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Surface defect lattice solitons in biased photovoltaic–photorefractive crystals
Optics & Laser Technology 75 (2015) 57–62
Contents lists available at ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Surface defect lattice solitons in biased photovoltaic–photorefractive crystals Juanli Hui, Keqing Lu n, Baoju Zhang, Jun Zhang, Haiying Xing School of Electronics and Information Engineering, Tianjin Polytechnic University, Tianjin 300387, China
art ic l e i nf o
a b s t r a c t
Article history: Received 8 April 2015 Received in revised form 12 June 2015 Accepted 15 June 2015
We show that surface defect lattice solitons (SDLSs) at the interface with a defect between the photonic lattices and the uniform photovoltaic–photorefractive (PP) crystals with external applied field are possible. These SDLSs exist only in the semi-infinite gap for positive defect strength, and in the semi-infinite, first, and second gaps for negative defect strength. In the semi-infinite gap, SDLSs are stable in the low power region and unstable in the high power region. In the first and second gaps, SDLSs are stable in the all power region. We find that these SDLSs are those studied previously in biased non-PP crystals when the bulk photovoltaic effect is negligible and are those in PP crystals when the external bias field is absent. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Surface waves Photonic lattices Photorefractive nonlinearity Nonlinear optics
1. Introduction Study of light propagation in optical periodic structures such as waveguide arrays, photonic crystals, and optically-induced photonic lattices has attracted growing interest due to both fundamental physics and applications. In such periodic structures, linear light propagation exhibits Bloch bands and forbidden bandgaps. When the forward- and backward-propagating waves experience Bragg scattering from the periodic structures, their nonlinear coupling can produce gap solitons [1], which exist in different bandgaps. Many branches of gap solitons in different gaps are known: fundamental solitons [2–9], dipole solitons [10,11], vortex solitons [12–14], quadrupole solitons [15], and defect solitons [16– 21], all of which form in bulk periodic mediums and were investigated in photorefractive crystals. The light induced refractive index change effect is shortened to the photorefractive effect, which is a nonlinear optical phenomenon of the refractive index change resulting from the spatial distribution of light intensity in irradiated photorefractive crystals [22]. In biased photorefractive crystals, the drift current dominates, and as a result the spacecharge field can induce an index waveguide by means of the Pockels effect. The latter process is capable of counteracting the effects of diffraction, and the end result is a photorefractive spatial soliton. In photovoltaic–photorefractive (PP) crystals, the uniform light irradiation can produce the open-circuit voltage or the n
Corresponding author. E-mail address: [email protected] (K. Lu).
http://dx.doi.org/10.1016/j.optlastec.2015.06.018 0030-3992/& 2015 Elsevier Ltd. All rights reserved.
equipollent short-circuit current, which is known as the photovoltaic effect producing photovoltaic solitons [22]. Surface waves are a special type of waves, are confined at the very boundary between two different media, and have been researched widely [23,24]. Gap solitons may also exist at periodically modulated surfaces [25]. Surface solitons at the interface between the uniform media and the periodic waveguide arrays [26–31], at the interface of two periodic media [32–35], and at the interface between the photonic lattices and the uniform photorefractive crystals [36,37] have been proposed and observed. On the other hand, the interface with a defect can support surface solitons [38,39]. Surface defect lattice solitons (SDLSs) in biased non-PP crystals have been predicted [39]. However, it would be of interest to explore whether SDLSs can be realized at the interface with a defect between the photonic lattices and the uniform PP crystals with external applied field as well. In this paper, we report that SDLSs are possible at the interface with a defect between the photonic lattices and the uniform PP crystals with external applied field. These SDLSs exist in different bandgaps due to the change of defect strength. For positive defect strength, SDLSs exist only in the semi-infinite gap and are stable in the low power region and unstable in the high power region. For negative defect strength, SDLSs exist in the semi-infinite, first, and second gaps. In the semi-infinite gap, SDLSs are stable in the low power region and unstable in the high power region. In the first and second gaps, SDLSs are stable in the all power region. When the bulk photovoltaic effect is negligible, these SDLSs are those studied previously in biased non-PP crystals. When the external bias field is absent, these SDLSs are those in PP crystals.
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2. Theoretical model To start, let us consider the physical situation in which an ordinarily polarized beam through a mask is launched into a biased PP crystal. The mask can control the distribution of optical intensity that forms the interface with a defect between the photonic lattices and the uniform biased PP crystal. Meanwhile, an extraordinarily polarized probe beam is launched into the defect site, propagating along the interface between the photonic lattices and the uniform biased PP crystal. The propagation of the probe beam in such a lattice is described by the nonlinear Schrödinger equation [1]. The nondimensionalized equation for the light field q is [17,19,40]
i
IL + q 2 ∂q ∂ 2q 1 =− + E0 q − Ep q, 2 2 ∂z ∂x IL + q + 1 IL + q 2 + 1
(1)
where q is the slowly varying amplitude of the probe beam, z is the normalized longitudinal coordinate (in units of 2kT 3/π 2), T is the lattice spacing, k = 2πne /λ is the optical wave number in the PP crystal, ne is the unperturbed extraordinary index of refraction, λ is the wavelength, x is the normalized transverse coordinate (in units of T /π ), E0 is the applied dc field [in units of π 2/(T 2k 2ne2 r33 )], r33 is the electro-optic coefficient, E p is the photovoltaic field constant [in units of π 2/(T 2k 2ne2 r33 )], IL is the intensity function of the photonic lattices described by
⎧ 2 ⎪I 0 cos (x)[1 + εg (x)], x ≥ −π /2 IL = ⎨ ⎪ x < −π /2 ⎩ 0,
(2)
Here I0 is the lattice peak intensity normalized by the dark irradiance Id , ε controls the strength of the defect, and g (x ) is a
localized function describing the shape of the defect. At this point, let us assume that the defect is restricted to a single lattice site at x = 0. Thus, we choose function g (x ) as g (x ) = exp ( − x8/128) and take −1 ≤ ε ≤ 1. For ε > 0, the lattice light intensity IL at the defect site is higher than that without defect. For ε < 0, the lattice intensity IL at the defect site is lower than that without defect. For ε = 0, the photonic lattices are uniform. The intensity distributions of optical lattices with ε = 0, 0.3, and −0.7 are shown in Fig. 1(b)– (d), respectively. The first term in the right-hand side of Eq. (1) describes the diffraction spreading of the beam, the second and the third terms describe the beam self-focusing or self-defocusing caused by the local drift component of nonlinear response and the photovoltaic effect of the photorefractive nonlinearity, respectively. Note that if the one is the self-defocusing, the other is the self-focusing, the value of the term with the beam self-focusing is greater than that with the beam self-defocusing, because the beam self-focusing gives rise to SDLSs in the system of Eq. (1). Evidently, the range of PP crystal use expands through an externally applied electric field. In this paper, let us consider a BaTiO3 PP crystal with the following parameters ne = 2.365 and r33 = 80 × 10−12m/V at a wavelength λ = 0.5μm . If T = 20μm , we find that one E0 or E p unit corresponds to 62V/mm , one x unit corresponds to 6.4μm , and one z unit corresponds to 2.4mm . We take I0 = 3, E0 = 6 , and E p = 18, which are typical experimental conditions [17]. In order to show the existent conditions for SDLSs, let us first understand the dispersion relation and bandgap structure of the linear version of Eq. (1). According to the Bloch theorem, eigenfunctions of the linear version of Eq. (1) can be sought in the form q = f (x ) exp (ik x x − ibz ), where f (x ) is the complex periodic function with the same periodicity as the lattices, k x is wave number in
Fig. 1. (a) Bandgap structure of a uniform lattice at I0 = 3 and E p = 18; the shaded regions are Bloch bands. (b)–(d) Lattice intensity profiles at ε = 0 , 0.3, and −0.7, respectively, when I0 = 3.
J. Hui et al. / Optics & Laser Technology 75 (2015) 57–62
59
the first Brillouin zone, and b is the Bloch-wave propagation constant. Substitution of this latter form of f (x ) into the linear version of Eq. (1) yields
eigenvalue [41]. If the growth rate Re (δ ) > 0, SDLSs are linearly unstable, otherwise they are linearly stable.
IL d2f df 1 + 2ik x − k x2 f − E0 f + Ep f = − bf , dx 1 + IL 1 + IL dx2
3. Numerical results
(3)
from which the bandgap diagram can be obtained by the plane wave expansion method. Fig. 1(a) shows the bandgap structure of the uniform photonic lattices when I0 = 3 and E p = 18. It reveals that there exist four complete gaps which are defined as the semiinfinite, first, second, and third gaps, respectively. The stationary soliton solutions in Eq. (1) are sough in the form q = f (x ) exp ( − ibz ), where f (x ) is the real-valued function. By substituting this form of q (x, z ) into Eq. (1), we find that f (x ) satisfies the nonlinear equation
IL + f 2 d2f 1 f + Ep f + bf = 0, − E0 2 2 dx 1 + IL + f 1 + IL + f 2
(4)
from which SDLSs f (x ) can be determined by the modified squared-operator iteration method [41]. The power of SDLSs is ∞ defined as P = ∫ f 2 (x )dx . −∞ To indicate the stability of SDLSs, we introduce perturbations in the form
q (x, z ) = [f (x) + W (x, z ) + iV (x, z )] exp ( − ibz ),
(5)
where W (x, z ) = w (x ) exp (δz ) and V (x, z ) = v (x ) exp (δz ) are the real and the imaginary parts of perturbations that can grow with a complex rate δ upon propagation. Substituting Eq. (5) into Eq. (1) and linearizing it, we obtain the following linear eigenvalue problem:
δw = −
δv =
IL + f 2 1 d2v − bv + E0 v − Ep v, 2 2 dx 1 + IL + f 1 + IL + f 2
(6a)
1 + IL − f 2 d2w + bw − E0 w 2 dx (1 + IL + f 2 )2 + Ep
(1 + I
L
+ f2
2
)
(1 + I
L
− IL + f 2 − 1 + f2
2
)
w, (6b)
from which the growth rate Re (δ ) can be determined by expanding functions (w, v ) into discrete Fourier series and then converting Eqs. (6a) and (6b) into a matrix eigenvalue problem with δ as the
We consider first SDLSs in Eq. (4) for positive defect strength (ε > 0). Fig. 2(a) shows the power diagram of SDLSs at I0 = 3, E0 = 6, and E p = 18 when ε = 0.3, where dotted curve represents unstable SDLSs and solid curve represents stable SDLSs. This figure demonstrates that SDLSs only exist in the semi-infinite gap and that the power of SDLSs decreases with an increase in the propagation constant. When b > − 10.85, SDLSs can not exist in the semi-infinite gap. Fig. 2(b) shows the real parts of perturbation growth rate Re (δ ) versus the propagation constant for ε = 0.3 when I0 = 3, E0 = 6, and E p = 18. In the range of propagation constant −16.34 < b < − 10.85 (corresponding to low power) in Fig. 2 (a), SDLSs are stable according to the diagram of growth rate Re (δ ) as shown in Fig. 2(b). When b < − 16.34 (corresponding to high power) in Fig. 2(a), SDLSs can not stably exist because of Re (δ ) > 0. When b = − 14.30 and −16.55 [points a, b in Fig. 2(a) and (b)], the profiles of SDLSs are displayed in Fig. 3(a) and (b). To further study the stability of SDLSs, Eq. (1) has been solved numerically using a beam propagation method. SDLSs given by Eq. (4) have been used as the input beam profiles. In all numerical simulations, we add a noise to the inputted SDLSs by multiplying them with [1 + ρ (x ) ], where ρ (x ) is the random function with Gaussian distribution and 2 2 variance σnoise (we choose that σnoise is equal to the white noise with 2 = 0.01). The evolution of the SDLSs corresponding to variance σnoise Fig. 3(a) and (b) are shown in Fig. 3(c) and (d). The stable behavior of the SDLS corresponding to Fig. 3(a) is evident in Fig. 3(c) since the SDLS retains its input structure. The SDLS corresponding to Fig. 3(b) can not stably propagate, as shown in Fig. 3(d). These results are in agreement with the growth rates Re (δ ) in Fig. 2(b). On the other hand, according to an effective particle approach the effective potential can be formed in the vicinity of the interface with a defect between the photonic lattices and the uniform PP crystals with external applied field [42]. Stable (unstable) SDLSs correspond to minima (maxima) of the effective potential. Close to the stable SDLS, the SDLS in Fig. 3(b) is trapped and oscillates in the potential well [see Figs. 2(b) and 3(d)] [42]. By the way, we also demonstrate that the oscillation period in the potential well is adjusted by the change of E0 . For ε < 0, we find that SDLSs exist not only in the semi-infinite gap but also in the first and second gaps. The power of SDLSs decreases with an increase in propagation constant, as shown in
Fig. 2. (a) Power P versus propagation constant b (blue regions are Bloch band) at I0 = 3, E0 = 6 , and E p = 18 when ε = 0.3. (b) Perturbation growth rate Re (δ ) versus the propagation constant b at I0 = 3, E0 = 6 , and E p = 18 when ε = 0.3. In (a), the solid curve indicates the stable SDLSs, and the dotted curve indicates the unstable SDLSs, see (b). Profiles of SDLSs at the circled points in (a) and (b) are displayed in Fig. 3. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 3. (a, b) Profiles of SDLSs at the circled points a, b in Fig. 2, with (a) b = − 14.30 and (b) −16.51, respectively. (c, d) Propagation of SDLSs corresponding to (a) and (b), respectively. The shaded stripes in (a) and (b) represent the locations of high intensities in the defected lattice.
Fig. 4(a). Moreover, SDLSs exist in the whole first gap and in the second gap when b < − 3.50. The real part of perturbation growth rate Re (δ ) at I0 = 3, E0 = 6 , and E p = 18 for ε = − 0.7 is shown in Fig. 4(b). In the semi-infinite gap, SDLSs in Fig. 4(a) are stable when −15.40 < b < − 9.68 (corresponding to low power) and unstable when b < − 15.40 (corresponding to high power) [see Fig. 4(b)]. In the first and second gaps, SDLSs are stable, as shown in Fig. 4(b). Fig. 5(a)–(c) shows the profiles of SDLSs when b = − 16.25, −6.10, and −3.81 [points a, b, and c in Fig. 4(a) and (b)]. The evolution of the SDLSs corresponding to Fig. 5(a)–(c) are
shown in Fig. 5(d)–(f). The SDLS corresponding to Fig. 5(a) is unstable and does not oscillate in the all effective potential because it is far from the stable SDLS [see Figs. 4(b) and 5(d)] [43]. The SDLSs corresponding to Fig. 5(e) and (f) are stable. The centers of the SDLSs in the semi-infinite gap reside in the first lattice channel [see Fig. 5(a)]. In the similitude of surface solitons in waveguide arrays [43], SDLSs in the second gap penetrate deeper inside the photonic lattices, oscillations on their profiles become more pronounced [see Fig. 5(b) and (c)]. Thus the interface effectively affects the finite low-gap SDLSs.
Fig. 4. (a) Power P versus propagation constant b (blue regions are Bloch band) at I0 = 3, E0 = 6 , and E p = 18 when ε = − 0.7. (b) Perturbation growth rate Re (δ ) versus the propagation constant b at I0 = 3, E0 = 6 , and E p = 18 when ε = − 0.7. In (a), the solid curve indicates the stable SDLSs, and the dotted curve indicates the unstable SDLSs, see (b). Profiles of SDLSs at the circled points in (a) and (b) are displayed in Fig. 5. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 5. (a)–(c) Profiles of SDLSs at the circled points a, b, and c in Fig. 4, with (a) b = − 16.2, (b) −6.1, and (c) −3.8, respectively. (d)–(f) Propagation of SDLSs corresponding to (a)–(c), respectively. The shaded stripes in (a)–(c) represent the locations of high intensities in the defected lattice.
Finally, it is interesting to discuss the properties of SDLSs in biased PP crystals. When the bulk photovoltaic effect is negligible, i.e., E p = 0, our physical system becomes the physical system of SDLSs studied previously at the interface with a defect between the photonic lattices and the uniform biased non-PP crystals. Substitution E p = 0 into Eqs. (1) and (4) leads to the same expression (1) and expression (3) in Ref. [39], respectively. When the external bias field is absent, i.e., E0 = 0, our physical system becomes the physical system of SDLSs at the interface with a defect between the photonic lattices and the uniform PP crystals. In this case, SDLSs in PP crystals can be obtained from Eqs. (1) to (6).
4. Conclusion In conclusion, we have found that SDLSs are possible at the interface with a defect between the photonic lattices and the uniform PP crystals with external applied field. We have shown that these SDLSs exist only in the semi-infinite gap for positive defect strength and in the semi-infinite, first, and second gaps for negative defect strength. In the semi-infinite gap, SDLSs are stable in the low power region and unstable in the high power region. In the first and second gaps, SDLSs are stable in the all power region. The properties of these SDLSs have been discussed and we have found that they are those studied previously in biased non-PP
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crystals when the bulk photovoltaic effect is negligible and are those in PP crystals when the external bias field is absent.
Acknowledgment The work was supported by the Natural Science Foundation of Chinese Tianjin (No. 13JCYBJC16400).
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Contents lists available at ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Surface defect lattice solitons in biased photovoltaic–photorefractive crystals Juanli Hui, Keqing Lu n, Baoju Zhang, Jun Zhang, Haiying Xing School of Electronics and Information Engineering, Tianjin Polytechnic University, Tianjin 300387, China
art ic l e i nf o
a b s t r a c t
Article history: Received 8 April 2015 Received in revised form 12 June 2015 Accepted 15 June 2015
We show that surface defect lattice solitons (SDLSs) at the interface with a defect between the photonic lattices and the uniform photovoltaic–photorefractive (PP) crystals with external applied field are possible. These SDLSs exist only in the semi-infinite gap for positive defect strength, and in the semi-infinite, first, and second gaps for negative defect strength. In the semi-infinite gap, SDLSs are stable in the low power region and unstable in the high power region. In the first and second gaps, SDLSs are stable in the all power region. We find that these SDLSs are those studied previously in biased non-PP crystals when the bulk photovoltaic effect is negligible and are those in PP crystals when the external bias field is absent. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Surface waves Photonic lattices Photorefractive nonlinearity Nonlinear optics
1. Introduction Study of light propagation in optical periodic structures such as waveguide arrays, photonic crystals, and optically-induced photonic lattices has attracted growing interest due to both fundamental physics and applications. In such periodic structures, linear light propagation exhibits Bloch bands and forbidden bandgaps. When the forward- and backward-propagating waves experience Bragg scattering from the periodic structures, their nonlinear coupling can produce gap solitons [1], which exist in different bandgaps. Many branches of gap solitons in different gaps are known: fundamental solitons [2–9], dipole solitons [10,11], vortex solitons [12–14], quadrupole solitons [15], and defect solitons [16– 21], all of which form in bulk periodic mediums and were investigated in photorefractive crystals. The light induced refractive index change effect is shortened to the photorefractive effect, which is a nonlinear optical phenomenon of the refractive index change resulting from the spatial distribution of light intensity in irradiated photorefractive crystals [22]. In biased photorefractive crystals, the drift current dominates, and as a result the spacecharge field can induce an index waveguide by means of the Pockels effect. The latter process is capable of counteracting the effects of diffraction, and the end result is a photorefractive spatial soliton. In photovoltaic–photorefractive (PP) crystals, the uniform light irradiation can produce the open-circuit voltage or the n
Corresponding author. E-mail address: [email protected] (K. Lu).
http://dx.doi.org/10.1016/j.optlastec.2015.06.018 0030-3992/& 2015 Elsevier Ltd. All rights reserved.
equipollent short-circuit current, which is known as the photovoltaic effect producing photovoltaic solitons [22]. Surface waves are a special type of waves, are confined at the very boundary between two different media, and have been researched widely [23,24]. Gap solitons may also exist at periodically modulated surfaces [25]. Surface solitons at the interface between the uniform media and the periodic waveguide arrays [26–31], at the interface of two periodic media [32–35], and at the interface between the photonic lattices and the uniform photorefractive crystals [36,37] have been proposed and observed. On the other hand, the interface with a defect can support surface solitons [38,39]. Surface defect lattice solitons (SDLSs) in biased non-PP crystals have been predicted [39]. However, it would be of interest to explore whether SDLSs can be realized at the interface with a defect between the photonic lattices and the uniform PP crystals with external applied field as well. In this paper, we report that SDLSs are possible at the interface with a defect between the photonic lattices and the uniform PP crystals with external applied field. These SDLSs exist in different bandgaps due to the change of defect strength. For positive defect strength, SDLSs exist only in the semi-infinite gap and are stable in the low power region and unstable in the high power region. For negative defect strength, SDLSs exist in the semi-infinite, first, and second gaps. In the semi-infinite gap, SDLSs are stable in the low power region and unstable in the high power region. In the first and second gaps, SDLSs are stable in the all power region. When the bulk photovoltaic effect is negligible, these SDLSs are those studied previously in biased non-PP crystals. When the external bias field is absent, these SDLSs are those in PP crystals.
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2. Theoretical model To start, let us consider the physical situation in which an ordinarily polarized beam through a mask is launched into a biased PP crystal. The mask can control the distribution of optical intensity that forms the interface with a defect between the photonic lattices and the uniform biased PP crystal. Meanwhile, an extraordinarily polarized probe beam is launched into the defect site, propagating along the interface between the photonic lattices and the uniform biased PP crystal. The propagation of the probe beam in such a lattice is described by the nonlinear Schrödinger equation [1]. The nondimensionalized equation for the light field q is [17,19,40]
i
IL + q 2 ∂q ∂ 2q 1 =− + E0 q − Ep q, 2 2 ∂z ∂x IL + q + 1 IL + q 2 + 1
(1)
where q is the slowly varying amplitude of the probe beam, z is the normalized longitudinal coordinate (in units of 2kT 3/π 2), T is the lattice spacing, k = 2πne /λ is the optical wave number in the PP crystal, ne is the unperturbed extraordinary index of refraction, λ is the wavelength, x is the normalized transverse coordinate (in units of T /π ), E0 is the applied dc field [in units of π 2/(T 2k 2ne2 r33 )], r33 is the electro-optic coefficient, E p is the photovoltaic field constant [in units of π 2/(T 2k 2ne2 r33 )], IL is the intensity function of the photonic lattices described by
⎧ 2 ⎪I 0 cos (x)[1 + εg (x)], x ≥ −π /2 IL = ⎨ ⎪ x < −π /2 ⎩ 0,
(2)
Here I0 is the lattice peak intensity normalized by the dark irradiance Id , ε controls the strength of the defect, and g (x ) is a
localized function describing the shape of the defect. At this point, let us assume that the defect is restricted to a single lattice site at x = 0. Thus, we choose function g (x ) as g (x ) = exp ( − x8/128) and take −1 ≤ ε ≤ 1. For ε > 0, the lattice light intensity IL at the defect site is higher than that without defect. For ε < 0, the lattice intensity IL at the defect site is lower than that without defect. For ε = 0, the photonic lattices are uniform. The intensity distributions of optical lattices with ε = 0, 0.3, and −0.7 are shown in Fig. 1(b)– (d), respectively. The first term in the right-hand side of Eq. (1) describes the diffraction spreading of the beam, the second and the third terms describe the beam self-focusing or self-defocusing caused by the local drift component of nonlinear response and the photovoltaic effect of the photorefractive nonlinearity, respectively. Note that if the one is the self-defocusing, the other is the self-focusing, the value of the term with the beam self-focusing is greater than that with the beam self-defocusing, because the beam self-focusing gives rise to SDLSs in the system of Eq. (1). Evidently, the range of PP crystal use expands through an externally applied electric field. In this paper, let us consider a BaTiO3 PP crystal with the following parameters ne = 2.365 and r33 = 80 × 10−12m/V at a wavelength λ = 0.5μm . If T = 20μm , we find that one E0 or E p unit corresponds to 62V/mm , one x unit corresponds to 6.4μm , and one z unit corresponds to 2.4mm . We take I0 = 3, E0 = 6 , and E p = 18, which are typical experimental conditions [17]. In order to show the existent conditions for SDLSs, let us first understand the dispersion relation and bandgap structure of the linear version of Eq. (1). According to the Bloch theorem, eigenfunctions of the linear version of Eq. (1) can be sought in the form q = f (x ) exp (ik x x − ibz ), where f (x ) is the complex periodic function with the same periodicity as the lattices, k x is wave number in
Fig. 1. (a) Bandgap structure of a uniform lattice at I0 = 3 and E p = 18; the shaded regions are Bloch bands. (b)–(d) Lattice intensity profiles at ε = 0 , 0.3, and −0.7, respectively, when I0 = 3.
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the first Brillouin zone, and b is the Bloch-wave propagation constant. Substitution of this latter form of f (x ) into the linear version of Eq. (1) yields
eigenvalue [41]. If the growth rate Re (δ ) > 0, SDLSs are linearly unstable, otherwise they are linearly stable.
IL d2f df 1 + 2ik x − k x2 f − E0 f + Ep f = − bf , dx 1 + IL 1 + IL dx2
3. Numerical results
(3)
from which the bandgap diagram can be obtained by the plane wave expansion method. Fig. 1(a) shows the bandgap structure of the uniform photonic lattices when I0 = 3 and E p = 18. It reveals that there exist four complete gaps which are defined as the semiinfinite, first, second, and third gaps, respectively. The stationary soliton solutions in Eq. (1) are sough in the form q = f (x ) exp ( − ibz ), where f (x ) is the real-valued function. By substituting this form of q (x, z ) into Eq. (1), we find that f (x ) satisfies the nonlinear equation
IL + f 2 d2f 1 f + Ep f + bf = 0, − E0 2 2 dx 1 + IL + f 1 + IL + f 2
(4)
from which SDLSs f (x ) can be determined by the modified squared-operator iteration method [41]. The power of SDLSs is ∞ defined as P = ∫ f 2 (x )dx . −∞ To indicate the stability of SDLSs, we introduce perturbations in the form
q (x, z ) = [f (x) + W (x, z ) + iV (x, z )] exp ( − ibz ),
(5)
where W (x, z ) = w (x ) exp (δz ) and V (x, z ) = v (x ) exp (δz ) are the real and the imaginary parts of perturbations that can grow with a complex rate δ upon propagation. Substituting Eq. (5) into Eq. (1) and linearizing it, we obtain the following linear eigenvalue problem:
δw = −
δv =
IL + f 2 1 d2v − bv + E0 v − Ep v, 2 2 dx 1 + IL + f 1 + IL + f 2
(6a)
1 + IL − f 2 d2w + bw − E0 w 2 dx (1 + IL + f 2 )2 + Ep
(1 + I
L
+ f2
2
)
(1 + I
L
− IL + f 2 − 1 + f2
2
)
w, (6b)
from which the growth rate Re (δ ) can be determined by expanding functions (w, v ) into discrete Fourier series and then converting Eqs. (6a) and (6b) into a matrix eigenvalue problem with δ as the
We consider first SDLSs in Eq. (4) for positive defect strength (ε > 0). Fig. 2(a) shows the power diagram of SDLSs at I0 = 3, E0 = 6, and E p = 18 when ε = 0.3, where dotted curve represents unstable SDLSs and solid curve represents stable SDLSs. This figure demonstrates that SDLSs only exist in the semi-infinite gap and that the power of SDLSs decreases with an increase in the propagation constant. When b > − 10.85, SDLSs can not exist in the semi-infinite gap. Fig. 2(b) shows the real parts of perturbation growth rate Re (δ ) versus the propagation constant for ε = 0.3 when I0 = 3, E0 = 6, and E p = 18. In the range of propagation constant −16.34 < b < − 10.85 (corresponding to low power) in Fig. 2 (a), SDLSs are stable according to the diagram of growth rate Re (δ ) as shown in Fig. 2(b). When b < − 16.34 (corresponding to high power) in Fig. 2(a), SDLSs can not stably exist because of Re (δ ) > 0. When b = − 14.30 and −16.55 [points a, b in Fig. 2(a) and (b)], the profiles of SDLSs are displayed in Fig. 3(a) and (b). To further study the stability of SDLSs, Eq. (1) has been solved numerically using a beam propagation method. SDLSs given by Eq. (4) have been used as the input beam profiles. In all numerical simulations, we add a noise to the inputted SDLSs by multiplying them with [1 + ρ (x ) ], where ρ (x ) is the random function with Gaussian distribution and 2 2 variance σnoise (we choose that σnoise is equal to the white noise with 2 = 0.01). The evolution of the SDLSs corresponding to variance σnoise Fig. 3(a) and (b) are shown in Fig. 3(c) and (d). The stable behavior of the SDLS corresponding to Fig. 3(a) is evident in Fig. 3(c) since the SDLS retains its input structure. The SDLS corresponding to Fig. 3(b) can not stably propagate, as shown in Fig. 3(d). These results are in agreement with the growth rates Re (δ ) in Fig. 2(b). On the other hand, according to an effective particle approach the effective potential can be formed in the vicinity of the interface with a defect between the photonic lattices and the uniform PP crystals with external applied field [42]. Stable (unstable) SDLSs correspond to minima (maxima) of the effective potential. Close to the stable SDLS, the SDLS in Fig. 3(b) is trapped and oscillates in the potential well [see Figs. 2(b) and 3(d)] [42]. By the way, we also demonstrate that the oscillation period in the potential well is adjusted by the change of E0 . For ε < 0, we find that SDLSs exist not only in the semi-infinite gap but also in the first and second gaps. The power of SDLSs decreases with an increase in propagation constant, as shown in
Fig. 2. (a) Power P versus propagation constant b (blue regions are Bloch band) at I0 = 3, E0 = 6 , and E p = 18 when ε = 0.3. (b) Perturbation growth rate Re (δ ) versus the propagation constant b at I0 = 3, E0 = 6 , and E p = 18 when ε = 0.3. In (a), the solid curve indicates the stable SDLSs, and the dotted curve indicates the unstable SDLSs, see (b). Profiles of SDLSs at the circled points in (a) and (b) are displayed in Fig. 3. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 3. (a, b) Profiles of SDLSs at the circled points a, b in Fig. 2, with (a) b = − 14.30 and (b) −16.51, respectively. (c, d) Propagation of SDLSs corresponding to (a) and (b), respectively. The shaded stripes in (a) and (b) represent the locations of high intensities in the defected lattice.
Fig. 4(a). Moreover, SDLSs exist in the whole first gap and in the second gap when b < − 3.50. The real part of perturbation growth rate Re (δ ) at I0 = 3, E0 = 6 , and E p = 18 for ε = − 0.7 is shown in Fig. 4(b). In the semi-infinite gap, SDLSs in Fig. 4(a) are stable when −15.40 < b < − 9.68 (corresponding to low power) and unstable when b < − 15.40 (corresponding to high power) [see Fig. 4(b)]. In the first and second gaps, SDLSs are stable, as shown in Fig. 4(b). Fig. 5(a)–(c) shows the profiles of SDLSs when b = − 16.25, −6.10, and −3.81 [points a, b, and c in Fig. 4(a) and (b)]. The evolution of the SDLSs corresponding to Fig. 5(a)–(c) are
shown in Fig. 5(d)–(f). The SDLS corresponding to Fig. 5(a) is unstable and does not oscillate in the all effective potential because it is far from the stable SDLS [see Figs. 4(b) and 5(d)] [43]. The SDLSs corresponding to Fig. 5(e) and (f) are stable. The centers of the SDLSs in the semi-infinite gap reside in the first lattice channel [see Fig. 5(a)]. In the similitude of surface solitons in waveguide arrays [43], SDLSs in the second gap penetrate deeper inside the photonic lattices, oscillations on their profiles become more pronounced [see Fig. 5(b) and (c)]. Thus the interface effectively affects the finite low-gap SDLSs.
Fig. 4. (a) Power P versus propagation constant b (blue regions are Bloch band) at I0 = 3, E0 = 6 , and E p = 18 when ε = − 0.7. (b) Perturbation growth rate Re (δ ) versus the propagation constant b at I0 = 3, E0 = 6 , and E p = 18 when ε = − 0.7. In (a), the solid curve indicates the stable SDLSs, and the dotted curve indicates the unstable SDLSs, see (b). Profiles of SDLSs at the circled points in (a) and (b) are displayed in Fig. 5. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 5. (a)–(c) Profiles of SDLSs at the circled points a, b, and c in Fig. 4, with (a) b = − 16.2, (b) −6.1, and (c) −3.8, respectively. (d)–(f) Propagation of SDLSs corresponding to (a)–(c), respectively. The shaded stripes in (a)–(c) represent the locations of high intensities in the defected lattice.
Finally, it is interesting to discuss the properties of SDLSs in biased PP crystals. When the bulk photovoltaic effect is negligible, i.e., E p = 0, our physical system becomes the physical system of SDLSs studied previously at the interface with a defect between the photonic lattices and the uniform biased non-PP crystals. Substitution E p = 0 into Eqs. (1) and (4) leads to the same expression (1) and expression (3) in Ref. [39], respectively. When the external bias field is absent, i.e., E0 = 0, our physical system becomes the physical system of SDLSs at the interface with a defect between the photonic lattices and the uniform PP crystals. In this case, SDLSs in PP crystals can be obtained from Eqs. (1) to (6).
4. Conclusion In conclusion, we have found that SDLSs are possible at the interface with a defect between the photonic lattices and the uniform PP crystals with external applied field. We have shown that these SDLSs exist only in the semi-infinite gap for positive defect strength and in the semi-infinite, first, and second gaps for negative defect strength. In the semi-infinite gap, SDLSs are stable in the low power region and unstable in the high power region. In the first and second gaps, SDLSs are stable in the all power region. The properties of these SDLSs have been discussed and we have found that they are those studied previously in biased non-PP
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crystals when the bulk photovoltaic effect is negligible and are those in PP crystals when the external bias field is absent.
Acknowledgment The work was supported by the Natural Science Foundation of Chinese Tianjin (No. 13JCYBJC16400).
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