CC BY
6Scientific Notes of Taurida National V. I. Vernadsky University
Series : Physics and Mathematics Sciences. Volume 26 (65). 2013. No. 2. P. 31-37
UDK 535.1
INFLUENCE OF A TWIST DEFECT IN A LAYERED HELICAL CORE FIBER ON SINGLE-CHARGED OPTICAL VORTEX GENERATION
Lapin B. P.
Taurida National V. I. Vernadsky University, 4 Vernadsky Ave., Simferopol 95007, Ukraine
E-mail: hwinborisitmiiuul com
It has been described how twist defect in a layered helical core fiber acts on generation of optical vortex with unity topological charge from the fundamental mode. Influence of the twist defect is maximal when it is located right in the middle of the fiber and leads to suppression of optical vortex generation. In addition it has been established that impact of twist defect is negligibly small if it is located near the ends of the fiber, which has the length equal to a zero order conversion length. Keywords: optical vortex, helical core fiber, optical fiber.
PACS: 42.81.Qb
INTRODUCTION
Amazing progress in data transfer by fiber-optic systems in past years [1, 3] suggests that future telecommunication systems would use light as information carrier. Indeed, data transfer rate in optical diapason (~ 1014 Hz) is higher by several orders than the one in the diapason, inherent for traditional wire communication lines (~ 1010 Hz). Moreover, another possibility to increase the data transfer rate exists for fiber-optic systems concerned with application of optical vortices (OVs) [4]. Indeed, in [5] it has been shown that a helical core fiber can maintain undamped propagation of OV with ±1 topological charge. According to [6, 7], OVs with different topological charges can be used for information encoding.
It is natural to suppose that more reliable methods of OV generation [8-14] for fiber-optic systems will be based on the use of optical fibers as generation and transporting elements. In the light of success in OV generation by microstructured helical core fiber (HCF) [14] it is desirable to know how diverse defects, which are inevitably present in real systems, will affect the generation processes. The aim of this paper is to show how the twist defect in a layered HCF with one-fold rotational symmetry affects the generation of 1-charged OV from the fundamental mode.
1. MODEL AND MODES OF LAYERED HELICAL CORE FIBER
The layered HCF with one-fold rotational symmetry is a set of dielectric layers, which centers lie on a spiral line with radius R (Fig.1a). In the center's area the refractive index has a maximal value nco . The linear size of this area is defined by r0 << R. Another region of a layer has refractive index value equal to nci < nco.
31
Let us now consider the HCF with a twist defect, which is placed at a distance dx from the input end of the fiber (Fig. 1 b). The twist defect appears where a fiber is transversely cut and the second part of the fiber is rotated around the mutual axis by an angle 9 .
a)
b)
Fig. 1. a) The model of layered helical core fiber, where R - offset, (X,Y,Z) and (X',Y', Z') - laboratory and local coordinate systems; b) helical core fiber with a twist defect: di - the distance from it to the input end of the fiber, d - the length of a whole fiber, 9 characterizes the degree of mutual twist of fiber's parts, H - the pitch of spiral line.
As is known, if the lattice vector q = 2n / H of the HCF satisfies the resonance condition q « q() = ¡3() - f3]. where [i{). are the scalar propagation constants of the HEn and LP11 modes, intensive hybridization of these modes takes place. The structure of coupled modes near the resonance is:
) = {q 11,0)^+0.5^ +C2 |Uy(A-0.5*)z Je/zr
|^1> = {d|1 |^2) = {-C2|1,0 )<M+0.5*)z
J1,1)
J{P-0.5s)z \ -izy
where c = cosx,C2 = sinx, tanx = Q/(V^2 + Q2 _£), 7 = 0.5yJ,
Q'
(1)
, B = q - qo, ( 1 ^
F
Q - coupling integral [12]. In the basic of linear polarizations \a,l) = Fj (r)
is the standard radial function. Expressions (1) are written in cylindrical-polar coordinates (r ,q>, z). Expressions for modes in the second part of the fiber can be obtained by making
the substitutions: l) l)e '(1+a)9 and z ^z-d^
32
2. INFLUENCE OF THE TWIST DEFECT ON GENERATION OF OPTICAL VORTEX FROM THE FUNDAMENTAL MODE
Let us consider how the defect's location affects the evolution of the fundamental mode 11,0) , which is known to converse into the optical vortex |l,l) in the HCF with the
length d = Sk =n / Q (l + 2k), k = 0,1,2... Decomposing the fundamental mode |l,0) in modes (1) and using matching conditions on the boundaries of the fibers one can obtain:
<bn (d) = -P2c2e^Ae<^5e^-di) |l,0) +
(2)
where P^icue^^ft^+te^y li^/^5^6 c^myd,. \At\2
defines the power of the corresponding field's components at the output end of the fiber.
Consider the case where ^ = ^0 = 632.8 nm (s = 0). Then the powers of the fundamental mode and the OV can be written as: 2 1 1
|^0i| = c°s Qdicos Q (d - di) + sin Qd^in Q (d - d ) cos ( qdi -6)]. (3)
Obviously, one has \A0f + = 1. If the length of the fiber d = S0 and 6 = q0d1 (defect is absent) then the incoming fundamental mode entirely converts into the OV 11,1) that coincides with the well known results [12].
|A1|2
2 B(rad) 4
Fig. 2. Dependence of the transformation coefficient [A^2 on 6 at s = 0 . The fiber parameters: nco = 1.5, A = 0.01, R / r0 = 0.05, r0 = 10^, Aq = 632.8 nm, q = 6444.24 m-1, dx = 0.5S0,d = S0 = 5 mm.
33
From (3) one can obtain that the influence of the defect is negligibly small if the defect is located near one of fiber's ends and d = S0. Indeed, at d -dx ^ 0 or
2
d - d1 ^n / Q (l + 2k) the power of the output OV A ^ 1. For example, the dependence of the OV's output power on 9 is shown in Fig. 2.
In the case where s^ 0 one can obtain similar expressions for |A |2. In Fig. 3 the curves for transformation coefficient are shown at d = n / Q and s ^ 0 obtained with the help of numerical calculations.
|Ao|
0.75
0.5
0.25
I Ai|
1
0.75 0.5 0.25
"-7
6.25 6.3 6.35 6.4 m) 6.25
a)
* -7
6.3 6.35 6.4 ^(10 m)
b)
i2
Fig. 3. Dependence of the transformation coefficients |A | at 9 = n + q0S0 / 2. The fiber parameters: nco = 1.5, A = 0.01, R/ = 0.05, rQ = 10^q, A = 632.8 nm, q = 6444.24 m-1, di = 0.5S0, d = S0 = 5 mm. Solid lines correspond to the envelopes
for the transformation coefficient |A |2. The fine structure of the transformation coefficients is shown on the insets.
As is seen from Fig. 3, at s ^ 0 and 9 = n + q0S0 / 2 part of energy of the incoming fundamental mode transfers into the OV 11,1) . It is connected with the dependence of the
conversion length Sk on X: Sk (A) = n(1 + 2k) / ^s2 + Q2 . The last expression makes it
evident that the fundamental mode with A^X needs larger distance for conversion into
the OV 11,1). From Fig. 4 it can be seen that at significant increase of the length
(d = S100) there are areas where appreciable part of energy is stored in the OV |1,1)
(Fig. 4 b). Nevertheless, at s « 0 the fundamental mode almost completely passes through the fiber (Fig. 4 a). It should be noted that these curves have a fine structure. It can be explained by the action of the whole fiber. Indeed, in (2) at the right boundary of the fiber
34
there are factors e d' '. Since /?o (Ao )d ~ 1 (f -f- 1 04 even small change of the
wavelength A leads to fast oscillations of these factors and hence of \At |2. Large scale oscillations can be explained by the presence of the factors e'r^d : /(Ao )So ~ 1.
2
|Ao|
6.2 7 6.4 6.6
A,(10 m)
a)
2
|A1|
6.2 7 6.4 6.6
M10m)
b)
Fig. 4. Dependence of the transformation coefficients \At f at d = n + qSioo / 2. The fiber parameters: nco = 1.5, A = 0.01, R/ ro = 0.05, r = 10Ao, A = 632.8 nm,
q = 6444.24 m"1, di = 0.5Sd0,d = S100 = 1.0178 m. Solid and dot lines are envelopes for the transformation coefficients. The fine structure of the transformation coefficients is shown on the insets.
35
CONCLUSION
In conclusion one can say that the twist defect can appreciably affect the optical vortex generation from the fundamental mode in a layered helical core fiber. Influence of twist defect is maximal where it is located right in the middle of the fiber. This leads to complete suppression optical vortex generation at resonance wavelength. If the defect is located near one of fiber's ends and the fiber's length is equal to S0 the influence of the defect is negligibly small.
ACKNOWLEDGEMENTS
This work was partially supported by the Grant № GP/F49/113 of the State Agency on Science, Innovations and Informatization of Ukraine.
References
1. Y. Ma, Q. Yang, Y. Tang, S. Chen, W. Shieh, Optics Express 17, 9421 (2009).
2. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, S. Ramachandran, Science 340, 1545 (2013).
3. V. A. J. M. Sleiffer, Y. Jung, V. Veljanovski, R. G. H. van Uden, M. Kuschnerov, H. Chen, B. Inan, L. G. Nielsen, Y. Sun, D. J. Richardson, S. U. Alam, F. Poletti, J. K. Sahu, A. Dhar, A. M. J. Koonen, B. Corbett, R. Winfield, A. D. Ellis, H. de Waardt, Optics Express 20, B428 (2012).
4. M. Vasnetsov and K. Staliunas, Optical Vortices (Nova Science Publishers, 1999).
5. C. N. Alexeyev, B. P. Lapin, M. A. Yavorsky, Phys. Rev. A 78, 013813 (2008).
6. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas'ko, S. M. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004);
7. Z. Bouchal and R. Chelechovsky, New J. Phys. 6, 131 (2004);
8. Ya. V. Izdebskaya, V. G. Shvedov, A. V. Volyar, Optics Letters 30, 2472 (2005).
9. I. Skab, Y. J. Vasylkiv, V. Savaryn, R. Vlokh, JOSA A 28, 633 (2011).
10. D. P. Ghai, Applied Optics 50, 1374 (2011).
11. S. Slussarenko, A. Murauski, T. Du, V. Chigrinov, L. Marrucci, E. Santamato, Optics Express 19, 4085 (2011).
12. C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapin, M. A. Yavorsky, Phys. Rev. A 83, 063820 (2011).
13. C. N. Alexeyev, Appl. Opt. 51, 6125 (2012).
14. H. Xu and L. Yang, Optics Letters 38, 1978 (2013).
Латн Б. П. Вплив дефекту скрутки в шаруватому волокш з сшральною серцевиною на генеращю оптичного вихору з одиничним зарядом / Б. П. Латн // Вчет записки Тавршського национального утверситету iменi В. I. Вернадського. Серш : Фiзико-математичнi науки. - 2013. -Т. 26 (65), № 2. - С. 31-37.
Описано, як дефект скрутки у шаруватому волокш з стральною серцевиною впливае на генеращю оптичного вихору з одиничним тополопчним зарядом з фундаментально! моди. Вплив дефекту скрутки максимальний, коли вш знаходиться точно посередит волокна, i призводить до повного пригшчення генераци оптичного вихору. Крiм того, встановлено, що вплив дефекту скрутки на генеращю оптичного вихору незначний, коли дефект розташований поблизу одного з кшщв волокна, а довжина волокна дорiвнюе нульово! довжиш конверси.
Ключовi слова оптичний вихор, волокно з сшрально! серцевиною, оптичне волокно.
36
Лапин Б. П. Влияние дефекта скрутки в слоистом волокне с геликоидальной сердцевиной на генерацию оптического вихря с единичным зарядом / Б. П. Лапин // Ученые записки Таврического национального университета имени В. И. Вернадского. Серия : Физико-математические науки. -2013. - Т. 26 (65), № 2. - С. 31-37.
Описано, как дефект скрутки в слоистом волокне с геликоидальной сердцевиной влияет на генерацию оптического вихря с единичным топологическим зарядом из фундаментальной моды. Влияние дефекта скрутки максимально, когда он находится точно посередине волокна, и приводит к полному подавлению генерации оптического вихря. Кроме того, было установлено, что влияние дефекта скрутки на генерацию оптического вихря пренебрежимо мало, когда дефект расположен вблизи одного из концов волокна, а длина волокна равна нулевой длине конверсии.
Ключевые слова: оптический вихрь, волокно с геликоидальной сердцевиной, оптическое волокно.
Список литературы
1. 1-Tb/s single-channel coherent optical OFDM transmission over 600-km SSMF fiber with subwavelength bandwidth access / Y. Ma, Q. Yang, Y. Tang, [et al.] // Optics Express. - 2009. - Vol. 17. - P. 9421.
2. Terabit-scale orbital angular momentum mode division multiplexing in fibers / N. Bozinovic, Y. Yue, Y. Ren, [et al.] // Science. - 2013. - Vol. 340. - P. 1545.
3. 73.7 Tb/s (96 x 3 x 256-Gb/s) mode-division-multiplexed DP-16QAM transmission with inline MM-EDFA / V. A. J. M. Sleiffer, Y. Jung, V. Veljanovski, [et al.] // Optics Express. - 2012. - Vol. 20. -P. B428.
4. Vasnetsov M. Optical Vortices / M. Vasnetsov and K. Staliunas. - Nova Science Publishers, 1999.
5. Alexeyev C. N. Helical core optical fibers maintaining propagation of a solitary optical vortex / C. N. Alexeyev, B. P. Lapin, M. A. Yavorsky // Phys. Rev. A. - 2008. - Vol. 78. - P. 013813.
6. Free-space information transfer using light beams carrying orbital angular momentum / G. Gibson, J. Courtial, M. J. Padgett, [et al.] // Opt. Express. - 2004. - Vol. 12. - P. 5448.
7. Bouchal Z. Mixed vortex states of light as information carriers / Z. Bouchal and R. Chelechovsky // New J. Phys. - 2004. - Vol. 6. - P. 131.
8. Izdebskaya Ya. V. Generation of higher-order optical vortices by the dielectric wedge / Ya. V. Izdebskaya, V. G. Shvedov, A. V. Volyar // Optics Letters. - 2005. - Vol. 30. - P. 2472.
9. Optical anisotropy induced by torsion stresses in LiNbO3 crystals: appearance of an optical vortex / I. Skab, Y. J. Vasylkiv, V. Savaryn, R. Vlokh // Journal of the Optical Society of America A. - 2011. -Vol. 28. - P. 633.
10. Ghai D. P. Generation of optical vortices with an adaptive helical mirror / D. P. Ghai // Applied Optics. -2011. - Vol. 50. - P. 1374.
11. Tunable liquid crystal q-plates with arbitrary topological charge / S. Slussarenko, A. Murauski, T. Du, [et al.] // Optics Express. - 2011. - Vol. 19. - P. 4085.
12. Generation of optical vortices in layered helical waveguides / C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapin, M. A. Yavorsky // Phys. Rev. A. - 2011. - Vol. 83. - P. 063820.
13. Alexeyev C. N. Generation of optical vortices in spun multihelicoidal optical fibers / C. N. Alexeyev // Appl. Opt. - 2012. - Vol. 51. - P. 6125.
14. Xu H. Conversion of orbital angular momentum of light in chiral fiber gratings / H. Xu and L. Yang // Optics Letters. - 2013. - Vol. 38. - P. 1978.
Received 21 June 2013.
37
