Anisotropic reflection oscillation in periodic multilayer structures of parity-time symmetry Xue-Feng Zhu,1,2,* Yu-Gui Peng,1,2 and De-Gang Zhao1,3 1
Department f Physics, Huazhong University of Science and Technology, Wuhan 430074, China 2 Innovation Institute, Huazhong University of Science and Technology, Wuhan 430074, China 3 [email protected] *[email protected]
Abstract: We show that periodic multilayer structures with parity-time (PT) symmetries imposed by a balanced arrangement of gain and loss can exhibit anisotropic reflection oscillation patterns as the number of unit-cells is increasing. At the minima of reflection oscillation patterns, the PT symmetric medium exhibits bidirectional transparency with the eigenvalues of the scattering matrix degenerated, where the PT symmetric set-up can still render directional responses due to the one-way field localization inside the system. With certain number of unit-cells, the PT symmetric crystal is unidirectionally invisible. More complicated reflection/transmission oscillations can be observed by segregating neighboring unit-cells with a uniform dielectric layer. Our results may pave the way towards a new class of functional optical devices with intriguing and unexpected directional responses. ©2014 Optical Society of America OCIS codes: (310.4165) Multilayer design; (290.5839) Scattering, invisibility.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009). C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of paritytime symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010). Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106(9), 093902 (2011). S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010). J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, and T. Kottos, “PT-symmetric electronics,” J. Phys. A: Math. Theor. 45(44), 444029 (2012). H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-symmetric Talbot effects,” Phys. Rev. Lett. 109(3), 033902 (2012). H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010). N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013). F. Nazari, N. Bender, H. Ramezani, M. K. Moravvej-Farshi, D. N. Christodoulides, and T. Kottos, “Optical isolation via PT-symmetric nonlinear Fano resonances,” Opt. Express 22(8), 9574–9584 (2014). B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014). L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in onedimensional PT-symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012). Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011). A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012). L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
#213157 - $15.00 USD (C) 2014 OSA
Received 30 May 2014; revised 7 Jul 2014; accepted 8 Jul 2014; published 22 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018401 | OPTICS EXPRESS 18401
15. L. Feng, X. Zhu, S. Yang, H. Zhu, P. Zhang, X. Yin, Y. Wang, and X. Zhang, “Demonstration of a large-scale optical exceptional point structure,” Opt. Express 22(2), 1760–1767 (2014). 16. X. Zhu, L. Feng, P. Zhang, X. Yin, and X. Zhang, “One-way invisible cloak using parity-time symmetric transformation optics,” Opt. Lett. 38(15), 2821–2824 (2013). 17. C. M. Bender and S. Böttcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998). 18. S. Longhi, “Invisibility in PT-symmetric complex crystals,” J. Phys. A: Math. Theor. 44(48), 485302 (2011). 19. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1997). 20. M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface Plasmon polaritons on planar metallic waveguides,” Opt. Express 12(17), 4072–4079 (2004). 21. C. Y. Qiu, Z. Y. Liu, J. Mei, and M. Z. Ke, “The layer multiple-scattering method for calculating transmission coefficiencts for 2D phononic crystals,” Solid State Commun. 134(11), 765–770 (2005). 22. D. G. Zhao, Y. T. Ye, S. J. Xu, X. F. Zhu, and L. Yi, “Broadband and wide-angle negative reflection at a phononic crystal boundary,” Appl. Phys. Lett. 104(4), 043503 (2014).
1. Introduction Wave phenomena under the framework of non-Hermitian Hamiltonians are often encountered in optical systems with gain and/or losses. In the past few years, a wide class of optical systems that respects parity-time (PT) symmetry was intensely studied with several exotic phenomena predicted and observed, such as loss induced transparency [1], power oscillations and nonreciprocity of light propagation [2], coherent perfect lasing and absorption [3–5], reconfigurable Talbot effect [6], on-chip optical isolation [7–10], and anisotropic transmission resonances [11], etc. Among the miscellaneous PT symmetric systems, the sinusoidal PT symmetric complex crystals with balanced gain/loss modulations show rather unusual scattering and transportation properties [12–16], viz. unidirectional invisibility, and the eigenvalues of Hamiltonian are completely real in the PT symmetric phase despite the nonHermiticity [17]. Intuitively, the periodic PT symmetric structure modulated in the form of δexp(ik·r) can be regarded as a complex grating providing a unidirectional wave vector k to the incident waves, where the diffraction mode can be selectively excited by satisfying the phase matching condition in one direction but not the other. Recently, Longhi [18] has demonstrated that unidirectional invisibility in sinusoidal PT symmetric complex crystals will break down in the case that the number of unit-cells surpasses a threshold by studying the exact analytical expressions of coupled mode theory, showing the importance of boundary effect in describing the Bragg scattering and coupling of counter-propagating waves in the sinusoidal PT symmetric system of a finite length. In this paper, we investigate the scattering and transportation properties of periodic multilayer structures with the rectangular PT symmetric modulation based on exact analytical expressions for transfer matrix method. The results show that periodic multilayer structure with rectangular PT symmetric modulation exhibits anisotropic reflection oscillation as crystal length or the number of unit-cells is increasing. In particular, at the lengths corresponding to the minima in the reflection/transmission oscillations, the complex photonic crystal is bidirectionally reflectionless, however with remarkably different field distributions inside the crystal when probed from opposite sides. At the minima, the system is operating in the PT symmetric phase, where all minima are de facto mapped to the central points of PT symmetric phase. Around the minima while still in the PT symmetric phase, the crystal is unidirectionally reflectionless due to the anisotropic reflection oscillations. In the PT broken phase, the transmission through the crystal is larger than unitary. With a dielectric layer inserted between the neighboring unit-cells, more complicated reflection/transmission oscillation patterns can be observed. The exploration of optical characteristics in periodic multilayer structures with the rectangular PT symmetric modulation shows rich physics and new functionalities, such as anisotropic reflection oscillations, where ingenious optical devices can be designed to react differently depending on the direction of wave propagation.
#213157 - $15.00 USD (C) 2014 OSA
Received 30 May 2014; revised 7 Jul 2014; accepted 8 Jul 2014; published 22 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018401 | OPTICS EXPRESS 18402
2. Schematic of the PT symmetric complex crystals In optics, PT symmetry requires the complex refractive index distributed in the form of n(z) = n*(-z) with respect to z = 0 [2]. In this paper, we consider an optical periodic structure with a PT symmetric refractive index distribution (see the schematic diagram in Fig. 1). For the loss regions, the refractive index distributions are n1(z) = n0 + Δn + Δni for mL≤z≤(m + 1/4)L and n2(z) = n0-Δn + Δni for (m + 1/4)L≤z≤(m + 1/2)L. For the gain regions, the refractive index distributions are n3(z) = n0-Δn-Δni for (m + 1/2)L≤z≤(m + 3/4)L and n4(z) = n0 + Δn-Δni for (m + 3/4)L≤z≤(m + 1)L. Here m = 1, 2, …, N, with N being the number of unit-cells in the PT symmetric structure, n0 the refractive index of the uniform background medium, and Δn the amplitude of both real and imaginary modulations. At the Bragg condition, L = 4Lj = λ/(2n0), with λ the wavelength of light in vacuum and j = 1, 2, 3, 4.
Fig. 1. Schematic diagram of PT symmetric complex crystals. Both real part (blue solid line) and imaginary part (red dashed line) of refractive index have rectangular modulations of equivalent amplitudes, where the modulations differ in phase by π/2.
3. Theory and formulation For describing the scattering of plane waves (wave number: k0 = 2πn0/λ) impinging on one unit-cell from the left, we can use a unimodular characteristic matrix as expressed into i − sin( k0 n j L j ) m11 m12 cos( k0 n j L j ) n = M j (Lj ) = j , m21 m22 −in sin( k n L ) cos( k0 n j L j ) 0 j j j
(1)
For the whole periodic medium with N unit-cells, we then have M(NL) = (M(L))N = [M11, M12; M21, M22]. From the theory of matrices, the Nth power of a unimodular matrix M(L) is [19] m12 uN −1 (a ) m u (a ) − uN − 2 (a ) [ M ( L)]N = 11 N −1 m21uN −1 ( a ) m22 uN −1 ( a ) − uN − 2 ( a )
(2)
where a = (m11 + m22)/2 and uN are the Chebyshev Polynomials of the second kind:
uN (a ) =
#213157 - $15.00 USD (C) 2014 OSA
sin(( N + 1) cos −1 a ) 1 − a2
.
(3)
Received 30 May 2014; revised 7 Jul 2014; accepted 8 Jul 2014; published 22 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018401 | OPTICS EXPRESS 18403
Fig. 2. Plots of a = (m11 + m22)/2 (blue solid line) and –cos(π/(11/9(n0/Δn)2)) (red dashed line) versus Δn/n0, where 0≤Δn/n0≤0.15.
As shown in Fig. 2, it is interesting to find out that a≈–cos(π/(11/9(n0/Δn)2)), for 0≤Δn/n0≤0.15. Therefore, under the condition that Δn/n0
Department f Physics, Huazhong University of Science and Technology, Wuhan 430074, China 2 Innovation Institute, Huazhong University of Science and Technology, Wuhan 430074, China 3 [email protected] *[email protected]
Abstract: We show that periodic multilayer structures with parity-time (PT) symmetries imposed by a balanced arrangement of gain and loss can exhibit anisotropic reflection oscillation patterns as the number of unit-cells is increasing. At the minima of reflection oscillation patterns, the PT symmetric medium exhibits bidirectional transparency with the eigenvalues of the scattering matrix degenerated, where the PT symmetric set-up can still render directional responses due to the one-way field localization inside the system. With certain number of unit-cells, the PT symmetric crystal is unidirectionally invisible. More complicated reflection/transmission oscillations can be observed by segregating neighboring unit-cells with a uniform dielectric layer. Our results may pave the way towards a new class of functional optical devices with intriguing and unexpected directional responses. ©2014 Optical Society of America OCIS codes: (310.4165) Multilayer design; (290.5839) Scattering, invisibility.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009). C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of paritytime symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010). Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106(9), 093902 (2011). S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010). J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, and T. Kottos, “PT-symmetric electronics,” J. Phys. A: Math. Theor. 45(44), 444029 (2012). H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-symmetric Talbot effects,” Phys. Rev. Lett. 109(3), 033902 (2012). H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010). N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013). F. Nazari, N. Bender, H. Ramezani, M. K. Moravvej-Farshi, D. N. Christodoulides, and T. Kottos, “Optical isolation via PT-symmetric nonlinear Fano resonances,” Opt. Express 22(8), 9574–9584 (2014). B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014). L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in onedimensional PT-symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012). Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011). A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012). L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
#213157 - $15.00 USD (C) 2014 OSA
Received 30 May 2014; revised 7 Jul 2014; accepted 8 Jul 2014; published 22 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018401 | OPTICS EXPRESS 18401
15. L. Feng, X. Zhu, S. Yang, H. Zhu, P. Zhang, X. Yin, Y. Wang, and X. Zhang, “Demonstration of a large-scale optical exceptional point structure,” Opt. Express 22(2), 1760–1767 (2014). 16. X. Zhu, L. Feng, P. Zhang, X. Yin, and X. Zhang, “One-way invisible cloak using parity-time symmetric transformation optics,” Opt. Lett. 38(15), 2821–2824 (2013). 17. C. M. Bender and S. Böttcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998). 18. S. Longhi, “Invisibility in PT-symmetric complex crystals,” J. Phys. A: Math. Theor. 44(48), 485302 (2011). 19. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1997). 20. M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface Plasmon polaritons on planar metallic waveguides,” Opt. Express 12(17), 4072–4079 (2004). 21. C. Y. Qiu, Z. Y. Liu, J. Mei, and M. Z. Ke, “The layer multiple-scattering method for calculating transmission coefficiencts for 2D phononic crystals,” Solid State Commun. 134(11), 765–770 (2005). 22. D. G. Zhao, Y. T. Ye, S. J. Xu, X. F. Zhu, and L. Yi, “Broadband and wide-angle negative reflection at a phononic crystal boundary,” Appl. Phys. Lett. 104(4), 043503 (2014).
1. Introduction Wave phenomena under the framework of non-Hermitian Hamiltonians are often encountered in optical systems with gain and/or losses. In the past few years, a wide class of optical systems that respects parity-time (PT) symmetry was intensely studied with several exotic phenomena predicted and observed, such as loss induced transparency [1], power oscillations and nonreciprocity of light propagation [2], coherent perfect lasing and absorption [3–5], reconfigurable Talbot effect [6], on-chip optical isolation [7–10], and anisotropic transmission resonances [11], etc. Among the miscellaneous PT symmetric systems, the sinusoidal PT symmetric complex crystals with balanced gain/loss modulations show rather unusual scattering and transportation properties [12–16], viz. unidirectional invisibility, and the eigenvalues of Hamiltonian are completely real in the PT symmetric phase despite the nonHermiticity [17]. Intuitively, the periodic PT symmetric structure modulated in the form of δexp(ik·r) can be regarded as a complex grating providing a unidirectional wave vector k to the incident waves, where the diffraction mode can be selectively excited by satisfying the phase matching condition in one direction but not the other. Recently, Longhi [18] has demonstrated that unidirectional invisibility in sinusoidal PT symmetric complex crystals will break down in the case that the number of unit-cells surpasses a threshold by studying the exact analytical expressions of coupled mode theory, showing the importance of boundary effect in describing the Bragg scattering and coupling of counter-propagating waves in the sinusoidal PT symmetric system of a finite length. In this paper, we investigate the scattering and transportation properties of periodic multilayer structures with the rectangular PT symmetric modulation based on exact analytical expressions for transfer matrix method. The results show that periodic multilayer structure with rectangular PT symmetric modulation exhibits anisotropic reflection oscillation as crystal length or the number of unit-cells is increasing. In particular, at the lengths corresponding to the minima in the reflection/transmission oscillations, the complex photonic crystal is bidirectionally reflectionless, however with remarkably different field distributions inside the crystal when probed from opposite sides. At the minima, the system is operating in the PT symmetric phase, where all minima are de facto mapped to the central points of PT symmetric phase. Around the minima while still in the PT symmetric phase, the crystal is unidirectionally reflectionless due to the anisotropic reflection oscillations. In the PT broken phase, the transmission through the crystal is larger than unitary. With a dielectric layer inserted between the neighboring unit-cells, more complicated reflection/transmission oscillation patterns can be observed. The exploration of optical characteristics in periodic multilayer structures with the rectangular PT symmetric modulation shows rich physics and new functionalities, such as anisotropic reflection oscillations, where ingenious optical devices can be designed to react differently depending on the direction of wave propagation.
#213157 - $15.00 USD (C) 2014 OSA
Received 30 May 2014; revised 7 Jul 2014; accepted 8 Jul 2014; published 22 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018401 | OPTICS EXPRESS 18402
2. Schematic of the PT symmetric complex crystals In optics, PT symmetry requires the complex refractive index distributed in the form of n(z) = n*(-z) with respect to z = 0 [2]. In this paper, we consider an optical periodic structure with a PT symmetric refractive index distribution (see the schematic diagram in Fig. 1). For the loss regions, the refractive index distributions are n1(z) = n0 + Δn + Δni for mL≤z≤(m + 1/4)L and n2(z) = n0-Δn + Δni for (m + 1/4)L≤z≤(m + 1/2)L. For the gain regions, the refractive index distributions are n3(z) = n0-Δn-Δni for (m + 1/2)L≤z≤(m + 3/4)L and n4(z) = n0 + Δn-Δni for (m + 3/4)L≤z≤(m + 1)L. Here m = 1, 2, …, N, with N being the number of unit-cells in the PT symmetric structure, n0 the refractive index of the uniform background medium, and Δn the amplitude of both real and imaginary modulations. At the Bragg condition, L = 4Lj = λ/(2n0), with λ the wavelength of light in vacuum and j = 1, 2, 3, 4.
Fig. 1. Schematic diagram of PT symmetric complex crystals. Both real part (blue solid line) and imaginary part (red dashed line) of refractive index have rectangular modulations of equivalent amplitudes, where the modulations differ in phase by π/2.
3. Theory and formulation For describing the scattering of plane waves (wave number: k0 = 2πn0/λ) impinging on one unit-cell from the left, we can use a unimodular characteristic matrix as expressed into i − sin( k0 n j L j ) m11 m12 cos( k0 n j L j ) n = M j (Lj ) = j , m21 m22 −in sin( k n L ) cos( k0 n j L j ) 0 j j j
(1)
For the whole periodic medium with N unit-cells, we then have M(NL) = (M(L))N = [M11, M12; M21, M22]. From the theory of matrices, the Nth power of a unimodular matrix M(L) is [19] m12 uN −1 (a ) m u (a ) − uN − 2 (a ) [ M ( L)]N = 11 N −1 m21uN −1 ( a ) m22 uN −1 ( a ) − uN − 2 ( a )
(2)
where a = (m11 + m22)/2 and uN are the Chebyshev Polynomials of the second kind:
uN (a ) =
#213157 - $15.00 USD (C) 2014 OSA
sin(( N + 1) cos −1 a ) 1 − a2
.
(3)
Received 30 May 2014; revised 7 Jul 2014; accepted 8 Jul 2014; published 22 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.018401 | OPTICS EXPRESS 18403
Fig. 2. Plots of a = (m11 + m22)/2 (blue solid line) and –cos(π/(11/9(n0/Δn)2)) (red dashed line) versus Δn/n0, where 0≤Δn/n0≤0.15.
As shown in Fig. 2, it is interesting to find out that a≈–cos(π/(11/9(n0/Δn)2)), for 0≤Δn/n0≤0.15. Therefore, under the condition that Δn/n0
