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Optik 122 (2011) 2039– 2043

Contents lists available at ScienceDirect

Optik

jou rna l homepage: www.elsev ier .de / i j leo

oupled tapered/uptapered optical beams

rpita Srivastavaa, Ram Krishna Sarkarb, Shraddha Prasada, S. Medhekara,∗

Department of Applied Physics, Birla Institute of Technology, Mesra, Ranchi 835215, IndiaDepartment of Applied Physics, Birla Institute of Technology, Deoghar Campus, Jasidih, Deoghar 814142, India

r t i c l e i n f o

rticle history:eceived 22 August 2010

a b s t r a c t

For the first time we propose the phenomenon of coupled tapering and uptapering in two mutuallyincoherent beams coaxially co-propagating in a nonlinear medium with small gain or loss. During tapering

ccepted 10 December 2010

eywords:elf-focusing

or uptapering, the widths and powers of the beams evolve in such a manner that they always satisfy thecondition of soliton pairing. It is shown that under certain condition one beam can taper/uptaper, while,other uptaper/taper during coupled tapering/uptapering which is quite counterintuitive.

© 2011 Elsevier GmbH. All rights reserved.

elf-taperingll-optical manipulation

. Introduction

There has been a lot of interest in the study of spatial soliton for-ation, soliton interaction and soliton pairing due to their potential

pplications in all-optical switching, all-optical interconnects andaveguide applications [1–6]. Particularly, spatial soliton pairing

s indispensable in all-optical switching devices (see for example,efs. [7–11]).

In addition to soliton pairing, self-tapering/uptapering of opticaleams is of great importance as it is the only means of all-opticalontrol of beam width without using any fabricated structure12]. A tapering/uptapering beam can induce a tapered/uptaperedaveguide for the other beam. When a solitonic beam enters into aonlinear medium which has small absorption or gain, it tapersr uptapers depending upon the initial conditions [13]. Taper-ng/uptapering of solitons has been predicted/investigated in pastut only for one solitonic beam [12–15].

In this paper, for the first time we have shown the possibilityf coupled tapering or uptapering of two co-propagating beams.nder certain condition one beam can taper/uptaper, while, otherptaper/taper during coupled tapering/uptapering which is quiteounterintuitive.

It is important to be mentioned that using present approach, thevolution of the co-propagating coupled beams with the distancef propagation could be obtained in a little time, for example, thegures of the paper have been obtained in the time of the order

f 30 s with an ordinary processor and requires ignorable memorypace. Investigation of the same problem using beam propagationethod (BPM) would require time in hours and requires enormous

∗ Corresponding author.E-mail address: sarangmedekar@rediffmail.com (S. Medhekar).

030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved.oi:10.1016/j.ijleo.2010.12.034

memory space. One can safely believe the results presented hereas the approach used in this paper has been thoroughly testedagainst beam propagation method (BPM) in Ref. [16] and shownthat both approaches predict same propagation features of twoco-propagating beams.

2. Physical model

We consider two coaxial co-propagating 1D Gaussian beams offrequencies ω1 and ω2 respectively along the z-axis. The intensitydistribution of these beams may be expressed as:

A21(z) = E2

01f1

exp

(− r2

r21 f 2

1

)(1)

A22(z) = E2

02f2

exp

(− r2

r22 f 2

2

)(2)

Here A1 and A2 are the space dependent real amplitudes of theelectric vectors of two beams of angular frequencies ω1 and ω2 andr is the variable coordinate. r1, r2 represent dimensions of thesebeams at z = 0. f1, f2 are the dimensionless beam width parametersof the beams which are initially equal to 1 (at z = 0). The values of f1and f2 change as per evolution of the beams with the propagationdistance and hence, r1f1 and r2f2 give dimensions of the beams at adistance z.

2.1. Dielectric constant of the medium

These beams modify dielectric constant of the medium as:

ε(ω1) = ε10 + ϕ1(A1, A2) (3)

ε(ω2) = ε20 + ϕ2(A1, A2) (4)

2 ptik 1

wfpa

ϕ

ϕ

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tt

ϕec

ϕ

H(

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2

oa

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E

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040 A. Srivastava et al. / O

here ε10 and ε20 are the linear part of the dielectric constants atrequencies ω1 and ω2 respectively and ϕ1 and ϕ2 are the nonlineararts. For Kerr type nonlinear medium, ϕ1 and ϕ2 may be expresseds [16]:

1 = (˛1A21 + �˛2A2

2) (5)

2 = (�˛1A21 + ˛2A2

2) (6)

n Eqs. (5) and (6), ˛1 and ˛2 are constants with their ratio equal tohe ratio of the nonlinear coefficients of the medium at frequencies1 and ω2, respectively (˛jA

2j; j = 1, 2 is the dimensionless elec-

ric field intensity), � is the coupling coefficient of the two beamshat depends on the experimental conditions.

In paraxial ray approximation, one can in general expand ϕ1 and2 around their value at r = 0. Employing A1 from Eq. (1), ϕ1 can bexpanded by Taylor’s expansion and terms beyond squares termsan be neglected so:

1(A1, A2) = ϕ1

[(˛1E2

01f1

+ �˛2E202

f2

)−(

˛1E201

r21 f 3

1

+ �˛2E202

r22 f 3

2

)r2

]

ere (((˛1E201)/f1) + ((�˛2E2

02)/f2)) � (((˛1E201)/r2

1 f 31 ) +

(�˛2E202)/r2

2 f 32 ))r2, therefore, one can write:

ϕ1 = ϕ1

(˛1E2

01f1

+ �˛2E202

f2

)−(

˛1E201r2

r21 f 3

1

+ �˛2E202r2

r22 f 3

2

)

ϕ′1

(˛1E2

01f1

+ �˛2E202

f2

)

here prime over ϕ1 signifies derivative with respect to the argu-ent, on simplifying above equation, we get:

1 = ϕ1

(˛1E2

01f1

+ �˛2E202

f2

)−(

˛1E201r2

r21 f 3

1

+ �˛2E202r2

r22 f 3

2

)(7)

ince ϕ′1(((˛1E2

01)/f1) + ((�˛2E202)/f2)) = 1.

Similarly

2 = ϕ2

(�˛1E2

01f1

+ ˛2E202

f2

)−(

�˛1E201r2

r21 f 3

1

+ ˛2E202r2

r22 f 3

2

)(8)

.2. Coupled propagation of beams

In a medium described by Eqs. (3) and (4), the electric vectorf the waves are governed by Maxwell’s equations, which in WKBpproximation reduce to the wave equation:

2E − ε

c2

∂2E

∂t2= 0 (9)

or slowly converging or slowly diverging beams, Eq. (9) can beatisfied by the following solutions:

= E1 exp[i(ω1t − k1z)] + E2 exp[i(ω2t − k2z)] (10)

here E1 and E2 are the space dependent complex amplitudesnd k1 = (ω1/c)

√ε10 and k2 = (ω2/c)

√ε20 are the propagation con-

tants.Eikonals can be introduced to describe E1 and E2 as:

1 = A1 exp[−ik1S1] (11)

2 = A2 exp[−ik2S2] (12)

n substituting Eqs. (10)–(12) in the wave equation, we get follow-

ng set of equations:

∂Sj

∂z+(

∂Sj

∂r

)2

= ϕj

εj0+ 1

k2jAj

(∂2Aj

∂r2

)(13)

22 (2011) 2039– 2043

∂A2j

∂z+ A2

j

(∂2Sj

∂r2

)+ ∂Sj

∂r

∂A2j

∂r= 0; j = 1, 2 (14)

With subscript 1 or 2 in the above equations, we get the relevantequations for the first or the second beam. To solve Eqs. (13) and(14), we assume that the nonlinear part of the dielectric constant ismuch smaller than the linear part, and therefore, nonlinearity maybe treated as a perturbation. One may therefore assume generalizedspherical wave solution for Eqs. (13) and (14):

Sj = r2

2ˇj(z) + �j(z) (15)

A2j =

E20j

fjexp

(− r2

r2j

f 2j

)(16)

ˇj = 1fj

∂fj∂z

(17)

Here ˇj represent inverse of radius of curvature of the beams’ fronts.Using Eqs. (15)–(17) in Eq. (13) and using paraxial ray approxi-

mation, i.e., (r/rjfj)4 � 1, we obtain:

r2

(1fj

∂2fj∂z2

)+ 2

∂�j

∂z= 1

k2jAj

(− Aj

r2j

fj2

+ Ajr2

r4j

fj4

)+ ϕj(A1, A2)

εj0(18)

On substituting ϕ1(A1, A2) from Eq. (7), Eq. (18) takes the form (forthe first beam):

r2

(1f1

∂2f1∂z2

)+ 2

∂�1

∂z= 1

k21A1

(− A1

r21 f1

2+ A1r2

r41 f1

4

)+(

1ε10

)

ϕ1

(˛1E2

01f1

+ �˛2E202

f2

)−(

1ε10

)(˛1E201r2

r21 f 3

1

+ �˛2E202r2

r22 f 3

2

)

Equating the coefficients of r2 on both sides of the above equa-tion, one obtains the following propagation equation for the firstbeam that governs the beam width parameter with the propagationdistance:

∂2f1∂z2

= 1

k21r4

1 f13

− C

ε10r21 f 2

1

− �Df1ε10r2

2 f 32

(19)

where C = ˛1E201 and D = ˛2E2

02.Similarly, the propagation equation for the second beam could

be obtained as:

∂2f2∂z2

= 1

k22r4

2 f23

− D

ε20r22 f 2

2

− �Cf2ε20r2

1 f 31

(20)

The set of coupled Eqs. (19) and (20) governs the evolution of beams’widths of the two beams with the propagation distance.

For self-trapped beams (spatial solitons), we must have(∂fj/∂z) = (∂2fj/∂z2) = 0. One can assume (∂fj/∂z) = 0 as the initial con-dition of the beams. To have (∂2fj/∂z2) = 0, we must have from Eqs.(19) and (20):

D = ε10r22

�k21r4

1

− Cr22

�r21

(21)

D = ε20

k2r2− �Cr2

2

r2(22)

2 2 1

Eqs. (21) and (22) provide conditions on beam powers and beamwidths to obtain the coupled propagation of the two beams as soli-tonic propagation (soliton pairing).

A. Srivastava et al. / Optik 122 (2011) 2039– 2043 2041

Fig. 1. (a) Using Eqs. (25) and (26), beams’ widths b1( = r1f1) and b2( = r2f2) have beenplotted with the propagation distance z in the figure (b1 and b2 are overlapping).The beams form solitonic pair in the loss less region I. The pair uptapers (dashedline) in the lossy region II with (K1, K2 = −0.5) and again becomes solitonic pair inthe loss less region III. Both the beams taper (solid line) when K1 and K2 are 0.5in region II (gain medium). (b) The variation of evolved beams’ widths b1 and b2with the evolved axial beams’ intensities I1 = C�1/f1 and I2 = D�2/f2 during the courseof coupled tapering/uptapering are shown in the figure. The dotted portion of thecDc

mr

A

A

wst

3

pεaw

Fig. 2. (a) The region II offers gain to the first beam (K1 = 0.5) and loss to the otherbeam (K2 = −0.5) in the figure. The coupled tapering/uptapering phenomenon insuch a case is dominated by the more intense beam. The second beam (thick line)uptapers due to losses and forces first beam (thin line) to uptaper. (b) The Ij versus bj

urves is obtained for uptapering beams, while solid portion is for tapering beams.uring coupled tapering/uptapering, the evolved intensities and widths are alwaysonsistent with Eqs. (21) and (22).

If the medium is not loss less for the two beams, i.e., if theedium is having finite losses or gains, Eqs. (1) and (2) may be

ewritten as:

21(z) = E2

01�1

f1exp

(− r2

r21 f 2

1

)(23)

22(z) = E2

02�2

f2exp

(− r2

r22 f 2

2

)(24)

here � j = exp(Kjz); j = 1, 2 is the loss/gain parameter, a positive Kjignifies gain, while a negative Kj signifies loss and follow the samereatment, Eqs. (19) and (20) may be modified as:

∂2f1∂z2

= 1

k21r4

1 f13

− C�1

ε10r21 f 2

1

− �D�2f1ε10r2

2 f 32

(25)

∂2f2∂z2

= 1

k22r4

2 f23

− D�2

ε20r22 f 2

2

− �C�1f2ε20r2

1 f 31

(26)

. Coupled tapering/uptapering

To examine Eqs. (21), (22), (25) and (26), following set of

arameters has been considered in this paper, � = 1, r1 = r2 = 10 �m,10 = ε20 = (1.6276)2, ω1 = 0.9 × ω2 and ω2 = 2.7148 × 1015 rad/s. Thexial beam intensities are chosen according to Eqs. (21) and (22)hich are C = 3.11 × 10−5, D = 5.9829 × 10−5. We mention here that

curves corresponding to (a) are shown in the figure. Here also the evolved intensitiesand widths are consistent with Eqs. (21) and (22).

the shown results are not limited to the chosen set of parametersand are valid for any other suitable parameters.

We propose the possibility of coupled tapering and upta-pering in soliton pairs. For the sake of clarity, we discuss theself-tapering/uptapering phenomenon in brief. It is known that asolitonic beam does not change its width (and also intensity) whilepropagating through a loss less nonlinear medium. However, whenthe beam enters into a nonlinear medium which has finite loss orgain, it tapers or uptapers depending upon the initial conditions[13]. The self-tapering/uptapering of beams is entirely differentthan the self-focusing/defocusing in the sense that, throughoutthe course of propagation, such a beam propagates as the fun-damental mode of the induced waveguide and when enters intoa loss less medium, becomes again a self-trapped beam, while, aself-focusing/defocusing beam forms a nonsolitonic (diverging orfocusing) beam in a loss less medium.

To investigate coupled tapering/uptapering, we consider threeregions along the beams’ propagation. Regions I and III are lossless/gain less with Kj = 0, while, region II is with finite gain (+Kj)or loss (−Kj).

We start numerical investigation by considering that themedium in region II is with finite loss with K1 and K2 equal to−0.5. We must add here that one may choose any other smallequal/unequal values of Kj. As the axial beam intensities (C and D)are chosen according to Eqs. (21) and (22), the beams form solitonic

pair in the region I as seen in Fig. 1a. In Fig. 1a, using propagationEqs. (25) and (26), evolved beams’ widths b1( = r1f1) and b2( = r2f2)have been plotted with the propagation distance z. b1 and b2 are

2042 A. Srivastava et al. / Optik 1

Fig. 3. (a) The coupled tapering/uptapering becomes counterintutive when theregion II is a gain medium for the second beam and lossy for the first along withthe axial intensities of the beams of the pair are equal (parameters mentioned in thetext). In this case the second beam tapers (solid line), while, the first beam uptapers(w

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[10] S. Medhekar, P.P. Paltani, Proposal for optical switch using nonlinear refraction,

dashed line) as shown. (b) The Ij versus bj curves corresponding to (a) are shownhich are consistent with Eqs. (21) and (22).

verlaping in this figure. When this pair enters into the region II,t uptapers (dashed line). The uptapered beam pair again becomesolitonic pair in the loss less region III. The magnitude of uptaperings same for both the beams of the pair. Both the beams taper (solidine) when K1 and K2 are 0.5 in region II (gain medium). The vari-tion of evolved beams’ widths b1 and b2 with the evolved axialeams’ intensities I1 = C�1/f1 and I2 = C�2/f2 during the course ofapering/uptapering is shown in Fig. 1b. The dotted portion of theurves is obtained for uptapering beams, while solid portion forapering beams. It is quite interesting to note that if beam widthsnd powers corresponding to all points of any one curve (say I2, b2)f the figure are used in Eqs. (21) and (22), the other curve (I1, b1)s produced. This means that during the course of coupled taper-ng/uptapering the axial intensities and beams’ widths of the twoeams vary in such a manner that they always satisfy the conditionf soliton pairing.

Next, we consider a situation where the region II offers gain tohe first beam (K1 = 0.5) and losses to the other beam (K2 = −0.5).uch a situation may occur with pump and signal beams in anctive medium where signal beam grows at the cost of the pumpeam. The result is shown in Fig. 2a. The small fluctuations in theeam widths are due to numerical oscillations. The coupled taper-

ng/uptapering phenomenon in such a case is dominated by theore intense beam. The second beam (thick line) uptapers due to

osses and forces first beam (thin line) to uptaper. The correspond-

ng Ij versus bj curves (Fig. 2b) are consistent with the Eqs. (21) and22).

Using Eqs. (21) and (22), we can find the condition for solitonairing with equal axial intensities of the two beams. If we choose

[

[

22 (2011) 2039– 2043

r1 = 10.2 �m and r2 = 9.75 �m, Eqs. (21) and (22) give common solu-tion at C = D = 4.5425 × 10−5. In this case if the region II is a gainmedium for the second beam and lossy for the first beam, then cou-pled tapering/uptapering phenomenon becomes quite interestingas shown in Fig. 3a. In this case the second beam tapers (solid line),while, the first beam uptapers (dashed line). Here also the corre-sponding Ij versus bj curves (Fig. 3b) are consistent with Eqs. (21)and (22).

We further mention here that, we have confirmed throughextensive numerical investigation that when the losses or gain issmall, throughout the course of coupled tapering or uptapering,the evolved peak powers ((Pj� j)/fj); j = 1, 2 and the evolved beamwidths rjfj always satisfy the conditions of Eqs. (21) and (22). Inother words, though the peak power of the beams change duringtapering or uptapering, their widths evolve in such a manner thatthey always remain a soliton pair.

Lastly, we must add that the governing coupled Eqs. (25) and(26) of this paper may appear similar to the governing equations fortapering/uptapering of elliptic beam of the paper of Ref. [14], how-ever, those are quite different. The governing equations of the paperof Ref. [14] are coupled through the widths along major and minoraxes of the elliptic beam, while, in the present paper, those are cou-pled through the axial intensities of the two co-propagating beams.This fact makes the phenomena of coupled tapering/uptaperingmuch more complicated than the tapering/uptapering of ellipticbeams.

4. Conclusions

In conclusion, we propose the phenomenon of coupled taper-ing and uptapering in two mutually incoherent beams coaxiallyco-propagating in a nonlinear medium with small gain or loss.The coupled beams remain a soliton pair throughout the courseof coupled tapering or uptapering. Under certain condition onebeam tapers/uptapers, while, other uptapers/tapers during coupledtapering/uptapering which is quite counterintuitive.

Acknowledgment

SP and SM thank University Grants Commission (UGC), Govern-ment of India for financial assistance through a research projectreference number: 34-24/2008 (SR).

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