Optics Communications 274 (2007) 66–73 www.elsevier.com/locate/optcom
A semi-classical approach to two-frequency solitons in a three-level cascade atomic system Nadeem A. Ansari, I.N. Towers *, Z. Jovanoski, H.S. Sidhu School of Physical, Mathematical and Environmental Sciences, University of New South Wales at ADFA 2600, ACT, Australia Received 25 October 2006; received in revised form 12 February 2007; accepted 14 February 2007
Abstract A semi-classical theory of two intense optical fields interacting with a third-order non-linear medium composed of a three-level cascade atomic system is presented. It is predicted that non-linear atom-field interactions allow the formation of two-frequency bright, dark and grey spatial solitons. We demonstrate through numerical simulations and analytic stability analysis that the bright and grey solitons are stable. Ó 2007 Published by Elsevier B.V. PACS: 42.65.Tg; 32.80.t; 42.65.k
1. Introduction Single or multi-frequency strong optical fields propagating through a Kerr-like medium exhibit some important non-linear effects such as self- and cross-phase modulation, modulational instability and multi-wave mixing [1]. In the case of self-phase modulation of a single frequency optical wave the medium can become self-focused or defocused depending upon the sign of the third-order susceptibility [2]. In a self-focusing medium a balance between the selffocusing and diffraction of the beam may cause the optical beam to propagate through the medium without changing its transverse profile. These sorts of waves are known as optical bright solitons. Conversely, a defocussing nonlinear medium can support the propagation of dark solitons. Solitons have some very important possible applications in optical switching and beam processing applications. Experimentally, bright [3–5] and dark solitons [6,7] have been studied in a variety of non-linear materials. In the case of multi-frequency optical wave propagation through a Kerr-like medium, the two waves not only interact with the medium but also with each other. Further, the *
Corresponding author. E-mail address:
[email protected] (I.N. Towers).
0030-4018/$ - see front matter Ó 2007 Published by Elsevier B.V. doi:10.1016/j.optcom.2007.02.019
waves induce cross-phase modulation of each other. In Ref. [8,9], such interactions have been considered in detail and they have shown that waves with two different frequencies and identical beam widths can propagate in the form of two-frequency spatial solitons. In the studies presented in [10,11] the collisions between two bright solitons have been considered and interesting results such as splitting, switching and steering of one beam by the other have been predicted. In this paper we investigate the propagation of two intense optical beams in a three-level atomic system in the cascade configuration. The three-level cascade atomic system has been studied in detail by many groups in quantum optics, non-linear optics and laser physics. In quantum-optics, a number of interesting phenomena such as, phase-sensitive [12] and super radiant amplifications [13], violation of classical effects [14], dipole amplitude square squeezing [15], and phase-dependent fluorescence linewidth narrowing [16], have all been analysed. In the area of nonlinear optics, the electromagnetically induced transparency for a probe field in the presence of a strong field [17,18] and field entropy [19] have been considered. In the theory of lasers a number of interesting and potentially important effects have been investigated, including, non-Markovian decay [20], dynamics of a two-photon laser with an injected
N.A. Ansari et al. / Optics Communications 274 (2007) 66–73
signal [21] and lasing without inversion [22]. Experimentally the three-level cascade atomic system has been studied for polarisation effects in Rubidium [23] and laser cooling and diffusion in metastable helium [24,25]. This paper deals with a Kerr-like medium composed of uniformly distributed three-level atomic systems in the cascade configuration for which the atoms are initially in the ground level (see Fig. 1). Two instense optical fields of different frequencies induce the top to intermediate and intermediate to ground atomic level dipole allowed transitions. The transition from the top to the ground levels are dipole forbidden transitions. We also consider a closed atomic system where the top level decays to the intermediate level and the intermediate level to the ground level with different decaying rates. We calculate the steady-state expressions for the third-order susceptibilities experienced by the two fields while propagating through the medium. We then use these susceptibility expressions to determine the nonlinear polarisation induced by the two fields. In Section 3, we consider the case when two optical fields propagate through a thin slab of medium that contains the three-level cascade atomic system described above. We use the expressions of non-linear polarisation experienced by the two waves to show that the two optical waves of the same width can propagate in the form of spatial solitons. We show that for different conditions of atomfield detuning and wavelengths of the two waves, a pairing of two bright, two dark and a bright-dark combination of beams (a grey soliton) are made possible because of the third-order non-linearity of the medium. In Section 4 we perform the analytical stability analysis of firstly, the plane waves solutions to our model. This is the well known modulational stability analysis. Secondly, we consider the linear stability analysis of our soliton solutions and examine the behaviour of the maximal eigenvalue.
67
Finally we analyse the stability of these solitons by performing numerical simulations of the governing equations. Our results show that bright and grey solitons can experience stable propagation through the medium. In the case of the dark soliton we consider two situations: (i) when both the fields have almost equal strength; (ii) when one field is much weaker than the other. In both cases the soliton is ultimately unstable, as predicted by the stability analysis of Section 4, but when different strength fields propagate together the stronger field can generate a waveguide for the weaker field thus enabling the latter field to propagate a longer distance before the soliton as a whole becomes unstable. 2. Two-field interaction in a Kerr-law medium In this section we will consider the interactions of two different frequency fields with a Kerr-law medium that is composed of a closed three-level atomic system in a cascade configuration as shown in Fig. 1. We also consider that the medium is in gaseous form and that the atoms do not interact with each other. The fields of frequencies x1 and x2 induce the dipole allowed transitions between levels jai and jbi, and levels jbi and jci, respectively. The levels jai and jci have the same parity and do not support dipole allowed transitions. We also consider that the medium is not perturbed externally and that initially most of the atom are at the ground level jci. The atom-field interaction can be described through Maxwell’s equation E P NL 1 o2 ~ 1 o2~ ¼ 2 ; ð1Þ 2 2 c ot c ot2 where in the case of two plane polarised fields of frequencies x1 and x2, we can define the total electric field as rr~ Eþ
Eðx; y; z; tÞ ¼ E1 ðx; y; z; tÞ þ E2 ðx; y; z; tÞ:
ð2Þ
For a Kerr-like medium the polarisation can be written as 2
P NL ðx; y; z; tÞ ¼ ðvð1Þ þ vð3Þ jEðx; y; z; tÞj ÞEðx; y; z; tÞ; ð3Þ (1) (3) where v and v are the first and third-order non-linear susceptibilities. Furthermore the refractive index experienced by the field is of the form 2
nðEÞ ¼ n0 þ n2 jEj ; or equivalently as
ð4Þ 2 1=2
nðEÞ ¼ ð1 þ vð1Þ þ vð3Þ jEj Þ
ð5Þ vð3Þ 2 jEj : 2n0 If we consider that the two fields interact with a homogeneously distributed three-level atomic system in the cascade configuration then, following [1], the polarisation induced by two fields will be the average dipole moment per unit volume and can be defined as P ðr; tÞ ¼ N TrðlqÞ; n0 þ
ð3Þ
Fig. 1. Three-level atomic system in cascade configuration.
ð3Þ ¼ N ½lac qca þ lbc qcb þ c:c:;
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N.A. Ansari et al. / Optics Communications 274 (2007) 66–73
where lij ¼ ehijrjji is the dipole moment matrix elements ð3Þ between states jii and jji and qij is the third-order density matrix elements between those states. According to Eq. (2) the total electric field is the superposition of two plane polarised fields, and each of which will induce polarisations of the form ð3Þ
ð3Þ
ð3Þ
ð3Þ
P 1 eix1 t ¼ N ½lba qab ¼ v1 E1 ; P 2 eix2 t ¼ N ½lcb qbc ¼ v2 E2 :
ð6Þ
In order to calculate the third-order susceptibilities experienced by the two fields while propagating into the medium, we have to calculate the density matrix elements between the states jai $ jbi and jbi $ jci. We now proceed to show how these density matrix elements can be calculated. The total Hamiltonian for the atom-field interaction can be defined as H ¼ H0 þ V ; where the unperturbed or the atomic part of the Hamiltonian is defined as H 0 ¼ hxjj
ðj ¼ a; b; cÞ;
and the atom-field interaction energy is V ¼ lEðtÞ:
V bc ¼ lbc E2 eix2 t :
ð8Þ
Here we have added the population decay rates ca and cb phenomenologically, as level jai decays to level jbi and level jbi decays to level jci with rates ca and cb, respectively. Since the total probability of finding the atoms in levels jai, jbi and jci is qaa þ qbb þ qcc ¼ 1, we have then qbb ¼ 1 qaa qcc . We can also define the dipole dephasing rates as cab ¼ bca =2, cbc ¼ aca =2 and cac ¼ ca =2 with a ¼ cb =ca and b ¼ 1 þ a. For the steady-state analysis, we set the left hand side of Eqs. (7) and (8) to zero which represents the situation long after the fields have been turned on. Then we have five simultaneous equations with five unknown density matrix elements. We can solve these equations algebraically and by retaining terms up to the cubic power of the field it is possible to work out the steady-state expressions for the transitional density matrix elements rab and rbc. After substituting these steady-state expressions of rab and rbc in Eq. (6) and by using Eq. (3), using a similar approach outlined in [26], we obtain an expression for the first-order steady-state susceptibilities ð1Þ
Under the rotating wave approximation we can define the atom-field interaction energy elements as V ab ¼ lab E1 eix1 t ;
i q_ aa ¼ ca qaa þ ½lab rba E1 lba rab E1 ; h i q_ cc ¼ cb qbb þ ½lcb rbc E2 lbc rcb E2 : h
v1 ¼ 0;
ð1Þ
v2 ¼ a20
2D02 ia ; a2 þ 4D02 2
ð9Þ
and likewise for the third-order steady-state susceptibilities ð3Þ
v1 ðE1 Þ ¼ 0;
As each level has a definite parity and levels jai and jbi have the same parity, then we have V jj ¼ 0 (j ¼ a; b; c) and V ac ¼ 0. The time dependent propagation equation for the transition density matrix elements can be obtained from the time-dependent Schro¨dinger equation i q_ ¼ ½H ; q: h We can define qab ¼ rab eix1 t and qbc ¼ rbc eix2 t in terms of slowly varying and fast varying components with respect to time. Subsequently we can write the propagation equation for these slowly varying density matrix elements as i r_ ab ¼ ðcab þ iD1 Þrab þ ½ðqbb qaa Þlab E1 lcb rac E2 ; h i r_ ac ¼ ðcac þ iðD1 þ D2 ÞÞrac þ ðrbc lab E1 rab lbc E2 Þ; h i r_ bc ¼ ðcbc þ iD2 Þrbc ½ðqbb qcc Þlbc E2 lba rac E1 : h
ð3Þ
v1 ðE2 Þ ¼ a10
; 0 0 2 ða2 þ 4D02 2 Þ½1 þ 4ðD1 þ D2 Þ 0 2 2aD02 þ ðD01 þ D02 Þða2 4D02 ð3Þ 2Þ @ h i v2 ðE1 Þ ¼ a20 0 0 2 ða2 þ 4D02 2 Þ 1 þ 4ðD1 þ D2 Þ 1 0 0 0 a2 þ 4D02 þ 8aD ðD þ D Þ 2h 2 1 2 iA þi ; 02 0 0 2 2 ða þ 4D2 Þ 1 þ 4ðD1 þ D2 Þ ð3Þ
v2 ðE2 Þ ¼ a20
In deriving the above equations we have added the dipole dephasing rates cab and cbc phenomenalogically. The atom-field detunings are D1 ¼ xa xb x1 and D2 ¼ xb xc x2 . The equation of motion for finding atoms at different levels in the cascade configuration can be written as
ð10Þ
4D02 þ ia ; a2 þ 4D02 2
where the line centre absorption coefficients take the form a10 ¼ N
ð7Þ
2ðD01 þ D02 Þ þ ia
2jla bj2 ; hca
a20 ¼ N
2jlb cj2 ; hca
ð11Þ
and a ¼ cb =ca is the ratio between atomic level decay rates. The dimensionless atom-field detunings are D0j ¼ Dj =ca (for j = 1, 2). It should be noted that the expressions in Eqs. (9) and (10) provide the complex first- and third-order susceptibilities. The imaginary parts of the susceptibilities decay 1=D0 faster than the real parts and for large values of atom-field detunings the imaginary parts make negligible contribution compared to the real parts and can be ignored. Thus at large detunings we can write the real susceptibilities as
N.A. Ansari et al. / Optics Communications 274 (2007) 66–73
a10 a20 ð3Þ ; v2 ðE1 Þ 02 0 ; 0 0 0 8D02 ðD þ D Þ 8D ðD 2 1 2 2 1 þ D2 Þ a20 ð3Þ ð12Þ v2 ðE2 Þ 03 : 4D2 From Eqs. (4), (5) and (12), we can write Eq. (3) as ð3Þ
v1 ðE2 Þ
2
P 1NL ¼ 2n1 n21 ðE2 ÞjE2 j ; 2
ð13Þ
2
P 2NL ¼ 2n2 ½n22 ðE1 ÞjE1 j þ n22 ðE2 ÞjE2 j ;
where the dimensionless electric field for the jth field is Ej ¼ Ej ðx; y; zÞ=Ejs (for j ¼ 1; 2). Here, the saturated field strengths of the two fields are c2a h2 2 ; jE j ¼ : ð14Þ 2s 2 2 4jlab j 4jlbc j To begin with all the atoms are initially in the ground state and only field E2 will initially interact with them and excite them to the intermediate level jbi after which field E1 can start interacting with the atoms. Thus field E2 will induce a non-linear response due to the Kerr effect and which field E1 will experience. On the other hand, field E1 will also contribute towards the third-order susceptibility of field E2. This is also clear from Eq. (12). 2
jE1s j ¼
c2a h2
3. Two-frequency bright and dark vector solitons
oE1 o2 E1 2 þ þ 2g2 k 21 n1 n21 ðE1 ÞjE2 j E1 ¼ 0; oZ oX 2 oE2 o2 E2 2 þ þ 2g2 k 22 n2 ½n22 ðE1 ÞjE1 j 2in2 k 2 g oZ oX 2 þ n22 ðE2 ÞjE2 j2 E2 ¼ 0;
2in1 k 1 g
ð15Þ
E1 ;
w2 ¼ gk 2 ð2n2 jn22 ðE2 ÞjÞ
1=2
E2 ;
we can write Eq. (15) as ow1 o2 w1 2 þ þ sgn½n21 ðE2 ÞM 1 jw2 j w1 ¼ 0; oZ oX 2 ow o2 w2 2 2 þ sgn½n22 ðE2 Þ½M 2 jw1 j þ jw2 j w2 ¼ 0; 2ik 2 n2 g 2 þ oZ oX 2 ð16Þ where 2ik 1 n1 g
ð3Þ
M1 ¼
n1 jn21 ðE2 Þjj2 jv1 ðE2 Þjj2 ¼ ; ð3Þ n2 jn22 ðE2 Þj jv2 ðE2 Þj ð3Þ
M 2 ¼ sgn½n22 ðE2 Þ
3.1. Two-frequency bright vector solitons ð3Þ
ð3Þ
When sgn½v1 ðE2 Þ > 0 and sgn½v2 ðE2 Þ > 0, the medium becomes focused for both the fields and makes it possible for the interaction of the two fields may yield a bright soliton. The solutions of Eq. (16) are the bright vector soliton whose components w1 and w2 are given by w1 ¼ A1 W sechðWX Þ exp½iW 2 Z=ð2k 1 n1 gÞ; w2 ¼ A2 W sechðWX Þ exp½iW 2 Z=ð2k 2 n2 gÞ; where W is the width and amplitudes are given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffi 2ðM 1 1Þ 2 A1 ¼ ; A2 ¼ : M 1M 2 M1
ð18Þ
ð3Þ
For v2 ðE2 Þ > 0 we should have D02 < 0 and v1 ðE2 Þ > 0 when D01 þ D02 > 0 or D01 > jD02 j. For these conditions ð3Þ v2 ðE1Þ > 0. Finally in order to ensure A1 real we have ð3Þ ð3Þ j2 jv1 ðE2 Þj > jv2 ðE2 Þj. This can be satisfied when j2 >
2a20 ðD01 þ D02 Þ : a10 D02
ð19Þ
If we define D01 ¼ ajD02 j where a > 1 and a20 ¼ ba10 , then we can write Eq. (19) as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð20Þ x1 > 2bða 1Þx2 : Bright vector solitons are realised if Eq. (20) is satisfied.
where we have scaled the x- and z-component as x ¼ gX , z ¼ gZ and g ¼ 106 m. This places the scale of problem in micrometres. Then by defining 1=2
and j2 ¼ k 21 =k 22 . In Eq. (16), the second term represents the diffraction experienced by the two fields. The third and fourth terms in the equation for w2 represent the crossand self-phase modulation respectively. As discussed before, the field w1 will not be able to induce self-phase modulation and only field w2 will induce cross-phase modulation in the equation of w1. Also the two fields are coupled through the coupling coefficients M1 and M2.
ð3Þ
If we consider that the two fields propagate into a thin slab of the medium, then the two fields are uniform in the transverse direction y. By substituting Eq. (13) into the steady-state expression of Maxwell’s equation (Eq. (1)) and applying the paraxial approximation [9] we get
w1 ¼ gk 1 ð2n1 jn21 ðE2 ÞjÞ
69
n2 n22 ðE1 Þ v ðE1 Þ ð3Þ ¼ sgn½v2 ðE2 Þ ð3Þ2 ; n1 jn21 ðE2 Þjj2 jv1 ðE2 Þjj2 ð17Þ
3.2. Two-frequency dark vector solitons ð3Þ
ð3Þ
When both sgn½v1 ðE2 Þ < 0 and sgn½v2 ðE2 Þ < 0, then two-frequency dark solitons of the form WX W 2Z w1 ¼ B1 W tanh pffiffiffi exp i ; 2k 1 n1 g 2 ð21Þ WX W 2Z w2 ¼ B2 W tanh pffiffiffi exp i ; 2k 2 n2 g 2 where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M1 1 1 B1 ¼ ; B2 ¼ pffiffiffiffiffiffiffi ð22Þ M 1M 2 M1 are possible in the system. ð3Þ In order to satisfy sgn½v2 ðE2 Þ < 0 we must have ð3Þ D02 > 0. Furthermore sgn½v1 ðE2 Þ < 0 can be satisfied 0 0 0 when D2 < jD1 j and D1 < 0. For these inequalities to hold ð3Þ we must have v2 ðE1 Þ < 0. For real valued B1, Eq. (20) must also be satisfied.
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N.A. Ansari et al. / Optics Communications 274 (2007) 66–73
3.3. Two-frequency gray vector solitons ð3Þ
may be stable as the background of the ‘‘dark’’ beam remains intact.
ð3Þ
When sgn½v1 ðE2 Þ < 0 and sgn½v2 ðE2 Þ < 0, then w1 and w2 may exist in the form of bright and dark soliton pair W 2Z w1 ¼ C 1 W sechðWX Þ exp i ; 2k 1 n1 g W 2Z ; w2 ¼ C 2 W tanhðWX Þ exp i k 2 n2 gM 1 where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð1 M 1 Þ ; C1 ¼ M 1M 2
sffiffiffiffiffiffiffi 2 C2 ¼ : M1
For C 1 to be real we have the condition 0 < j2 < 2bða 1Þ;
for a > 1:
ð23Þ
4.2. Linear stability analysis of solitons To consider the stability of the soliton solutions we assume a perturbation of the form w1 ¼ ðu0 þ u1 eXz Þeibz ;
w2 ¼ ðv0 þ v1 eXz Þeibz ;
ð24Þ
where u0 ¼ u0 ðxÞ and v0 ¼ v0 ðxÞ are the soliton solutions, u1 ¼ u1 ðxÞ and v1 ¼ v1 ðxÞ are the complex perturbation eigenmodes, and X is the perturbation eigenvalue. Substituting (24) into Eq. (16) and keeping to first-order in eXz we find the system of equations for u1 and v1. This system is expanded into a system of four linear equations by rewriting the eigenmodes so that the real and imaginary parts are explicit. We are now able to solve an eigenvalue problem of the form
4. Stability analysis
A~ x ¼ X~ x;
In this section we consider two types of stability analysis. Firstly, we investigated the so-called modulational stability of the plane wave solutions to Eq. (16). Secondly, we performed a standard linear stability analysis on the bright, dark and grey soliton solutions of Eq. (16).
where ~ x ¼ ðuR ; uI ; vR ; vI ÞT , the subscripts R and I stand for real and imaginary respectively, and A is the coefficient matrix. Using the ARPACK software package [29], the eigenvalues of largest magnitude and with largest real part were calculated. In Fig. 2 the maximum real part of the eigenvalues for dark solitons with various values of k2 are displayed. The figure has two distinct parts: a region of almost zero maximum real eigenvalues to the left and; a region of clearly increasing eigenvalues to the right. The region to the left appears to show eigenvalues of zero magnitude but in fact all eigenvalues are greater than zero to the limits of the computer code’s accuracy. Simulations were run for dark solitons deep within both regions. The
Analysis of the modulational stability of plane waves is a well known practice [27]. The basic procedure is to find a plane wave to the system in question and then perturbs them with a modulating perturbation. If a plane wave is modulationally stable then oscillating perturbations of all frequencies will fail to grow and break up the plane wave. In the context of solitons, modulational stability analysis can be used as a necessary (but not sufficient) condition for the stability of the soliton itself. The filimentation that occurs when a plane wave is modulationally unstable produces localised fine structures which may be a precursor to the formation of stable bright solitons. This exact same process destroys the background which is part of the structure of a dark soliton. An examination of the modulational stability of plane waves can rule out the possibility of stable dark solitons. The modulational stability analysis of Eq. (16) showed that the background of the dark solitons is always unstable to a modulation of some frequency. It was found that the growth rate of the perturbation was greatest when the plane wave amplitudes equivalent to w1 and w2 were comparable. The growth rate could be driven to zero in the limit that the w1 wave tends to zero. In this limit the model becomes a single non-linear Schro¨dinger equation and its well known that dark solitons are stable in this equation because of its integrable nature [28]. Further, combinations of a zero background for w1 and a non-zero background for w2 were found to be always modulational stable. This indicates that a grey soliton
1.4 1.2
Maximum real eigenvalue
4.1. Modulational stability
1 0.8 0.6 0.4 0.2 0 1.5
2
2.5
3
λ2 Fig. 2. Maximum real part of the eigenvalues for dark solitons as the wavelength k2 is varied. When the two fields are comparable in amplitude the value of k2 is large and in the unstable part of the diagram. The large dots represent k2 ¼ 1:6 lm and k2 ¼ 2:55 lm. These are the values used in the numerical simulations.
N.A. Ansari et al. / Optics Communications 274 (2007) 66–73 2 1.5
|ψ1|
examples chosen for Section 5 are indicated in Fig. 2 by large dots. For k2 ¼ 2:55 lm the dark soliton fits into the region on the right and we expect it to become unstable quickly. While, for k2 ¼ 1:6 lm the soliton has a near-zero eigenvalues therefore we expect the it to be long lived and likely to be physically observable. The choices of values for k1;2 are somewhat arbitrary, however, these values give roughly equal amplitudes and any deviations from the stationary solutions are clearly visible. The simulations presented in the next section bears out these predictions.
71
1 0.5
0 1000 800
10
600
5
400 200
5. Numerical simulations
For the two-frequency bright vector soliton case, we select the atom-field detunings as D01 ¼ 15 and D02 ¼ 10, with equal line centre absorption coefficients for both fields. Under these conditions Eq. (20) requires ð25Þ
and provides a simple condition for the selection of the wavelengths of two interacting optical fields. Here we consider k1 ¼ 1:54 lm and k2 ¼ 1:92 lm, and equal beam width W ¼ 1:5. We initially perturb the steady state amplitudes of the vector solitons as given in Eq. (18) by 1%. We then numerically solve Eq. (16) along the Z-axis. In Fig. 3, we have plotted the propagation of jw1 j through the medium. As is evident from the plots the soliton is stable while propagating through 1000 units through the medium. 5.2. Two-frequency dark vector soliton In the case of two-frequency dark vector solitons, we consider the atom-field detunings of the fields E1 and E2 as D01 ¼ 15 and D02 ¼ 10, respectively, equal line centre absorption coefficients for both the fields and W ¼1.5. Under these conditions Eq. (25) is satisfied. It is clear from Eq. (22), that the amplitude of the field w1 is directly proportional to the ratio of the wavelengths of the two fields and the field w2 is inversely proportional to the ratio of the wavelengths of the two fields. Thus a selection of the wavelength that satisfies Eq. (25) will also play an impor-
-10
X
Fig. 3. Propagation of a perturbed jw1 j through the medium with D01 ¼ 15, D02 ¼ 10, a10 ¼ a20 , W ¼ 1:5, k1 ¼ 1:54 lm and k2 ¼ 1:92 lm.
tant role in determining the amplitudes of each of the components of the dark vector soliton. 5.2.1. Dark vector solitons with components of comparable strength In this section we consider a medium that supports the propagation of two fields of wavelengths k1 ¼ 1:54 lm and k2 ¼ 1:6 lm under the above mentioned atom-field detunings for fields w1 and w2. Under these selections of the wavelengths the two dark beams have comparable amplitudes. We initially perturbed the amplitudes of the two fields by 1% and numerically propagate the solutions given in Eq. (21) through the medium along the propagation axes Z. Fig. 4 presents the propagation of jw1 j through the medium. It shows that jw1 j becomes unstable just after propagating a distance Z ¼ 40 units. Thus the two-frequency dark vector soliton is not stable. 5.2.2. A weak beam in the presence of a strong beam As mentioned before the wavelengths of the two fields also determine the amplitudes of the two dark beams. We consider the medium that supports the propagation of two fields of wavelengths k1 ¼ 1:54 lm and k2 ¼ 2:55 lm,
1.5
1
|ψ1|
5.1. Two-frequency bright soliton
-5 0
Z
In Section 2 we calculated the steady-state solutions for the two-frequency bright, dark and grey vector solitons and worked out the different choices of atom-field detunings and wavelengths of the two waves under which these solutions are physically feasible. In this section we will numerically analyse these solutions for bright, dark and grey vector solitons. These solutions were obtained using an adaptive finite element package FLEXPDETM [30]. Due to its adaptive nature, errors are minimised as mesh points are added in regions of large gradients. All calculations were performed to a relative error of tolerance less than 0.1%.
k2 > k1
0
0.5
0 60 10
40
5 0
20
Z
-5 0
-10
X
Fig. 4. Propagation of jw1 j along Z-axis with D01 ¼ 15, D02 ¼ 10, a10 ¼ a20 , W ¼ 1:5, k1 ¼ 1:54 lm and k2 ¼ 2:55 lm.
N.A. Ansari et al. / Optics Communications 274 (2007) 66–73
with the same atom-field detunings as discussed in the previous case. In this case the amplitude of the field w1 becomes much smaller than w2. This represents the configuration of a weak dark field propagating in the presence of a strong dark field. In Fig. 5a, we have plotted the result of our numerical simulation of the propagation of the weak beam jw1 j, where we have initially perturbed the amplitudes of both beams by 1%. The figure shows that the weaker field w1 will propagate a longer distance, as compared to the case when the amplitude of the fields are comparable, before becoming unstable. In Fig. 5b, we have presented the propagation of the stronger field. It is clear from the figure that the strong field still maintains its form after Z ¼ 1000 units. Consequently in this case the strong field forms a waveguide for the weak field enabling the weak field to travel a longer distance before becoming unstable.
1.5
1
|ψ1|
72
0.5
0 1000 800
10
600
5
400
0
200
-5 0
Z
-10
X
3
5.3. Grey vector solitons 2
ð3Þ
|ψ |
2 ð3Þ
When sgn½v1 ðE2 Þ < 0 and sgn½v2 ðE2 Þ < 0 the medium can support two fields w1 and w2 in the form of bright and dark beams. In this section we will perform the numer-
1
0 1000 800
10
600
5
400
0.4
0
200
Z
|ψ1|
0.3
-5 0
-10
X
Fig. 6. Two-frequency grey soliton propagation through the medium with D01 ¼ 15, D02 ¼ 10, a10 ¼ a20 , W ¼ 1:5, k1 ¼ 2:55 lm and k2 ¼ 1:54 lm. (a) Propagation of bright beam jw1 j along Z-axis. (b) Propagation of dark beam jw2 j along Z-axis.
0.2 0.1
0 1000 800
10
600
5
400
0
200
-5 0
Z
-10
X
ical simulations in order to check the stability of these beams while propagating in the medium. We consider the same values of the parameters of atomfield detunings, a10 and a20 as we used in Fig. 4. Under these conditions Eq. (23) requires that k1 > k2 : Thus we consider the medium that supports the propagation of two intense fields of wavelengths k1 ¼ 2:55 and k2 ¼ 1:54 for the given atom-field detunings D01 and D02 for the fields w1 and w2. In Fig. 6, we have plotted the propagation of the bright beam jw1 j and the dark beam jw2 j each with a 1% perturbation in the amplitude. The figure illustrates the two solitons propagating stably through the medium.
1.5
|ψ2|
1
0.5
0 1000
6. Conclusions 800
10
600
5
400
0
200
Z
-5 0
-10
X
Fig. 5. Two-frequency dark soliton propagation through the medium with D01 ¼ 15, D02 ¼ 10, a10 ¼ a20 , W ¼ 1:5, k1 ¼ 1:54 lm and k2 ¼ 1:6 lm. (a) Propagation of weak dark field jw1 j along the Z-axis. (b) Propagation of strong dark field jw2 j along the Z-axis.
Within the semi-classical approach the propagation problem was described by a coupled system of non-linear Schro¨dinger equations for the field amplitudes of the electric fields. The steady-state expressions were obtained for third-order non-linear susceptibilities experienced by the two fields while propagating through the medium. The calculations showed that in the absence of the field E2 the field
N.A. Ansari et al. / Optics Communications 274 (2007) 66–73
E1 will not be able to experience any non-linear effects. An intense field E2, cannot only experience non-linear susceptibility but it will also induce the susceptibility to field E1. The field E1 may then interact with the medium and will only contribute towards the susceptibility experience by the field E2 in a Kerr-like medium. We used the steady-state expression of the non-linear susceptibilities to find the expressions for the non-linear polarisation experience by the two fields. Our results showed that for different choices of atom-field detunings the medium can become either focused or defocused. By substituting these non-linear polarisation expressions into Maxwell’s equations we described the steady-state solutions for a two-frequency bright, dark and grey vector solitons and determined the different conditions for which solutions exist. Our analysis showed that in a defocused medium we can as well as get a steady-state solution for a two-frequency dark vector soliton as well as a two-frequency grey soliton with a bright component for field E1 and a dark component for E2. Using a combination of analytical stability analysis and numerical simulations of the governing equations we have shown that the bright solitons and grey solitons are stable while the dark solitons are all unstable albeit long lived in some cases. References [1] See for example R.W. Boyd, Non-Linear Optics, Academic Press, NY, 1992; P. Meystre, M. Sargent III, Elements of Quantum Optics, SpringerVerlag, Berlin, 1990. [2] R.Y. Chiao, E. Garmire, C.H. Townes, Phys. Rev. Lett. 13 (1964) 479. [3] J.S. Aitchison, A.M. Weiner, Y. Silberberg, D.E. Leaird, M.K. Oliver, J.L. Jackel, P.W.E. Smith, Opt. Lett. 16 (1991) 15. [4] A. Barthelemy, S. Maneuf, C. Froehly, Opt. Commun. 55 (1985) 201.
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