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A new type of nonlinear waveguides, photorefractive surface optical waveguides is suggested, which can be induced by photorefractive surface waves on the boundary of photorefractive crystal. The disturbed refractive index distribution of such waveguides behaves as a periodic lattice with apodized envelope, thus we call them photorefractive surface apodized waveguide arrays. Moreover, the dispersion relation and corresponding modes are analyzed. It is very interesting that the dispersion curves of index-guided modes and Bragg-guided modes couple and intertwine with each other, and anti-crossings instead of crossings between them hence generate some mini-gaps. Moreover there exists a type of extraordinary modes constituted by the splice of index-guided modes and Bragg-guided modes.
© 2015 Optical Society of America
1. Introduction
Surface waves (SWs), localized waves propagating at the interface between two media with different optical properties, are among the most intriguing phenomena in optics [1]. A new type of surface waves, namely, photorefractive surface waves (PR SWs) was suggested by Garcia Quirino et al in 1995 [2]. Self-guided SWs along the surface of a nonlinear medium ensure the concentration of light energy and result in strong enhancement of nonlinear surface optical phenomena, such as surface second harmonic generation [3,4 ].
Optical waveguide arrays have an enormous range of applications in molding the flow of light [5]. Due to the applications in optical breathers, optical filters and all-optical switching etc., tunable waveguides become a hotspot in the past years [6,7 ]. Owing to its photosensitive, real time and recyclable properties, photorefractive materials are often used to fabricate various tunable waveguide arrays by optical induction [8]. In 1995, the waveguides induced by photorefractive solitons were presented [9]. In 2003, J. W. Fleischer et al reported a 2D square photonic lattice induced by two interfering plane waves in SBN:75, and each waveguide has a diameter of about 7 μm [8]. However, introducing defects handily is also the major problem. Much like PR solitons, the waveguides can be also formed while PR SWs propagate near the boundary of PRCs and can be also tuned conveniently, in that the PR SWs can be adjusted by external electrical field, background illumination and incident beam etc [10]. However, such promising waveguides have not been fully investigated yet, we find this type of waveguides is provided with periodicity and abundant defects so that lots of intriguing phenomena are generated.
In this report we suggest a new type of waveguides induced by PR SWs for the first time to our knowledge. PR SWs induce a disturbed refractive index distribution from surface to bulk of the crystal, which behaves as a periodic lattice with apodized envelope. In consequence, this waveguide can essentially be recognized as a type of apodized waveguide arrays (AWGAs), a waveguide arrays modulated by apodized function. The apodized function has been broadly applied in fiber gratings [11,12 ]. Owing to their periodicity and abundant defects, lots of intriguing phenomena are generated, such as the coupled and intertwined dispersion curves of index-guided modes (IGMs) and Bragg-guided modes (BGMs), the anti-crossings and the mini-gaps between them and the extraordinary modes (EMs) constituted by the splice of IGMs and BGMs.
2. The formation of the apodized waveguide arrays
To begin with, we analyze theoretically how AWGAs are induced by PR SWs. Consider a slit e-polarized laser beam (transverse extent of the beam along the y axis greatly exceeds that along the x axis) propagating along the interface between air and a PR crystal, with optical c axis oriented along the x axis. The complex amplitude E(x,z) satisfies the nonlinear scalar wave equation:
In the air (x < 0), k = k 0 n 0 = 2π/λ 0, n 0 = 1 and λ 0 is the wavelength in vacuum. In PRC (x > 0), k = k 0(n + ∆n), n is the refractive index of e-polarized beam in the PRC, ∆n is the disturbed refractive index induced by nonlinearity, (n + ∆n)2 = n 2 − n 4 r eff E sc, r eff = r 33 is the effective electro-optical coefficient, E sc is the space-charge field. As is known that, the PR diffusion nonlinearity is the essential cause for PR SWs, other nonlinearity such as drift and photovoltaic only modulates PR SWs, and also PR dark SWs (PR DSWs) [10,13 ]. Thus, for convenience, only consider diffusion nonlinearity in this letter. Under open-circuit conditions, E sc can be written as:
Where k B is Boltzman constant, T is the temperature, q is the charge of carriers, (negative for the electrons and positive for the holes); I′(x) = I(x)/(Ib + Id) is the normalized intensity of light beam, I(x) is the light intensity of the PR surface, Ib and Id are the background illumination and the dark irradiance, respectively.We look for the stationary solution as E(x,z) = A(x)exp(ißz) = A′(x)(Ib + Id)1/2exp(ißz), where β is the propagating constant, A′(x) = I′(x)1/2 is the normalized real amplitude of PR SWs. Then the envelope equation can be written as:
The refractive index profile of the waveguide induced by PR SWs can be written as, according to (n + ∆n)2 = n2 − n4r eff Esc :
The crystal here is taken SBN:75 as sample, we solved Eq. (3) numerically and the parameters used in the calculation are: λ = 532 nm, n = 2.3117, r eff = 1340 × 10−12 m/V, q = −1.6 × 10−19 C, kB = 1.38 × 10−23 J/K, T = 300 K, P == 456.5 m. The continuity conditions at the interface of PRC-air are A′(x = 0) = A 0 and dA′(x = 0)/dx = A 0(β 2 − k 0 2)1/2 (A 0 is an arbitrary constant). The modes of PR SWs are determined by corresponding propagation constant. One of the typical periodical oscillating decayed PR SW mode is shown in Fig. 1(a) with n eff = 0.999890119, where n eff = β/nk 0 is the normalized effective refractive index.
Fig. 1 (a) PR SW mode with neff = 0.999890119 and P = 456.5 m, (b) the induced refractive index change ∆n and (c) the fine-structure of (b).
From the Eq. (4), we get the disturbed refractive index distribution as shown in Fig. 1(b). An interesting waveguide array called photorefractive surface apodized waveguide array (PR SAWGA) is induced, and this structure can be fixed by the photo-fixation and electro-fixation method, or directly guide weaker light or other light insensitive to the PR materials. Obviously, the AWGA has a stepped periodical oscillating decayed structure [8]. The ∆n along the x-axis has a step boundary between adjacent periods with continuously varying within a period, that is, it exhibits discrete characteristic.
3. The guided-wave properties of the apodized waveguide arrays
Then we adopt the transfer matrix method to analyze the guided-wave properties of such interesting AWGA, taking the TE modes for example. Consider a probe beam propagating in the AWGA, the amplitude of TE modes satisfies the scalar wave Eq. (5).
Obviously, Eq. (5) is equivalent to Eq. (3) provided Ey(x) = CA′(x), where C is an arbitrary constant. That is, there is a self-guided PR SW mode of in Fig. 1(a) for λ = 532 nm. In this article, the self-guided mode with neff = 0.9998905 < 1 is called Bragg-guided mode (BGM), it can be slso called Forbidden band mode (FBM), or Surface mode (SM) etc., as shown in Fig. 1(a). This indeed occurs only when the light wave is incident on the interface under conditions corresponding to that of a forbidden gap or Bragg reflection (BR) [14], a new transverse bound mechanism completely different from conventional total internal reflection (TIR).
In the end, we investigate the dispersion relation represented in Figs. 2(a)-2(c) and corresponding modes of the AWGA. For convenience, the modes will be designated as TEmn and TEmn with m as the mode index and n as the bound mechanism index, where n = 0, 1, 2 stand for index-guided modes (IGMs), first BGMs and second BGMs, respectively. Besides, ω/ω 0 is the normalized optical frequency, where ω 0 is optical frequency for λ = 532 nm. Figures 2(d) and 2(e) show the mode profiles of TE80 and TE70 for ω/ω 0 = 2 and 2.4630 marked with star in Fig. 2(b), respectively. The modes are plot in the same color with the corresponding dispersion curves, and so do other modes. Figures 2(f) and 2(g) show the modes of TE70 and TE80 for ω/ω 0 = 2.3964, corresponding to the first and third intersections of the dispersion curves and the right vertical dotted line in Fig. 2(b) from top to bottom, respectively; and the modes corresponding to the remaining intersections are orderly shown in Figs. 3(a1)-3(a12) .
Fig. 2 Dispersion relation of the AWGAs and corresponding mode profiles; (a) Dispersion relation of the TE00-TE60 modes; (b) and (c)the fine-structure of the dashed box in the (a) and (b), respectively; (d) TE80 with neff = 1.000005499 and (e) TE70 with neff = 1.000029190 are the mode profiles of marked point 1 and 2 in (b) for ω/ω0 = 2 and 2.4630, respectively; (f) TE70 with neff = 1.000028415 and (g) TE80 with neff = 1.000011877 are the mode profiles for ω/ω0 = 2.3964, corresponding to the first and third intersections of the dispersion curves and the right vertical dotted line in (b) from top to bottom, respectively.
Fig. 3 The remaining mode profiles for ω/ω0 = 2.3964, a(1)-(12) corresponding to intersections of dispersion curves and the vertical dotted line in Fig. 2(b) from top to bottom, respectively; index-guided modes: a(3) TE90 with neff = 1.000004330; a(4) TE10,0 with neff = 1.000001282; a(5) TE11,0 with neff = 1.000000298; first Bragg-guided modes: a(1) TE01 with neff = 1.000026972; a(2) TE11 with neff = 1.000009384; a(6) TE21 with neff = 0.999993666; a(7) TE31 with neff = 0.999964745; a(8) TE41 with neff = 0.999975992; Second Bragg-guided modes: a(9) TE02 with neff = 0.999923192; a(10) TE12 with neff = 0.999913323; a(11) TE22 with neff = 0.999902351; a(12) TE32 with neff = 0.999887535.
Moreover, we have found a lot of intriguing phenomena. Firstly, in general, there is exactly m zero crossings (nodes) of the x axis and the m order IGM. Compared with the TE70 for ω/ω 0 = 2.3964 [Fig. 2(f)], the TE70 for ω/ω 0 = 2.4630 [Fig. 2(e)] has an additional node, that is, the number of nodes is equal to TE80. However, it has a greatly different mode profile and nodes distribution with TE80, such as the TE80 for ω/ω 0 = 2 [Fig. 2(d)]. Similarly, the TE80 for ω/ω 0 = 2.3964 [Fig. 2(g)] has an analogous field distribution with the TE80 for ω/ω 0 = 2 [Fig. 2(d)] but it also has an additional node in the first period. Moreover, the phenomenon is generally found in other higher-order IGMs, such as the modes shown in Figs. 3(a3)-3(a5). This drastic change is due to the anti-crossing of dispersion curves, as depicted in Fig. 2(b): two curves that are expected to intersect rather than couple to one another (unless some symmetry prevents it) and the curves repel. Because of the periodicity in the waveguides, the gaps called mini-gaps or mode gaps opens up in the dispersion relation, and the dispersion curves couple at their intersection point, split and form two new hybrid curves composed of the related dispersion curves [5]. In addition, such phenomenon happens where IGMs and BGMs intersect. The inset in Fig. 2(b) shows the local picture of the mini-gap of TE70 and TE01. S. Olivier et al had also reported anti-crossing phenomenon in 2D photonic crystal waveguides and hold that the mini-gaps associated to the coupling between the modes around mini-gaps so that a dip or mini-stop band (MSB) was generated in the transmission spectrum of guided modes [15,16 ]. MSBs are characterized by the central frequency and bandwidth of the mini-gap. Moreover, MSBs have vast potential applications, such as filter, sensor, light amplifying, wavelength monitoring and broadband switch etc [17–19 ].
Secondly, there are some extraordinary modes (EMs) constituted by the splice of index-guided modes and Bragg-guided modes, such as Figs. 3(a2), 3(a6)-3(a8), and 3(a10)-3(a12). There is a boundary (the green dotted line) in the mode profile where the right part exhibits BGM characteristic and the left part presents the characteristic of IGM. We would classify the index (m) of these EMs according to their boundary locations. For illustration, the index of the first to fourth EMs [Figs. 3(a2), 3(a6)-3(a8)] are m = 1-4, respectively, whose boundaries are located at x = 8.22, 16.44, 24.66, 32.88 μm, corresponding to the 1-4th periodic boundaries of AWGAs from surface to bulk, respectively. These EMs essentially are canonical BGMs, bound by Bragg reflection, so that we would recognize them as BGMs. The appearing of such EMs is due to the stepped periodical oscillating decayed profile of the AWGAs. As the scalar guided-wave equation Eq. (5) is mathematically equivalent to the Schrödinger equation, we can consider the light propagating in a waveguide using the classical concept of solid-state physics. I. Tamm, E. T. Goodwin and W. Shockley had demonstrated that there exists the surface electronic state localized at crystal surface as consequence of the interruption of the periodic field [20,21 ]. In the AWGAs, each period interface can be regarded as an interruption of the periodic distribution, and the right region of interruption site can be seen as a new AWGA profile. Consequently, under the conditons of the simultaneous fulfillment of the mode eigen-equation in the IGM region and the Bragg condition in the posterior layered media, it will naturally led to the so called EMs mentioned above.
4. Conclusion
We have reported a new type of nonlinear waveguides, apodized waveguide arrays (AWGAs), induced by PR SWs for the first time, and analyzed its guided-wave properties. The AWGAs are of many intriguing guided-wave properties, such as the intertwined dispersion relation, the mini-gaps and the extraordinary guided modes etc. Moreover, owing to its convenience in fixing, scrubbing, adjusting and controlling, such AWGAs have a great prospect for the development and application of integrated optical components. We will continue to study the influence of external electric field, background illumination and incident beam on the AWGAs and its guided-wave properties in detail in the future.
Acknowledgments
This work was supported by the Chinese National Key Basic Research Special Fund (CNKBRSF) (2011CB922003), National Natural Science Foundation of China (NSFC) (61178005, J1103208) and Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP) (20120031110030).
References and links
1. B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, “Nonlocal surface-wave solitons,” Phys. Rev. Lett. 98(21), 213901 (2007). [CrossRef] [PubMed]
2. G. S. Garcia Quirino, J. J. Sanchez-Mondragon, and S. Stepanov, “Nonlinear surface optical waves in photorefractive crystals with a diffusion mechanism of nonlinearity,” Phys. Rev. A 51(2), 1571–1577 (1995). [CrossRef] [PubMed]
3. I. I. Smolyaninov, C. H. Lee, and C. C. Davis, “Giant enhancement of surface second harmonic generation in BaTiO3 due to photorefractive surface wave excitation,” Phys. Rev. Lett. 83(12), 2429–2432 (1999). [CrossRef]
4. H. Z. Kang, T. H. Zhang, H. H. Ma, C. B. Lou, S. M. Liu, J. G. Tian, and J. J. Xu, “Giant enhancement of surface second-harmonic generation using photorefractive surface waves with diffusion and drift nonlinearities,” Opt. Lett. 35(10), 1605–1607 (2010). [CrossRef] [PubMed]
5. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystal: Molding the Flow of Light (Princeton University, 2008).
6. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “All-optical switching and beam steering in tunable waveguide arrays,” Appl. Phys. Lett. 86(5), 051112 (2005). [CrossRef]
7. M. Morin, G. Duree, G. Salamo, and M. Segev, “Waveguides formed by quasi-steady-state photorefractive spatial solitons,” Opt. Lett. 20(20), 2066–2068 (1995). [CrossRef] [PubMed]
8. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003). [CrossRef] [PubMed]
9. M. Morin, G. Duree, G. Salamo, and M. Segev, “Waveguides formed by quasi-steady-state photorefractive spatial solitons,” Opt. Lett. 20(20), 2066–2068 (1995). [CrossRef] [PubMed]
10. T. H. Zhang, X. K. Ren, B. H. Wang, C. B. Lou, Z. J. Hu, W. W. Shao, Y. H. Xu, H. Z. Kang, J. Yang, D. P. Yang, L. Feng, and J. J. Xu, “Surface waves with photorefractive nonlinearity,” Phys. Rev. A 76(1), 013827 (2007). [CrossRef]
11. K. Ennser, M. N. Zervas, and R. I. Laming, “Optimization of apodized linearly chirped fiber gratings for optical communications,” IEEE J. Quantum Electron. 34(5), 770–778 (1998). [CrossRef]
12. N. Yokoi, T. Fujisawa, K. Saitoh, and M. Koshiba, “Apodized photonic crystal waveguide gratings,” Opt. Express 14(10), 4459–4468 (2006). [CrossRef] [PubMed]
13. Z. Luo, F. Liu, Y. Xu, H. Liu, T. Zhang, J. Xu, and J. Tian, “Dark surface waves in self-focusing media with diffusion and photovoltaic nonlinearities,” Opt. Express 21(13), 15075–15080 (2013). [CrossRef] [PubMed]
14. P. Yeh, A. Yariv, and C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67(4), 423–438 (1977). [CrossRef]
15. S. Olivier, M. Rattier, H. Benisty, C. Weisbuch, C. J. M. Smith, R. M. De La Rue, T. F. Krauss, U. Oesterle, and R. Houdré, “Mini-stopbands of a one-dimensional system: The channel waveguide in a two-dimensional photonic crystal,” Phys. Rev. B 63(11), 113311 (2001). [CrossRef]
16. S. Olivier, H. Benisty, C. Weisbuch, C. Smith, T. Krauss, and R. Houdré, “Coupled-mode theory and propagation losses in photonic crystal waveguides,” Opt. Express 11(13), 1490–1496 (2003). [CrossRef] [PubMed]
17. E. Viasnoff-Schwoob, C. Weisbuch, H. Benisty, C. Cuisin, E. Derouin, O. Drisse, G. Duan, L. Legouézigou, O. Legouézigou, F. Pommereau, S. Golka, H. Heidrich, H. J. Hensel, and K. Janiak, “Compact wavelength monitoring by lateral out-coupling in wedged photonic crystal multimode waveguides,” Appl. Phys. Lett. 86(10), 101107 (2005). [CrossRef]
18. N. Shahid, M. Amin, S. Naureen, and S. Anand, “Mini-stop bands in single heterojunction photonic crystal waveguides,” AIP Adv. 3(3), 032136 (2013). [CrossRef]
19. K. Y. Cui, X. Feng, Y. D. Huang, Q. Zhao, Z. L. Huang, and W. Zhang, “Broadband switching functionality based on defect mode coupling in W2 photonic crystal waveguide,” Appl. Phys. Lett. 101(15), 151110 (2012). [CrossRef]
20. I. E. Tamm, “On the possible bound states of electrons on a crystal surface,” Phys. Z. Sowjetunion 1, 733–735 (1932).
21. W. Shockley, “On the surface states associated with a periodic potential,” Phys. Rev. 56(4), 317–323 (1939). [CrossRef]
