Soliton-induced waveguides in photorefractive

photovoltaic materials

KEQING LU*y, WEI ZHAOy, YANLONG YANGy,YANG YANGy, XUEMING LIUy, YANPENG ZHANGz

and JINGJUN XUx

yState Key Laboratory of Transient Optics and Photonics,Xi’an Institute of Optics and Precision Mechanics,Chinese Academic of Sciences, Xi’an 710068, ChinazDepartment of Electronic Science and Technology,Xi’an Jiaotong University, Xi’an 710049, China

xDepartment of Physics, Nankai University, Tianjin 300071, China

(Received 6 August 2005)

Waveguides induced by one-dimensional spatial photovoltaic solitons areinvestigated in both self-defocusing-type and self-focusing-type photorefractivephotovoltaic materials. The number of possible guided modes in a waveguideinduced by a bright photovoltaic soliton is obtained using numerical techniques.This number of guided modes increases monotonically with increasingintensity ratio, which is the ratio between the peak intensity of the soliton andthe sum of the background illumination and the dark irradiance. On theother hand, waveguides induced by dark photovoltaic solitons are always singlemode for all intensity ratios, and the higher the intensity ratio, the moreconfined is the optical energy near the centre of the dark photovoltaic soliton.Relevant examples are provided where photorefractive photovoltaic materials areof self-defocusing and self-focusing types. The properties of soliton-inducedwaveguides in both self-defocusing-type and self-focusing-type materials are alsodiscussed.

1. Introduction

Photorefractive (PR) spatial solitons have attracted substantial research interest

because of their possible applications for optical switching and routing. They are

generally classified into three generic types: quasisteady-state solitons [1–3], screening

solitons [4–7] and photovoltaic (PV) solitons [8–12]. Following this, attention has

been paid to dipole-mode vector solitons [13, 14] and multi-hump optical solitons

[15, 16]. Recently, we have shown theoretically that the application of an external

*Corresponding author. Email: kq.lu@263.net

Journal of Modern OpticsVol. 53, No. 15, 15 October 2006, 2137–2151

Journal of Modern OpticsISSN 0950–0340 print/ISSN 1362–3044 online # 2006 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/09500340600809145

field enables steady-state solitons [17], soliton pairs [18], and vector solitons [19, 20]

in photorefractive photovoltaic crystals. Of particular interest is the physical

system of solitons in biased photorefractive photovoltaic crystals, which changes

into a physical system of screening solitons when the bulk PV effect is neglected

and a physical system of PV solitons when the external field is absent [17]. On the

other hand, photorefractive spatial solitons induce waveguides that offer potential

applications, some of which are truly unique [21, 22]: three-dimensional optical

circuitry [23], reconfigurable near-field interconnects [22], nonlinear frequency

conversion [21, 22], directional coupling [24], and beam steering [25]. Recent

theoretical and experimental work have shown waveguides induced by

quasisteady-state solitons [26], by screening solitons [27, 28], by photovoltaic (PV)

solitons [29], and by solitons in anisotropic nonlinear materials [30]. Here we focus

on waveguides induced by PV solitons, which exist in PR media that exhibit a strong

bulk PV effect. In these crystals in an open-circuit configuration, the sign of the index

change �n cannot be changed in a given material because it is determined by the

sign of the product riijEp (riij is the electro-optic coefficient and Ep is the PV field

constant) [31]. Waveguides induced by dark PV solitons are investigated in a LiNbO3

photovoltaic crystal [29], in which �n < 0. It is justified in [31] that the pototovoltaic

nonlinearity that gives rise to spatial PV solitons can be switched from self-

defocusing to self-focusing (or vice versa) by using background illumination.

In this case, waveguides induced by bright and dark PV solitons also deserve special

consideration.In this paper, we analyse waveguides induced by one-dimensional spatial photo-

voltaic solitons in both self-defocusing-type and self-focusing-type photorefractive

photovoltaic materials. We obtain the number of possible guided modes in a

waveguide induced by a bright photovoltaic soliton by using numerical techniques,

which increases monotonically with increasing intensity ratio between the peak

intensity of the soliton and the sum of the background illumination and the dark

irradiance. Subsequently, we find that waveguides induced by dark photovoltaic

solitons are always single mode for all intensity ratios, of which the confined energy

increases monotonically with increasing intensity ratio. Moreover, the properties of

soliton-induced waveguides in both self-defocusing-type and self-focusing-type

photorefractive photovoltaic materials are also discussed. Relevant examples are

provided where photovoltaic crystals are assumed to be of self-defocusing and

self-focusing types.

2. General treatment

We start with the standard set of rate and continuity equations and Gauss’ law,

which describe the photorefractive effect in a medium in which electrons are the sole

charge carriers, plus the wave equation for the slowly varying amplitude of the

optical field. The crystal is illuminated uniformly by a background beam of intensity

Ib, of which the polarization is in the i direction, giving rise to a current in

the i direction. Moreover, we assume that the polarization of the optical beam

2138 K. Lu et al.

is in the j crystalline direction, giving rise to a PV current in the i direction. In steadystate and two dimensions these equations are [31]

s Aj j2þsIb þ �T

� �Nd �Ni

d

� �� �nNi

d ¼ 0, ð1Þ

r � J ¼ r � q�nEþ kBT�rnþ �ijjs Nd �Nid

� �Aj j2þ�Ib

� �i

h i¼ 0, ð2Þ

r � Eþ q="sð Þ nþNA �Nid

� �¼ 0, ð3Þ

@

@z�

i

2k

@2

@x2

� �Aðx, zÞ ¼

ik

nb�nðEÞAðx, zÞ, ð4Þ

where �nðEÞ ¼ �ð1=2Þn3briijE is the perturbation in the refractive index, nb is theunperturbed refractive index, z is the propagation axis, x is the transverse coordinate,Ni

d is the number density of ionized donors, n is the electron number density, E ¼ Eiis the space-charge field inside the crystal, J ¼ Ji is the current density, A is theslowly varying amplitude of the optical field defined by Eoptðx, z, tÞ ¼Aðx, zÞ expðikz� i!tÞ þ c:c: (k ¼ 2pnb=�, where � is the wavelength in vacuo, and! is the frequency), NA is the number density of negatively charged acceptors, Nd isthe total donor number density, �T is the dark generation rate, s is the photo-ionization cross-section, � is the recombination rate, "s is the low-frequencydielectric constant, � is the electron mobility, � ¼ �iii=�ijj is the ratio between thephotovoltaic constants that can attain any value, �q is the electron charge, kB isBoltzmann’s constant, and T is the absolute temperature. In equations (1)–(4), therelation between the space-charge field and the current density satisfies the potentialcondition V ¼ �

Ð l=2�l=2 Edl ¼ RSJ, where V is the potential between the crystal’s

electrodes, l is the width of the crystal between the electrodes, S is the surface areaof the electrodes and R is the external resistance. Finally, we define the opticalintensity as I ¼ jAj2 and the dark irradiance as Id ¼ �T=s that is very small ascompared to Ib in typical photorefractive materials.

We seek solutions of the form

Aðx, zÞ ¼ uðxÞ expði�zÞ Id þ Ibð Þ1=2, ð5Þ

where � is the soliton propagation constant and u(x) is the normalizedamplitude. From equations (1)–(3) the space-charge field can be determined and isgiven by [8, 31]

E ¼J� Ep Iþ �Ibð Þ

Iþ Ib þ Id, ð6Þ

where J ¼ J= q�sNd=�NAð Þ and Ep ¼ �ijj�NA=ðq�Þ. We transform equation (4) intodimensionless form by � ¼ x=Ls, where Ls ¼ 1=ð�2kbÞ1=2 and b ¼ k=nbð Þ �

ð1=2Þn3briijEp ¼ k=nbð Þ�n0 is the parameter that characterizes the strength and thesign of the optical nonlinearity. Thus equations (4)–(6) yield the following results:

d2u

d�2¼ �

Jp � �Ir þ u2� �1þ u2

þ�

b

� �u, ð7Þ

Soliton-induced waveguides in photorefractive photovoltaic materials 2139

where Jp ¼ J= Id þ Ibð ÞEp

� �, Ir ¼ Ib= Id þ Ibð Þ and the upper (lower) sign indicates

�n0 > 0 (< 0). Using quadrature, we obtain the first integral of equation (7):

p2 � p20 ¼ ��

b� 1

� �u2 � u20� �

þ Jp � �Ir þ 1� �

lnu2 þ 1

u20 þ 1

� �� �, ð8Þ

where p ¼ du=d�, p0 ¼ p � ¼ 0ð Þ and u0 ¼ uð0Þ. The solutions of equation (8), u(�)with the appropriate boundary conditions, are dark and bright solitons. Nonlinearself-defocusing (focusing) �n gives rise to dark (bright) solitons. Next, let us showthe refractive index perturbation �n in photorefractive photovoltaic crystals.Substituting equation (6) and � ¼ 1þ � into �nðEÞ ¼ �ð1=2Þn3briijE yields

�n ¼ �n0 1þ�Ib � Id � JpIþ Ib þ Id

� �: ð9Þ

In order to explain that whether or not the current reverses the polarity of therefractive index perturbation, we evaluate the maximum current in equation (9).In this case, where R¼ 0, by substituting equation (6) into the potential conditionand by using the optically induced current equation [8] leads to J ¼ sNdIb�iii.Therefore equation (6) and �nðEÞ ¼ �ð1=2Þn3briijE yield �n ¼ �n0I= Iþ Ib þ Idð Þ.This is either a self-defocusing �n or a self-focusing �n, of which the signcannot be changed in a given photovoltaic photorefractive material [31]. Underan open-circuit condition, it is justified in [31] that when equation (9) is ofself-defocusing (focusing) nonlinearity for Ib ¼ 0, adding the background beamturns equation (9) into self-focusing (defocusing) nonlinearity. After the solitonhas formed, the space-charge field is present in the photorefractive photovoltaicmaterial, which gives rise to a change in the refractive index. Let us consider a probebeam that propagates in this soliton-induced waveguide. Under these conditions,the perturbed refractive index �n of the probe beam is given by [27]

�nðxÞ ¼ �nb þ��nðxÞ ¼ �n3b �riijEðxÞ, ð10Þ

where �nb is the unperturbed index of refraction and �riij is the electro-optic coefficient.On the other hand, the electric-field component �Eopt of the probe beam satisfies thefollowing Helmholtz equation:

r2 �Eopt þ�k0 �n

� �2 �Eopt ¼ 0, ð11Þ

where �k0 ¼ 2p= �� is the wave vector in vacuum and �� is the common free-spacewavelength used. By expressing �Eopt in terms of a slowly varying amplitude v(x),i.e. �Eoptðx, z, tÞ ¼ ivðxÞ exp i ��z� i �$t

� �þ c:c:, where �$ is the frequency of the

probe beam and �� is the propagation constant, we find that equation (11) leads tothe following wave equation

d2v

dx2þ �k20 �n

2ðxÞ � ��2� �

v ¼ 0: ð12Þ

The solutions of equation (12) give the guided modes of the waveguide induced byspatial solitons in photorefractive photovoltaic materials.

2140 K. Lu et al.

3. Soliton-induced waveguides in self-defocusing-type materials

For self-defocusing-type (�n0 < 0) materials and under open-circuit conditions,equation (7) takes the form

d2u

d�2¼ �

�

b��Ir þ u2

1þ u2

� �u: ð13Þ

When �Ir > 1 the solutions of equation (13) are bright solitons, which will beexplained in what follows. The boundary conditions for a bright soliton areuð1Þ ¼ u0ð1Þ ¼ u00ð1Þ ¼ 0, u0ð0Þ ¼ 0, and u00ð0Þ=uð0Þ < 0, where u0ð�Þ ¼ du=d�, etc.Taking the limit � ! 1 and Jp ¼ 0 in equation (8) leads to G=b ¼

1� 1� �Irð Þ ln u20 þ 1� �

=u20. Substituting � ¼ 0 and the expression we have justfound for G=b into equation (13) yields u00ð0Þ=uð0Þ ¼ 1� �Irð Þ ln u20 þ 1

� �=

u20 � 1� �Irð Þ= 1þ u20� �

, which satisfies just boundary condition u00ð0Þ=uð0Þ < 0when �Ir > 1. Under open-circuit conditions, equations (5), (6) and (10) yield thefollowing results:

�nðxÞ ¼ �nb þ1

2�n3b �riijEP

�Ir þ u2ðxÞ

1þ u2ðxÞ: ð14Þ

Substituting equation (14) into equation (12) leads to

d2v

d ��2¼ � �� �

�Ir þ u2 ��=12� �

1þ u2 ��=12� �

" #v, ð15Þ

where the normalized propagation constant ��, the new normalized length �� andnormalization factors 1 and 2 are defined as

�� ¼��2 � 2

1k2

2kb21

22

, ð16Þ

�� ¼ 12�, ð17Þ

1 ¼�nbnb

���, ð18Þ

2 ¼�nbnb

�riijriij

� �1=2

: ð19Þ

When 1 ¼ 2 ¼ 1, equation (15) becomes

d2v

d ��2¼ �

��2 � k2

2kb��Ir þ u2 ��

� �1þ u2 ��

� �" #

v: ð20Þ

By comparing equation (20) with equation (13), we find that one of the eigenfunc-tions (guided modes) v(�) of equation (20) is identical to u(�) of equation (13) with theeigenvalue ��2 � k2

� �=2kb ¼ 1� 1� �Irð Þ lnðu20 þ 1Þ=u20. In other words, the bright

soliton is a guided mode of the waveguide it induces. In what follows, we will discussthe possible modes, v ��

� �, in a waveguide induced by the bright soliton u(�). First,

let us solve equation (13) numerically for various values of u0 and find the soliton

Soliton-induced waveguides in photorefractive photovoltaic materials 2141

wave forms u(�). We then solve equation (15) for v ��� �

by the shooting method.In equations (15) and (16), the eigenvalue, ��, must be in the range between�Ir þ u20� �

= 1þ u20� �

and �Ir to fulfill the condition that the propagation constant ofa guided mode, ��, lies between the maximum and the minimum values of therefractive index �nðxÞ times the wave vector �k0. As an example, let us consider a self-defocusing-type crystal (LiNbO3) illuminated uniformly by the background beam.When 1 ¼ 2 ¼ 1 in the self-defocusing-type crystal, we find that for u0 ¼ 3at �Ir ¼ 1:5 there are three guided modes: the first mode coinciding with u(�), andthe second and third modes shown in figure 1(a). Figure 2 shows the normalizedpropagation constants of the guided modes as functions of u0 for 0 � u0 � 20at �Ir ¼ 1:5 in the self-defocusing-type crystal. The number of guided modes for u0is given by the number of intersections of the vertical line of u0 with the curves ofthe propagation constant in figure 2. The two dashed lines correspond to u0 ¼ 3(the example given in figure 1(a)) and 10. This figure demonstrates that thenumber of guided modes increases monotonically with increasing u0. For differentvalues of 12, we obtain similar results by using the same approach. Figure 1(b)shows the amplitude of the bright soliton and the normalized amplitude of theguided modes for u0 ¼ 3 at �Ir ¼ 1:5 when 12 ¼ 0:697. Figure 3 showsthe number of guided modes and the propagation constants as a function of u0 inthe self-defocusing-type crystal when 12 ¼ 0:697, which corresponds to a brightsoliton at � ¼ 0:5 mm in a self-defocusing-type LiNbO3 crystal withriij ¼ 30� 10�12 m/V and guiding within it a probe beam at � ¼ 0:7 mm wavelengthwith �riij ¼ 28:5� 10�12 m/V, assuming negligible difference between nb and �nb. Fromfigures 2 and 3, we find that the number of guided modes at any given u0 for12 ¼ 0:697 is smaller than that for 1 ¼ 2 ¼ 1 and u0 > 1:3, and the spacingbetween adjacent modes is also wider.

In the absence of the background beam of intensity Ib, we can rewriteequation (13) as

d2u

d�2¼ �

Gb�

u2

1þ u2

� �u, ð21Þ

of which solutions are dark solitons. The boundary conditions for a dark soliton areuð1Þ ¼ u1, u0ð1Þ ¼ u00ð1Þ ¼ 0, uð0Þ ¼ 0, and a real u0ð0Þ. Using boundary conditionu00ð1Þ ¼ 0 and substituting � ! 1 into equation (21) leads to G=b ¼ u21= 1þ u21

� �.

Under open-circuit conditions and in the absence of the background beam ofintensity Ib, equations (5), (6) and (10) yield the following results:

�nðxÞ ¼ �nb þ1

2�n3b �riijEP

u2ðxÞ

1þ u2ðxÞ: ð22Þ

By using the same normalized propagation constant ��, the normalized length ��, andnormalization factors 1 and 2 as in equations (16)–(19), and by substitutingequation (22) into equation (12), we find that

d2v

d ��2¼ � �� �

u2 ��=12� �

1þ u2 ��=12� �

" #v: ð23Þ

2142 K. Lu et al.

When 1 ¼ 2 ¼ 1, equation (23) takes the form

d2v

d ��2¼ �

��2 � k2

2kb�

u2 ��� �

1þ u2 ��� �

" #v: ð24Þ

By comparing equation (24) with equation (21), we can see that one of the eigen-functions of equation (24) is identical to u(�) with the eigenvalue ��2 � k2

� �=2kb ¼

u21= 1þ u21� �

. We now wish to find the modes of the waveguide, v ��� �

, in a waveguide

−40 −20 0 20 40−1

0

1

ξ

u/u

0; v

= v

max

(a)

−40 −20 0 20 40−1

0

1

ξ

u/u

0; v

= v

max

(b)

Figure 1. Normalized amplitude profiles of the bright PV soliton uð�Þ=u0 (solid curve) andof the guided modes [v1ð�Þ=v1,max (dashed curve), v2ð�Þ=v2,max (dotted curve) and v3ð�Þ=v3,max

(dash-dot curve)] of the soliton-induced waveguide for u0 ¼ 3 at �Ir ¼ 1:5 when(a) 1 ¼ 2 ¼ 1 and (b) 12 ¼ 0:697. For 1 ¼ 2 ¼ 1, the first mode v1ð�Þ=v1, max coincideswith uð�Þ=u0.

Soliton-induced waveguides in photorefractive photovoltaic materials 2143

induced by the dark soliton u(�). We integrate equation (21) numerically for various

values of u1 and obtain the waveforms. Several particular cases are shown by the

dashed curves in figure 4. The solutions of equation (24), v ��� �

, can then be obtained by

the shooting method. Here �� must be in the range between unity and u21= 1þ u21� �

to fulfill the condition that the propagation constant of a guided mode, ��,

0 5 10 15 20

1.1

1.2

1.3

1.4

1.5

u0

1st mode

2nd

3rd

4th

5th

6th

7th

8th9th

10th

Nor

mal

ized

pro

paga

tion

cons

tant

Figure 2. Propagation constants of the guided modes of the waveguide induced by thebright PV solitons at �Ir ¼ 1:5 when 1 ¼ 2 ¼ 1. The dashed lines correspond to u0 ¼ 3 offigure 1(a) and u0 ¼ 10, at which the soliton-induced waveguides have three and six guidedmodes, respectively.

0 5 10 15 20

1.1

1.2

1.3

1.4

1.5

u0

1st mode

2nd

3rd

4th

5th

6th

7th

Nor

mal

ized

pro

paga

tion

cons

tant

Figure 3. Propagation constants of the guided modes of the waveguide induced by thebright PV solitons at �Ir ¼ 1:5 when 12 ¼ 0:697. The dashed lines correspond to u0 ¼ 3 offigure 1(b) and u0 ¼ 10, at which the soliton-induced waveguides have two and four guidedmodes, respectively.

2144 K. Lu et al.

lies between the maximum and the minimum values of the refractive index �nðxÞ times

the wave vector �k0. Figure 4 shows the normalized amplitude of the dark soliton

uð�Þ=u1 (dashed curve) and the normalized amplitude of the single guided mode

vð�Þ=v0 (solid curve) for three different values of u1 in the self-defocusing-type

−40 0 40−1

0

1

ξ

(a)u

/u∞

; v/v

max

−40 0 40−1

0

1

ξ

u/u

∞; v

/vm

ax

(b)

Figure 4. Normalized amplitude profiles of the dark PV soliton uð�Þ=u1 (solid curve) andof the single guided mode vð�Þ=v1 (dashed curve) of the soliton-induced waveguideat (a) u1 ¼ 0:5, (b) u1 ¼ 1 and (c) u1 ¼ 10 for �Ir ¼ 0 when 12 ¼ 0:745.

Soliton-induced waveguides in photorefractive photovoltaic materials 2145

crystal when 12 ¼ 0:745. This figure demonstrates that the optical energy near the

centre of the dark soliton can be confined by u1. Figure 5 plots the propagation

constant as a function of u1 for 1 ¼ 2 ¼ 1 and 12 ¼ 0:745. The latter

case corresponds to a dark soliton at � ¼ 0:5 mm in the self-defocusing-type LiNbO3

crystal with riij ¼ 30� 10�12 m/V and guiding within it a probe beam at � ¼ 0:68 mmwavelength with �riij ¼ 29:2� 10�12 m/V, assuming negligible difference between

nb and �nb. Obviously, the propagation constant is monotonically increasing with u1.

4. Soliton-induced waveguides in self-focusing-type materials

For self-focusing-type (�n0 > 0) materials and under open-circuit conditions,

equation (7) takes the form

d2u

d�2¼

Gb��Ir þ u2

1þ u2

� �u, ð25Þ

of which solutions are bright solitons for �Ir ¼ 0. Using boundary conditionsu0ð1Þ ¼ 0 and u0ð0Þ ¼ 0, and substituting � ! 1, �Ir ¼ 0 and Jp ¼ 0 into

equation (8) leads to G=b ¼ 1� ln ðu20 þ 1Þ=u20. When the background beam of

intensity is absent for the open circuit, equations (5), (6) and (10) yield

�nðxÞ ¼ �nb þ1

2�n3b �riijEP

u2ðxÞ

1þ u2ðxÞ: ð26Þ

u/u

∞; v

/vm

ax

−200 0 200−1

0

1

ξ

(c)

Figure 4. Continued.

2146 K. Lu et al.

By using the same normalized propagation constant ��, the normalized length ��,and normalization factors 1 and 2 as in equations (16)–(19), and by substitutingequation (26) into equation (12), we find that

d2v

d ��2¼ �� �

u2 ��=12� �

1þ u2 ��=12� �

" #v: ð27Þ

When 1 ¼ 2 ¼ 1, one of the eigenfunctions of equation (27) is identical to u(�)of equation (25) for �Ir ¼ 0 with the eigenvalue �� ¼ 1� ln u20 þ 1

� �=u20. When the

background beam of intensity is absent, we integrate equation (25) numerically forvarious values of u0. The solutions of equation (27), v ��

� �, can then be obtained by

the shooting method. In real space, �� must be in the range between unity andu20= 1þ u20

� �. Figure 6 shows the normalized amplitude of the bright soliton uð�Þ=u0

and the normalized amplitude of the guided modes of the soliton-induced waveguideat u0 ¼ 5 in a self-focusing-type crystal when 12 ¼ 0:697. On the other hand,figure 7 plots the number of guided modes and the propagation constants as afunction of u0 in the self-focusing-type crystal when 12 ¼ 0:697. Moreover, thenumber of guided modes at u0 ¼ 5 in figure 7 is given in figure 6. Figure 7 alsoillustrates that the number of guided modes increases monotonically with u0.

When �Ir > 1, the solutions of equation (25) are dark solitons. Imposingboundary condition u00ð1Þ ¼ 0 and substituting � ! 1 into equation (25) leads toG=b ¼ �Ir þ u21

� �= 1þ u21� �

. Taking the upper sign and substituting conditionsp1 ¼ 0 and u0 ¼ 0, the expression we have just found for G=b, � ! 1, and Jp ¼ 0into equation (8) yield p20 ¼ � �Ir � 1ð Þu21= 1þ u21

� �� �Ir � 1ð Þ ln u21 þ 1

� �� �. The

reality of p0 can be obtained for �Ir > 1. Under open-circuit conditions, equations (5),(6) and (10) yield

�nðxÞ ¼ �nb þ1

2�n3b �riijEP

�Ir þ u2ðxÞ

1þ u2ðxÞ: ð28Þ

0 5 10 15 200

0.2

0.4

0.6

0.8

1

u∞

Nor

mal

ized

pro

paga

tion

cons

tant

Figure 5. Propagation constants of the guided modes of the waveguide induced by thedark PV solitons for �Ir ¼ 0 when 1 ¼ 2 ¼ 1 (solid curve) and 12 ¼ 0:745 (dashed curve).

Soliton-induced waveguides in photorefractive photovoltaic materials 2147

Using the same normalized propagation constant ��, the normalized length ��and normalization factors 1 and 2 as in equations (16)–(19), and substitutingequation (28) into equation (12) leads to

d2v

d ��2¼ �� �

�Ir þ u2 ��=12� �

1þ u2 ��=12� �

" #v: ð29Þ

−40 −20 0 20 40 −1

−0.5

0

0.5

1

ξ

u/u

0; v

/vm

ax

Figure 6. Normalized amplitude profiles of the bright PV soliton uð�Þ=u0 (solid curve) andof the guided modes [v1ð�Þ=v1,max (dashed curve), v2ð�Þ=v2,max (dotted curve) and v3ð�Þ=v3, max

(dash-dot curve)] of the soliton-induced waveguide for u0 ¼ 5 at �Ir ¼ 0 when 12 ¼ 0:697.

0 5 10 15 200

0.2

0.4

0.6

0.8

1

u0

1st mode

2nd

3rd

4th

5th

6th

7th

Nor

mal

ized

pro

paga

tion

cons

tant

Figure 7. Propagation constants of the guided modes of the waveguide induced by thebright PV solitons at �Ir ¼ 0 when 12 ¼ 0:697. The dashed lines correspond to u0 ¼ 5 offigure 6, at which the soliton-induced waveguides have three guided modes.

2148 K. Lu et al.

When 1 ¼ 2 ¼ 1, one of the eigenfunctions of equation (29) is identical to u(�) ofequation (25) with the eigenvalue �� ¼ �Ir þ u21

� �= 1þ u21� �

. We solve equation (29)by the shooting method after solving equation (25) numerically. The normalizedpropagation constant �� must be in the range between �Ir and �Ir þ u21

� �= 1þ u21� �

.Figure 8 shows the normalized amplitude of the dark soliton uð�Þ=u1 andthe normalized amplitude of the single guided mode vð�Þ=v0 when 12 ¼ 0:697at various values of u1 (0.5, 1 and 10) for �Ir ¼ 1:5 in the self-focusing-type crystal.Figure 9 plots the propagation constant as a function of u1 for 1 ¼ 2 ¼ 1 and12 ¼ 0:697. Evidently, a waveguide-induced dark PV soliton has a single guidedmode, of which the confined energy is increasing with u1. On the other hand, thepropagation constant of the guided mode is decreasing with u1.

Finally, it is important to discuss the properties of soliton-induced waveguidesin both self-defocusing-type and self-focusing-type photorefractive photovoltaic

−40 0 40−1

0

1

ξ

u/u

∞; v

/vm

ax(a)

−40 0 40−1

0

1

ξ

u/u

∞; v

/vm

ax

(b)

−200 0 200−1

0

1

ξ

u/u

∞; v

/vm

ax

(c)

Figure 8. Normalized amplitude profiles of the dark PV soliton uð�Þ=u1 (dashed curve)and of the single guided mode vð�Þ=v1 (solid curve) of the soliton-induced waveguideat (a) u1 ¼ 0:5, (b) u1 ¼ 1 and (c) u1 ¼ 10 for �Ir ¼ 0 when 12 ¼ 0:745.

Soliton-induced waveguides in photorefractive photovoltaic materials 2149

materials. Let us first consider waveguides induced by bright PV solitons. When

u0<1:3 in figures 3 and 7, the induced waveguide is single mode, and the

corresponding normalized propagation constant �� decreases with increasing u0 in

self-defocusing-type materials and increases with increasing u0 in self-focusing-type

photovoltaic materials. When u0>1:3 in figures 3 and 7, there are many guided

modes, and �j increases with increasing vj in self-defocusing-type photovoltaic

materials and decreases with increasing vj in self-focusing-type photovoltaic

materials, where vj is the jth mode and �j is the jth normalized propagation constant.

Similarly, the properties of waveguides induced by dark PV solitons can be

analysed in figures 4, 5, 8 and 9. When u0 is increased in self-defocusing-type

photovoltaic materials, the normalized propagation constant increases

monotonically in the range between 0 and 1, which is shown in figure 5. When u0is increased in self-focusing-type photovoltaic materials, the normalized propagation

constant decreases monotonically in the range between 1 and �Ir < 1 (�Ir ¼ 1:5 is

given in figure 9), which is shown in figure 9. From figures 4 and 8 we find

that when we fix u1, the optical energy near the centre of the dark soliton in

self-focusing-type photovoltaic materials is more than in self-defocusing-type

photovoltaic materials.

5. Conclusions

In conclusion, we have investigated waveguides induced by one-dimensional

spatial photovoltaic solitons in both self-defocusing-type and self-focusing-type

photorefractive photovoltaic materials. We have obtained the number of possible

0 5 10 15 201

1.1

1.2

1.3

1.4

1.5

1.6

u∞

Nor

mal

ized

pro

paga

tion

cons

tant

Figure 9. Propagation constants of the guided modes of the waveguide induced bythe dark PV solitons for �Ir ¼ 1:5 when 1 ¼ 2 ¼ 1 (solid curve) and 12 ¼ 0:745(dashed curve).

2150 K. Lu et al.

guided modes in a waveguide induced by a bright photovoltaic soliton by using

numerical techniques, which increases monotonically with increasing intensity ratio.

On the other hand, we have found that waveguides induced by dark photovoltaic

solitons are always single mode for all intensity ratios, of which the confined

energy increases monotonically with increasing intensity ratio. The properties of

soliton-induced waveguides in both self-defocusing-type and self-focusing-type

photorefractive photovoltaic materials were also discussed.

Acknowledgments

This work was supported by the National Natural Science Foundation of China

(No. 10474136).

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