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Engineering wavefront caustics trajectories in PT -symmetric lattices Nicholas Bender,1 Hamidreza Ramezani,2 and Tsampikos Kottos* 1

2

Department of Physics, Wesleyan University, Middletown Connecticut 06457, USA NSF Nanoscale Science and Engineering Center, University of California, Berkeley, California 94720, USA *Corresponding author: [email protected] Received March 16, 2015; revised April 4, 2015; accepted April 13, 2015; posted April 13, 2015 (Doc. ID 236128); published May 1, 2015

We utilize caustic theory in PT -symmetric lattices to design focusing and curved beam dynamics. We show that the gain and loss parameter in these systems provides an additional degree of freedom that allows for the design of the same caustics trajectories with different intensity distribution in the individual waveguides. Moreover we can create aberration-free focal points at any paraxial distance zf , with anomalously large focal intensity. © 2015 Optical Society of America OCIS codes: (130.0130) Integrated optics; (130.2790) Guided waves; (230.0230) Optical devices; (350.5500) Propagation. http://dx.doi.org/10.1364/OL.40.002138

Diffraction management of beam propagation and the possibility of designing initial wavefronts that lead to abrupt energy focusing have attracted the attention of researchers over the years [1–7]. Not only is the fundamental side of this problem both charming and challenging for researchers, but also its applied side has attracted attention. Perhaps the most pronounced example comes from the field of medical lasers where one needs to have an abrupt beam focusing at a specific point, without affecting nearby tissues [8]. Other applications include particle manipulation [9], generation of self-bending plasma channels [10], and optical bullets [11], etc. In this Letter, we will investigate caustics trajectory dynamics and abrupt focusing in parity-time (PT )symmetric discrete array settings, like the one shown in Fig. 1. In contrast to other focusing schemes [12], we will utilize curved trajectory dynamics and caustics design in discrete elements and thus extend the analysis for passive lattices [7] to lattices with PT -symmetry. This type of optical systems have been recently introduced [13] and, during the last years, have gained a lot of attention due to the wealth of exotic properties that they posses [13–32]. The main characteristic of these systems is that the optical potential (index of refraction) ϵx  ϵR x  iγx is complex and satisfies the PT symmetric property ϵx  ϵ −x. Since the optical potential is complex, these systems are described by an effective non-Hermitian Hamiltonian with eigenvalues (propagation constants in optics language) that are real if the gain/loss parameter is below some critical value, i.e., γ ≤ γ PT . Above this value, the eigenvalues become complex, and the system is unstable. The transition point γ  γ PT has the properties of an exceptional point (EP) singularity, i.e., both eigenvalues and eigenvectors coalesce. This PT -symmetric phase transition leads to a number of interesting features: asymmetric transmission [16–20], unidirectional invisibility [21–23], novel lasing schemes [24,33,34], non-reciprocal Bloch oscillations [30,31], and reconfigurable Talbot effects [32]. We consider an array of one-dimensional (1D) coupled waveguides. The array is composed of two types of waveguides: the first with gain (A) and the other with 0146-9592/15/092138-04$15.00/0

equivalent loss (B). We further assume that each waveguide is supporting only one propagating mode. The nearby waveguides are assumed to be coupled evanescently. The waveguides are arranged in a way that they form N coupled A-B dimers with intra-dimer coupling k and inter-dimer coupling c. In the slowly varying envelope approximation, the electric field amplitude Ψn  an ; bn T at the nth dimer evolves (along the propagation direction z) according to the equation: dan z  ϵan z  kbn z  cbn−1 z  0 dz db z  ϵ bn z  kan z  can1 z  0; i n dz i

(1)

where the refractive index of the nth waveguide is ϵ  ϵ0  iγ. ϵ0 is the background index of refraction, and γ is the gain-loss parameter. Without any loss of generality, we will assume below that ϵ0  0. Equation (1) is invariant under a parity-time (PT ) symmetric operation

Fig. 1. PT -symmetric lattice consisting of dimers with local PT -symmetry. Each dimer consists of two waveguides, one attenuating (type “a” waveguides, indicated with green), while the other one (type “b” waveguides, indicated with red) has equivalent amplification. The coupling between two waveguides of a specific dimer is k, while the coupling between different dimers is c < k. © 2015 Optical Society of America

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[28,35]. The parity-symmetry operator P is defined as a spatial inversion (around an axis of symmetry of the array) while the time-reversal symmetry operator T is associated with a complex conjugation. The propagation of the electric field along the paraxial direction z is conveniently R evaluatedinqin the Fourier 1 π ˜ q space where an z  2π (similarly for −π dqaq ze bn ). Specifically, the translational invariance of the system allows us to decouple, for each value of q, the equations of motion into 2 × 2 blocks: d i dz



a˜q z b˜q z



  Hq

 a˜q z ; b˜q z

 Hq 

−iγ vq

 vq ; (2) iγ

where vq  −k  c · e−iq . Equation (2) can be solved analytically, thus allowing us to evaluate the wave packet evolution in the Fourier space. The wavepacket in the real space is then an inverse Fourier transP obtained by −iqm (similarly for b˜ z). form a˜q z  ∞ q m−∞ am ze The dispersion relation of the dimeric lattice is calculated from Eq. (2) by substituting the stationary form an ; bn T  exp−iEzA; BT [28]. We get: q E  q   k − c2  4kc cos2 q∕2 − γ 2 :

(3)

For γ < γ PT  k − c, the dispersion relation consists of two bands that are separated by a gap. In this parameter domain, all the eigenvalues are real, and the system is stable. The maximum gap size 2k − c occurs for γ  0. For larger values of γ, the gap becomes smaller until it disappears at γ PT . At γ  γ PT , the levels associated with q  π and their corresponding eigenvectors become degenerate resulting in an exceptional point (EP) singularity. For γ > γ PT , the spectrum becomes partially complex [28]. Below we assume that k > c. Using the dispersion relation Eq. (3) we evaluate the electric field amplitudes associated with the nth dimer Ψn  an z; bn zT at any distance z: 

an z bn z



Z ∞  π 1 X am 0  eiqn−m−iE  z dq: 2π m−∞ bm 0 −π

(4)

Next we extend the integer index m, defining the dimer number, to a continuous variable ξ, which can be interpreted as the transverse spatial coordinate variable of the incident (i.e., z  0) field. Similarly, we express the integer index n as a general continuous transverse spatialcoordinate x. This allows us to define the continuous smooth functions aξ; 0 and bξ; 0 with the property that ajξj → ∞; 0 → 0 and similarly bjξj → ∞; 0 → 0. Next we express Eq. (4) in the following integral form: 

ax; z bx; z

 

1 2π

Z

Z π ∞ −∞

−π



αξ iϕξ qx−ξ−E  z e dqdξ; βξ

(5)

where we have used the polar representation aξ; 0 ≡ αξ eiϕξ and bξ; 0 ≡ βξ eiϕξ . The subindex ξ means that α, β and ϕ are functions of ξ. It turns out that the caustics formation is independent of the amplitudes αξ ; βξ [see Eqs. (6)–(8)]. Thus we will assume that αξ  βξ  1.

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In order to enforce caustic beam dynamics, we impose a stationary phase (first- and second-order) condition on both integration variables q and ξ in Eq. (5): ∂ϕξ ∂Φ  0 → qξ  ∂ξ ∂ξ

(6a)

∂Φ ckz sin qξ 0→xξ− ∂q E  qξ 

(6b)

Along with the second-order stationary condition  3 qξ E   2 2 2 2 ; ckqξ0 ∂ Φ∂ Φ ∂ ∂Φ  →z 2 2 cos q  ck3  cos 2q  ∂ξ ∂q 2C ∂ξ ∂q 1 ξ ξ 7 

∂q

where C 1  c2  k2 − γ 2 , q0ξ ≡ ∂ξξ , and Φ  ϕξ  qx− ξ − E  z. Equations (6), (7) define the coordinates x and z of the ray trajectory associated with the ξth dimer. They allow us to design various types of caustics associated with Eqs. (1) of a beam propagating through the lattice of Fig. 1. Solving Eq. (6b) for qξ leads to: 0

s1 C η2 qξ  cos−1 @−η2  1 − 1  η4 A; ck

ξ−x η ≡ p : (8) ckz

Further integration of Eq. (8) with respect to ξ [see Eq. (6a)] results in a relationship between the lattice parameters, the ray trajectories, and the initial phases of the waveform ϕξ . This enables us to tailor the initial phases ϕξ in order to achieve a desired caustics dynamic. The condition that ϕξ must be real imposes a constraint to qξ allowing for the calculation of the maximum number of dimers ξ ≤ jξ0 j that participate in the caustic dynamics. We start with designing a focusing point at a predefined transverse coordinate position xf and at a paraxial distance zf . Below, we keep zf as a free parameter while, for the sake of presentation, we assert that xf  0, i.e., it coincides with the origin of the transverse coordinate. The requirement p for focusing at xf  0 (corresponding to ηf  ξ∕ ckzf ) simplifies Eq. (8), which can be integrated [see Eq. (6a)] in order to evaluate the initial phase ϕξ of the wavefronts. We get: ϕ ξ  

ξzf jE q j  ξqf  zf C 0 jξj  f

s1 C 1 η2f  η4f A; qfξ  cos−1 @−η2f  1 − ck

(9)

0

(10)

p ξ where C 0  c  k2 − γ 2 and ηf  p . The existence ckzf of a focal point for zf > 0 is achieved when the initial phases are ϕ− for ξ < 0 (ξ > 0). The number of dimers ξ jξ0 j that participate in the creation of the focal point can be calculated from Eq. (8). We get

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r q ξ0  −

zf

C 1 − C 0 γ 2PT − γ 2 p . 2

(11)

Equations (9)–(11) allow us to gain a better understanding of the role of the gain and loss parameter γ in the formation of focusing points. Let us consider, for example, the two scenarios of a case with γ  0 and a PT symmetric case with γ  γ PT . In the former case, we find from Eq. (11) that jξ0 j  czf dimer trajectories (and thus dimer waveguides) contribute for the construction of a focal point at a paraxial distance zf from the plane of preparation. In contrast, when γ → γ PT , we get jξ0 j  p ckzf , i.e., more waveguides are contributing to a focal point. In Fig. 2, we show the evolution of an initial beam that is designed to create an ideal (i.e., aberration-free) focal point for the two different values of γ. The phase engineered of the initial beam is dictated by the rules given by Eqs. (9)–(11) for each γ value. We find that increasing the gain/loss parameter γ leads to an increase of the number of waveguides ξ0 , which contribute to the creation of the focal point. At the same time, the total beam intensity at the focal point increases dramatically. This growth is a consequence of the EP dynamics and has been predicted theoretically in [28] and observed experimentally in [23]. This growth is manifested here as a giant intensity at a pre-engineered focal point. Next we investigate the management of curved caustic trajectories in PT -symmetric lattices. We start with Eqs. (6)–(8). For theoretical simplicity, the ensuing calculation will focus on designing caustics associated with an initial wave-front involving only dimers with index

Fig. 2. Design of an aberration-free focal point at zf  20 of an initial wavefront for a PT -symmetric lattice with two different γ-values. In (a) and (c), the intensity profile of the evolving engineered beam is presented. In (b) and (d), we show the associated ray trajectories. Each individual ray corresponds to a dimer. The initial wavefront in (a) and (c) has the form aξ; 0  eiϕξ and bξ; 0  eiϕξ for ξ0 ≤ ξ ≤ jξ0 j, where the initial phases ϕξ are given from Eq. (9). The coupling constants are k  5, and c  1. In (a) and (b), γ  0, and therefore we use jξ0 j ≈ 20. In (c) and (d), γ  3.999 ≈ γ PT and hence jξ0 j  44. In this case, the field intensity at the focal point is an order of magnitude larger than in (a).

ξ ≤ 0. We will assume a power law caustic trajectory x  αzδ . Furthermore, we relate the caustic propagation distance z, with the dimer index ξ using Eq. (6b) which can be rewritten compactly as ξ  x − z dx dz . One then uses these two relations to write the variables x and z as functions of the dimer index ξ, the exponent δ, and the scaling constant α. The resulting expressions are: x

ξ ; 1 − δ

 z

1∕δ ξ . α1 − δ

(12)

The above relations encode the information of the caustic trajectory into the ray trajectory coordinates for the ξth dimer. Substituting these expressions back into Eq. (8) enables us to calculate the initial phase distribution ϕξ producing dynamics following the caustic trajectory defined by Eq. (12). Furthermore using Eq. (8), we estimate the number of dimers ξ0 that are used to construct the propagating caustics. In Fig. 3, we present two examples of different caustic trajectories for two different δ values. Our results confirm that power-law caustic trajectories can be supported in PT -symmetric lattices. Obviously, this agreement is valid up to some paraxial propagation distance z and transverse coordinate variable x, which are given by Eq. (12) for ξ  ξ0 . We also studied the effect of the gain/loss parameter γ on a specific caustic trajectory. In Fig. 4, we demonstrate the beam dynamics originated by the same caustic trajectory for two different values of gain/loss parameter. We see that increase of γ, while also adjusting the initial phase accordingly, augments the amplitude of the caustic wave without altering its trajectory. In conclusion we have investigated the possibility of implementing caustics design in PT -symmetric lattices.

Fig. 3. Two representative examples (a) and (c) of designed caustic propagation juxtaposed with the associated caustic trajectory of the form x  αzδ . In both cases, we used k  3, c  1, and γ  1.95. The initial condition for the system is aξ; 0  bξ; 0  eiϕξ for ξ0 ≤ ξ ≤ 0 and zero elsewhere. In (a) δ  3, α  0.0003 and consequently ξ0 ≈ −40, while in (c) δ  5∕4, α  0.4 and consequently ξ0 ≈ −25. The associated ray trajectory for (a) is shown in (b). The associated ray trajectory for (c) is shown in (d). Green lines in (b) and (d) are the caustic trajectories.

May 1, 2015 / Vol. 40, No. 9 / OPTICS LETTERS

Fig. 4. Two examples of designed caustic propagation juxtaposed with the desired caustic trajectory of the form x  αzδ , where α  0.1 and δ  3∕2, for (a) γ  0.5 and (c) γ  1.8. In both cases we used k  3, c  1. The initial wavefront is aξ; 0  bξ; 0  eiϕξ for ξ0 ≤ ξ ≤ 0 and zero elsewhere. In (a) ξ0  −15 while in (c) ξ0  −36. (b) is the associated ray trajectory for (a), while (d) is the associated trajectory for (c). In (a),(b), zξ0  45 and xξ0  30 while in (c),(d), zξ0  80 and xξ0  72. Furthermore, the point of maximal intensity along the caustic in (c) occurs approximately at the same coordinates as the end of the caustic in (a). The green lines in (b),(d) are the desired caustic trajectories.

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Engineering wavefront caustics trajectories in PT-symmetric lattices.

We utilize caustic theory in PT-symmetric lattices to design focusing and curved beam dynamics. We show that the gain and loss parameter in these syst...
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