of the potential V is an even function of position x, whereas the imaginary part is odd; that is, the Hamiltonian must have the form , where VR,I are the symmetric and antisymmetric components of V, respectively12,13,14. Clearly, if ɛ=0, this Hamiltonian is Hermitian. It turns out that, even if the antisymmetric imaginary component is finite, this class of potentials can still allow for both bound and radiation states, all with entirely real spectra. This is possible as long as ɛ is below some threshold, ɛ<ɛth. If, on the other hand, this limit is crossed (ɛ>ɛth), the spectrum ceases to be real and starts to involve imaginary eigenvalues. This signifies the onset of a spontaneous P T symmetry-breaking, that is, a ‘phase transition’ from the exact to broken-P T phase7,20.

In optics, several physical processes are known to obey equations that are formally equivalent to that of Schrödinger in quantum mechanics. Spatial diffraction and temporal dispersion are perhaps the most prominent examples. In this work we focus our attention on the spatial domain, for example optical beam propagation in P T-symmetric complex potentials. In fact, such P T ‘optical potentials’ can be realized through a judicious inclusion of index guiding and gain/loss regions7,12,13,14. Given that the complex refractive-index distribution n(x)=nR(x)+i nI(x) plays the role of an optical potential, we can then design a P T-symmetric system by satisfying the conditions nR(x)=nR(−x) and nI(x)=−nI(−x).

In other words, the refractive-index profile must be an even function of position x whereas the gain/loss distribution should be odd. Under these conditions, the electric-field envelope E of the optical beam is governed by the paraxial equation of diffraction13:

where k0=2π/λ, k=k0n0, λ is the wavelength of light in vacuum and n0 represents the substrate index.

Ultimately, it will be of interest to synthesize artificial periodic optical systems showing unusual features stemming from P T symmetry13,14. Yet, it is first imperative to understand P T behaviour at a single-cell level. In integrated optics, such a single P T element can be realized in the form of a coupled system7,12, with only one of the two parallel channels being optically pumped to provide gain γG for the guided light, whereas the neighbour arm experiences loss γL (Fig. 1a). Under these conditions and by using the coupled-mode approach, the optical-field dynamics in the two coupled waveguides are described by

where E1,2 represent field amplitudes in channels 1 and 2, κ=π/(2Lc) is the coupling constant with coupling length Lc and the effective gain coefficient is γGeff=γGγL. From previous considerations, P T symmetry demands that γGeff=γL=γ. The behaviour of this non-Hermitian system can be explained by considering the structure of its eigenvectors, above and below the phase-transition point γ/(2κ)=1. Below this threshold the supermodes are given by |1,2〉=(1,±exp(±i θ)), with corresponding eigenvalues being ±cosθ, where sinθ=γ/2κ. At phase transition (the ‘exceptional point’), the modes coalesce to |1,2〉=(1,i), where the amplitudes in the two channels have the same magnitude23,24. Above threshold, that is, for γ>2κ, , where in this range coshθ=γ/2κ and the two eigenvalues are . We emphasize that, unlike Hermitian systems, these eigenmodes are no longer orthogonal. Instead, the basis is now skewed. This in turn has important implications for optical-beam dynamics including a non-reciprocal response and power oscillations. For a conventional Hermitian system (γ=0), any superposition of the two (symmetric and antisymmetric, see Fig. 1b) eigenmodes leads to reciprocal wave propagation: obviously, the light distribution in Fig. 1c (top) obeys left–right symmetry. This situation changes when the coupled system involves a gain/loss dipole. If the gain increases but is still below threshold, the relative phase differences ϑ between the two field components increase from their initial values at 0 and π, respectively, and finally, at threshold, the two modes coalesce at ϑ=π/2 (see Fig. 1b). More interestingly, light propagation is now obviously non-reciprocal: by exchanging the input channel from 1 to 2 in Fig. 1c (middle) we obtain an entirely different output state. This behaviour is altered drastically above threshold (Fig. 1c, bottom). In this regime light always leaves the sample from channel 1, irrespective of the input—again in a non-reciprocal fashion. This can be explained by noting that, above threshold, the system’s eigenvalues are complex, with the corresponding amplitudes either exponentially increasing or decaying. Thus only one supermode effectively survives. Here it is worth noting that any coupled system with an asymmetric gain–loss profile can be mathematically transformed into a P T-symmetric one. In particular, this is true for an asymmetric loss/loss-type potential (coupled states with low/high losses), showing a ‘passive’ P T system25. Very recently, for such a system, loss-induced optical transparency was experimentally demonstrated.

Figure 1: Conventional and P T-symmetric coupled optical systems.
figure 1

a, Real (nR, red line) and imaginary (nI, green line) parts of the complex refractive-index distribution. b, Supermodes of a conventional system, and of a P T-symmetric arrangement below and above threshold. c, Optical wave propagation when the system is excited at either channel 1 or channel 2. For the conventional case, wave propagation is known to be reciprocal, whereas in a P T-symmetric system light propagates in a non-reciprocal manner both below and above threshold.

Here we observe non-reciprocal wave propagation in an ‘active’ P T-symmetric coupled waveguide system based on Fe-doped LiNbO3. As such, this structure shows richer dynamics and enables us to explore a wider range of behaviour not previously accessible because of fixed losses. We use Ti in-diffusion to form the symmetric index profile nR(x). Optical gain γG (the typical magnitude is a few cm−1 in Fe-doped LiNbO3) is provided through two-wave mixing using the material’s photorefractive nonlinearity26,27. A mask on top of the sample is used to partially block the pump light, to provide amplification in only one channel (see the experimental set-up in Fig. 2). Both the output intensity and the phase relation between the two channels (using interference with a plane reference wave) are monitored by a CCD (charge-coupled device) camera. In our system, losses arise from the optical excitation of electrons from Fe2+ centres to the conduction band. On the other hand, the optical two-wave mixing gain (which is proportional to the concentration of Fe3+ centres) has a finite response time26. Assuming an exponential temporal build-up, γG(t)=γmax[1−exp(−t/τ)] with Maxwell time constant τ, the evolution of the intensity distribution at the output facet can be monitored as a function of time t. In other words, the state of the system below, at and above threshold can be directly observed at different instants t (refs 7, 13).

Figure 2: Experimental set-up.
figure 2

An Ar+ laser beam (wavelength 514.5 nm) is coupled into the arms of the structure fabricated on a photorefractive LiNbO3 substrate. An amplitude mask blocks the pump beam from entering channel 2, thus enabling two-wave mixing gain only in channel 1. A CCD camera is used to monitor both the intensity and phases at the output.

Although equations (1) can be solved analytically, we here obtain the output intensities I1|E1|2, I2|E2|2 as a function of gain γG(t) by numerical integration. Figure 3a shows results of two such simulations when γL=2κ, γmax=2.5γL, where channel 1 (Fig. 3a, left-hand side) or channel 2 (Fig. 3a, right-hand side) has been excited. At t=0 the system starts from γG=0 and shows a reciprocal response. However, as the gain builds up at t>0 and the P T structure is tuned below threshold (which is reached for t/τ≈1.6), wave propagation becomes strongly non-reciprocal (with different numbers of zero-crossings, depending on the ratio L/Lc; see Fig. 3a). At the threshold the two supermodes become degenerate; however, the intensities of the two fields are slightly different. The reason lies in the limited length L of our sample: at threshold, the pure coalesced eigenstate |1,2〉=(1,i) of our system (excited by an input state (1, 0) or (0, 1), respectively) is approached adiabatically only for infinitely long propagation, . Above threshold the output of the P T system is no longer sensitive to the input conditions. In this regime, one supermode is exponentially amplified whereas the other decays.

Figure 3: Computed and experimentally measured response of a P T-symmetric coupled system.
figure 3

a, Numerical solution of the coupled equations (1) describing the P T-symmetric system. The left (right) panel shows the situation when light is coupled into channel 1 (2). Red dashed lines mark the symmetry-breaking threshold. Above threshold, light is predominantly guided in channel 1 experiencing gain, and the intensity of channels 1 and 2 depends solely on the magnitude of the gain. b, Experimentally measured (normalized) intensities at the output facet during the gain build-up as a function of time.

The experimental response of this LiNbO3 P T-symmetric optical system (with κ=1.9 cm−1 and L=2 cm) is shown in Fig. 3b. In all our experiments we used low input power levels (signal power 25 nW and pump intensity Ip=0.5 mW cm−2 when exciting channel 2, and twice these values when exciting channel 1) to avoid any index perturbations (including the situation above threshold, where intensity increases rapidly), which may in turn spoil the symmetry in nR(x) (refs 28, 29). Figure 3b (left-hand side and right-hand side) depicts the temporal behaviour of the output intensity distribution when channel 1 and 2 is excited, respectively. We note that the build-up time constants τ in these two situations are different owing to different intensities used during excitation. As a result, the threshold is reached faster (tth,1≈10 min) in the first case as compared with the latter (tth,2≈70 min). By taking this into account, we find an excellent agreement between our experiments and numerical simulations.

Another manifestation of P T symmetry is the relative phase difference ϑ between the two elements of the same eigenmode—which can be measured at the output facet of the sample. These results are depicted in Fig. 4. For γG=0, the phases corresponding to the even and odd supermodes are ϑ=0 and ϑ=π, respectively, as in conventional arrangements (Fig. 4a,b). When the gain is further increased and the system is below threshold, the two eigenstates are not orthogonal and their phases can be anywhere (depending on γ/2κ) in the interval [0,π]. An example is given in Fig. 4c, where a phase difference of ϑ≈2π/3 was estimated from our measurements. Finally, Fig. 4d illustrates the situation slightly above the exceptional point. In this case the phase is fixed at ϑ=π/2, irrespective of γ/2κ, again in good agreement with theoretical predictions.

Figure 4: P T supermode phase measurements.
figure 4

ad, Intensity distribution (upper panels) and phase relation (lower panels) of conventional (a,b) and P T-symmetric (c,d) systems. a,b, Measured relative phases of an even (a) and odd (b) eigenstate associated with a conventional system. c,d, Phase relation corresponding to a P T-symmetric system below (c) and above (d) threshold. Although below threshold (c) the phase difference lies in the interval [0,π], depending on the magnitude of gain, above threshold this value is fixed at π/2, as shown in d.

Our results can be easily extended to transversely periodic media, enabling new intriguing effects such as P T solitons, double-refraction or synthetic systems with tailored transverse flow of optical energy, and thus pave the way for developing new non-reciprocal optical components, where light is propagating forward and backward in a different fashion. This letter has made the simplest demonstration of P T effects: just a coupled two-channel system. However, the vision is to incorporate nonlinearities and construct sophisticated P T systems, such as P T optical lattices, P T-based solitons and so on. Last but not least, the phase-transition or exceptional point has been intriguing researchers for a long time, because the eigenmodes associated with that point are self-orthogonal, and as such their amplitudes should diverge7. In addition, in optical P T lattices, anomalous transport or discrete diffraction can occur in the neighbourhoods of such points, as indicated in ref. 13. Is this self-orthogonality a physical property with truly unique and experimentally observable quantities? This and related questions are now within experimental reach.