Unidirectional invisibility in a two-layer non-PTsymmetric slab Yun Shen,1,* Xiao Hua Deng,2 and Lin Chen3 1

Department of Physics, Nanchang University, Nanchang 330031, China Institute of Space Science and Technology, Nanchang University, Nanchang 330031, China 3 Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan 430074, China * [email protected] 2

Abstract: Recently, unidirectional invisibility has been demonstrated in parity-time (PT) symmetric periodic structures and has attracted great attention. Nevertheless, fabrication of a complex periodic structure may not be practically easy. In this paper, a simple two-layer non-PT-symmetric slab structure is proposed to realize unidirectional invisibility. We numerically show that in such conventional structure consisting of two slabs with different real parts of refractive indices, unidirectional invisibility can be achieved as proper imaginary parts of refractive indices and thicknesses of the slabs are satisfied. Moreover, the unidirectional invisibility can be converted to unidirectional reflection when the imaginary parts of the refractive indices are tuned to their odd symmetric forms. ©2014 Optical Society of America OCIS codes: (230.7400) Waveguides, slab; (310.6860) Thin films, optical properties; (290.5839) Scattering, invisibility.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

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#211064 - $15.00 USD Received 29 Apr 2014; revised 16 Jul 2014; accepted 17 Jul 2014; published 4 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019440 | OPTICS EXPRESS 19440

19. C. C. Baker, J. Heikenfeld, Z. Yu, and A. J. Steckl, “Optical amplification and electroluminescence at 1.54 μm in Er-doped zinc silicate germanate on silicon,” Appl. Phys. Lett. 84(9), 1462–1464 (2004). 20. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. AP-14, 302–307 (1966). 21. Y. Shen and G. P. Wang, “Optical bistability in metal gap waveguide nanocavities,” Opt. Express 16(12), 8421– 8426 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-12-8421.

1. Introduction In the past few years, systems exhibiting parity-time (PT) symmetry have attracted considerable attention in various areas, ranging from quantum field theory [1], mathematical physics [2,3], solid-state [4,5] and atomic physics [6], to linear and nonlinear optics [7–9]. In PT symmetric systems, physical properties of quantum mechanics and quantum field theories can be created. By exploiting optical modulation of the complex refractive index, the constructed optical PT structures can lead to a series of intriguing optical phenomena, such as double refraction, power oscillations [10,11], absorption enhanced transmission [12], nonreciprocity of light propagation [10], and coherent perfect absorbing-lasing [13]. Recently, unidirectional invisibility has also been demonstrated in PT symmetric periodic structures [14], which shows that at the exceptional points, reflection of a PT symmetric periodic structure is diminished from one end while it is enhanced from the other end. This exotic property is of practical significance and it opens the door to a new generation of photonic devices, for example, chip-scale optical network analyzers based on unidirectional photonic implement [15]. Additionally, the general design principle in the optical domain can also be extended to other frequencies and classical wave systems, such as unidirectional microwave invisibility for military applications and ultrasonic equipment for marine exploration.   Nevertheless, as the complex refractive index obeying n( r ) = n* ( − r ) , i.e., even/odd symmetry for real/imaginary part of the refractive index, is strictly demanded in a PT structure and it is not easy to control it precisely in practice. Fabrication of such complex periodic structure still remains challenging. In this paper, a simple two-layer slab structure is proposed to realize unidirectional invisibility. Theoretical analysis and numerical calculations illustrate that unidirectional invisibility can be achieved in such conventional non-PT-symmetric structure, and no special relationship between the real parts of the refractive indices are demanded except they cannot be in same values. Particularly, unidirectional invisibility can be converted into unidirectional reflection when the imaginary parts of the refractive indices are converted into the odd symmetric forms. 2. Design and theory The proposed one-dimensional two-layer slab structure is schematically shown in Fig. 1(b) inset, where n1 , n2 and l1 , l2 represent the refractive indices and thicknesses of the slabs respectively. As light propagates along the z direction, the transmission and reflection properties of such structure can be calculated by transfer matrix method (TMM) [16,17], and the transfer matrices are read as cos δ i Mi =   − jηi sin δ i

− j sin δ i / ηi   (i = 1, 2), cos δ i 

(1)

with δ i = ω / c ⋅ ni li and ηi = (ε 0 / μ0 )1/2 ni . In those equations, ω is the frequency, c , ε 0 and μ0 are velocity of light in vacuum, permittivity and permeability of free space, respectively. As light propagates along the positive z direction, i.e., incident from the left side in Fig. 1(b) inset, the corresponding transmission coefficient tL and reflection coefficient rL can be deduced from M 1 ⋅ M 2 . Conversely, if light propagates along the negative z direction, i.e., incident from the right side, then tR and rR are offered by M 2 ⋅ M 1 . In this way, results of

#211064 - $15.00 USD Received 29 Apr 2014; revised 16 Jul 2014; accepted 17 Jul 2014; published 4 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019440 | OPTICS EXPRESS 19441

tR / tL = 1 and rR / rL being the function of ( η0 ,η1 ,η2 , δ1 , δ 2 ) are obtained when background

refractive index n0 for left and right sides are considered, where η0 = (ε 0 / μ0 )1/2 n0 . In other words, tL is always equal to tR no matter what n1 and n2 might be, but rL is generally different with rR and their ratio rR / rL is a complicated fraction formula. When numerator of this fraction equals zero and simultaneously the denominator does not, rR / rL ( rL / rR ) will come up to zero (infinite), which implies that only light incident from the left side of the two-layer structure has reflection. Similarly, as the denominator equals zero and numerator does not, with rL / rR ( rR / rL ) coming up to zero (infinite), only the right side has reflection. On every account, the phenomenon of unidirectional invisibility, which shows property of reflectlessness, can happen. To simplify the discussion and clearly demonstrate the unidirectional invisibility phenomena, in our paper we set δ1 satisfying

δ1 = 2 pπ + π / 2,( p = 0,1, 2...),

(2)

i.e., lossless n1 and suitable l1 are chosen. We note that in practice the requirement for a lossless n1 can be met by selecting active material with fitting gain coefficient, or ordinary material working in transparent regions, for example, SiO2 at 1550nm [18]. Under such δ1 conditions, rR / rL can be written as: rR ( n2 / n1 − n1 / n2 ) sin δ 2 + j ( n1 / n0 − n0 / n1 ) cos δ 2 = . rL ( n1 / n2 − n2 / n1 ) sin δ 2 + j ( n1 / n0 − n0 / n1 ) cos δ 2

(3)

The solutions for numerator and denominator polynomial of Eq. (3) being zero can be expressed as:

e j 2 L2 n2 =

± ( n22 − n12 ) + n2 ( n12 − 1) . ±( n22 − n12 ) − n2 ( n12 − 1)

(4)

In this equation, plus and minus signs correspond to the solution of numerator and denominator of Eq. (3) being zero respectively. In this calculation, L2 = ωl2 / c is defined as effective length and background refractive index n0 = 1 is assumed. For solution expression Eq. (4), we can further write it to L2 =

±( n22 − n12 ) + n2 ( n12 − 1) 1 [log + j 2mπ ], ( m = 0, 1, 2...) . j 2 n2 ±( n22 − n12 ) − n2 ( n12 − 1)

(5)

Consequently, when effective length L2 , refractive indices n1 and n2 together satisfy the relationship of Eq. (5), rR / rL of Eq. (3) possibly comes up to zero or infinite, leading to unidirectional invisibility for light incident from right side and left side of two-layer structure shown in Fig. 1(b) inset, respectively. It is important to note that in practice the length l2 , and so L2 must be real values. Due to this, relationships among L2 , n2 and n1 can be eventually determined.

#211064 - $15.00 USD Received 29 Apr 2014; revised 16 Jul 2014; accepted 17 Jul 2014; published 4 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019440 | OPTICS EXPRESS 19442

Fig. 1. (a)

n2

derived based on Eq. (5) for unidirectional invisibility of one-dimensional two-

layer slab in (b) inset as m = 10, 20, 30 and n1 = 1.444 . Groups (i) and (ii) correspond to solutions of unidirectional invisibility for left and right incidence, respectively; at singular point A, reflectionlessness happens for both right and left incidences. Based on corresponding effective length (b)

L2

n2

, the

can be obtained from Eq. (5). It is shown that no

matter n2 belongs to group (i) or group (ii) in (a), the same

n2'

will correspond to a same L2 .

3. Simulation and discussion

Specifically, in this paper we choose n1 ≈ 1.444 for example, which corresponds to the refractive index of SiO2 for 1550 nm [18] and meets the assumed lossless condition for Eq. (2). Then n2 that satisfies the demand of L2 being real values can be derived according to Eq. (5), and the relations between its real and imaginary parts for m = 10, 20, 30 are illustrated in Fig. 1(a). In this figure, the group (i) is deduced from Eq. (5) with the plus signs taken, which leads to numerator of Eq. (3) being zero and implies the reflectionless property for light incident from right side of two-layer structure [Fig. 1(b) inset]. The group (ii) is deduced with minus signs taken and leads to the denominator being zero, implying the reflectionless property of light from the left side. Here, it is a remarkable fact that the group (i) and (ii) are symmetric by line of n2'' = 0, while n2 = n2' + in2'' . Namely, as a n2 [see, group (i)] makes rR = 0, rL ≠ 0 [see, Eq. (3)] and offers unidirectional invisibility (reflection) for right (left) incidence, its conjugate value [see, group (ii)] can make rR ≠ 0, rL = 0 [see Eq. (3)] and offers unidirectional invisibility (reflection) for left (right) incidence. This also implies that the unidirectional invisibility for one side can be converted into unidirectional reflection when we tune n2'' to its opposite value. Next, due to n2 , the corresponding effective length L2 can be obtained from Eq. (5), and the dependences of L2 on n2' for m = 10, 20, 30 are illustrated in Fig. 1(b). It is shown that no matter n2 belongs to group (i) or group (ii) in Fig. 1(a), same n2'

#211064 - $15.00 USD Received 29 Apr 2014; revised 16 Jul 2014; accepted 17 Jul 2014; published 4 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019440 | OPTICS EXPRESS 19443

will correspond to a same L2 . Particularly, we note that in Fig. 1(a), singular points exist at n2 = n1 for both group (i) and (ii). Here, it is at n2 = 1.444, which combines with its corresponding L2 in Fig. 1(b) and leads to rR = 0, rL = 0 [see, Eq. (3)], and provides reflectionless property for both right and left incidences. Additionally, in our calculation to derive n2 based on Eq. (5), we assume that the demand of L2 being real values is satisfied when imaginary part of L2 is less than 10−5 . In other words, the truncation error of L2 we set for transcendental Eq. (5) is j10−5 . To verify the above results, the following takes the reflections of structure with n2 and corresponding L2 severally located at m = 10 in Fig. 1(a) (i) and Fig. 1(b) as an instance. As we know, in this case the numerator of Eq. (3), except for n2 = 1.444, should be zeros and leads to rR = 0, rL ≠ 0, resulting in the unidirectional invisibility for right incidence. For n2 = 1.444, the numerator and denominator of Eq. (3) should be both zeros, and rR = 0, rL = 0 are simultaneously provided. With TMM method, the calculated reflections being dependent on n2' are depicted in Fig. 2. In which, reflections for right and left incidence, i.e., | rR |2 and | rL |2 , are illustrated by the blue dotted and red circle curves respectively. The results on linear scale are plotted in Fig. 2 inset. From Fig. 2 we can see that the values of right reflection (blue dotted) in the considered n2' region are all less than 10−10 and approximated to

zero, except that region of n2' near 1.444, | rR |2 / | rL |2 < 10−8 are provided in the considered region. In other words, the reflections of the structure for right and left incidence have enormous difference, indicating unidirectional invisibility (right side) and reflection (left side) respectively. At n2' = 1.444, | rR |2 =| rL |2 in Fig. 2 both are approximately zeros. Those results well verify what expected from Fig. 1.

'

Fig. 2. Reflections depend on n2 of structure with n2 and corresponding L2 located at m = 10 in Fig. 1(a) (i) and Fig. 1(b) respectively. Reflections for right and left incidences, i.e.,

| rR |2 and | rL |2 ,

are illustrated by the blue dotted and red circle curves respectively. The results on linear scale are plotted in inset.

Furthermore, we can choose Si [18] and active material Er-doped Zn2Si0.5Ge0.5O4 (ZSG:Er) [19] as dielectrics n1 and n2 respectively, as an instance performed by finitedifference time-domain [17,20,21] to demonstrate the realization of unidirectional invisibility in practice. For 1550 nm, refractive index n1 of Si is approximately 1.444, and n2 of ZSG is n2' = 1.7514 with tunable n2'' able to be tuned by external field. In this instance from the curves of m = 10 in Fig. 1(a) (i) and Fig. 1(b) we can know, as n2'' = −0.03034, and L2 =

#211064 - $15.00 USD Received 29 Apr 2014; revised 16 Jul 2014; accepted 17 Jul 2014; published 4 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019440 | OPTICS EXPRESS 19444

18.79797 (corresponding to l2 = 4.6373 μm), the unidirectional invisibility for right incidence will happen. We note that the length of dielectric n1 need to be selected to make Eq. (2) satisfied, and it is artificially set as l1 = 3.4886 μm here. Figure 3(a) shows the optical intensity distribution (normalized to the incident intensity) along z direction when light from left is incident on the structure with the above parameters. In this simulation, the left unidirectional incident plane is located at z = −10.083 μm, which has a 6.5943 μm distance from the left face of n1 , and it is denoted with pink dashed line. Thus, we can detect the left reflected light intensity in the region of z11.232 μm and obtain | rR |2 ≈ 0.0125. Evidently, | rR |2 4.6373 μm and z

Unidirectional invisibility in a two-layer non-PT-symmetric slab.

Recently, unidirectional invisibility has been demonstrated in parity-time (PT) symmetric periodic structures and has attracted great attention. Never...
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