Experimental demonstration of a unidirectional ... - Semantic Scholar
LETTERS PUBLISHED ONLINE: 25 NOVEMBER 2012 | DOI: 10.1038/NMAT3495
Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies Liang Feng1 *† , Ye-Long Xu2† , William S. Fegadolli1,3,4† , Ming-Hui Lu2 *, José E. B. Oliveira3 , Vilson R. Almeida3,4 , Yan-Feng Chen2 and Axel Scherer1 Invisibility by metamaterials is of great interest, where optical properties are manipulated in the real permittivity– permeability plane1,2 . However, the most effective approach to achieving invisibility in various military applications is to absorb the electromagnetic waves emitted from radar to minimize the corresponding reflection and scattering, such that no signal gets bounced back. Here, we show the experimental realization of chip-scale unidirectional reflectionless optical metamaterials near the spontaneous parity-time symmetry phase transition point where reflection from one side is significantly suppressed. This is enabled by engineering the corresponding optical properties of the designed paritytime metamaterial in the complex dielectric permittivity plane. Numerical simulations and experimental verification consistently exhibit asymmetric reflection with high contrast ratios around a wavelength of of 1,550 nm. The demonstrated unidirectional phenomenon at the corresponding parity-time exceptional point on-a-chip confirms the feasibility of creating complicated on-chip parity-time metamaterials and optical devices based on their properties. Parity-time-symmetric material, also called synthetic matter, is a new class of metamaterials in which parity and time reversal symmetries are combined in such a way that non-Hermitian Hamiltonians may still possess real energy spectra3–5 , creating physical properties of quantum mechanics and quantum field theories in classical systems6 . Owing to the equivalence between the Schrödinger equation in quantum mechanics and the wave equation in optics, the combined parity-time symmetry has been intensively studied in classical optical systems with non-Hermitian optical potentials. By exploiting optical modulation of the refractive index in the complex dielectric permittivity plane and engineering both optical absorption and amplification, parity-time metamaterials can lead to a series of intriguing optical phenomena and devices, such as dynamic power oscillations of light propagation7–13 and coherent perfect absorber-lasers14–16 . Another important aspect of such parity-time optical materials is that the evolution of parity-time symmetry becomes measurable through the quantum-optical analogue17–19 . The threshold of parity-time symmetry breaking was clearly visualized as the
corresponding energy spectra change dramatically from real to complex after this phase transition point. This threshold is therefore called an exceptional point or spontaneous parity-time symmetry breaking point, where amplitudes of the real and imaginary parts of the modulated refractive index are identical. An interesting phenomenon named unidirectional invisibility was theoretically proposed at this exceptional point in parity-time metamaterials, where reflection from one direction is diminished20–23 . With the analysis using the scattering matrix (S-matrix)15,24–26 , in the combined parity-time system, the eigenvalues of the Smatrix are all unimodular in the exact parity-time phase but become a pair of reciprocal moduli in the parity-time symmetry broken phase. In particular, the eigenstates in the parity-time metamaterials become degenerate at the exceptional point, causing unidirectional invisibility where it is reflectionless when probed from one side. Most recently, it has been experimentally demonstrated using gain/loss balanced parity-time optical fibre networks in the temporal domain27 , but the realization of the spatial analogue still remains challenging28 and its on-chip implementation is expected to underpin a new generation of photonic devices29 . However, optical gain is difficult to achieve using conventional complementary metal-oxide-semiconductor silicon (Si) technology. In this Letter, therefore, in contrast to previous studies20–23,27 , we theoretically propose and experimentally realize an Si-based optical non-Hermitian parity-time system with only absorptive media on the Si-on-insulator (SOI) platform. Because gain and loss are no longer compensated, the corresponding S-matrix is always not unitary. However, a similar exceptional point still exists where the parity-time phase becomes degenerate as the loss in the system is represented by an additional term of attenuation30 . Remarkably, unidirectional reflection can still be expected at this exceptional point of the proposed passive parity-time metamaterial. In other words, the implemented passive parity-time metamaterial can be viewed as creating a normal parity-time-symmetric potential in a lossy background whose spectra are invariant in the corresponding Fourier space. As depicted in Fig. 1a, the studied passive parity-time metamaterial embedded inside SiO2 is an 800-nm-wide and 220-nm-thick Si waveguide with periodic modulation of its dielectric permittivity.
1 Department
of Electrical Engineering and Kavli Nanoscience Institute, California Institute of Technology, Pasadena, California 91125, USA, 2 National Laboratory of Solid State Microstructures and Department of Materials Science and Engineering, Nanjing University, Nanjing, Jiangsu 210093, China, 3 Department of Electronic Engineering, Instituto Tecnológico de Aeronáutica, São José dos Campos, São Paulo 12229-900, Brazil, 4 Division of Photonics, Instituto de Estudos Avançados, São José dos Campos, São Paulo 12229-900, Brazil. † These authors contributed equally to this work. *e-mail: [email protected]; [email protected]. NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2012 Macmillan Publishers Limited. All rights reserved.
1
NATURE MATERIALS DOI: 10.1038/NMAT3495
LETTERS a
b
1.0
0
0.9 0.8 0.7 Δ ε = cos [q(z-z0)]¬iδ sin[q(z-z0)]
δ
Rf
0.6 0.5 0.4
x
1
0.3
y
0.2
z
5
0.1 0.0 0
1.0
d
0
0.9
0.8
0.8
0.7 0.5
δ
Rb
0.6 0.4 0.3 0.1
e
30 40 50 60 70 80 90 100 L (μm)
0.7 0.6 0.5 0.4 0.3 0.2
0.2 0.0 0
20
1.0
0.9
Contrast ratio
c
10
10
20
0.1
5 1 30 40 50 60 70 80 90 100 L (μm)
0.0 0
1
2
3
4
5
δ
1 0 ¬1 ¬9
¬8
¬7
¬6
¬5
¬4
¬3
¬2
¬1
1 0 z (μm)
2
3
4
5
6
7
8
9
¬8
¬7
¬6
¬5
¬4
¬3
¬2
¬1
1 0 z (μm)
2
3
4
5
6
7
8
9
1 0 ¬1 ¬9
Figure 1 | Characteristics of evolution of parity-time symmetry in the proposed passive parity-time metamaterial. a, Schematic of the passive parity-time metamaterial on an SOI platform. Periodically arranged parity-time optical potentials with the modulated dielectric permittivity of 1ε = cos(qz) − iδsin(qz) (4nπ/q + π/q ≤ z ≤ 4nπ/q + 2π/q) are introduced into an 800-nm-wide and 220-nm-thick Si waveguide embedded in SiO2 . The period is 4π/q = 575.5 nm. b,c, Dependences of reflectance of the passive parity-time metamaterial on the modulation length with different values of δ at the wavelength of 1,550 nm in forward (b) and backward (c) directions, respectively. In contrast to the monotonic decrease in the forward direction, reflection in the backward direction first reaches a minimum value of Rb = 0 at δ = 1 and then increases as δ becomes larger. d, The corresponding contrast ratio of reflectivities in both directions at different values of δ. e, Electric field amplitude distribution of light in the parity-time metamaterial at its exceptional point, where δ = 1 for both forward (upper) and backward (lower) light propagation at a wavelength of 1,550 nm. The 3D FDTD simulation (FDTD Solutions 7.5, Lumerical) consists of 25 parity-time optical potentials with a period of 575.5 nm. Guided light is set to be incident along +z/ − z in the forward/backward direction at the boundaries of the parity-time metamaterial (marked with dashed white lines). Therefore light behind the lines is attributed only to reflection.
The waveguide supports a fundamental mode with a wave vector of k1 = 2.69k0 (k0 is the wave vector in air) at a wavelength of 1,550 nm. Therefore to form a Bragg grating to reflect the fundamental mode of guided light, the introduced modulation of the dielectric permittivity is 1ε = cos(qz)−iδsin(qz), where q = 2k1 and 4nπ/q+π/q ≤ z ≤ 4nπ/q + 2π/q. The period is chosen to be 4π/q, corresponding to the second Bragg order, for ease of experimental implementation as described below, but it is worth noting that our structure contains the same parity-time characteristics as the first Bragg order with a period of 2π/q. In the modulated regime, the electric field can be written as E(x,y,z) = A(z)E(x,y)eik1 z + B(z)E(x,y)e−ik1 z , where A(z) and B(z) are the amplitudes of forward and backward 2
fundamental modes, respectively. With slowly varying approximation, the coupled-mode equations can be derived as δ 1−δ dA(z) dz = − 2π αA(z) + i 8 κB(z) dB(z) = −i 1 + δ κA(z) + δ αB(z) dz 8 2π where α and κ denote attenuation and mode coupling between forward and backward fundamental modes (see Methods and Supplementary Information). The transfer matrix for the optical
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials © 2012 Macmillan Publishers Limited. All rights reserved.
NATURE MATERIALS DOI: 10.1038/NMAT3495 modulation from z = 0 to z = L can be written as A(L) M11 M12 A(0) = B(L) M21 M22 B(0) where M11 = cosh(γ L) − M21 = −i
δ α sinh(γ L), 2π γ
1+δ κ sinh(γ L), 8 γ
M12 = i
1−δ κ sinh(γ L), 8 γ
M22 = cosh(γ L) +
δ α sinh(γ L) 2π γ
and p γ = (δα/2π)2 + (1 − δ 2 )κ 2 /64 The corresponding S-matrix of the parity-time system (notice that the S-matrix here is not the same as the convention in electromagnetics)23–26 is √ 1 + M12 M21 t rb S= = aS0 = rf t M22 1 M12 √ √ 1 + M12 M21 1 + M12 M21 × −M21 1 √ √ 1 + M12 M21 1 + M12 M21 √ where a = 1 + M12 M21 /M22 , t is the transmission amplitude, and rf and rb are the amplitudes of reflection in forward and backward√directions, respectively. Because the eigenvalues of S (sn = t ± rf rb ) correspond to different phases of parity-time, the parity-time phase and its dependence on δ are analysed accordingly. When 0 ≤ δ < 1, the eigenvalues √ 1 ± i 8γκ sinh(γ L) 1 − δ 2 0 sn = asn = a √ 1 + M12 M21 where sn 0 is unimodular. It is therefore evident that the parity-time symmetry of the studied passive system is similar to the parity-time symmetric phase in the balanced gain/loss system but with an additional attenuation term |a| (see Supplementary Information for detailed comparison with typical parity-time systems with balanced gain/loss). When δ > 1, √ 1 ± 8γκ sinh(γ L) δ 2 − 1 0 sn = asn = a √ 1 + M12 M21 corresponds to the broken parity-time phase in the gain/loss system. Remarkably, at δ = 1, sn = as0n = exp(−αL/2π), where sn 0 = 1, and therefore the eigenvalues of the S-matrix become degenerate, indicating that the phase transition takes place at this exceptional point. The evolution of parity-time symmetry can be macroscopically observed via forward and backward reflection from the Bragg grating. The corresponding reflectivities of the Bragg grating for forward and backward directions are, respectively, 2 M21 2 , Rb = M12 Rf = M22 M22 Figure 1b,c shows the calculated reflection spectra as a function of modulation lengths at different values of δ for forward and backward directions, respectively. Because the parity-time metamaterial is only absorptive, the reflectivity reaches an asymptotic value after enough modulation. However, forward and backward
LETTERS directions show significantly distinguishable tendencies of the corresponding maximum reflection when δ increases. It is evident that δ = 1 is the exceptional point corresponding to the phase transition of parity-time symmetry where the Bragg grating is unidirectional reflectionless. The exceptional characteristics owing to the parity-time phase transition can be clearly visualized from the contrast ratio between the forward and backward reflectivities (C = |(Rf − Rb )/(Rf + Rb )| = (2δ/(1 + δ 2 ))), shown in Fig. 1d, where the Bragg grating is assumed long enough such that the reflectivities in both directions approach their corresponding asymptotic values. At δ = 1, the extreme of the dispersion is reached, which is direct evidence of the parity-time phase transition. Even with the frequency detuning, the system is unidirectional reflectionless in a broad band, with a contrast ratio of 1 at the exceptional point (see Supplementary Information). Consistent with the above theoretical analysis, mappings of light intensity in the waveguide using three-dimensional (3D) finite difference time domain (FDTD) simulations only show a reflected field in the forward direction (Fig. 1e). Moreover, the interference pattern between incidence and reflection can be observed only in the forward direction, further confirming that the system is unidirectional reflectionless at its exceptional point. To realize the proposed unidirectional reflectionless paritytime metamaterial at its exceptional point (δ = 1), an equivalent guided-mode modulation is designed using additional structures on top of the waveguide to mimic the microscopic parity-time modulation on a macroscopic scale. However, the balance of the modulation amplitude in the real part 1εreal = cos(qz) in an entire period is broken, because additional structures on top can only result in a higher effective index. To be consistent with the final design, regions of 1εreal are extracted from original parity-time optical potentials and the corresponding cosine modulation is shifted 5π/2q in the z direction to become a positive sinusoidal modulation, whereas the imaginary part modulation remains at the same location to provide in-phase modulation to guided light together with the shifted real part. Although the real and imaginary part modulations are now separated, our in-phase arrangement creates an equivalent optical modulation to guided light in both amplitude and phase as if the original complex parity-time optical potential still existed (see Supplementary Information). Finally, to achieve these sinusoidal-function modulated optical potentials using microscopically homogeneous materials, sinusoidal-shaped combo structures (combination of a sinusoidal-shaped structure and its mirror image to the transverse direction) are adopted on top of the Si waveguide (Fig. 2a). Mode effective indices with additional Si and germanium (Ge)/chrome (Cr) bilayer combo structures are consistent with the real and imaginary part modulations, respectively, showing the equivalence of the designed structure to the parity-time metamaterial (Fig. 2b). The characteristics of this parity-time metamaterial have also been verified using FDTD simulations. The reflectivities are significantly distinguished in the forward and backward directions, with about 11 dB of extinction ratio in the studied wavelength range from 1,520 to 1,580 nm (Fig. 2c). It is worth noting that the resonance peak of the Bragg grating moves to around 1,560 nm, because the modulation in the real part is only positive, which creates a higher effective index and thus red-shifts the resonance peak compared with the case in Fig. 1 where the real part modulation has balanced positive and negative amplitudes (see Supplementary Information). The contrast ratio between the forward and backward directions is plotted in Fig. 2d, showing that reflection in the backward direction is significantly suppressed. The asymmetry in reflection from the designed parity-time metamaterial can also be visualized from mappings of light propagating inside the waveguide (Fig. 2e): forward propagating light and its reflection form strong interference, whereas reflection is barely seen with
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2012 Macmillan Publishers Limited. All rights reserved.
3
NATURE MATERIALS DOI: 10.1038/NMAT3495
LETTERS a
51 nm Si 14 nm Ge/24 nm Cr Si SiO2 x
b
Real effective index
Effective index
Si
2.8 2.6
2.5
0.8 0.6
Δε = isin (qΔz)
1.5
Ge/Cr 0.4
1.0
0.2
0.5 0
20
40
60 80 Δz (nm)
100
120
0.0 0
140
d
0.40 0.35 0.25
Contrast ratio
Reflection
Re (neff)
2.0
0.30 Forward
0.20 0.15 0.10 Backward
0.05 0.00 1,520
e
1.0
1,530
1,540 1,550 1,560 Wavelength (nm)
1,570
1,580
Im (neff) 20
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1,520
40
1,530
60 80 Δz (nm)
100
1,540 1,550
0.0 140
120
1,560
Imaginary effective index
Δε = sin (qΔz)
3.0
c
z
3.0
3.2
2.4
y
1,570
1,580
Wavelength (nm)
1 0 ¬1 ¬9
¬8
¬7
¬6
¬5
¬4
¬3
¬2
¬1
0 1 z (μm)
2
3
4
5
6
7
8
9
¬8
¬7
¬6
¬5
¬4
¬3
¬2
¬1
0 1 z (μm)
2
3
4
5
6
7
8
9
1 0 ¬1 ¬9
Figure 2 | Optical properties of the designed passive unidirectional reflectionless parity-time metamaterial. a, Periodically arranged 760-nm-wide sinusoidal shaped combo structures are applied on top of an 800-nm-wide Si waveguide embedded inside SiO2 to mimic parity-time optical potentials, in which imaginary part modulation is implemented with 14 nm Ge/24 nm Cr structures and 51 nm Si layers are for real part modulation. The designed parity-time metamaterial consists of 25 sets of top-modulated combo structures with a period of 575.5 nm and a width of 143.9 nm for each sinusoidal-shaped combo. b, Mode effective index with the real (left) and imaginary (right) parts of the modulated dielectric permittivity (red lines) as well as comparisons with their corresponding sinusoidal-shaped Si (left) and Ge/Cr (right) combos (blue dots). c, Simulated reflection spectra of the device in forward (red) and backward (blue) directions. d, Spectrum of contrast ratio of reflectivities, showing high contrast ratios over the studied wavelength range from 1,520 to 1,580 nm. e, Simulated electric field amplitude distribution of light in the device, where incidence is set at boundaries of the parity-time metamaterial (marked with white lines) along +z/ − z in the forward/backward direction.
backward incidence. It is thus evident that the designed on-chip waveguide system successfully mimics the unidirectional effect inherently associated with the exceptional point. In experiments, however, measuring in-line reflection would require the use of external components such as optical circulators or directional couplers, which in turn cause additional insertion 4
loss and extra noise in the measurements, making it difficult to perform. Therefore, we propose a strategy using on-chip waveguide directional couplers to measure the corresponding reflectance in a way similar to the transmission measurement as shown in Fig. 3a. The sample was fabricated using overlay electron beam lithography, followed by evaporation and lift-off of Si and Ge/Cr as well as dry
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials © 2012 Macmillan Publishers Limited. All rights reserved.
NATURE MATERIALS DOI: 10.1038/NMAT3495
LETTERS
a
Forward
on
reflecti
ence
cid rd in kwa
Bac
riods
25 pe
ce ciden rd in a w r Fo
w Back
ard r
eflec
tion
b
c
Back Forw
ward
ard
Si Ge/Cr 2 µm
1 µm
Figure 3 | Experimental implementation of the passive unidirectional reflectionless parity-time metamaterial. a, Configuration of waveguides on the SOI platform to measure reflection from the device. A single-mode waveguide directional coupler with about 3 dB of coupling ratio is designed to maximize the detected signal (see Supplementary Information). Tapered waveguides are also introduced to transfer light from the single mode in the 400-nm-wide Si waveguides to the fundamental mode in the proposed waveguide device and vice versa. b, SEM picture of the whole device before deposition of SiO2 cladding. The fabricated device consists of 25 periods of Ge/Cr and Si sinusoidal combo structures with a period of 575.5 nm on top of the Si waveguide. c, Zoom-in SEM picture of the device, where the boxed area indicates a unit cell. The remaining HSQ resist and two kinds of sinusoidal combos can be seen on top of the waveguide.
a
b
¬30
0.8
Forward
0.6
Contrast ratio
Reflection (dBm)
0.7 ¬35
¬40
0.4 0.3 0.2
¬45 Backward ¬50 1,520
0.5
1,530
1,540 1,550 1,560 Wavelength (nm)
1,570
0.1 0.0 1,520
1,580
1,530
1,540 1,550 1,560 Wavelength (nm)
1,570
1,580
Figure 4 | Measured optical properties of the parity-time metamaterial. a, Measured reflection spectra of the device through the waveguide coupler for both directions over a broad band of telecom wavelengths from 1,520 to 1,580 nm. Red and blue curves are Gaussian fits of raw data in forward (black) and backward (green) directions, respectively. b, Spectrum of contrast ratio of reflectivities obtained from the fitting data in a.
etching to form the Si waveguide (see Methods). The pictures of the device before deposition of the SiO2 cladding are shown in Fig. 3b,c. In experiments, tapered fibres were used to couple light from fibres to waveguides and vice versa. The reflection spectra of the fabricated device have been measured for both forward and backward directions, as shown in Fig. 4a, respectively. Consistent with simulations, red-shifts of the resonance peaks are also observed in experimental measurements for both directions. The measured reflection spectra also show significantly distinguished characteristics in reflection: the reflectivity in the forward direction is about 7.5 dB stronger than that in the backward direction, indicating asymmetric optical properties owing to the parity-time symmetry phase transition. The corresponding high-contrast ratios
over a broad band of telecom wavelengths in Fig. 4b further confirm the asymmetric reflection of the parity-time metamaterial associated with the exceptional point. It is therefore evident that the presented parity-time metamaterial can successfully mimic the complicated quantum phenomena in classical on-chip optical waveguide systems. Similarly to previously investigated balanced gain/loss systems, proper engineering of the complex dielectric permittivity in a passive system also makes the exceptional point as well as its associated phase transition of parity-time symmetry become measurable quantities. The chip-scale parity-time metamaterial fabricated by conventional complementary metaloxide-semiconductor fabrication procedures clearly confirms
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2012 Macmillan Publishers Limited. All rights reserved.
5
NATURE MATERIALS DOI: 10.1038/NMAT3495
LETTERS the expected unidirectional and asymmetric characteristics in experiments. Moreover, the demonstrated unidirectional reflection can be converted to unidirectional invisibility by adding an additional linear amplifier to the presented paritytime metamaterial (see Supplementary Information). Further investigation of engineering the complex refractive index (linear and nonlinear parts) as well as geometric arrangement of these optical potentials is expected to create novel parity-time metamaterials with even more counterintuitive physical responses that once belonged only to the field of quantum theory, thus paving an approach to a new generation of photonic devices. For example, a four-port broadband unidirectional photonic device may construct backbones of chip-scale optical network analysers as its circuit counterpart in state-of-the-art microwave network analysers. Furthermore, the general design principle in our device in the optical domain can be extended to other frequencies and classical wave systems, such as unidirectional microwave invisibility for military applications and ultrasonic equipment for marine exploration31 .
Methods Derivation of attenuation and coupling coefficients in coupled mode equations. The attenuation coefficient α indicates the system absorption due to the imaginary part of the parity-time potentials and the coupling coefficient κ is the mode overlap between forward and backward propagating light. Both of them can be derived at the exceptional point, δ = 1, where the calculation is most simplified. At the exceptional point, the modulated mode effective index can be approximately written as neff ≈ 2.69 + 0.15cos(qz) − i0.15sin(qz). Therefore the attenuation −1 constant is calculated as α = 0 = 0.61 µm , while the coupling R nimag k0 = 0.15k R coefficient is κ = (k02 /2k1 )( E ∗ 1εEds/ |E|2 ds) = 0.48 µm−1 . Both α and κ are validated using FDTD simulations. At the exceptional point, the coupled mode equations can be written as α dA(z) = − A(z) dz 2π κ α dB(z) = −i A(z) + B(z) dz 4 2π The corresponding transmission and reflection coefficients are T = exp(−αL/π), Rf = (π2 κ 2 /4α 2 )sinh2 (αL/2π)exp(−αL/π), and Rb = 0, which can be used to fit the obtained reflection spectra from FDTD. α and κ in the best fitting results are about 0.61 µm−1 and 0.49 µm−1 (see Supplementary Information), consistent with the above analytical derivations. Sample fabrication. The fabrication starts with an SOI wafer. Two layers of periodically arranged sinusoidal-shaped combo structures are first patterned with accurate alignment in polymethyl methacrylate by electron beam lithography, followed by electron beam evaporation of Ge/Cr and Si and lift-off in acetone, respectively, in two steps. Then the Si waveguide is defined by aligned electron beam lithography using hydrogen silsesquioxane (HSQ), followed by dry etching with mixed gases of SF6 and C4 F8 . Because the developed HSQ becomes porous SiO2 and its refractive index is similar to SiO2 , it remains on top of the waveguide and therefore the topology of the sinusoidal combos on top of the waveguide became obscure in scanning electron microscope (SEM) images (Fig. 3b,c). Finally, plasma enhanced chemical vapour deposition with mixed gases of SiH4 and N2 O is used to deposit the cladding of SiO2 on the entire wafer to increase the light coupling efficiency from tapered fibres to waveguides.
Received 22 August 2012; accepted 22 October 2012; published online 25 November 2012
References 1. Leonhardt, U. Optical conformal mapping. Science 312, 1777–1780 (2006). 2. Pendry, J. B., Schurig, D. & Smith, D. R. Controlling electromagnetic fields. Science 312, 1780–1782 (2006). 3. Bender, C. M. & Böttcher, S. Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998). 4. Bender, C. M. Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947–1018 (2007). 5. Bender, C. M., Brody, D. C. & Jones, H. F. Extension of PT-symmetric quantum mechanics to quantum field theory with cubic interaction. Phys. Rev. D 70, 025001 (2004). 6. Longhi, S. & Della Valle, G. Photonic realization of PT-symmetric quantum field theories. Phys. Rev. A 85, 012112 (2012). 6
7. Makris, K. G., El-Ganainy, R., Christodoulides, D. N. & Musslimani, Z. H. Beam dynamics in PT-symmetric optical lattices. Phys. Rev. Lett. 100, 103904 (2008). 8. Musslimani, Z. H., El-Ganainy, R., Makris, K. G. & Christodoulides, D. N. Optical solitons in PT periodic potentials. Phys. Rev. Lett. 100, 030402 (2008). 9. Klaiman, S., Guenther, U. & Moiseyev, N. Visualization of branch points in PT symmetric waveguides. Phys. Rev. Lett. 101, 080402 (2008). 10. Longhi, S. Bloch oscillations in complex crystals with PT symmetry. Phys. Rev. Lett. 103, 123601 (2009). 11. Zheng, M. C., Christodoulides, D. N., Fleischmann, R. & Kottos, T. PT optical lattices and universality in beam dynamics. Phys. Rev. A 82, 010103(R) (2010). 12. Graefe, E. M. & Jones, H. F. PT-symmetric sinusoidal optical lattices at the symmetry-breaking threshold. Phys. Rev. A 84, 013818 (2011). 13. Miroshnichenko, A. E., Malomed, B. A. & Kivshar, Y. S. Nonlinearly PT-symmetric systems: Spontaneous symmetry breaking and transmission resonances. Phys. Rev. A 84, 012123 (2011). 14. Longhi, S. PT-symmetric laser absorber. Phys. Rev. A 82, 031801 (2010). 15. Chong, Y. D., Ge, L. & Stone, A. D. PT-symmetry breaking and laser-absorber modes in optical scattering systems. Phys. Rev. Lett. 106, 093902 (2011). 16. Ge, L. et al. Unconventional modes in lasers with spatially varying gain and loss. Phys. Rev. A 84, 023820 (2011). 17. Rüter, C. E. et al. Observation of parity–time symmetry in optics. Nature Phys. 6, 192–195 (2010). 18. Guo, A. et al. Observation of PT-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902 (2009). 19. Schindler, J., Li, A., Zheng, M. C., Ellis, F. M. & Kottos, T. Experimental study of active LRC circuits with PT symmetries. Phys. Rev. A 84, 040101(R) (2011). 20. Kulishov, M., Laniel, J., Bélanger, N., Azaña, J. & Plant, D. Nonreciprocal waveguide Bragg gratings. Opt. Express 13, 3068–3078 (2005). 21. Lin, Z. et al. Unidirectional invisibility induced by PT-symmetric periodic structures. Phys. Rev. Lett. 106, 213901 (2011). 22. Longhi, S. Invisibility in PT-symmetric complex crystals. J. Phys. A 44, 485302 (2011). 23. Jones, H. F. Analytic results for a PT-symmetric optical structure. J. Phys. A 45, 135306 (2012). 24. Mostafazadeh, A. Spectral singularities of complex scattering potentials and infinite reflection and transmission coefficients at real energies. Phys. Rev. Lett. 102, 220402 (2009). 25. Muga, J. G., Palaob, J. P., Navarroa, B. & Egusquizac, I. L. Complex absorbing potentials. Phys. Rep. 395, 357–426 (2004). 26. Cannata, F., Dedonder, J-P. & Ventura, A. Scattering in PT-symmetric quantum mechanics. Ann. Phys. 322, 397–433 (2007). 27. Regensburger, A. et al. Parity–time synthetic photonic lattices. Nature 488, 167–171 (2012). 28. Razzari, L. & Morandotti, R. Gain and loss mixed in the same cauldron. Nature 488, 163–164 (2012). 29. Kottos, T. Optical physics: Broken symmetry makes light work. Nature Phys. 6, 166–167 (2010). 30. Berry, M. V. Optical lattices with PT symmetry are not transparent. J. Phys. A 41, 244007 (2008). 31. Zhang, S., Xia, C. & Fang, N. Broadband acoustic cloak for ultrasound waves. Phys. Rev. Lett. 106, 024301 (2011).
Acknowledgements We acknowledge critical support and infrastructure provided for this work by the Kavli Nanoscience Institute at Caltech. This work was supported by the NSF ERC Center for Integrated Access Networks (no. EEC-0812072), the National Basic Research of China (no. 2012CB921503 and no. 2013CB632702), the National Nature Science Foundation of China (no. 11134006), the Nature Science Foundation of Jiangsu Province (no. BK2009007), the Priority Academic Program Development of Jiangsu Higher Education, and CAPES and CNPQ—Brazilian Foundations. M-H.L. also acknowledges the support of FANEDD of China.
Author contributions L.F. and M-H.L. conceived the idea. L.F., Y-L.X. and M-H.L. designed the device. Y-L.X., L.F. and M-H.L. performed the theoretical analysis of parity-time symmetry. W.S.F. and L.F. designed the chip and carried out fabrications and measurements. All the authors contributed to discussion of the project. Y-F.C. and A.S. guided the project. L.F. wrote the manuscript with revisions from other authors.
Additional information Supplementary information is available in the online version of the paper. Reprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to L.F. (for general design and experiment detail) or M-H.L. (for theoretical analysis).
Competing financial interests The authors declare no competing financial interests.
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials © 2012 Macmillan Publishers Limited. All rights reserved.
Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies Liang Feng1 *† , Ye-Long Xu2† , William S. Fegadolli1,3,4† , Ming-Hui Lu2 *, José E. B. Oliveira3 , Vilson R. Almeida3,4 , Yan-Feng Chen2 and Axel Scherer1 Invisibility by metamaterials is of great interest, where optical properties are manipulated in the real permittivity– permeability plane1,2 . However, the most effective approach to achieving invisibility in various military applications is to absorb the electromagnetic waves emitted from radar to minimize the corresponding reflection and scattering, such that no signal gets bounced back. Here, we show the experimental realization of chip-scale unidirectional reflectionless optical metamaterials near the spontaneous parity-time symmetry phase transition point where reflection from one side is significantly suppressed. This is enabled by engineering the corresponding optical properties of the designed paritytime metamaterial in the complex dielectric permittivity plane. Numerical simulations and experimental verification consistently exhibit asymmetric reflection with high contrast ratios around a wavelength of of 1,550 nm. The demonstrated unidirectional phenomenon at the corresponding parity-time exceptional point on-a-chip confirms the feasibility of creating complicated on-chip parity-time metamaterials and optical devices based on their properties. Parity-time-symmetric material, also called synthetic matter, is a new class of metamaterials in which parity and time reversal symmetries are combined in such a way that non-Hermitian Hamiltonians may still possess real energy spectra3–5 , creating physical properties of quantum mechanics and quantum field theories in classical systems6 . Owing to the equivalence between the Schrödinger equation in quantum mechanics and the wave equation in optics, the combined parity-time symmetry has been intensively studied in classical optical systems with non-Hermitian optical potentials. By exploiting optical modulation of the refractive index in the complex dielectric permittivity plane and engineering both optical absorption and amplification, parity-time metamaterials can lead to a series of intriguing optical phenomena and devices, such as dynamic power oscillations of light propagation7–13 and coherent perfect absorber-lasers14–16 . Another important aspect of such parity-time optical materials is that the evolution of parity-time symmetry becomes measurable through the quantum-optical analogue17–19 . The threshold of parity-time symmetry breaking was clearly visualized as the
corresponding energy spectra change dramatically from real to complex after this phase transition point. This threshold is therefore called an exceptional point or spontaneous parity-time symmetry breaking point, where amplitudes of the real and imaginary parts of the modulated refractive index are identical. An interesting phenomenon named unidirectional invisibility was theoretically proposed at this exceptional point in parity-time metamaterials, where reflection from one direction is diminished20–23 . With the analysis using the scattering matrix (S-matrix)15,24–26 , in the combined parity-time system, the eigenvalues of the Smatrix are all unimodular in the exact parity-time phase but become a pair of reciprocal moduli in the parity-time symmetry broken phase. In particular, the eigenstates in the parity-time metamaterials become degenerate at the exceptional point, causing unidirectional invisibility where it is reflectionless when probed from one side. Most recently, it has been experimentally demonstrated using gain/loss balanced parity-time optical fibre networks in the temporal domain27 , but the realization of the spatial analogue still remains challenging28 and its on-chip implementation is expected to underpin a new generation of photonic devices29 . However, optical gain is difficult to achieve using conventional complementary metal-oxide-semiconductor silicon (Si) technology. In this Letter, therefore, in contrast to previous studies20–23,27 , we theoretically propose and experimentally realize an Si-based optical non-Hermitian parity-time system with only absorptive media on the Si-on-insulator (SOI) platform. Because gain and loss are no longer compensated, the corresponding S-matrix is always not unitary. However, a similar exceptional point still exists where the parity-time phase becomes degenerate as the loss in the system is represented by an additional term of attenuation30 . Remarkably, unidirectional reflection can still be expected at this exceptional point of the proposed passive parity-time metamaterial. In other words, the implemented passive parity-time metamaterial can be viewed as creating a normal parity-time-symmetric potential in a lossy background whose spectra are invariant in the corresponding Fourier space. As depicted in Fig. 1a, the studied passive parity-time metamaterial embedded inside SiO2 is an 800-nm-wide and 220-nm-thick Si waveguide with periodic modulation of its dielectric permittivity.
1 Department
of Electrical Engineering and Kavli Nanoscience Institute, California Institute of Technology, Pasadena, California 91125, USA, 2 National Laboratory of Solid State Microstructures and Department of Materials Science and Engineering, Nanjing University, Nanjing, Jiangsu 210093, China, 3 Department of Electronic Engineering, Instituto Tecnológico de Aeronáutica, São José dos Campos, São Paulo 12229-900, Brazil, 4 Division of Photonics, Instituto de Estudos Avançados, São José dos Campos, São Paulo 12229-900, Brazil. † These authors contributed equally to this work. *e-mail: [email protected]; [email protected]. NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2012 Macmillan Publishers Limited. All rights reserved.
1
NATURE MATERIALS DOI: 10.1038/NMAT3495
LETTERS a
b
1.0
0
0.9 0.8 0.7 Δ ε = cos [q(z-z0)]¬iδ sin[q(z-z0)]
δ
Rf
0.6 0.5 0.4
x
1
0.3
y
0.2
z
5
0.1 0.0 0
1.0
d
0
0.9
0.8
0.8
0.7 0.5
δ
Rb
0.6 0.4 0.3 0.1
e
30 40 50 60 70 80 90 100 L (μm)
0.7 0.6 0.5 0.4 0.3 0.2
0.2 0.0 0
20
1.0
0.9
Contrast ratio
c
10
10
20
0.1
5 1 30 40 50 60 70 80 90 100 L (μm)
0.0 0
1
2
3
4
5
δ
1 0 ¬1 ¬9
¬8
¬7
¬6
¬5
¬4
¬3
¬2
¬1
1 0 z (μm)
2
3
4
5
6
7
8
9
¬8
¬7
¬6
¬5
¬4
¬3
¬2
¬1
1 0 z (μm)
2
3
4
5
6
7
8
9
1 0 ¬1 ¬9
Figure 1 | Characteristics of evolution of parity-time symmetry in the proposed passive parity-time metamaterial. a, Schematic of the passive parity-time metamaterial on an SOI platform. Periodically arranged parity-time optical potentials with the modulated dielectric permittivity of 1ε = cos(qz) − iδsin(qz) (4nπ/q + π/q ≤ z ≤ 4nπ/q + 2π/q) are introduced into an 800-nm-wide and 220-nm-thick Si waveguide embedded in SiO2 . The period is 4π/q = 575.5 nm. b,c, Dependences of reflectance of the passive parity-time metamaterial on the modulation length with different values of δ at the wavelength of 1,550 nm in forward (b) and backward (c) directions, respectively. In contrast to the monotonic decrease in the forward direction, reflection in the backward direction first reaches a minimum value of Rb = 0 at δ = 1 and then increases as δ becomes larger. d, The corresponding contrast ratio of reflectivities in both directions at different values of δ. e, Electric field amplitude distribution of light in the parity-time metamaterial at its exceptional point, where δ = 1 for both forward (upper) and backward (lower) light propagation at a wavelength of 1,550 nm. The 3D FDTD simulation (FDTD Solutions 7.5, Lumerical) consists of 25 parity-time optical potentials with a period of 575.5 nm. Guided light is set to be incident along +z/ − z in the forward/backward direction at the boundaries of the parity-time metamaterial (marked with dashed white lines). Therefore light behind the lines is attributed only to reflection.
The waveguide supports a fundamental mode with a wave vector of k1 = 2.69k0 (k0 is the wave vector in air) at a wavelength of 1,550 nm. Therefore to form a Bragg grating to reflect the fundamental mode of guided light, the introduced modulation of the dielectric permittivity is 1ε = cos(qz)−iδsin(qz), where q = 2k1 and 4nπ/q+π/q ≤ z ≤ 4nπ/q + 2π/q. The period is chosen to be 4π/q, corresponding to the second Bragg order, for ease of experimental implementation as described below, but it is worth noting that our structure contains the same parity-time characteristics as the first Bragg order with a period of 2π/q. In the modulated regime, the electric field can be written as E(x,y,z) = A(z)E(x,y)eik1 z + B(z)E(x,y)e−ik1 z , where A(z) and B(z) are the amplitudes of forward and backward 2
fundamental modes, respectively. With slowly varying approximation, the coupled-mode equations can be derived as δ 1−δ dA(z) dz = − 2π αA(z) + i 8 κB(z) dB(z) = −i 1 + δ κA(z) + δ αB(z) dz 8 2π where α and κ denote attenuation and mode coupling between forward and backward fundamental modes (see Methods and Supplementary Information). The transfer matrix for the optical
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials © 2012 Macmillan Publishers Limited. All rights reserved.
NATURE MATERIALS DOI: 10.1038/NMAT3495 modulation from z = 0 to z = L can be written as A(L) M11 M12 A(0) = B(L) M21 M22 B(0) where M11 = cosh(γ L) − M21 = −i
δ α sinh(γ L), 2π γ
1+δ κ sinh(γ L), 8 γ
M12 = i
1−δ κ sinh(γ L), 8 γ
M22 = cosh(γ L) +
δ α sinh(γ L) 2π γ
and p γ = (δα/2π)2 + (1 − δ 2 )κ 2 /64 The corresponding S-matrix of the parity-time system (notice that the S-matrix here is not the same as the convention in electromagnetics)23–26 is √ 1 + M12 M21 t rb S= = aS0 = rf t M22 1 M12 √ √ 1 + M12 M21 1 + M12 M21 × −M21 1 √ √ 1 + M12 M21 1 + M12 M21 √ where a = 1 + M12 M21 /M22 , t is the transmission amplitude, and rf and rb are the amplitudes of reflection in forward and backward√directions, respectively. Because the eigenvalues of S (sn = t ± rf rb ) correspond to different phases of parity-time, the parity-time phase and its dependence on δ are analysed accordingly. When 0 ≤ δ < 1, the eigenvalues √ 1 ± i 8γκ sinh(γ L) 1 − δ 2 0 sn = asn = a √ 1 + M12 M21 where sn 0 is unimodular. It is therefore evident that the parity-time symmetry of the studied passive system is similar to the parity-time symmetric phase in the balanced gain/loss system but with an additional attenuation term |a| (see Supplementary Information for detailed comparison with typical parity-time systems with balanced gain/loss). When δ > 1, √ 1 ± 8γκ sinh(γ L) δ 2 − 1 0 sn = asn = a √ 1 + M12 M21 corresponds to the broken parity-time phase in the gain/loss system. Remarkably, at δ = 1, sn = as0n = exp(−αL/2π), where sn 0 = 1, and therefore the eigenvalues of the S-matrix become degenerate, indicating that the phase transition takes place at this exceptional point. The evolution of parity-time symmetry can be macroscopically observed via forward and backward reflection from the Bragg grating. The corresponding reflectivities of the Bragg grating for forward and backward directions are, respectively, 2 M21 2 , Rb = M12 Rf = M22 M22 Figure 1b,c shows the calculated reflection spectra as a function of modulation lengths at different values of δ for forward and backward directions, respectively. Because the parity-time metamaterial is only absorptive, the reflectivity reaches an asymptotic value after enough modulation. However, forward and backward
LETTERS directions show significantly distinguishable tendencies of the corresponding maximum reflection when δ increases. It is evident that δ = 1 is the exceptional point corresponding to the phase transition of parity-time symmetry where the Bragg grating is unidirectional reflectionless. The exceptional characteristics owing to the parity-time phase transition can be clearly visualized from the contrast ratio between the forward and backward reflectivities (C = |(Rf − Rb )/(Rf + Rb )| = (2δ/(1 + δ 2 ))), shown in Fig. 1d, where the Bragg grating is assumed long enough such that the reflectivities in both directions approach their corresponding asymptotic values. At δ = 1, the extreme of the dispersion is reached, which is direct evidence of the parity-time phase transition. Even with the frequency detuning, the system is unidirectional reflectionless in a broad band, with a contrast ratio of 1 at the exceptional point (see Supplementary Information). Consistent with the above theoretical analysis, mappings of light intensity in the waveguide using three-dimensional (3D) finite difference time domain (FDTD) simulations only show a reflected field in the forward direction (Fig. 1e). Moreover, the interference pattern between incidence and reflection can be observed only in the forward direction, further confirming that the system is unidirectional reflectionless at its exceptional point. To realize the proposed unidirectional reflectionless paritytime metamaterial at its exceptional point (δ = 1), an equivalent guided-mode modulation is designed using additional structures on top of the waveguide to mimic the microscopic parity-time modulation on a macroscopic scale. However, the balance of the modulation amplitude in the real part 1εreal = cos(qz) in an entire period is broken, because additional structures on top can only result in a higher effective index. To be consistent with the final design, regions of 1εreal are extracted from original parity-time optical potentials and the corresponding cosine modulation is shifted 5π/2q in the z direction to become a positive sinusoidal modulation, whereas the imaginary part modulation remains at the same location to provide in-phase modulation to guided light together with the shifted real part. Although the real and imaginary part modulations are now separated, our in-phase arrangement creates an equivalent optical modulation to guided light in both amplitude and phase as if the original complex parity-time optical potential still existed (see Supplementary Information). Finally, to achieve these sinusoidal-function modulated optical potentials using microscopically homogeneous materials, sinusoidal-shaped combo structures (combination of a sinusoidal-shaped structure and its mirror image to the transverse direction) are adopted on top of the Si waveguide (Fig. 2a). Mode effective indices with additional Si and germanium (Ge)/chrome (Cr) bilayer combo structures are consistent with the real and imaginary part modulations, respectively, showing the equivalence of the designed structure to the parity-time metamaterial (Fig. 2b). The characteristics of this parity-time metamaterial have also been verified using FDTD simulations. The reflectivities are significantly distinguished in the forward and backward directions, with about 11 dB of extinction ratio in the studied wavelength range from 1,520 to 1,580 nm (Fig. 2c). It is worth noting that the resonance peak of the Bragg grating moves to around 1,560 nm, because the modulation in the real part is only positive, which creates a higher effective index and thus red-shifts the resonance peak compared with the case in Fig. 1 where the real part modulation has balanced positive and negative amplitudes (see Supplementary Information). The contrast ratio between the forward and backward directions is plotted in Fig. 2d, showing that reflection in the backward direction is significantly suppressed. The asymmetry in reflection from the designed parity-time metamaterial can also be visualized from mappings of light propagating inside the waveguide (Fig. 2e): forward propagating light and its reflection form strong interference, whereas reflection is barely seen with
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2012 Macmillan Publishers Limited. All rights reserved.
3
NATURE MATERIALS DOI: 10.1038/NMAT3495
LETTERS a
51 nm Si 14 nm Ge/24 nm Cr Si SiO2 x
b
Real effective index
Effective index
Si
2.8 2.6
2.5
0.8 0.6
Δε = isin (qΔz)
1.5
Ge/Cr 0.4
1.0
0.2
0.5 0
20
40
60 80 Δz (nm)
100
120
0.0 0
140
d
0.40 0.35 0.25
Contrast ratio
Reflection
Re (neff)
2.0
0.30 Forward
0.20 0.15 0.10 Backward
0.05 0.00 1,520
e
1.0
1,530
1,540 1,550 1,560 Wavelength (nm)
1,570
1,580
Im (neff) 20
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1,520
40
1,530
60 80 Δz (nm)
100
1,540 1,550
0.0 140
120
1,560
Imaginary effective index
Δε = sin (qΔz)
3.0
c
z
3.0
3.2
2.4
y
1,570
1,580
Wavelength (nm)
1 0 ¬1 ¬9
¬8
¬7
¬6
¬5
¬4
¬3
¬2
¬1
0 1 z (μm)
2
3
4
5
6
7
8
9
¬8
¬7
¬6
¬5
¬4
¬3
¬2
¬1
0 1 z (μm)
2
3
4
5
6
7
8
9
1 0 ¬1 ¬9
Figure 2 | Optical properties of the designed passive unidirectional reflectionless parity-time metamaterial. a, Periodically arranged 760-nm-wide sinusoidal shaped combo structures are applied on top of an 800-nm-wide Si waveguide embedded inside SiO2 to mimic parity-time optical potentials, in which imaginary part modulation is implemented with 14 nm Ge/24 nm Cr structures and 51 nm Si layers are for real part modulation. The designed parity-time metamaterial consists of 25 sets of top-modulated combo structures with a period of 575.5 nm and a width of 143.9 nm for each sinusoidal-shaped combo. b, Mode effective index with the real (left) and imaginary (right) parts of the modulated dielectric permittivity (red lines) as well as comparisons with their corresponding sinusoidal-shaped Si (left) and Ge/Cr (right) combos (blue dots). c, Simulated reflection spectra of the device in forward (red) and backward (blue) directions. d, Spectrum of contrast ratio of reflectivities, showing high contrast ratios over the studied wavelength range from 1,520 to 1,580 nm. e, Simulated electric field amplitude distribution of light in the device, where incidence is set at boundaries of the parity-time metamaterial (marked with white lines) along +z/ − z in the forward/backward direction.
backward incidence. It is thus evident that the designed on-chip waveguide system successfully mimics the unidirectional effect inherently associated with the exceptional point. In experiments, however, measuring in-line reflection would require the use of external components such as optical circulators or directional couplers, which in turn cause additional insertion 4
loss and extra noise in the measurements, making it difficult to perform. Therefore, we propose a strategy using on-chip waveguide directional couplers to measure the corresponding reflectance in a way similar to the transmission measurement as shown in Fig. 3a. The sample was fabricated using overlay electron beam lithography, followed by evaporation and lift-off of Si and Ge/Cr as well as dry
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials © 2012 Macmillan Publishers Limited. All rights reserved.
NATURE MATERIALS DOI: 10.1038/NMAT3495
LETTERS
a
Forward
on
reflecti
ence
cid rd in kwa
Bac
riods
25 pe
ce ciden rd in a w r Fo
w Back
ard r
eflec
tion
b
c
Back Forw
ward
ard
Si Ge/Cr 2 µm
1 µm
Figure 3 | Experimental implementation of the passive unidirectional reflectionless parity-time metamaterial. a, Configuration of waveguides on the SOI platform to measure reflection from the device. A single-mode waveguide directional coupler with about 3 dB of coupling ratio is designed to maximize the detected signal (see Supplementary Information). Tapered waveguides are also introduced to transfer light from the single mode in the 400-nm-wide Si waveguides to the fundamental mode in the proposed waveguide device and vice versa. b, SEM picture of the whole device before deposition of SiO2 cladding. The fabricated device consists of 25 periods of Ge/Cr and Si sinusoidal combo structures with a period of 575.5 nm on top of the Si waveguide. c, Zoom-in SEM picture of the device, where the boxed area indicates a unit cell. The remaining HSQ resist and two kinds of sinusoidal combos can be seen on top of the waveguide.
a
b
¬30
0.8
Forward
0.6
Contrast ratio
Reflection (dBm)
0.7 ¬35
¬40
0.4 0.3 0.2
¬45 Backward ¬50 1,520
0.5
1,530
1,540 1,550 1,560 Wavelength (nm)
1,570
0.1 0.0 1,520
1,580
1,530
1,540 1,550 1,560 Wavelength (nm)
1,570
1,580
Figure 4 | Measured optical properties of the parity-time metamaterial. a, Measured reflection spectra of the device through the waveguide coupler for both directions over a broad band of telecom wavelengths from 1,520 to 1,580 nm. Red and blue curves are Gaussian fits of raw data in forward (black) and backward (green) directions, respectively. b, Spectrum of contrast ratio of reflectivities obtained from the fitting data in a.
etching to form the Si waveguide (see Methods). The pictures of the device before deposition of the SiO2 cladding are shown in Fig. 3b,c. In experiments, tapered fibres were used to couple light from fibres to waveguides and vice versa. The reflection spectra of the fabricated device have been measured for both forward and backward directions, as shown in Fig. 4a, respectively. Consistent with simulations, red-shifts of the resonance peaks are also observed in experimental measurements for both directions. The measured reflection spectra also show significantly distinguished characteristics in reflection: the reflectivity in the forward direction is about 7.5 dB stronger than that in the backward direction, indicating asymmetric optical properties owing to the parity-time symmetry phase transition. The corresponding high-contrast ratios
over a broad band of telecom wavelengths in Fig. 4b further confirm the asymmetric reflection of the parity-time metamaterial associated with the exceptional point. It is therefore evident that the presented parity-time metamaterial can successfully mimic the complicated quantum phenomena in classical on-chip optical waveguide systems. Similarly to previously investigated balanced gain/loss systems, proper engineering of the complex dielectric permittivity in a passive system also makes the exceptional point as well as its associated phase transition of parity-time symmetry become measurable quantities. The chip-scale parity-time metamaterial fabricated by conventional complementary metaloxide-semiconductor fabrication procedures clearly confirms
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2012 Macmillan Publishers Limited. All rights reserved.
5
NATURE MATERIALS DOI: 10.1038/NMAT3495
LETTERS the expected unidirectional and asymmetric characteristics in experiments. Moreover, the demonstrated unidirectional reflection can be converted to unidirectional invisibility by adding an additional linear amplifier to the presented paritytime metamaterial (see Supplementary Information). Further investigation of engineering the complex refractive index (linear and nonlinear parts) as well as geometric arrangement of these optical potentials is expected to create novel parity-time metamaterials with even more counterintuitive physical responses that once belonged only to the field of quantum theory, thus paving an approach to a new generation of photonic devices. For example, a four-port broadband unidirectional photonic device may construct backbones of chip-scale optical network analysers as its circuit counterpart in state-of-the-art microwave network analysers. Furthermore, the general design principle in our device in the optical domain can be extended to other frequencies and classical wave systems, such as unidirectional microwave invisibility for military applications and ultrasonic equipment for marine exploration31 .
Methods Derivation of attenuation and coupling coefficients in coupled mode equations. The attenuation coefficient α indicates the system absorption due to the imaginary part of the parity-time potentials and the coupling coefficient κ is the mode overlap between forward and backward propagating light. Both of them can be derived at the exceptional point, δ = 1, where the calculation is most simplified. At the exceptional point, the modulated mode effective index can be approximately written as neff ≈ 2.69 + 0.15cos(qz) − i0.15sin(qz). Therefore the attenuation −1 constant is calculated as α = 0 = 0.61 µm , while the coupling R nimag k0 = 0.15k R coefficient is κ = (k02 /2k1 )( E ∗ 1εEds/ |E|2 ds) = 0.48 µm−1 . Both α and κ are validated using FDTD simulations. At the exceptional point, the coupled mode equations can be written as α dA(z) = − A(z) dz 2π κ α dB(z) = −i A(z) + B(z) dz 4 2π The corresponding transmission and reflection coefficients are T = exp(−αL/π), Rf = (π2 κ 2 /4α 2 )sinh2 (αL/2π)exp(−αL/π), and Rb = 0, which can be used to fit the obtained reflection spectra from FDTD. α and κ in the best fitting results are about 0.61 µm−1 and 0.49 µm−1 (see Supplementary Information), consistent with the above analytical derivations. Sample fabrication. The fabrication starts with an SOI wafer. Two layers of periodically arranged sinusoidal-shaped combo structures are first patterned with accurate alignment in polymethyl methacrylate by electron beam lithography, followed by electron beam evaporation of Ge/Cr and Si and lift-off in acetone, respectively, in two steps. Then the Si waveguide is defined by aligned electron beam lithography using hydrogen silsesquioxane (HSQ), followed by dry etching with mixed gases of SF6 and C4 F8 . Because the developed HSQ becomes porous SiO2 and its refractive index is similar to SiO2 , it remains on top of the waveguide and therefore the topology of the sinusoidal combos on top of the waveguide became obscure in scanning electron microscope (SEM) images (Fig. 3b,c). Finally, plasma enhanced chemical vapour deposition with mixed gases of SiH4 and N2 O is used to deposit the cladding of SiO2 on the entire wafer to increase the light coupling efficiency from tapered fibres to waveguides.
Received 22 August 2012; accepted 22 October 2012; published online 25 November 2012
References 1. Leonhardt, U. Optical conformal mapping. Science 312, 1777–1780 (2006). 2. Pendry, J. B., Schurig, D. & Smith, D. R. Controlling electromagnetic fields. Science 312, 1780–1782 (2006). 3. Bender, C. M. & Böttcher, S. Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998). 4. Bender, C. M. Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947–1018 (2007). 5. Bender, C. M., Brody, D. C. & Jones, H. F. Extension of PT-symmetric quantum mechanics to quantum field theory with cubic interaction. Phys. Rev. D 70, 025001 (2004). 6. Longhi, S. & Della Valle, G. Photonic realization of PT-symmetric quantum field theories. Phys. Rev. A 85, 012112 (2012). 6
7. Makris, K. G., El-Ganainy, R., Christodoulides, D. N. & Musslimani, Z. H. Beam dynamics in PT-symmetric optical lattices. Phys. Rev. Lett. 100, 103904 (2008). 8. Musslimani, Z. H., El-Ganainy, R., Makris, K. G. & Christodoulides, D. N. Optical solitons in PT periodic potentials. Phys. Rev. Lett. 100, 030402 (2008). 9. Klaiman, S., Guenther, U. & Moiseyev, N. Visualization of branch points in PT symmetric waveguides. Phys. Rev. Lett. 101, 080402 (2008). 10. Longhi, S. Bloch oscillations in complex crystals with PT symmetry. Phys. Rev. Lett. 103, 123601 (2009). 11. Zheng, M. C., Christodoulides, D. N., Fleischmann, R. & Kottos, T. PT optical lattices and universality in beam dynamics. Phys. Rev. A 82, 010103(R) (2010). 12. Graefe, E. M. & Jones, H. F. PT-symmetric sinusoidal optical lattices at the symmetry-breaking threshold. Phys. Rev. A 84, 013818 (2011). 13. Miroshnichenko, A. E., Malomed, B. A. & Kivshar, Y. S. Nonlinearly PT-symmetric systems: Spontaneous symmetry breaking and transmission resonances. Phys. Rev. A 84, 012123 (2011). 14. Longhi, S. PT-symmetric laser absorber. Phys. Rev. A 82, 031801 (2010). 15. Chong, Y. D., Ge, L. & Stone, A. D. PT-symmetry breaking and laser-absorber modes in optical scattering systems. Phys. Rev. Lett. 106, 093902 (2011). 16. Ge, L. et al. Unconventional modes in lasers with spatially varying gain and loss. Phys. Rev. A 84, 023820 (2011). 17. Rüter, C. E. et al. Observation of parity–time symmetry in optics. Nature Phys. 6, 192–195 (2010). 18. Guo, A. et al. Observation of PT-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902 (2009). 19. Schindler, J., Li, A., Zheng, M. C., Ellis, F. M. & Kottos, T. Experimental study of active LRC circuits with PT symmetries. Phys. Rev. A 84, 040101(R) (2011). 20. Kulishov, M., Laniel, J., Bélanger, N., Azaña, J. & Plant, D. Nonreciprocal waveguide Bragg gratings. Opt. Express 13, 3068–3078 (2005). 21. Lin, Z. et al. Unidirectional invisibility induced by PT-symmetric periodic structures. Phys. Rev. Lett. 106, 213901 (2011). 22. Longhi, S. Invisibility in PT-symmetric complex crystals. J. Phys. A 44, 485302 (2011). 23. Jones, H. F. Analytic results for a PT-symmetric optical structure. J. Phys. A 45, 135306 (2012). 24. Mostafazadeh, A. Spectral singularities of complex scattering potentials and infinite reflection and transmission coefficients at real energies. Phys. Rev. Lett. 102, 220402 (2009). 25. Muga, J. G., Palaob, J. P., Navarroa, B. & Egusquizac, I. L. Complex absorbing potentials. Phys. Rep. 395, 357–426 (2004). 26. Cannata, F., Dedonder, J-P. & Ventura, A. Scattering in PT-symmetric quantum mechanics. Ann. Phys. 322, 397–433 (2007). 27. Regensburger, A. et al. Parity–time synthetic photonic lattices. Nature 488, 167–171 (2012). 28. Razzari, L. & Morandotti, R. Gain and loss mixed in the same cauldron. Nature 488, 163–164 (2012). 29. Kottos, T. Optical physics: Broken symmetry makes light work. Nature Phys. 6, 166–167 (2010). 30. Berry, M. V. Optical lattices with PT symmetry are not transparent. J. Phys. A 41, 244007 (2008). 31. Zhang, S., Xia, C. & Fang, N. Broadband acoustic cloak for ultrasound waves. Phys. Rev. Lett. 106, 024301 (2011).
Acknowledgements We acknowledge critical support and infrastructure provided for this work by the Kavli Nanoscience Institute at Caltech. This work was supported by the NSF ERC Center for Integrated Access Networks (no. EEC-0812072), the National Basic Research of China (no. 2012CB921503 and no. 2013CB632702), the National Nature Science Foundation of China (no. 11134006), the Nature Science Foundation of Jiangsu Province (no. BK2009007), the Priority Academic Program Development of Jiangsu Higher Education, and CAPES and CNPQ—Brazilian Foundations. M-H.L. also acknowledges the support of FANEDD of China.
Author contributions L.F. and M-H.L. conceived the idea. L.F., Y-L.X. and M-H.L. designed the device. Y-L.X., L.F. and M-H.L. performed the theoretical analysis of parity-time symmetry. W.S.F. and L.F. designed the chip and carried out fabrications and measurements. All the authors contributed to discussion of the project. Y-F.C. and A.S. guided the project. L.F. wrote the manuscript with revisions from other authors.
Additional information Supplementary information is available in the online version of the paper. Reprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to L.F. (for general design and experiment detail) or M-H.L. (for theoretical analysis).
Competing financial interests The authors declare no competing financial interests.
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials © 2012 Macmillan Publishers Limited. All rights reserved.
