## Abstract

Dark surface waves with photorefractive diffusion and photovoltaic nonlinearities are predicted for the first time. We find it is extraordinary that this type of dark surface waves should be in self-focusing media, which is very different from the surface dark solitons or other nonlinear dark surface waves. An oscillator model is proposed by which the above extraordinary phenomenon is demonstrated. In this model an equivalent force function is established, whose form determines the varieties of surface waves (bright surface waves, dark surface waves or others).

©2013 Optical Society of America

## 1. Introduction

Self-guided waves along the surface of a non-linear medium are among the most intriguing phenomena in nonlinear optics and may result in very strong enhancement of nonlinear surface optical phenomena. From one aspect these can be attributed to the natural line path supported by surface, which provides the fine situation for the phase-matching condition [1].

In general, there are three types of optical surface waves (SWs): surface plasmon polaritons, surface electromagnetic modes and nonlinear optical surface waves. The earlier studies for nonlinear optical surface waves were mainly focused on the diffusive Kerr nonlinearity [2,3]. At the end of last century the possibility of propagation of the nonlinear surface waves near the boundary of the photorefractive (PR) medium was considered, which were named photorefractive surface waves (PR SWs) [4–6]. PR SWs have been demonstrated to solve the phase-mismatching problem caused by beam self-bending and to support giant enhancement of second-harmonic generation [1,7].

Another type of nonlinear optical surface waves is named surface solitons. The earlier surface solitons were achieved in nonlinear optical lattice and were named surface lattice solitons [8–11], which utilized periodic structures and near surface defects induced by nonlinearity. Surface solitons in uniform nonlinear medium were proposed by Barak Alfassi et al. in 2007 utilizing nonlocal self-focusing type thermal nonlinearity, where prefabrication of periodical structures was not needed [12]. In 2009, we demonstrated surface solitons in virtue of the cooperation of nonlocal diffusion and local drift PR nonlinearities in uniform PR medium [13]. In 2010 Jassem Safioui et al. demonstrated surface-wave pyroelectric photorefractive solitons in LiNbO_{3} [14]. Surface solitons are very different from the above mentioned nonlinear SWs. Surface solitons are solitons propagating near surface, while the above mentioned nonlinear SWs are the diffraction-free modes due to diffusion nonlinearity. It is well-known that self-focusing nonlinearity supports bright solitons and self-defocusing nonlinearity supports dark solitons [15–18]. So do the surface solitons (surface bright solitons and surface dark solitons) [12–14, 19]. While the above mentioned nonlinear SWs are supported by diffusion nonlinearity whether the medium is self-focusing or self-defocusing. In 2007 Kartashov et al. have predicted surface waves in defocusing thermal media [20].

The dark surface waves (DSWs) in self-defocusing media were proposed soon afterwards those bright SWs with the diffusive Kerr nonlinearity, which is normally associated with trapping at the interface between a linear medium and a self-defocusing medium or between two self-defocusing nonlinear media [21–23]. In practice, these DSWs can be considered as surface dark solitons propagating along interface, since the forms of their steady solutions are more similar to that of solitons. So far the DSWs corresponding to bright nonlinear SWs are not reported to our knowledge.

We have proved that the photorefractive diffusion nonlinearity is the essential cause for PR SWs, which ensures the light field of bright PR SWs to be confined near surface and converging to zero from surface to bulk with oscillating [24]. Can this light field converge to a non-zero value? That is to ask if a photorefractive dark surface wave (PR DSW) corresponding to the above bright PR SWs can also propagate along the surface of a PR crystal and how to realize it if possible? What property of nonlinearity is demanded?

In this paper we report on the existence and realization method of photorefractive dark surface waves (PR DSWs) for the first time, in which diffusion and photovoltaic components of the photorefractive nonlinearity are considered and coherent uniform background illumination is used. It is very extraordinary that this type of DSWs should be in self-focusing nonlinear media.

## 2. Theory and model

Generally, surface wave should be of (1 + 1)D form. Considering an *e*-polarized light beam with intensity *I*(*x*) propagating along the interface between air and a PR crystal (PRC), the complex amplitude *E*(*x*,*z*) satisfies the nonlinear scalar wave equation:

In the air (*x*<0), *k* = *k*_{0}*n*_{0} = 2*π*/*λ*_{0}, *n*_{0} = 1and *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} is the effective electro-optical coefficient, *E*_{sc} is the space-charge field. With an *o*-polarized coherent uniform background illumination, under open-circuit conditions *E*_{sc} can be written as [25]:

*k*is the Boltzman constant,

_{B}*T*is the temperature,

*q*is the charge of carriers, (negative for the electrons and positive for the holes),

*E*is the the photovoltaic field,

_{p}*I*is the equivalent dark irradiance,

_{d}*I*is the background illumination normalized by

_{b}*I*,

_{d}*κ*=

*β*

_{31}

*/β*

_{33},

*β*

_{33}and

*β*

_{31}are the photovoltaic constant for

*e*-polarized light and

*o*-polarized light, respectively. The first and the second terms in the right side of Eq. (2) describe the effects of the diffusion and the photovoltaic components of PR nonlinearity, respectively. For photovoltaic medium, such as LiNbO

_{3}a typical photovoltaic crystal, the effect of photovoltaic component is self-defocusing. Here the

*o*-polarized coherent uniform background illumination is used for self-defocusing-to-self-focusing transition [25]. In the following one can find the roles of self-defocusing and self-focusing for PR SWs and PR DSWs.

We look for the stationary PR SW solution as *E*(*x*,*z*) = *A*(*x*)exp(*ißz*), where *β* is the propagation constant and *A*(*x*) = [*I*(*x*)/(*I _{d}*)]

^{1/2}is the normalized amplitude. Equation (1) can be rewritten as:

*γ*= −2

*k*

_{0}

^{2}

*n*

_{e}^{4}

*r*

_{eff}

*k*/

_{B}T*q*,

*b*=

*k*

_{0}

^{2}

*n*

_{e}^{4}

*r*

_{eff}

*E*,

_{P}*g*=

*k*

_{0}

^{2}

*n*

_{e}^{2}–

*β*

^{2},

*β*is the propagation constant. In Eq. (3a), the first term indicates the diffraction spreading of the light beam, the second term describes the effect of diffusion mechanism, and the third term states the influence of photovoltaic component of the photorefractive nonlinearity.

Equation (3a) is a typical damped oscillation equation, so we propose an oscillator model, in which the profile of PR SW *A*(*x*) along *x* can be treated as the damped oscillation of an oscillator with an external force. Under this model, the second term, the third term and the fourth term on the right side of Eq. (3a) can be regarded as the damping force, the external force and the restoring force, respectively. Without diffusion component of the photorefractive nonlinearity (*γ* = 0), the profile of PR SW will always oscillate without decaying, just as the oscillation of a harmonic oscillator without damping. Therefore, diffusion component is necessary to the formation of PR SW, which permits the amplitude of PR SW to decay from surface to bulk. The decaying rate mainly depends on *γ*. Larger *γ* will let the amplitude decay faster. The restoring force and external force constitute the equivalent force function *F*

*g*and

*I*in a certain medium. In the following we can know that the form of equivalent force function determines the varieties of PR SWs (bright PR SWs, PR DSWs or others).

_{b}## 3. Numerical simulation

#### 3.1 Modes of PR SWs

*(*1*)* g > b, bκI_{b} –g(I_{b} + 1) > 0

The equivalent force *F* as a function of amplitude *A*(*x*) is sketched in Fig. 1(a1)-1(c1). *F* has three points of intersection with *x*-axis at *A*_{1}(*x*) = 0 and *A*_{2,3}(*x*) = ± {[*bκI _{b}* –

*g*(

*I*+ 1)]/(

_{b}*g*−

*b*)}

^{1/2}.

*F*always exhibits repulsive force around

*A*

_{1}(

*x*) while

*F*always exhibits attractive force around

*A*

_{2,3}(

*x*). That is to say

*A*

_{1}(

*x*) is not a stable balance position and

*A*

_{2,3}(

*x*) are two stable balance positions. So the oscillation can only converge to the couple of nonzero values

*A*

_{2,3}(

*x*) rather than at

*A*

_{1}(

*x*) = 0, which indicates the existence of PR DSWs.

For DSWs the nonzero stable balance positions are demanded, where *F* always exhibits attractive force. That means *F* = 0 and *dF*/*dA*(*x*) < 0 should be satisfied at these positions, consequently based on Eq. (4) one can get

*κI*> (

_{b}*I*+ 1) should be satisfied. From Eq. (3a) one can see that in this case the effect of photovoltaic component is self-focusing. That is to say PR DSWs should be supported by self-focusing nonlinearities. When

_{b}*κI*< (

_{b}*I*+ 1), the effect of photovoltaic component is self-defocusing and PR DSWs cannot exist; instead, bright PR SWs may occur.

_{b}*g*>

*b*and

*bκI*–

_{b}*g*(

*I*+ 1) > 0 also means

_{b}*κI*> (

_{b}*I*+ 1).

_{b}Figures 1(a2) and 1(a6), 1(b2) and 1(b6), 1(c2) and 1(c6) show the modes of PR DSWs for lower *g* (higher *β*), moderate *g* (moderate *β*) and higher *g* (lower *β*), respectively. All the modes behave like damped oscillation and converge to the nonzero values corresponding to *A*_{2,3}(*x*) in Fig. 1(a1)-1(c1). From Eq. (3a) one can see that the spatial frequency of the PR DSW modes mainly depends on *F* and diminishes with increasing of *g*, as shown in Figs. 1(a2)-1(c2) and 1(a5)-1(c5). Base on the damping oscillation model, larger amplitude of PR DSWs means higher energy of the oscillator, and more intense oscillation will occur. So the decaying oscillation of PR DSWs for same *g* larger amplitude of PR DSW responds to more intense oscillation, and out-of phase profile will occur more possible.

### (2) g > b, bκI_{b} –g(I_{b} + 1) < 0

*F* has only one point of intersection with *x*-axis at *A*(*x*) = 0, as shown in Fig. 2(a1), which is a stable balance position. The oscillation will converge at 0, corresponding to bright PR SW, as shown in Fig. 2(a2). In this case both self-focusing and self-defocusing are permitted.

### (3) g < b, bκI_{b} –g(I_{b} + 1) < 0

*F* has three points of intersection with *x*-axis at *A*_{1}(*x*) = 0 and *A*_{2,3}(*x*) = ± {[*bκI _{b}* –

*g*(

*I*+ 1)]/(

_{b}*g*−

*b*)}

^{1/2}, as shown in Fig. 2(b1).

*A*

_{1}(

*x*) is a stable balance position, while

*A*

_{2,3}(

*x*) are two unstable balance positions. So the oscillation can converge at 0 and the oscillator is steady only in a limited range of amplitude. That indicates bright PR SWs with lower amplitude, as shown in Fig. 2(b2). In this case,

*g*<

*b*,

*bκI*–

_{b}*g*(

*I*+ 1) < 0 means

_{b}*κI*> (

_{b}*I*+ 1) and the effect of photovoltaic component is self-defocusing.

_{b}### (4) g < b, bκI_{b} –g(I_{b} + 1) > 0

*F* has only one point of intersection with *x*-axis at *A*(*x*) = 0, where *F* always exhibits repulsive force. In this case, the solutions are corresponding to evanescent waves, as shown in Fig. 2(c2).

#### 3.2 Stabilities of PR SWs

To investigate the stability of PR SWs, we used the beam propagation method (BPM) to simulate the evolution of the stationary PR SW solution with a random noise (10%). Figures 1(a4)-1(c4) and 1(a7)-1(c7) show the evolutions for the perturbed PR DSW modes of Figs. 1(a2)-1(c2) and 1(a5)-1(c5), respectively. Figures 2(a4) and 2(b4) exhibit the evolutions for the perturbed bright PR SW modes of Figs. 2(a2) and 2(b2), respectively. All the perturbed PR SWs maintain their shapes quite well, indicating the PR SWs are stable.

In the above simulation, Fe:LiNbO_{3} is taken as sample and the material parameters at *λ* = 633 nm are *n _{e}* = 2.2,

*r*

_{eff}=

*r*

_{33}= 31 × 10

^{−12}m/V,

*κ*=

*β*

_{31}/

*β*

_{33}= 1.2,

*E*= −1.67 × 10

_{P}^{4}V/m. At room temperature, there are

*γ*= 3.7 × 10

^{3}m

^{−1}, and

*b*= 1.1949 × 10

^{9}m

^{−2},

*q*= −1.6 × 10

^{−19}C.

## 4. Conclusion

We predict a type of Dark surface waves, which are really corresponding to the bright surface waves for the first time. We find and demonstrate that this type of dark surface waves should be in self-focusing media, while bright surface waves have not this demand. It is very different from the surface dark solitons or other nonlinear dark surface waves.

## Acknowledgments

This work was supported by CNKBRSF (2011CB922003), NSFC (61078014, 61178005, J1103208), SRFDP (20100031110007, 20120031110030), NCET-11-0263, and the 111 Project (B07013), and NUIT (111005537).

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