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Temperature effects of Hamiltonian dark solitons on the stability of holographic solitons in a separate holographic–Hamiltonian soliton pair
Optik 124 (2013) 4912–4916
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Optik journal homepage: www.elsevier.de/ijleo
Temperature effects of Hamiltonian dark solitons on the stability of holographic solitons in a separate holographic–Hamiltonian soliton pair Xin Cai ∗ , Bin Guo School of Science, Wuhan University of Technology, Wuhan 430070, China
a r t i c l e
i n f o
Article history: Received 22 September 2012 Accepted 1 March 2013
Keywords: Spatial optical solitons Photorefractive effects Soliton pair
a b s t r a c t We investigate theoretically the temperature effects on the evolution and stability of a separate holographic–Hamiltonian dark–dark or bright–dark soliton pair formed in an unbiased serial photorefractive crystal circuit. Our numerical results show that, for a stable dark–dark or bright–dark soliton pair originally formed in a crystal circuit at given temperatures, when the crystal in which formed a Hamiltonian dark soliton changes, the holographic dark or bright soliton supported by the other crystal tends to evolve into another stable soliton or experiences larger cycles of compression or breaks up into beam filaments or exhibit a common decaying process. The holographic dark soliton is more sensitive to the temperature change than the holographic bright one. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction Hamiltonian solitons and holographic solitons are two types of spatial solitons in photorefractive (PR) media, which have aroused much interest in the past few years [1–12]. Up to date, three different kinds of Hamiltonian PR solitons (screening solitons [2–5], photovoltaic (PV) solitons [6–9] and screening-photovoltaic solitons [1,10]) which are supported by the self-phase-modulation self-focusing mechanism have been predicted and found experimentally. Combining the holographic focusing mechanism with the self-phase-modulation self-focusing mechanism, holographic photovoltaic solitons, holographic screening solitons and holographic screening-photovoltaic solitons have been predicted [11] and the first one has been found experimentally [12]. Recently, Cai and Liu [13] predicted a new type of separate spatial solitons pair, named separate holographic–Hamiltonian soliton pair, which is formed in a crystal circuit in which one PR crystal and one PV-PR crystal are connected in a chain by electrode leads. There are two types of the separate solitons pair: dark–dark and bright–dark, the two solitons in such a solitons pair are coupling with each other by parameters, and the most interesting phenomenon is that the change of the intensity of Hamiltonian solitons can affect the other solitons whereas the holographic one cannot. As is known, the PR effects are dependent on the dark irradiance of the crystal, and the latter is dependent on the temperature [14]. Therefore, the temperature of crystals has an obvious effect on the
∗ Corresponding author. E-mail address: [email protected] (X. Cai). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.03.037
spatial shape, evolution and stability of a PR soltion [15,16]. A natural question to address is whether the temperature of one crystal can affect the characteristics of the other soliton formed in another crystal in a separate holographic–Hamiltonian soliton pair or not. In this paper, we investigate the temperature effects on the stability of a separate holographic–Hamiltonian solitons pair in an unbiased PR crystal circuit as shown in Fig. 1.
2. Theoretical model To study the separate holographic–Hamiltonian solitons pair in an unbiased serial PR circuit, we use one PR crystal and one PV-PR crystal to compose the circuit as shown in Fig. 1. For the two crysˆ two electrodes are made on the surfaces tals, denoted by P and P, with their normal parallel to the crystal’s axis and they are connected electronically in a chain by electrodes leads. The two optical beams I and ˆI are made to propagate in the two crystals along z and zˆ axes and are permitted to diffract only along the x and xˆ directions, respectively, and Ip , an optical beam with a uniform spatial distribution in both transverse dimensions as the pump beam, is also made to propagate in the crystal along the z axis and makes a small angle in the crystal with the signal beam I. Moreover, let us assume that the two incident optical beams I and ˆI are both linearly polarized along the x and xˆ directions, respectively. The pump beam Ip is also linearly polarized light and its polarized direction makes an angle with the x axis. In the limit in which the optical wave has a spatial extent x much less than the x width of the crystal (the x-direction being taken parallel to the c-axis), we predict that each crystal can support a spatial bright (dark) solitons.
X. Cai, B. Guo / Optik 124 (2013) 4912–4916
(a)
x
C
xˆ z
P
y
xˆ z
x
zˆ
Pˆ
yˆ
Iˆ
Ip
I
(b) C
Cˆ
P
y Ip
I
Cˆ
zˆ
Pˆ
Iˆ
yˆ
As usual, the two optical fields are expressed in terms of slowly ˆ In the slowly varying approximation, varying envelopes and . the two optical beams satisfy the following paraxial equations of evolution [13]:
1 ∂2 k0 3 ∂ i − +i n r33 Esc + 0 − 2 e 2 2k ∂x2 ∂z
I p
Ip + I
+i
Esc = E0
I∞ + Ip I∞ − I + Ep I + Ip I + Ip
(3)
Eˆ sc = Eˆ 0
ˆI∞ + ˆId ˆI∞ − ˆI + Eˆ p ˆI + ˆId ˆI + ˆId
(4)
where I∞ = I(x → ± ∞ , z), Ep is the photovoltaic field constant, ˆId is the so-called dark irradiance for crystal Pˆ which depends strongly on the temperature. Here, we use the one-level and one-carrier model presented by Cheng and Partovi [14] to describe the temperature dependence of ˆId as the following:
exp
Eˆ 0 = − Eˆ p
(7)
ˆ ˆI∞ + ˆId )], ˆ ˚/W , = ıˆ ˆI∞ W ˚ = 1/[ı(I∞ + Ip ) + ı( ˆD − N ˆ A )/(ˆrR N ˆ AW ˆ ). The ˆ si (N ı = S si (ND − NA )/(rR NA W) and ıˆ = Sˆ ˆ sign ± is corresponding to the dark–dark or bright–dark soliton pair. S denotes the electrode surfaces, is the electron mobility, si is the photo excitation cross section, ND denotes the donor density, NA is the acceptor density, rR is the carrier recombination rate. Then the evolution equations of the two solitons formed in the crystal P and Pˆ can now be established by substituting Eqs. (3) and (4) into Eqs. (1) and (2). It proves more convenient to study this equation in a normalized fashion. To do so, we adopt the following dimensionless coordinates and variables; let = z/(kx02 ), s = x/x0 ,
Eˆ t kB
1
300
−
1 Tˆ
ˆ = (2 0 ˆId /nˆ e ) U. ˆ Here x0 is an arbitrary = (2 0 Ip /ne )1/2 U and spatial width. We can obtain the dynamical evolution equations of ˆ as follows: the normalized envelope U and U 1 U Uss − [ˇ(1 + ) − g0 + ig] + iϑU = 0 2 1 + |U|2
(8)
(2)
where k = ne k0 , k0 = 2/ 0 , 0 is the free-space wavelength of the lightwave employed, ne is the unperturbed extraordinary refractive index, ϑ0 is the absorption coefficient of the crystal, r33 and is the electro-optical coefficient, and 0 are the intensity and phase coupling coefficients respectively of the two-wave mixing, Esc is the induced space-charge field. I(x, z) = (ne /2 0 )|(x, z)|2 , 0 = ( 0 /ε0 )1/2 and Ip = (ne /2 0 )|p |2 . It should be emphasized that, in this paper, the parameters with the symbol ˆ. have the same physical meaning as those without it. The expression of Esc (Eˆ sc ) can be obtained from the standard set of rate and continuity equations and Gauss’s law, which describe the photorefractive effects in a medium in which the photovoltaic current is non-zero and the electrons are the sole charge carriers. Here, the same approach was adopted as in Ref. [13] and in the limit of the spatial extent x ( ˆx) of the optical wave being much less ˆ ) of the crystal P (P), ˆ that is x W than that the x width W (W ˆ ), and we obtain the expression of Esc (Eˆ sc ) as follows: (ˆx W
(6)
iU +
ˆ ˆ 1 ∂2 ∂ kˆ 0 3 ˆ =0 +i (nˆ rˆ33 Eˆ sc ) − 2 2 e ∂z ˆ ∂x 2k
Tˆ 300
E0 = ± Eˆ p
1/2
ϑ0 =0 2 (1)
that the Hamiltonian dark solitons formed in the crystal Pˆ and the holographic dark or bright solitons formed in the crystal P. The expressions for E0 and Eˆ 0 necessary for a soliton pair are as follows:
where
Fig. 1. Illustration of an unbiased series PR crystal circuit consisting of one PV-PR crystal and one PR crystal to support a dark–dark soliton pair in (a) and a bright–dark soliton pair in (b). P and Pˆ denote the two crystals. C and Cˆ denote the two caxes. Ip denotes the incident pump beam. I and ˆI denote the incident bright or dark solitonlike one-dimensional laser beams.
ˆId = ˆId0
4913
(5)
where ˆId0 is the value of ˆId at Tˆ = 300 K, and Eˆ t is the level location in the gap. For photovoltaic photorefractive crystals, such as LiNbO3 , Eˆ t ∼10−19 J [14], in this paper, for simplicity, we take Eˆ t = 10−19 J. We emphasize our attention on the effect of Hamiltonian dark solitons on the holographic one, so we discuss two types of soliton pairs, i.e., dark–dark and bright–dark soliton pairs. Let us assume
ˆ ˆ + 1U ˆ − ˇ( iU ˆ + 1) 2 ss
ˆ U ˆ 2 1+|U|
ˆ 2 ˆ
ˆ U| )U −˛ ˆ (−| =0 ˆ 2 1+|U|
(9)
where U = ∂U/∂, Uss = ∂2 U/∂s2 , = I∞ /Ip , g0 = kx02 0 , g = kx02 /2, 2 ˆ = ˇ = r33 E0 , = (k0 x0 ) (n4e /2), ϑ = kx2 ϑ0 /2, ˇ ˆ Eˆ 0 , ˛ ˆ = ˆ Eˆ p and 0
2
ˆ = (kˆ 0 xˆ 0 ) (nˆ 4e rˆ33 /2). It is important to note that the value of ˆId changes with the crystal’s temperature, therefore the value of ˆ = ˆI∞ /ˆId also changes with the crystal’s temperature. Letting denotes the value of ˆ at Tˆ = 300 K, using Eq. (5), the quantity ˆ = ˆI∞ /ˆId can be expressed as follow:
ˆ = ˆ0
Tˆ 300
−3/2
exp −
Eˆ t kB
1 300
−
1 Tˆ
(10)
From PR theory [17], we can obtain the expressions for g and g0 2 + E 2 ) and g = (r E E 2 )/(E 2 + as g = [reff Es (E02 + Ed Eds )]/(Eds 0 eff 0 s 0 ds 2 E0 ), where Es = eNA 0 /(4ε0 εr sin ) is the saturation field, Ed = 4kB T sin /( 0 e) is the diffusion field, and Eds = Ed + Es . In this letter soliton formation of the signal beam in the crystal P is studied under the condition |U|2 = I/Ip 1, thus allowing the approximation that (1 + |U|2 )−1 ≈ 1 − |U|2 . As a result, Eq. (8) becomes iU +
1 Uss − [Pd + iG]U + [Pd + ig]U|U|2 = 0 2
(11)
where G = g − ϑ and Pd = (1 + )ˇ − g0 . The Hamiltonian dark soliton solution which formed in crystal ˆ in Pˆ can be derived from Eq. (9) by expressing the beam envelop U ˆ where ˆ represents a nonˆ = the usual fashion: U ˆ 1/2 yˆ (ˆs) exp(iˆ), linear shift of the propagation constant and yˆ (ˆs) is a normalized real function bounded between 0 ≤ yˆ (ˆs) ≤ 1. Using the boundary condition of the dark soliton yˆ (0) = 0, yˆ˙ (∞) = 0 and yˆ (ˆs → ±∞) = 1, the normalized Hamiltonian dark field profile yˆ (ˆs) obey the following equation: ˆ [−2G]
1/2
0
sˆ = ± yˆ (ˆs)
ˆ ˆ =˛ where G ˆ + ˇ.
dy 1/2
[(y 2 − 1) − ((1 + )/ ˆ ) ˆ ln((1 + y ˆ 2 )/(1 + ))] ˆ (12)
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X. Cai, B. Guo / Optik 124 (2013) 4912–4916
The holographic dark soliton solution in the crystal P of Eq. (11) has been found as the following [11]: U = D tanh(Hs) exp{id ln[cosh(Hs)]} exp(−i˝)
(13) 1/2
where D = (G/g)1/2 , H = [2G/(3d)]1/2 , d = [3Pd + (9Pd2 + 8g 2 ) ]/(2g) and ˝ = 2G/(3d) + Pd . For the holographic bright soliton, the optical beam intensity is expected to vanish at infinity (s → ±∞), i.e., I∞ = 0 and then = I∞ /Ip = 0. From Eq. (11), we can obtain the bright soliton solution as the following [11]: U = F sech(Bs) exp{ib ln[sech(Bs)]} exp(−i) where (9Pd2
F = [3G/(2g)]1/2 ,
1/2 + 8g 2 ) ]/(2g),
B = (G/b)1/2 ,
(14) b = [−3Pd +
= (b2
− 1)G/(2b) + Pd . Eqs. (12)–(14) indicate that a Hamiltonian dark soliton and a holographic dark or bright soliton can co-exist in the crystal circuit ˆ ≺ 0. Although the two dynamical evolution equation provided G have a similar form to that for a single PR crystal [1,11], they couple each other by the coupling coefficient . If we keep the temperature of crystal P unchanged, and adjust the temperature of Pˆ or the maximum intensity ˆI∞ , the Hamiltonian dark soliton can affect the characteristic of the holographic dark or bright soliton. In this paper, we emphasize out attention on the temperature of Hamiltonian dark soliton on the evolution and stability of holographic one. 3. Temperature effects on the stability in separate holographic–Hamiltonians soliton pairs In the following calculation, we take a LiNbO3 (PV-PR) and a SBN (PR) crystal as an example to compose the circuit. The two crystals have the following parameters: ne = 2.33, n0 = 2.36, εr = 880, r33 = 220 pm/V, NA = 1.2 × 1017 cm−3 , = 2◦ , ϕ = 87◦ , ϑ0 = 0.27 cm−1 , ˆ W =W ˆ , Ip = mˆId0 nˆ e = 2.2, rˆ33 = 30 pm/V, Eˆ p = −105 V/m, ı = ı, and reff = (ne /n0 )3 r33 cos cos(/2) cos ϕ. The arbitrary scales are ˆ0 = x0 = xˆ 0 = 40 m and the free–space wavelengths are 0 = 0.5 m. For these set of values, we have = 3.72 × 106 m/V, ˆ = ˆ = −8.8781. The dependence of coupling 8.88 × 10−5 m/V and ˛ coefficient on temperature Tˆ can be obtained as following: =
Fig. 2. The effect of the temperature of crystal Pˆ on the normalized intensity profile in the separate holographic–Hamiltonian dark–dark soliton pair: (a) holographic dark soliton: (b) Hamiltonian dark soliton. Solid curve denotes the case of Tˆ = 300 K, and the dashed and dotted curves denote the case of Tˆ = 290 K and Tˆ = 310 K, respectively.
ˆ mϕ ˆ + mϕ ˆ + ˆ +1
We now investigate the effect of the temperature of crystal Pˆ on the evolution and stability of the holographic dark soliton. In doing so, a holographic dark soliton is obtained from Eq. (13) with the system parameters chosen as T = 300 K, Tˆ = 300 K, D = 0.5645, d = 0.0944 and H = 1.4457. Taking the solitary state as the input beam, we solve Eq. (11) numerically at different temperature Tˆ to obtain the dynamical evolution of the holographic dark soliton, as shown in Fig. 3. The results indicate that changing the temperature Tˆ can obviously affect the dynamical evolution and the stability of the holographic dark soliton in a separate holographic–Hamiltonian dark–dark soliton pair. First, we examine the case of Tˆ = 300 K, as shown in Fig. 3(a). As expected, our numerical results confirm that the holographic dark soliton beam remains invariant with increasing propagation distance in the crystal. The reason is that the dark soliton is just the dark solitary solution supported by crystal P at Tˆ = 300 K, thus the dark soliton can evolve in an adiabatic fashion. Then we examine the case that the temperature Tˆ deviates from Tˆ = 300 K. We can see that the incident beam reshapes itself and tends to evolve into another stable holographic dark soliton when the temperature Tˆ is higher than Tˆ = 300 K as shown in Fig. 3(b). Adjusting the temperature Tˆ to be lower than Tˆ = 300 K, we can see from Fig. 3(c) and (d) that the holographic dark soliton cannot evolve into a stable dark soliton but instead tends to break up into beam filaments as the temperature is lowered.
(15)
−3/2
where ϕ ˆ = (Tˆ /300) exp[−(Eˆ t /kB )((1/300) − (1/Tˆ ))]. If we keep the temperature of crystal P unchanged, and adjust the temperature ˆ the Hamiltonian dark soliton can affect the characteristic of the of P, holographic one. 3.1. Dark–dark soliton pair Firstly, we investigate the effect of temperature of crystal Pˆ on the holographic dark soliton. We can adjust the temperature of crystal P to T = 300 K and then keep it unchanged. We choose the input light beam to be of = 0.03, ˆ = 1, m = 10 and Tˆ = 300 K, ˆ = −8.1563, D = 0.5645, d = 0.0944 and which result in = 1/12.3, G H = 1.4457. Using the above parameters, we obtain the normalized intensities of the two solitons from Eqs. (12) and (13), as show in Fig. 2 (solid curve). Then we change the temperature Tˆ to investigate the temperature effect on the normalized intensities, as shown in Fig. 2 (dashed and dotted curves). From the results we can see that the change of temperature Tˆ can affect the normalized intensities of the two dark solitons. The higher the temperature of crystal ˆ the bigger the intensity in full width at half maximum (FWHM) P, of the dark solitons.
Fig. 3. The effect of the temperature of crystal Pˆ on the dynamical evolution of the holographic dark solution in the separate holographic–Hamiltonian dark–dark soliton pair: (a) Tˆ = 300 K, (b) Tˆ = 350 K, (c) Tˆ = 280 K, (d) Tˆ = 270 K. The input incident holographic dark solitary state is obtained from Eq. (13) with T = 300 K, D = 0.5645, d = 0.0944 and H = 1.4457.
X. Cai, B. Guo / Optik 124 (2013) 4912–4916
4915
Fig. 4. The effect of the temperature of crystal Pˆ on the normalized intensity profile in the separate holographic–Hamiltonian bright–dark soliton pair: (a) holographic bright soliton; (b) Hamiltonian dark soliton. Solid curve denotes the case of Tˆ = 300 K, and the dashed and dotted curves denote the case of Tˆ = 290 K and Tˆ = 310 K, respectively.
ton; at the temperature p the dark soliton deviates slightly from Tˆ = 300 K. However, if the temperature of crystal Pˆ further deviates from Tˆ = 300 K, such as Tˆ = 290 K and Tˆ = 310 K, as shown in Fig. 5(c) and (d), the bright soliton cannot evolve into a stable bright soliton but instead tends to experience larger cycles of compression, and its maximum amplitude oscillates with propagation distance at Tˆ = 290 K or exhibit a common decaying process in the crystal P at Tˆ = 310 K. By a close examination of the above results, in contrast to that of holographic dark soliton in Fig. 3, we can see that the holographic dark soliton has a narrower stable temperature region, and it is more sensitive to the temperature change than the holographic bright one in a separate holographic–Hamiltonian soliton pair.
3.2. Bright–dark soliton pair
4. Conclusions
Next we investigate the effect of the temperature of crystal Pˆ on the evolution and stability of the holographic bright soliton formed in crystal P. In doing so, we adjust the temperature of the crystal P to T = 300 K, and keep it unchanged. We choose the input light beam to be of = 0, ˆ = 1, m = 10 and Tˆ = 300 K, which result in = 1/2, ˆ = −8.1383, F = 0.6914, b = 0.0950 and B = 1.7652. Using the above G parameters, we obtain the normalized intensities of the two solitons from Eqs. (12) and (14), as show in Fig. 4 (solid curve). Then we change the temperature Tˆ to investigate the temperature effect on the normalized intensities, as shown in Fig. 4 (dashed and dotted curves). From the results we can see that the change of temperature Tˆ can affect the normalized intensities of the Hamiltonian dark soliton and holographic bright soliton. The higher the temperature ˆ the bigger the intensity in FWHM of the two solitons. of crystal P, The effect of the temperature of crystal Pˆ on the evolution and stability of the holographic bright soliton in such separate soliton pair is shown in Fig. 5. The fundamental holographic bright solitary state is obtained from Eq. (14) with the system parameters chosen as T = 300 K, Tˆ = 300 K, F = 0.6914, b = 0.0950 and B = 1.7652. The results indicate that changing the temperature of crystal Pˆ can sharply affect the dynamical evolution and the stability of the holographic bright soliton. When Tˆ = 300 K, the bright soliton is just the fundamental solitary solution supported by the crystal, therefore, the soliton can evolve in an adiabatic fashion during its propagation in the crystal; Fig. 5(b) shows that the bright soliton can reshape itself and tend to evolve into another stable bright soli-
The temperature effects on the evolution and stability of a separate holographic–Hamiltonian soliton pair formed in an unbiased serial photorefractive crystal circuit have been investigated under steady-state conditions for the case of neglecting the diffusion process. Our numerical results show that the Hamiltonian dark soliton can affect the holographic dark or bright soliton in a stable dark–dark or bright–dark soliton pair formed in a crystal circuit at given temperatures. When the temperature of the crystal in which formed a Hamiltonian dark soliton changes, the holographic dark supported by the other crystal tends to evolve into another stable holographic dark soliton when the temperature is higher than the original temperature, whereas it tends to break up into beam filaments as the temperature is lowered. On the other hand, the holographic bright soliton tends to be unstable and experiences larger cycles of compression if the temperature difference is big enough, whereas it tends to evolve into another stable soliton when the temperature difference is small enough The holographic dark soliton has a narrower stable temperature range than the holographic bright one in such a separate holographic–Hamiltonian soliton pair, therefore the dark solitons, therefore the holographic dark soliton is more sensitive to the temperature change. Acknowledgement This work is supported by the Fundamental Research Funds for the Central Universities no. 2011-la-001. References
Fig. 5. The effect of the temperature of crystal Pˆ on the dynamical evolution of the holographic bright solution in the separate holographic–Hamiltonian bright–dark soliton pair: (a) Tˆ = 300 K, (b) Tˆ = 305 K, (c) Tˆ = 290 K, (d) Tˆ = 310 K. The input incident holographic bright solitary state is obtained from Eq. (14) with T = 300 K, F = 0.6914, b = 0.0950 and B = 1.7652.
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Optik journal homepage: www.elsevier.de/ijleo
Temperature effects of Hamiltonian dark solitons on the stability of holographic solitons in a separate holographic–Hamiltonian soliton pair Xin Cai ∗ , Bin Guo School of Science, Wuhan University of Technology, Wuhan 430070, China
a r t i c l e
i n f o
Article history: Received 22 September 2012 Accepted 1 March 2013
Keywords: Spatial optical solitons Photorefractive effects Soliton pair
a b s t r a c t We investigate theoretically the temperature effects on the evolution and stability of a separate holographic–Hamiltonian dark–dark or bright–dark soliton pair formed in an unbiased serial photorefractive crystal circuit. Our numerical results show that, for a stable dark–dark or bright–dark soliton pair originally formed in a crystal circuit at given temperatures, when the crystal in which formed a Hamiltonian dark soliton changes, the holographic dark or bright soliton supported by the other crystal tends to evolve into another stable soliton or experiences larger cycles of compression or breaks up into beam filaments or exhibit a common decaying process. The holographic dark soliton is more sensitive to the temperature change than the holographic bright one. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction Hamiltonian solitons and holographic solitons are two types of spatial solitons in photorefractive (PR) media, which have aroused much interest in the past few years [1–12]. Up to date, three different kinds of Hamiltonian PR solitons (screening solitons [2–5], photovoltaic (PV) solitons [6–9] and screening-photovoltaic solitons [1,10]) which are supported by the self-phase-modulation self-focusing mechanism have been predicted and found experimentally. Combining the holographic focusing mechanism with the self-phase-modulation self-focusing mechanism, holographic photovoltaic solitons, holographic screening solitons and holographic screening-photovoltaic solitons have been predicted [11] and the first one has been found experimentally [12]. Recently, Cai and Liu [13] predicted a new type of separate spatial solitons pair, named separate holographic–Hamiltonian soliton pair, which is formed in a crystal circuit in which one PR crystal and one PV-PR crystal are connected in a chain by electrode leads. There are two types of the separate solitons pair: dark–dark and bright–dark, the two solitons in such a solitons pair are coupling with each other by parameters, and the most interesting phenomenon is that the change of the intensity of Hamiltonian solitons can affect the other solitons whereas the holographic one cannot. As is known, the PR effects are dependent on the dark irradiance of the crystal, and the latter is dependent on the temperature [14]. Therefore, the temperature of crystals has an obvious effect on the
∗ Corresponding author. E-mail address: [email protected] (X. Cai). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.03.037
spatial shape, evolution and stability of a PR soltion [15,16]. A natural question to address is whether the temperature of one crystal can affect the characteristics of the other soliton formed in another crystal in a separate holographic–Hamiltonian soliton pair or not. In this paper, we investigate the temperature effects on the stability of a separate holographic–Hamiltonian solitons pair in an unbiased PR crystal circuit as shown in Fig. 1.
2. Theoretical model To study the separate holographic–Hamiltonian solitons pair in an unbiased serial PR circuit, we use one PR crystal and one PV-PR crystal to compose the circuit as shown in Fig. 1. For the two crysˆ two electrodes are made on the surfaces tals, denoted by P and P, with their normal parallel to the crystal’s axis and they are connected electronically in a chain by electrodes leads. The two optical beams I and ˆI are made to propagate in the two crystals along z and zˆ axes and are permitted to diffract only along the x and xˆ directions, respectively, and Ip , an optical beam with a uniform spatial distribution in both transverse dimensions as the pump beam, is also made to propagate in the crystal along the z axis and makes a small angle in the crystal with the signal beam I. Moreover, let us assume that the two incident optical beams I and ˆI are both linearly polarized along the x and xˆ directions, respectively. The pump beam Ip is also linearly polarized light and its polarized direction makes an angle with the x axis. In the limit in which the optical wave has a spatial extent x much less than the x width of the crystal (the x-direction being taken parallel to the c-axis), we predict that each crystal can support a spatial bright (dark) solitons.
X. Cai, B. Guo / Optik 124 (2013) 4912–4916
(a)
x
C
xˆ z
P
y
xˆ z
x
zˆ
Pˆ
yˆ
Iˆ
Ip
I
(b) C
Cˆ
P
y Ip
I
Cˆ
zˆ
Pˆ
Iˆ
yˆ
As usual, the two optical fields are expressed in terms of slowly ˆ In the slowly varying approximation, varying envelopes and . the two optical beams satisfy the following paraxial equations of evolution [13]:
1 ∂2 k0 3 ∂ i − +i n r33 Esc + 0 − 2 e 2 2k ∂x2 ∂z
I p
Ip + I
+i
Esc = E0
I∞ + Ip I∞ − I + Ep I + Ip I + Ip
(3)
Eˆ sc = Eˆ 0
ˆI∞ + ˆId ˆI∞ − ˆI + Eˆ p ˆI + ˆId ˆI + ˆId
(4)
where I∞ = I(x → ± ∞ , z), Ep is the photovoltaic field constant, ˆId is the so-called dark irradiance for crystal Pˆ which depends strongly on the temperature. Here, we use the one-level and one-carrier model presented by Cheng and Partovi [14] to describe the temperature dependence of ˆId as the following:
exp
Eˆ 0 = − Eˆ p
(7)
ˆ ˆI∞ + ˆId )], ˆ ˚/W , = ıˆ ˆI∞ W ˚ = 1/[ı(I∞ + Ip ) + ı( ˆD − N ˆ A )/(ˆrR N ˆ AW ˆ ). The ˆ si (N ı = S si (ND − NA )/(rR NA W) and ıˆ = Sˆ ˆ sign ± is corresponding to the dark–dark or bright–dark soliton pair. S denotes the electrode surfaces, is the electron mobility, si is the photo excitation cross section, ND denotes the donor density, NA is the acceptor density, rR is the carrier recombination rate. Then the evolution equations of the two solitons formed in the crystal P and Pˆ can now be established by substituting Eqs. (3) and (4) into Eqs. (1) and (2). It proves more convenient to study this equation in a normalized fashion. To do so, we adopt the following dimensionless coordinates and variables; let = z/(kx02 ), s = x/x0 ,
Eˆ t kB
1
300
−
1 Tˆ
ˆ = (2 0 ˆId /nˆ e ) U. ˆ Here x0 is an arbitrary = (2 0 Ip /ne )1/2 U and spatial width. We can obtain the dynamical evolution equations of ˆ as follows: the normalized envelope U and U 1 U Uss − [ˇ(1 + ) − g0 + ig] + iϑU = 0 2 1 + |U|2
(8)
(2)
where k = ne k0 , k0 = 2/ 0 , 0 is the free-space wavelength of the lightwave employed, ne is the unperturbed extraordinary refractive index, ϑ0 is the absorption coefficient of the crystal, r33 and is the electro-optical coefficient, and 0 are the intensity and phase coupling coefficients respectively of the two-wave mixing, Esc is the induced space-charge field. I(x, z) = (ne /2 0 )|(x, z)|2 , 0 = ( 0 /ε0 )1/2 and Ip = (ne /2 0 )|p |2 . It should be emphasized that, in this paper, the parameters with the symbol ˆ. have the same physical meaning as those without it. The expression of Esc (Eˆ sc ) can be obtained from the standard set of rate and continuity equations and Gauss’s law, which describe the photorefractive effects in a medium in which the photovoltaic current is non-zero and the electrons are the sole charge carriers. Here, the same approach was adopted as in Ref. [13] and in the limit of the spatial extent x ( ˆx) of the optical wave being much less ˆ ) of the crystal P (P), ˆ that is x W than that the x width W (W ˆ ), and we obtain the expression of Esc (Eˆ sc ) as follows: (ˆx W
(6)
iU +
ˆ ˆ 1 ∂2 ∂ kˆ 0 3 ˆ =0 +i (nˆ rˆ33 Eˆ sc ) − 2 2 e ∂z ˆ ∂x 2k
Tˆ 300
E0 = ± Eˆ p
1/2
ϑ0 =0 2 (1)
that the Hamiltonian dark solitons formed in the crystal Pˆ and the holographic dark or bright solitons formed in the crystal P. The expressions for E0 and Eˆ 0 necessary for a soliton pair are as follows:
where
Fig. 1. Illustration of an unbiased series PR crystal circuit consisting of one PV-PR crystal and one PR crystal to support a dark–dark soliton pair in (a) and a bright–dark soliton pair in (b). P and Pˆ denote the two crystals. C and Cˆ denote the two caxes. Ip denotes the incident pump beam. I and ˆI denote the incident bright or dark solitonlike one-dimensional laser beams.
ˆId = ˆId0
4913
(5)
where ˆId0 is the value of ˆId at Tˆ = 300 K, and Eˆ t is the level location in the gap. For photovoltaic photorefractive crystals, such as LiNbO3 , Eˆ t ∼10−19 J [14], in this paper, for simplicity, we take Eˆ t = 10−19 J. We emphasize our attention on the effect of Hamiltonian dark solitons on the holographic one, so we discuss two types of soliton pairs, i.e., dark–dark and bright–dark soliton pairs. Let us assume
ˆ ˆ + 1U ˆ − ˇ( iU ˆ + 1) 2 ss
ˆ U ˆ 2 1+|U|
ˆ 2 ˆ
ˆ U| )U −˛ ˆ (−| =0 ˆ 2 1+|U|
(9)
where U = ∂U/∂, Uss = ∂2 U/∂s2 , = I∞ /Ip , g0 = kx02 0 , g = kx02 /2, 2 ˆ = ˇ = r33 E0 , = (k0 x0 ) (n4e /2), ϑ = kx2 ϑ0 /2, ˇ ˆ Eˆ 0 , ˛ ˆ = ˆ Eˆ p and 0
2
ˆ = (kˆ 0 xˆ 0 ) (nˆ 4e rˆ33 /2). It is important to note that the value of ˆId changes with the crystal’s temperature, therefore the value of ˆ = ˆI∞ /ˆId also changes with the crystal’s temperature. Letting denotes the value of ˆ at Tˆ = 300 K, using Eq. (5), the quantity ˆ = ˆI∞ /ˆId can be expressed as follow:
ˆ = ˆ0
Tˆ 300
−3/2
exp −
Eˆ t kB
1 300
−
1 Tˆ
(10)
From PR theory [17], we can obtain the expressions for g and g0 2 + E 2 ) and g = (r E E 2 )/(E 2 + as g = [reff Es (E02 + Ed Eds )]/(Eds 0 eff 0 s 0 ds 2 E0 ), where Es = eNA 0 /(4ε0 εr sin ) is the saturation field, Ed = 4kB T sin /( 0 e) is the diffusion field, and Eds = Ed + Es . In this letter soliton formation of the signal beam in the crystal P is studied under the condition |U|2 = I/Ip 1, thus allowing the approximation that (1 + |U|2 )−1 ≈ 1 − |U|2 . As a result, Eq. (8) becomes iU +
1 Uss − [Pd + iG]U + [Pd + ig]U|U|2 = 0 2
(11)
where G = g − ϑ and Pd = (1 + )ˇ − g0 . The Hamiltonian dark soliton solution which formed in crystal ˆ in Pˆ can be derived from Eq. (9) by expressing the beam envelop U ˆ where ˆ represents a nonˆ = the usual fashion: U ˆ 1/2 yˆ (ˆs) exp(iˆ), linear shift of the propagation constant and yˆ (ˆs) is a normalized real function bounded between 0 ≤ yˆ (ˆs) ≤ 1. Using the boundary condition of the dark soliton yˆ (0) = 0, yˆ˙ (∞) = 0 and yˆ (ˆs → ±∞) = 1, the normalized Hamiltonian dark field profile yˆ (ˆs) obey the following equation: ˆ [−2G]
1/2
0
sˆ = ± yˆ (ˆs)
ˆ ˆ =˛ where G ˆ + ˇ.
dy 1/2
[(y 2 − 1) − ((1 + )/ ˆ ) ˆ ln((1 + y ˆ 2 )/(1 + ))] ˆ (12)
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The holographic dark soliton solution in the crystal P of Eq. (11) has been found as the following [11]: U = D tanh(Hs) exp{id ln[cosh(Hs)]} exp(−i˝)
(13) 1/2
where D = (G/g)1/2 , H = [2G/(3d)]1/2 , d = [3Pd + (9Pd2 + 8g 2 ) ]/(2g) and ˝ = 2G/(3d) + Pd . For the holographic bright soliton, the optical beam intensity is expected to vanish at infinity (s → ±∞), i.e., I∞ = 0 and then = I∞ /Ip = 0. From Eq. (11), we can obtain the bright soliton solution as the following [11]: U = F sech(Bs) exp{ib ln[sech(Bs)]} exp(−i) where (9Pd2
F = [3G/(2g)]1/2 ,
1/2 + 8g 2 ) ]/(2g),
B = (G/b)1/2 ,
(14) b = [−3Pd +
= (b2
− 1)G/(2b) + Pd . Eqs. (12)–(14) indicate that a Hamiltonian dark soliton and a holographic dark or bright soliton can co-exist in the crystal circuit ˆ ≺ 0. Although the two dynamical evolution equation provided G have a similar form to that for a single PR crystal [1,11], they couple each other by the coupling coefficient . If we keep the temperature of crystal P unchanged, and adjust the temperature of Pˆ or the maximum intensity ˆI∞ , the Hamiltonian dark soliton can affect the characteristic of the holographic dark or bright soliton. In this paper, we emphasize out attention on the temperature of Hamiltonian dark soliton on the evolution and stability of holographic one. 3. Temperature effects on the stability in separate holographic–Hamiltonians soliton pairs In the following calculation, we take a LiNbO3 (PV-PR) and a SBN (PR) crystal as an example to compose the circuit. The two crystals have the following parameters: ne = 2.33, n0 = 2.36, εr = 880, r33 = 220 pm/V, NA = 1.2 × 1017 cm−3 , = 2◦ , ϕ = 87◦ , ϑ0 = 0.27 cm−1 , ˆ W =W ˆ , Ip = mˆId0 nˆ e = 2.2, rˆ33 = 30 pm/V, Eˆ p = −105 V/m, ı = ı, and reff = (ne /n0 )3 r33 cos cos(/2) cos ϕ. The arbitrary scales are ˆ0 = x0 = xˆ 0 = 40 m and the free–space wavelengths are 0 = 0.5 m. For these set of values, we have = 3.72 × 106 m/V, ˆ = ˆ = −8.8781. The dependence of coupling 8.88 × 10−5 m/V and ˛ coefficient on temperature Tˆ can be obtained as following: =
Fig. 2. The effect of the temperature of crystal Pˆ on the normalized intensity profile in the separate holographic–Hamiltonian dark–dark soliton pair: (a) holographic dark soliton: (b) Hamiltonian dark soliton. Solid curve denotes the case of Tˆ = 300 K, and the dashed and dotted curves denote the case of Tˆ = 290 K and Tˆ = 310 K, respectively.
ˆ mϕ ˆ + mϕ ˆ + ˆ +1
We now investigate the effect of the temperature of crystal Pˆ on the evolution and stability of the holographic dark soliton. In doing so, a holographic dark soliton is obtained from Eq. (13) with the system parameters chosen as T = 300 K, Tˆ = 300 K, D = 0.5645, d = 0.0944 and H = 1.4457. Taking the solitary state as the input beam, we solve Eq. (11) numerically at different temperature Tˆ to obtain the dynamical evolution of the holographic dark soliton, as shown in Fig. 3. The results indicate that changing the temperature Tˆ can obviously affect the dynamical evolution and the stability of the holographic dark soliton in a separate holographic–Hamiltonian dark–dark soliton pair. First, we examine the case of Tˆ = 300 K, as shown in Fig. 3(a). As expected, our numerical results confirm that the holographic dark soliton beam remains invariant with increasing propagation distance in the crystal. The reason is that the dark soliton is just the dark solitary solution supported by crystal P at Tˆ = 300 K, thus the dark soliton can evolve in an adiabatic fashion. Then we examine the case that the temperature Tˆ deviates from Tˆ = 300 K. We can see that the incident beam reshapes itself and tends to evolve into another stable holographic dark soliton when the temperature Tˆ is higher than Tˆ = 300 K as shown in Fig. 3(b). Adjusting the temperature Tˆ to be lower than Tˆ = 300 K, we can see from Fig. 3(c) and (d) that the holographic dark soliton cannot evolve into a stable dark soliton but instead tends to break up into beam filaments as the temperature is lowered.
(15)
−3/2
where ϕ ˆ = (Tˆ /300) exp[−(Eˆ t /kB )((1/300) − (1/Tˆ ))]. If we keep the temperature of crystal P unchanged, and adjust the temperature ˆ the Hamiltonian dark soliton can affect the characteristic of the of P, holographic one. 3.1. Dark–dark soliton pair Firstly, we investigate the effect of temperature of crystal Pˆ on the holographic dark soliton. We can adjust the temperature of crystal P to T = 300 K and then keep it unchanged. We choose the input light beam to be of = 0.03, ˆ = 1, m = 10 and Tˆ = 300 K, ˆ = −8.1563, D = 0.5645, d = 0.0944 and which result in = 1/12.3, G H = 1.4457. Using the above parameters, we obtain the normalized intensities of the two solitons from Eqs. (12) and (13), as show in Fig. 2 (solid curve). Then we change the temperature Tˆ to investigate the temperature effect on the normalized intensities, as shown in Fig. 2 (dashed and dotted curves). From the results we can see that the change of temperature Tˆ can affect the normalized intensities of the two dark solitons. The higher the temperature of crystal ˆ the bigger the intensity in full width at half maximum (FWHM) P, of the dark solitons.
Fig. 3. The effect of the temperature of crystal Pˆ on the dynamical evolution of the holographic dark solution in the separate holographic–Hamiltonian dark–dark soliton pair: (a) Tˆ = 300 K, (b) Tˆ = 350 K, (c) Tˆ = 280 K, (d) Tˆ = 270 K. The input incident holographic dark solitary state is obtained from Eq. (13) with T = 300 K, D = 0.5645, d = 0.0944 and H = 1.4457.
X. Cai, B. Guo / Optik 124 (2013) 4912–4916
4915
Fig. 4. The effect of the temperature of crystal Pˆ on the normalized intensity profile in the separate holographic–Hamiltonian bright–dark soliton pair: (a) holographic bright soliton; (b) Hamiltonian dark soliton. Solid curve denotes the case of Tˆ = 300 K, and the dashed and dotted curves denote the case of Tˆ = 290 K and Tˆ = 310 K, respectively.
ton; at the temperature p the dark soliton deviates slightly from Tˆ = 300 K. However, if the temperature of crystal Pˆ further deviates from Tˆ = 300 K, such as Tˆ = 290 K and Tˆ = 310 K, as shown in Fig. 5(c) and (d), the bright soliton cannot evolve into a stable bright soliton but instead tends to experience larger cycles of compression, and its maximum amplitude oscillates with propagation distance at Tˆ = 290 K or exhibit a common decaying process in the crystal P at Tˆ = 310 K. By a close examination of the above results, in contrast to that of holographic dark soliton in Fig. 3, we can see that the holographic dark soliton has a narrower stable temperature region, and it is more sensitive to the temperature change than the holographic bright one in a separate holographic–Hamiltonian soliton pair.
3.2. Bright–dark soliton pair
4. Conclusions
Next we investigate the effect of the temperature of crystal Pˆ on the evolution and stability of the holographic bright soliton formed in crystal P. In doing so, we adjust the temperature of the crystal P to T = 300 K, and keep it unchanged. We choose the input light beam to be of = 0, ˆ = 1, m = 10 and Tˆ = 300 K, which result in = 1/2, ˆ = −8.1383, F = 0.6914, b = 0.0950 and B = 1.7652. Using the above G parameters, we obtain the normalized intensities of the two solitons from Eqs. (12) and (14), as show in Fig. 4 (solid curve). Then we change the temperature Tˆ to investigate the temperature effect on the normalized intensities, as shown in Fig. 4 (dashed and dotted curves). From the results we can see that the change of temperature Tˆ can affect the normalized intensities of the Hamiltonian dark soliton and holographic bright soliton. The higher the temperature ˆ the bigger the intensity in FWHM of the two solitons. of crystal P, The effect of the temperature of crystal Pˆ on the evolution and stability of the holographic bright soliton in such separate soliton pair is shown in Fig. 5. The fundamental holographic bright solitary state is obtained from Eq. (14) with the system parameters chosen as T = 300 K, Tˆ = 300 K, F = 0.6914, b = 0.0950 and B = 1.7652. The results indicate that changing the temperature of crystal Pˆ can sharply affect the dynamical evolution and the stability of the holographic bright soliton. When Tˆ = 300 K, the bright soliton is just the fundamental solitary solution supported by the crystal, therefore, the soliton can evolve in an adiabatic fashion during its propagation in the crystal; Fig. 5(b) shows that the bright soliton can reshape itself and tend to evolve into another stable bright soli-
The temperature effects on the evolution and stability of a separate holographic–Hamiltonian soliton pair formed in an unbiased serial photorefractive crystal circuit have been investigated under steady-state conditions for the case of neglecting the diffusion process. Our numerical results show that the Hamiltonian dark soliton can affect the holographic dark or bright soliton in a stable dark–dark or bright–dark soliton pair formed in a crystal circuit at given temperatures. When the temperature of the crystal in which formed a Hamiltonian dark soliton changes, the holographic dark supported by the other crystal tends to evolve into another stable holographic dark soliton when the temperature is higher than the original temperature, whereas it tends to break up into beam filaments as the temperature is lowered. On the other hand, the holographic bright soliton tends to be unstable and experiences larger cycles of compression if the temperature difference is big enough, whereas it tends to evolve into another stable soliton when the temperature difference is small enough The holographic dark soliton has a narrower stable temperature range than the holographic bright one in such a separate holographic–Hamiltonian soliton pair, therefore the dark solitons, therefore the holographic dark soliton is more sensitive to the temperature change. Acknowledgement This work is supported by the Fundamental Research Funds for the Central Universities no. 2011-la-001. References
Fig. 5. The effect of the temperature of crystal Pˆ on the dynamical evolution of the holographic bright solution in the separate holographic–Hamiltonian bright–dark soliton pair: (a) Tˆ = 300 K, (b) Tˆ = 305 K, (c) Tˆ = 290 K, (d) Tˆ = 310 K. The input incident holographic bright solitary state is obtained from Eq. (14) with T = 300 K, F = 0.6914, b = 0.0950 and B = 1.7652.
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