March 15, 2014 / Vol. 39, No. 6 / OPTICS LETTERS
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Bound states in the continuum in PT -symmetric optical lattices 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 ([email protected]) Received January 17, 2014; revised February 14, 2014; accepted February 17, 2014; posted February 18, 2014 (Doc. ID 204989); published March 14, 2014 This Letter shows that bound states in the continuum (BIC), which are normalizable modes with energy embedded in the continuous spectrum of scattered states, exist in certain optical waveguide lattices with PT -symmetric defects. There are two distinct types of BIC modes: BIC states that exist in the broken PT phase, and correspond to exponentially localized modes with either exponentially damped or amplified optical power; and BIC modes with sub-exponential spatial localization that also can exist in the unbroken PT phase. These two types of BIC modes at the PT symmetry breaking point behave differently: in the former case, spatial localization is lost and the defect coherently radiates outgoing waves with an optical power that linearly increases with the propagation distance; in the latter case, localization is maintained and the optical power increase is quadratic. © 2014 Optical Society of America OCIS codes: (230.7370) Waveguides; (000.1600) Classical and quantum physics. http://dx.doi.org/10.1364/OL.39.001697
Von Neumann and Wigner suggested [1] in 1929 that certain potentials could support spatially localized states with energy embedded into the continuum spectrum of scattered states, so-called bound states in the continuum (BIC). However, BIC states are generally fragile, require specially tailored potentials, and thus are difficult to observe in quantum systems. Photonic structures, which allow robust control of parameters, have turned out to be very promising for the observation of BIC states [2–5]. Photonic BIC state observations recently have been reported by a few groups using waveguide arrays [6–8] and photonic crystals [9]. Most previous studies on BIC states have been limited to considering Hermitian (i.e., nondissipative) systems. In recent years, there has been a growing interest in studying light propagation in so-called PT -symmetric optical structures, which comprise balanced regions of gain and loss. The notion of PT symmetry, originated within the context of quantum field theories [10], has been introduced in optics as a new paradigm to mold the flow of light [11–14], enabling unusual and previously unattainable light propagation features [12,14–19]. In the framework of PT optics, linear bound states sustained by defects in otherwise periodic lattices have been investigated in a few recent works [20,21]. Such states generally fall in a forbidden lattice band [20], like in Hermitian lattices. However, a recent work [21] shows, both theoretically and experimentally, that BIC states can arise in the broken PT phase. Such states, that we will refer to as type-I BIC, are exponentially localized in space and are exponentially damped or amplified along the propagation distance. As the PT symmetry breaking point is approached from above, type-I BIC modes cease to be localized; however, they can coherently radiate outgoing waves [21] as a signature of a spectral singularity, like in a lasing medium at threshold [16,22]. BIC states in the unbroken PT phase have not yet been predicted. This Letter shows that, in addition to the most common type-I BIC states found in the broken PT phase, a novel type of BIC mode, referred to as type-II BIC, can arise in specially tailored optical lattices. Unlike type-I BIC, 0146-9592/14/061697-04$15.00/0
type-II BIC exist above and below the PT symmetry breaking, show a lower than exponential spatial localization. They also behave differently than type-I BIC modes at the symmetry breaking point. I considered light propagation in an array of evanescently coupled optical waveguides, which is described by standard coupled mode equations: dcn κ n cn−1 κn1 cn1 V n cn ≡ ℋcn ; (1) dz where cn cn z is the amplitude of the guided mode at lattice site n and propagation distance z, κn is the coupling constant between waveguides n and n − 1, and V n is the complex optical potential at site n. The potential V n accounts for both propagation constant shift (the real part of V n ) and gain/loss (the imaginary part of V n ) in waveguide n. PT invariance requires κ −n κn1 and V −n V n . In this case, eigenvalues of the matrix system ℋ are either real-valued or pairs of complex-conjugate numbers. In fact, if c¯ n is an eigenstate of ℋ with eigenvalue E, ℋc¯ n E c¯ n , then c¯ −n is also an eigenstate of ℋ with eigenvalue E . The Hamiltonian ℋ is assumed to depend on a control parameter g, which measures the strength of non-Hermiticity (g 0 for the Hermitian case). While at g 0 the spectrum of ℋ is entirely real, as g is increased a threshold value g gth is generally found, above which pairs of complex conjugate energies emerge. Such a point is usually referred to as the PT symmetry breaking point. Here, I consider the case where a PT -symmetric defect is introduced in an otherwise homogeneous and Hermitian lattice (i.e., I assume that V n → 0 and κn → κ as n → ∞, where κ is the homogeneous real-valued coupling constant far from the defective region). Defective type-I BIC modes, such as those studied in [21], are common and can be found in simple lattice models. Less common are type-II BIC states, which exist below, at and above the symmetry breaking point. In this Letter we will consider two examples of waveguide lattices with PT -symmetric defects, which show very different scattering and amplifying properties as a result of the different nature of their BIC states. i
© 2014 Optical Society of America
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First example. Defective lattice with type-I BIC. As a first example, let us consider one of the simplest lattice model with a PT -symmetric defect, corresponding to a uniform lattice with two site potential defects [23,24], namely V n σδn;−1 σ δn;1 ;
κ n κ;
(2)
where σ Δ ig is the complex potential, accounting for propagation constant mismatch Δ and gain/loss g of waveguides at the defective sites n 1 [Fig. 1(a)]. For such a simple lattice model, bound states with exponential spatial localization, either inside or outside the lattice band −2κ; 2κ of Bloch modes, can be found in
(a)
κ
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O
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κ
O
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0 0
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(d)
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mode amplitude |cn |
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transmittance T
energy
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a closed form by standard methods. For a fixed and relatively small value of the normalized index detuning Δ∕κ, the spectrum of ℋ for increasing values of normalized gain g∕κ shows the typical behavior depicted in Fig. 1(b). For low values of g∕κ, there is a single bound state with a real energy outside the lattice band (i.e., an ordinary bound state outside the continuum, BOC). As g∕κ is increased, the spectrum remains real-valued but the BOC disappears until the PT -symmetry breaking threshold is reached at g∕κ gth ∕κ. Above the symmetry breaking threshold, a couple of bound states with complex conjugate energies, embedded in the lattice band, emanate at some real energy E 0 2κ cos q0 , with a localization length that diverges as g approaches gth from above [Fig. 1(c)]. These are precisely type-I BIC modes, previously observed in [21] in a defective mesh lattice. For an initial excitation of the lattice at site n 0, for g > gth light trapping is observed near n 0 owing to the existence of the bound states, and the optical power exponential increases with the propagation distance as a result of the broken PT phase (complex energies). Interestingly, at the PT symmetry breaking point light is not trapped near n 0 anymore because of the disappearance of type-I BIC modes; however, undamped coherent emission of outgoing waves from the defective region is observed [Fig. 1(e)]. Note that the local intensity jcn zj2 remains bounded while the optical beam power Pz P 2 n jcn zj increases linearly with z as a result of outgoing wave emission [Fig. 1(f)]. This type of result, which was observed in [21], is typical behavior for PT symmetry breaking associated with the appearance of spectral singularity [22,25]. This can be seen by looking at the behavior of the spectral transmission and reflection coefficients of the lattice across the defective region. The spectral transmission coefficient t for a Bloch wave with wave number q can be readily calculated by standard methods and reads: tq
20 10 0 0
10
20
30
40
propagation distance κz
Fig. 1. (a) Schematic of a homogeneous lattice with two site impurities at waveguides n 1, [Eq. (2) in the text]. (b) Behavior of the energy spectrum (eigenvalues of the matrix ℋ normalized to the coupling constant κ) for Δ∕κ 0.3 and for increasing values of g∕κ. Shaded region depicts the energy band of scattered states (real eigenvalues). Solid and dashed curves are the real and imaginary parts, respectively, of bound state energies. For small values of g∕κ (below < ∼ 0.457), there is one BOC state with a real-valued energy that falls just outside the upper band edge. PT symmetry breaking occurs at gth ∕κ ≃ 0.752, above which two type-I BIC modes with complex conjugate energies emerge. (c) Amplitude distribution (j¯cn j) of the amplified type-I BIC mode for g∕gth 1.08 (curve 1), g∕gth 1.2 (curve 2), and g∕gth 1.5 (curve 3). (d) Lattice spectral transmittance for g∕gth 0.5 (curve 1), g∕gth 0.8 (curve 2) and g∕gth 0.9 (curve 3). Resonance peak occurs at q q0 ≃ 0.161π. (e) Evolution of beam intensity and (f) of normalized total beam power Pz∕P0 for initial excitation of waveguide n 0 at the symmetry breaking transition.
i sin q exp2iq : Δ i sin q − jσj2 cos q exp2iq
(3)
Figure 1(d) shows a typical behavior of the spectral transmittance Tq jtqj2 for a few values of g∕gth < 1. Note that a strong resonance peak near q q0 is found, which narrows and diverges as the symmetry breaking point is approached. This is a clear signature of a spectral singularity [25] and is a typical scenario of a lasing instability that arises in an active optical cavity [22]. Second example. Defective lattice with type-II BIC. Type-II BIC modes, which exist in the unbroken PT phase, have not been discussed so far. Similar to Hermitian lattices [5,7], such states are expected to show a lower than exponential localization and to arise in specially engineered lattices (i.e., they are much less common than type-I BIC). An example of a PT symmetric lattice that sustains type-II BIC is shown below. I assume a lattice with vanishing potential, where V n 0, and with inhomogeneous coupling constants given by p κn n 1∕n − 1 n even; n ≠ 0 κ0 −ig p ; (4) κ n − 2∕n n odd; n ≠ 1 κ1 ig
March 15, 2014 / Vol. 39, No. 6 / OPTICS LETTERS
coupling constant
κn /κ
where g > 0 is a real-valued parameter. Figure 2(a) shows the behavior of κ n . Since κn ∕κ → 1 as n → ∞, the lattice is asymptotically homogeneous. It also satisfies the PT symmetry requirement κ −n κn1 . The nonHermitian nature of the lattice arises, in this case, by the imaginary value of the coupling constants κ0 and κ1 . In coupled optical waveguides, an effective imaginary coupling constant can be realized by suitable longitudinal modulation of index and gain/loss terms, as discussed in [26]. The inhomogeneous coupling constants κ n for n ≠ 0; 1 can be readily obtained by judicious waveguide spacing, as demonstrated in [7]. Numerical analysis of the eigenvalues of the matrix ℋ shows that the energy spectrum is real-valued for g ≤ gth κ; that is, PT symmetry
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-20
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15
(f)
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5
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Fig. 2. (a) Behavior of the hopping rates κ n (for n ≠ 0; 1) of the inhomogeneous lattice defined by Eq. (4). (b) Behavior of the energy spectrum of ℋ for increasing values of g∕κ. The shaded region depicts the energy band of scattered states (real eigenvalues). The solid and dashed curves are the real and imaginary parts, respectively, of bound state energies. PT symmetry breaking occurs at gth ∕κ 1, above which two type-I BIC modes with complex conjugate energies emerge. The type-II BIC mode with energy E 0 exists for any value of g. (c) Amplitude distribution (j¯cn j) of type-II BIC mode at g gth . (d) Lattice spectral transmittance Tq for g∕gth 0.5 (curve 1), g∕gth 0.9 (curve 2) and g∕gth 0.99 (curve 3). (e) Evolution of beam intensity and (f) Evolution of square root of normalized p total beam power (i.e., Pz∕P0) for initial excitation of waveguide n 0 at the symmetry breaking transition.
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breaking occurs at gth κ. In the unbroken PT phase (g < gth ), the energy spectrum comprises, in addition to the continuous spectrum −2κ; 2κ of scattered states of the asymptotic homogeneous lattice, one BIC mode with algebraic localization at the energy E 0 [i.e., a type-II BIC, as shown in Fig. 2(b)]. Using Eqs. (1) and (4), it is possible to check that the BIC mode is given by (apart from a normalization factor):
c¯ n
8 > > < > > :
0
n odd
κ∕g
n0
n1 n pi jnj n2 −1
n even; n ≠ 0
:
(5)
In the broken PT phase (g > gth ), two additional type-I BIC modes, with complex conjugate energies and a vanishing real part that emanate from E 0, are found [Fig. 2(b)]. Such additional BIC states have the same properties as those found in the previous example, and are thus not discussed here anymore. Note that type-II BIC mode [Eq. (5)] does exist below, at and above the symmetry breaking point, and that by varying the g parameter the amplitude of the n 0 site solely is modified. Figure 2(c) shows a typical behavior of the amplitude distribution j¯cn j of type-II BIC. At g gth , it can be shown that the lattice (4) belongs to a rather general class of non-Hermitian lattices that can be synthesized from the homogeneous lattice κ n κ, V n 0 by a double discrete Darboux (supersymmetric) transformation; mathematical details are rather cumbersome and will be given elsewhere [27]. Here we point out two important results of the analysis, which show that the scattering and amplifying properties of lattice (4) at the symmetry breaking point are distinct from those of lattice (2): (i) The supersymmetric transformation ensures that the spectral transmission across the defective region, at any energy E ≠ 0 (i.e., Bloch wave number q ≠ π∕2), approaches unity as g → g−th [tq 1 for q ≠ π∕2]; that is, the lattice becomes reflectionless as the symmetry breaking point is approached. Figure 2(d) clearly shows this with the numerically computed spectral transmittance Tq for a few values of g∕gth < 1. Note that Tq shows a dip (rather than a resonance, as in the previous example) near q π∕2. As the symmetry breaking point is attained, the dip is narrower and one obtains Tq 1 for q ≠ π∕2. (ii) At the symmetry breaking point, the energy E 0 is an exceptional point in the continuous spectrum [28]. Exceptional points in the continuous spectrum (not to be confused with more common exceptional points in finite-dimensional, non-Hermitian systems and with spectral singularities in non-Hermitian systems with purely continuous spectrum) and their mathematical properties have been discussed for the continuous Schrödinger equation by Andrianov and Sokolov in [28]. The circumstance that E 0 is an exceptional point in the continuum means that there exists an associated function f n to type-II BIC mode c¯ n , which is limited at n → ∞ and given by f n −i∕2 sinnπ∕2, satisfying the equation ℋf n c¯ n [28]. Hence, at the symmetry breaking
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point Eq. (1) is satisfied not only by the stationary solution cn z c¯ n , but also by any (nonstationary) solution cn z 1 − iεz¯cn εf n
(6)
for any arbitrary value of ε. Equation (6) has an important physical implication: it shows that at the symmetry breaking point a small perturbation shaped like the associated function f n leads to a secular growth with propagation distance of the type-II BIC state c¯ n , with an optical power Pz that increases quadratically (rather than linearly) with z. This is clearly shown in Figs. 2(e) and 2(f), which depict the evolution along the array of the beam intensity and total beam power for excitation of waveguide n 0 at the input plane. In conclusion, this Letter has shown that different types of BIC states can exist in optical lattices with PT symmetric defects, and that their nature can deeply modify the scattering and amplifying properties of the lattice at the symmetry breaking point. In particular, we have introduced a defective waveguide lattice, which close to the symmetry breaking threshold, is reflectionless and shows a rather uncommon quadratic amplification law for the optical power in the guided (BIC) mode, owing to the appearance of an exceptional point in the continuum. The author acknowledges hospitality at the IFISC (CSIC-UIB), Palma de Mallorca. References 1. J. von Neumann and E. Wigner, Z. Phys. 30, 465 (1929). 2. E. N. Bulgakov and A. F. Sadreev, Phys. Rev. B 78, 075105 (2008). 3. D. C. Marinica and A. G. Borisov, Phys. Rev. Lett. 100, 183902 (2008). 4. N. Prodanovic, V. Milanovic, and J. Radovanovic, J. Phys. A 42, 415304 (2009). 5. M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, Phys. Rev. Lett. 108, 070401 (2012). 6. Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, Phys. Rev. Lett. 107, 183901 (2011).
7. G. Corrielli, G. Della Valle, A. Crespi, R. Osellame, and S. Longhi, Phys. Rev. Lett. 111, 220403 (2013). 8. S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, Phys. Rev. Lett. 111, 240403 (2013). 9. C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, Nature 499, 188 (2013). 10. C. M. Bender, Rep. Prog. Phys. 70, 947 (2007). 11. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632 (2007). 12. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008). 13. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010). 14. A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Nature 488, 167 (2012). 15. S. Longhi, Phys. Rev. Lett. 103, 123601 (2009). 16. S. Longhi, Phys. Rev. A 82, 031801 (2010). 17. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011). 18. L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, Nat. Mater. 12, 108 (2012). 19. T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, Nat. Commun. 4, 2533 (2013). 20. K. Zhou, Z. Guo, J. Wang, and S. Liu, Opt. Lett. 35, 2928 (2010). 21. A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Phys. Rev. Lett. 110, 223902 (2013). 22. S. Longhi, Phys. Rev. Lett. 105, 013903 (2010). 23. O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 103, 030402 (2009). 24. Y. N. Joglekar, D. Scott, and M. Babbey, Phys. Rev. A 82, 030103(R) (2010). 25. A. Mostafazadeh, Phys. Rev. Lett. 102, 220402 (2009). 26. S. Longhi, Phys. Rev. A 82, 032111 (2010). 27. S. Longhi and G. Della Valle, “Optical lattices with exceptional points in the continuum,” Phys. Rev. A, submitted for publication. 28. A. A. Andrianov and A. V. Sokolov, SIGMA 7, 111 (2011).
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Bound states in the continuum in PT -symmetric optical lattices 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 ([email protected]) Received January 17, 2014; revised February 14, 2014; accepted February 17, 2014; posted February 18, 2014 (Doc. ID 204989); published March 14, 2014 This Letter shows that bound states in the continuum (BIC), which are normalizable modes with energy embedded in the continuous spectrum of scattered states, exist in certain optical waveguide lattices with PT -symmetric defects. There are two distinct types of BIC modes: BIC states that exist in the broken PT phase, and correspond to exponentially localized modes with either exponentially damped or amplified optical power; and BIC modes with sub-exponential spatial localization that also can exist in the unbroken PT phase. These two types of BIC modes at the PT symmetry breaking point behave differently: in the former case, spatial localization is lost and the defect coherently radiates outgoing waves with an optical power that linearly increases with the propagation distance; in the latter case, localization is maintained and the optical power increase is quadratic. © 2014 Optical Society of America OCIS codes: (230.7370) Waveguides; (000.1600) Classical and quantum physics. http://dx.doi.org/10.1364/OL.39.001697
Von Neumann and Wigner suggested [1] in 1929 that certain potentials could support spatially localized states with energy embedded into the continuum spectrum of scattered states, so-called bound states in the continuum (BIC). However, BIC states are generally fragile, require specially tailored potentials, and thus are difficult to observe in quantum systems. Photonic structures, which allow robust control of parameters, have turned out to be very promising for the observation of BIC states [2–5]. Photonic BIC state observations recently have been reported by a few groups using waveguide arrays [6–8] and photonic crystals [9]. Most previous studies on BIC states have been limited to considering Hermitian (i.e., nondissipative) systems. In recent years, there has been a growing interest in studying light propagation in so-called PT -symmetric optical structures, which comprise balanced regions of gain and loss. The notion of PT symmetry, originated within the context of quantum field theories [10], has been introduced in optics as a new paradigm to mold the flow of light [11–14], enabling unusual and previously unattainable light propagation features [12,14–19]. In the framework of PT optics, linear bound states sustained by defects in otherwise periodic lattices have been investigated in a few recent works [20,21]. Such states generally fall in a forbidden lattice band [20], like in Hermitian lattices. However, a recent work [21] shows, both theoretically and experimentally, that BIC states can arise in the broken PT phase. Such states, that we will refer to as type-I BIC, are exponentially localized in space and are exponentially damped or amplified along the propagation distance. As the PT symmetry breaking point is approached from above, type-I BIC modes cease to be localized; however, they can coherently radiate outgoing waves [21] as a signature of a spectral singularity, like in a lasing medium at threshold [16,22]. BIC states in the unbroken PT phase have not yet been predicted. This Letter shows that, in addition to the most common type-I BIC states found in the broken PT phase, a novel type of BIC mode, referred to as type-II BIC, can arise in specially tailored optical lattices. Unlike type-I BIC, 0146-9592/14/061697-04$15.00/0
type-II BIC exist above and below the PT symmetry breaking, show a lower than exponential spatial localization. They also behave differently than type-I BIC modes at the symmetry breaking point. I considered light propagation in an array of evanescently coupled optical waveguides, which is described by standard coupled mode equations: dcn κ n cn−1 κn1 cn1 V n cn ≡ ℋcn ; (1) dz where cn cn z is the amplitude of the guided mode at lattice site n and propagation distance z, κn is the coupling constant between waveguides n and n − 1, and V n is the complex optical potential at site n. The potential V n accounts for both propagation constant shift (the real part of V n ) and gain/loss (the imaginary part of V n ) in waveguide n. PT invariance requires κ −n κn1 and V −n V n . In this case, eigenvalues of the matrix system ℋ are either real-valued or pairs of complex-conjugate numbers. In fact, if c¯ n is an eigenstate of ℋ with eigenvalue E, ℋc¯ n E c¯ n , then c¯ −n is also an eigenstate of ℋ with eigenvalue E . The Hamiltonian ℋ is assumed to depend on a control parameter g, which measures the strength of non-Hermiticity (g 0 for the Hermitian case). While at g 0 the spectrum of ℋ is entirely real, as g is increased a threshold value g gth is generally found, above which pairs of complex conjugate energies emerge. Such a point is usually referred to as the PT symmetry breaking point. Here, I consider the case where a PT -symmetric defect is introduced in an otherwise homogeneous and Hermitian lattice (i.e., I assume that V n → 0 and κn → κ as n → ∞, where κ is the homogeneous real-valued coupling constant far from the defective region). Defective type-I BIC modes, such as those studied in [21], are common and can be found in simple lattice models. Less common are type-II BIC states, which exist below, at and above the symmetry breaking point. In this Letter we will consider two examples of waveguide lattices with PT -symmetric defects, which show very different scattering and amplifying properties as a result of the different nature of their BIC states. i
© 2014 Optical Society of America
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First example. Defective lattice with type-I BIC. As a first example, let us consider one of the simplest lattice model with a PT -symmetric defect, corresponding to a uniform lattice with two site potential defects [23,24], namely V n σδn;−1 σ δn;1 ;
κ n κ;
(2)
where σ Δ ig is the complex potential, accounting for propagation constant mismatch Δ and gain/loss g of waveguides at the defective sites n 1 [Fig. 1(a)]. For such a simple lattice model, bound states with exponential spatial localization, either inside or outside the lattice band −2κ; 2κ of Bloch modes, can be found in
(a)
κ
O
O
κ
O
κ
κ
O
n=1
n=0
n=-3 n=-2 n=-1
O
κ
O
κ
O
n=3
n=2
(b) BOC type-I BIC
1 0 -1
unbroken PT phase
-2
1
0.1
0.3
0.8 0.6 0.4 0.2
40
0.4
0.5
g/κ
60
(c)
2
1
0 -200
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0.2
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-100
3 0
100
20
2 0.2 40
20 0
0
1
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(f)
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lattice site n
0.9
wave number q/π
0.6
0 -100
0.8
3
lattice site n
(e)
0.7
40
0 0
200
0.6
(d)
beam power
mode amplitude |cn |
0
transmittance T
energy
E/κ
2
100
30
a closed form by standard methods. For a fixed and relatively small value of the normalized index detuning Δ∕κ, the spectrum of ℋ for increasing values of normalized gain g∕κ shows the typical behavior depicted in Fig. 1(b). For low values of g∕κ, there is a single bound state with a real energy outside the lattice band (i.e., an ordinary bound state outside the continuum, BOC). As g∕κ is increased, the spectrum remains real-valued but the BOC disappears until the PT -symmetry breaking threshold is reached at g∕κ gth ∕κ. Above the symmetry breaking threshold, a couple of bound states with complex conjugate energies, embedded in the lattice band, emanate at some real energy E 0 2κ cos q0 , with a localization length that diverges as g approaches gth from above [Fig. 1(c)]. These are precisely type-I BIC modes, previously observed in [21] in a defective mesh lattice. For an initial excitation of the lattice at site n 0, for g > gth light trapping is observed near n 0 owing to the existence of the bound states, and the optical power exponential increases with the propagation distance as a result of the broken PT phase (complex energies). Interestingly, at the PT symmetry breaking point light is not trapped near n 0 anymore because of the disappearance of type-I BIC modes; however, undamped coherent emission of outgoing waves from the defective region is observed [Fig. 1(e)]. Note that the local intensity jcn zj2 remains bounded while the optical beam power Pz P 2 n jcn zj increases linearly with z as a result of outgoing wave emission [Fig. 1(f)]. This type of result, which was observed in [21], is typical behavior for PT symmetry breaking associated with the appearance of spectral singularity [22,25]. This can be seen by looking at the behavior of the spectral transmission and reflection coefficients of the lattice across the defective region. The spectral transmission coefficient t for a Bloch wave with wave number q can be readily calculated by standard methods and reads: tq
20 10 0 0
10
20
30
40
propagation distance κz
Fig. 1. (a) Schematic of a homogeneous lattice with two site impurities at waveguides n 1, [Eq. (2) in the text]. (b) Behavior of the energy spectrum (eigenvalues of the matrix ℋ normalized to the coupling constant κ) for Δ∕κ 0.3 and for increasing values of g∕κ. Shaded region depicts the energy band of scattered states (real eigenvalues). Solid and dashed curves are the real and imaginary parts, respectively, of bound state energies. For small values of g∕κ (below < ∼ 0.457), there is one BOC state with a real-valued energy that falls just outside the upper band edge. PT symmetry breaking occurs at gth ∕κ ≃ 0.752, above which two type-I BIC modes with complex conjugate energies emerge. (c) Amplitude distribution (j¯cn j) of the amplified type-I BIC mode for g∕gth 1.08 (curve 1), g∕gth 1.2 (curve 2), and g∕gth 1.5 (curve 3). (d) Lattice spectral transmittance for g∕gth 0.5 (curve 1), g∕gth 0.8 (curve 2) and g∕gth 0.9 (curve 3). Resonance peak occurs at q q0 ≃ 0.161π. (e) Evolution of beam intensity and (f) of normalized total beam power Pz∕P0 for initial excitation of waveguide n 0 at the symmetry breaking transition.
i sin q exp2iq : Δ i sin q − jσj2 cos q exp2iq
(3)
Figure 1(d) shows a typical behavior of the spectral transmittance Tq jtqj2 for a few values of g∕gth < 1. Note that a strong resonance peak near q q0 is found, which narrows and diverges as the symmetry breaking point is approached. This is a clear signature of a spectral singularity [25] and is a typical scenario of a lasing instability that arises in an active optical cavity [22]. Second example. Defective lattice with type-II BIC. Type-II BIC modes, which exist in the unbroken PT phase, have not been discussed so far. Similar to Hermitian lattices [5,7], such states are expected to show a lower than exponential localization and to arise in specially engineered lattices (i.e., they are much less common than type-I BIC). An example of a PT symmetric lattice that sustains type-II BIC is shown below. I assume a lattice with vanishing potential, where V n 0, and with inhomogeneous coupling constants given by p κn n 1∕n − 1 n even; n ≠ 0 κ0 −ig p ; (4) κ n − 2∕n n odd; n ≠ 1 κ1 ig
March 15, 2014 / Vol. 39, No. 6 / OPTICS LETTERS
coupling constant
κn /κ
where g > 0 is a real-valued parameter. Figure 2(a) shows the behavior of κ n . Since κn ∕κ → 1 as n → ∞, the lattice is asymptotically homogeneous. It also satisfies the PT symmetry requirement κ −n κn1 . The nonHermitian nature of the lattice arises, in this case, by the imaginary value of the coupling constants κ0 and κ1 . In coupled optical waveguides, an effective imaginary coupling constant can be realized by suitable longitudinal modulation of index and gain/loss terms, as discussed in [26]. The inhomogeneous coupling constants κ n for n ≠ 0; 1 can be readily obtained by judicious waveguide spacing, as demonstrated in [7]. Numerical analysis of the eigenvalues of the matrix ℋ shows that the energy spectrum is real-valued for g ≤ gth κ; that is, PT symmetry
2
(a)
1.5 1 0.5 0 -40
-30
3
E/κ
-10
0
10
20
unbroken PT phase type-II BIC
0
type-II BIC
-1
type-I BIC
-2
(c)
0.8 0.6 0.4 0.2 0 -20
80
0
lattice site n
0.4
0
lattice site n
1
0 0
20
0
-50
2
0.2
1
0
3
0.6
2
40
(d)
1.5
0.8
x 104
(e)
1
g/κ 1
transmittance T
mode amplitude |cn |
0.5
50
0.2
0.4
0.6
0.8
1
20
40
60
80
wave number q/π
beam power (square root)
0
1
propagation distance κz
40
broken PT phase type-I BIC
1
-3
30
lattice site n
2
energy
-20
(b)
15
(f)
10
5
0 0
propagation distance κz
Fig. 2. (a) Behavior of the hopping rates κ n (for n ≠ 0; 1) of the inhomogeneous lattice defined by Eq. (4). (b) Behavior of the energy spectrum of ℋ for increasing values of g∕κ. The shaded region depicts the energy band of scattered states (real eigenvalues). The solid and dashed curves are the real and imaginary parts, respectively, of bound state energies. PT symmetry breaking occurs at gth ∕κ 1, above which two type-I BIC modes with complex conjugate energies emerge. The type-II BIC mode with energy E 0 exists for any value of g. (c) Amplitude distribution (j¯cn j) of type-II BIC mode at g gth . (d) Lattice spectral transmittance Tq for g∕gth 0.5 (curve 1), g∕gth 0.9 (curve 2) and g∕gth 0.99 (curve 3). (e) Evolution of beam intensity and (f) Evolution of square root of normalized p total beam power (i.e., Pz∕P0) for initial excitation of waveguide n 0 at the symmetry breaking transition.
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breaking occurs at gth κ. In the unbroken PT phase (g < gth ), the energy spectrum comprises, in addition to the continuous spectrum −2κ; 2κ of scattered states of the asymptotic homogeneous lattice, one BIC mode with algebraic localization at the energy E 0 [i.e., a type-II BIC, as shown in Fig. 2(b)]. Using Eqs. (1) and (4), it is possible to check that the BIC mode is given by (apart from a normalization factor):
c¯ n
8 > > < > > :
0
n odd
κ∕g
n0
n1 n pi jnj n2 −1
n even; n ≠ 0
:
(5)
In the broken PT phase (g > gth ), two additional type-I BIC modes, with complex conjugate energies and a vanishing real part that emanate from E 0, are found [Fig. 2(b)]. Such additional BIC states have the same properties as those found in the previous example, and are thus not discussed here anymore. Note that type-II BIC mode [Eq. (5)] does exist below, at and above the symmetry breaking point, and that by varying the g parameter the amplitude of the n 0 site solely is modified. Figure 2(c) shows a typical behavior of the amplitude distribution j¯cn j of type-II BIC. At g gth , it can be shown that the lattice (4) belongs to a rather general class of non-Hermitian lattices that can be synthesized from the homogeneous lattice κ n κ, V n 0 by a double discrete Darboux (supersymmetric) transformation; mathematical details are rather cumbersome and will be given elsewhere [27]. Here we point out two important results of the analysis, which show that the scattering and amplifying properties of lattice (4) at the symmetry breaking point are distinct from those of lattice (2): (i) The supersymmetric transformation ensures that the spectral transmission across the defective region, at any energy E ≠ 0 (i.e., Bloch wave number q ≠ π∕2), approaches unity as g → g−th [tq 1 for q ≠ π∕2]; that is, the lattice becomes reflectionless as the symmetry breaking point is approached. Figure 2(d) clearly shows this with the numerically computed spectral transmittance Tq for a few values of g∕gth < 1. Note that Tq shows a dip (rather than a resonance, as in the previous example) near q π∕2. As the symmetry breaking point is attained, the dip is narrower and one obtains Tq 1 for q ≠ π∕2. (ii) At the symmetry breaking point, the energy E 0 is an exceptional point in the continuous spectrum [28]. Exceptional points in the continuous spectrum (not to be confused with more common exceptional points in finite-dimensional, non-Hermitian systems and with spectral singularities in non-Hermitian systems with purely continuous spectrum) and their mathematical properties have been discussed for the continuous Schrödinger equation by Andrianov and Sokolov in [28]. The circumstance that E 0 is an exceptional point in the continuum means that there exists an associated function f n to type-II BIC mode c¯ n , which is limited at n → ∞ and given by f n −i∕2 sinnπ∕2, satisfying the equation ℋf n c¯ n [28]. Hence, at the symmetry breaking
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OPTICS LETTERS / Vol. 39, No. 6 / March 15, 2014
point Eq. (1) is satisfied not only by the stationary solution cn z c¯ n , but also by any (nonstationary) solution cn z 1 − iεz¯cn εf n
(6)
for any arbitrary value of ε. Equation (6) has an important physical implication: it shows that at the symmetry breaking point a small perturbation shaped like the associated function f n leads to a secular growth with propagation distance of the type-II BIC state c¯ n , with an optical power Pz that increases quadratically (rather than linearly) with z. This is clearly shown in Figs. 2(e) and 2(f), which depict the evolution along the array of the beam intensity and total beam power for excitation of waveguide n 0 at the input plane. In conclusion, this Letter has shown that different types of BIC states can exist in optical lattices with PT symmetric defects, and that their nature can deeply modify the scattering and amplifying properties of the lattice at the symmetry breaking point. In particular, we have introduced a defective waveguide lattice, which close to the symmetry breaking threshold, is reflectionless and shows a rather uncommon quadratic amplification law for the optical power in the guided (BIC) mode, owing to the appearance of an exceptional point in the continuum. The author acknowledges hospitality at the IFISC (CSIC-UIB), Palma de Mallorca. References 1. J. von Neumann and E. Wigner, Z. Phys. 30, 465 (1929). 2. E. N. Bulgakov and A. F. Sadreev, Phys. Rev. B 78, 075105 (2008). 3. D. C. Marinica and A. G. Borisov, Phys. Rev. Lett. 100, 183902 (2008). 4. N. Prodanovic, V. Milanovic, and J. Radovanovic, J. Phys. A 42, 415304 (2009). 5. M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, Phys. Rev. Lett. 108, 070401 (2012). 6. Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, Phys. Rev. Lett. 107, 183901 (2011).
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