Highly nonlinear layered spiral microstructured optical fiber

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Photonics and Nanostructures – Fundamentals and Applications 11 (2013) 226–233 www.elsevier.com/locate/photonics

Highly nonlinear layered spiral microstructured optical fiber Sı´lvia M. Rodrigues, Margarida M. Faca˜o, Sofia C. Latas, Ma´rio F. Ferreira * I3N-Institute of Nanostructures, Nanomodelling and Nanofabrication, Department of Physics, University of Aveiro, 3810-193 Aveiro, Portugal Received 5 October 2012; received in revised form 30 January 2013; accepted 2 March 2013 Available online 26 March 2013

Abstract A layered spiral microstructured optical fiber (LS-MOF) is presented, which offers the possibility of a good control of both the dispersion and the nonlinear properties. The proposed design is analyzed using a finite element method considering silica and air as the materials. Zero dispersion, low confinement loss, and a record value of g = 70.0 W1/km for the LS-MOF nonlinear parameter are simultaneously obtained at 1.55 mm, whereas a higher value g = 169.4 W1/km can be achieved at 1.06 mm. Our results demonstrate the great potential of the LS-MOF for several nonlinear applications, namely for an efficient generation of the supercontinuum. # 2013 Elsevier B.V. All rights reserved. Keywords: Silica glass fibers; Chromatic dispersion; Highly non-linear fibers; Microstructured optical fibers

1. Introduction Microstructured optical fibers (MOFs) with a matrix of air holes around a solid central core can provide not only a tightly confined optical mode with small effective area and consequent large nonlinearity, but also anomalous dispersion at shorter wavelengths, namely in the visible region [1,2]. The availability of optical fibers with anomalous dispersion close to 800 nm, where high energy femtosecond pulses are readily available from a Ti-sapphire system, was actually responsible for a renewed interest in supercontinuum generation during the last decade [3]. However, since Ti-sapphire laser sources are relatively expensive and bulky, much effort has been expended recently to generate the supercontinuum pumped at other wavelengths, namely in the regions around 1 mm, where

* Corresponding author. Tel.: +351 234370279. E-mail addresses: [email protected] (S.M. Rodrigues), [email protected] (M.M. Faca˜o), [email protected] (M.F. Ferreira). 1569-4410/$ – see front matter # 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.photonics.2013.03.001

Nd3+ or Yb3+ doped solid state and fiber lasers are available, as well as around 1.55 mm, where Er3+-based fiber sources can be used. One aspect of the effort to enhance the supercontinuum generation is to increase the fiber nonlinear parameter by decreasing the effective mode area, together with the use of soft glasses with higher nonlinearities than silica. These developments have led to extensive research on MOFs with optimized designs, as well as on suspended core fibers using compound glasses [4–8]. The so-called jacket air-suspended rod fiber [9] design has the smallest possible effective mode area, but it is mainly a theoretical concept, which can only be approximately realized in practice. On of such approximations is provided by the triangular core fiber [5]. Extremely broad band supercontinuum generation has been recently demonstrated in soft glass MOFs [9–13]. Although soft glasses offer substantially higher nonlinear coefficients, silica fibers usually provide better overall figure of merit due to the two or four orders of magnitude longer fiber that can be used for the

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same insertion loss [14]. For this reason, it becomes actually important to investigate optimized silica fiber designs that are able to efficiently generate the supercontinumm. In this paper we present and investigate a novel spiral-like microstructured silica fiber, which will be able to provide both a high nonlinear parameter and a great versatility concerning its dispersion properties. The proposed fiber design is intended to be used for supercontinuum generation using practical and readily available fiber laser sources. The novel fiber design is described in Section 2, whereas its dispersion and nonlinear properties are discussed in Sections 3 and 4, respectively. The main conclusions are summarized in Section 5. 2. Layered spiral microstructured optical fiber design For an efficient generation of the supercontinuum, it is desirable to obtain a good confinement of the optical field, and therefore an enhanced nonlinear parameter, a good control of the dispersion characteristics, and the possibility of achieving simultaneously zero dispersion and high nonlinearity at specific wavelengths, namely those where practical laser sources are available. In order to achieve these objectives, we propose the layered spiral microstructured optical fiber (LS-MOF) represented in Fig. 1.

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In the design of the LS-MOF we used two Archimedean spirals and one equiangular spiral, both spinning around the center. The first inner turning in the center of Fig. 1 is an equiangular spiral (ES), that starts at point ri and ends at point rf = ri + D  dair. The equiangular spiral is a special kind of spiral curve which often appears in nature, as is the case of the arms of a spiral galaxy, the bands of tropical cyclones, and of many biological structures, namely the shells of mollusks. Since the radius of this kind of spirals increases exponentially with the angle u of polar coordinates, it varies slowly close to ri and more quickly close to rf ; as a consequence, the curve becomes more distant from a circle only close to rf . The radius of the enclosed circle in this fiber’s core is equal to ri, which is the minimum value for the ES radius. The equiangular spiral’s equation that matches the boundary conditions of this design, i.e. that passes through ri and rf , is given by: rðuÞ ¼ aebu

(1)

where a = ri and b = [ln(rf /ri)]/2p. We will consider that the ES in this fiber has only one turn, so u 2 [0, 2p]. The two other spirals used in the fiber design are Archimedean spirals (AS) and we call them the inner Archimedean spiral (ASin) and the outer Archimedean spiral (ASout), according to their relative position. The spirals for the surrounding cladding were chosen to be Archimedean spirals because the displacement between consecutive turnings of this kind of spirals, D, is constant, as consequence of the fact that the radius of an AS varies linearly with u, in polar coordinates. It should be noted that the first turning of the inner Archimedean spiral was replaced by the ES previously mentioned. The radius of the inner AS is given by the following relation: rðuÞ ¼ r f þ

D u; 2p

u 2 ½0; ðn  1Þ2p

(2a)

whereas that of the outer AS is given by: rðuÞ ¼ r i þ

Fig. 1. Illustration of the proposed fiber geometry and its main parameters: core radius (ri), air layer thickness (dair), and spatial period (D). Another controllable parameter is the number of spiral turns. The white (gray) region corresponds to the air (glass) region.

D u; 2p

u 2 ½0; n2p

(2b)

where n is the number of layers. These spiral curves all together divide the optical fibers’ cross section into two regions, as represented in Fig. 1: the air and the glass (e.g. silica) regions, in such a way that we can think in a sole resulting spiral which constitutes the fiber. The proposed spiral fiber geometry has some similarities to solid-core Bragg fibers, which consist on a core and a cladding formed by concentric layers of two different materials. In fact, the fabrication of an

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exact Bragg fiber using air and silica (or any other material) is not possible, since the material layers would be suspended in air. An example of such fiber was reported in Ref. [15] and consists on a silica core and a multilayer cladding of silica up-doped and down-doped. However, in his case, the contrast index is not as high as in silica-air fibers, which determines a weaker field confinement and, consequently, a lower nonlinear parameter. Compared with a conventional microstructured fiber with a regular distribution of air holes, the proposed configuration provides a higher air filling fraction (AFF). In the case of the LS-MOF, we have AFF = dair/D, which can achieve a theoretical limit of 100%. On the other hand, considering a solid-core MOF with a hexagonal pattern of circular airpholes ffiffiffiffiffiffiffiffiffiffiffi of diameter d, spaced of L, we have AFF ¼ ð 3p=6Þ  ðd=LÞ2 [16], which has a limit of 90.7%. For d/L = 0.9 it is AFF = 73.5% [9]. As mentioned above, a higher AFF leads to a stronger confinement of the optical field in the core and to a higher nonlinear parameter. In the following sections we will present some numerical results concerning the optical properties of the LS-MOF described above, which were obtained by vector modal analysis by using a finite element method through the commercial software COMSOL Multiphysics. In order to obtain the solutions for the fiber propagation modes, we solve numerically the following Helmholtz equation for the magnetic field: ! 1 r  HðrÞ ¼ k02 HðrÞ (3a) r nðrÞ2 where d/L = 0.9 is the magnetic field, n(r) is the refractive index in each point of the fiber’s cross section, and k0 is the free-space wavenumber. Eq. (3a) was obtained from Maxwell’s equation considering that the charge density and the current density in the optical fiber are both zero. The corresponding electric field E(r) is given by: EðrÞ ¼

i r  HðrÞ veðrÞ

(3b)

where v is the optical frequency and e(r) is the electric permittivity. 3. Dispersion properties of the LS-MOF In computing the dispersion of the LS-MOF, material dispersion has been taken into account using the

Sellmeier equation [17]: n2 ðvÞ ¼ 1 þ

m X A j l2 2 2 j¼1 l  l j

(4)

where l is the vacuum wavelength of the propagating light, Aj and lj are the strength and the wavelength corresponding to the jth resonance of the material. In the case of pure silica, we have the following values [18]: A1 = 0.6961663, A2 = 0.4079426, A3 = 0.8974794, l21 ¼ 0:004679148, l22 ¼ 0:01351206, and l23 ¼ 97:93400. The total dispersion of a guided mode in an optical fiber depends not only on the material dispersion, as given above, but also on the waveguide dispersion. The mode-propagation constant, b(v), obtained numerically, contains both contributions. Mathematically, the effects of fiber dispersion are accounted for by expanding b(v) in a Taylor series about the carrier frequency v0 at which the pulse spectrum is centered: 1 bðvÞ ¼ b0 þ b1 ðv  v0 Þ þ b2 ðv  v0 Þ2 þ    2 where  j  db bj ¼ dv j v¼v0

ð j ¼ 0; 1; 2; . . .Þ

(5)

(6)

For pulse propagation purposes, one is interested on the group velocity dispersion (GVD) that is characterized by the parameter b2. However, in practice, the GVD is often characterized by another parameter, D, given by D¼

2 p c b2 l d 2 ne f f ¼  ; c dl2 l2

(7)

where c is the velocity of light in vacuum and neff is the effective refractive index seen by the optical mode, which is related to b(v) as, bðvÞ ¼

vne f f ðvÞ c

(8)

The waveguide dispersion becomes especially important in cases of fibers exhibiting a large difference in refractive indexes of the core and cladding materials. Increasing such refractive index difference affects not only the dispersion characteristics, but provides also an enhancement of the nonlinear parameter of the fiber, since the mode effective area is reduced. Fig. 2 shows the dispersion curves for the fundamental mode of the LS-MOF assuming that D = 1 mm, dair/D = 0.7 and considering different values for the

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Fig. 2. Dispersion curves for the fundamental mode of the LS-MOF, assuming D = 1 mm, dair/D = 0.7 and several values for the core radius ri.

core radius ri. The dispersive properties are shown to be very sensitive to the core size. By decreasing this size, the zero dispersion wavelength (ZDW) is shifted toward shorter wavelengths. Moreover, a decrease of the overall GVD is observed at long wavelengths and a second ZDW appears, defining a wavelength window with anomalous GVD. Meanwhile, the first zero dispersion points do not exceed a wavelength of 800 nm for a core radius below 1 mm. SC generation generally requires that the pump wavelength is close to the ZDW. Fig. 3 shows that this can occur both in the visible and in the near IR regions. In the first case the dispersion slope is positive, which means that the dispersion is anomalous at longer wavelengths, where soliton propagation is possible. Mode-locked Ti-sapphire lasers at 800 nm and

mode-locked krypton ion lasers at 647 nm can be considered for this purpose, using LS-MOFs with a core radius sufficiently reduced. Furthermore, ZDWs with negative dispersion slopes can also be obtained in the regions around 1 mm, where Nd3+ or Yb3+ doped solid state and fiber lasers are available, as well as around 1.55 mm, where Er3+-based sources can be used. The open circle in Fig. 3 indicates that the latter case can be achieved with a core radius of 0.6 mm. Fig. 3 shows the dispersion curves for the fundamental mode of the LS-MOF assuming that D = 1 mm, ri = 0.6 mm and considering different values for the relative air layer thickness dair/D. The red curve is for a microfiber of silica with radius r = ri + nD = 7.6 mm (for n = 7 rings), which corresponds approximately to the case dair/D = 0. This curve is similar to the

Fig. 3. Dispersion curves for the fundamental mode of the LS-MOF, assuming D = 1 mm, ri = 0.6 mm and several values for the relative air layer thickness dair/D.

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Fig. 4, assuming ri = 0.6 mm and different values for dair/D. It can be seen that the dispersion increases monotonously with D and that zero dispersion at 1.55 mm can be always achieved by choosing appropriately the values of ri, dair/D and D. We also observed in our simulations that the dispersion is particularly sensitive to variations of the radius ri. 4. Fundamental modes and nonlinear parameter The fiber nonlinear parameter g is defined as [17]: g¼

v0 n2 ðv0 Þ c Ae f f

(9)

Fig. 4. Dispersion curves for the fundamental mode of the LS-MOF against D at 1.55 mm, assuming ri = 0.6 mm and several values for dair/D.

where n2 the Kerr parameter and RR 2 jFðx; y; vo Þj2 dxdy Ae f f ¼ RR jFðx; y; v0 Þj4 dxdy

dispersion curve of bulk silica, given by the black curve. On the other extreme, the pink curve is for dair/D = 1, which corresponds to a nano-jacket air suspended rod of radius r = ri = 0.6 mm. We observe from Fig. 3 that a ZDW is achieved at 800 nm for dair/D  0.4. On the other hand, a nearly zero and flat dispersion is achieved around 1 mm for dair/D  0.3. This fact is particularly important for applications making use of the four-wave mixing effect. Another significant feature is the occurrence of nearly zero dispersion around 1.55 mm for dair/D = 0.7  0.9. The dependence of the fundamental mode dispersion on the spatial period D at 1.55 mm is illustrated in

is the effective mode area, F(x, y, v0) representing the spatial distribution of the transverse electric field mode in the fiber’s cross section. Eq. (9) shows that, in order to increase the nonlinear parameter, it is important to reduce the effective mode area. Fig. 5 shows the effective mode area as a function of the wavelength for the same cases considered in Fig. 2. We observe that reducing the core radius produces a monotonous reduction of the effective mode area for wavelengths shorter than 1 mm, as a result of the increased confinement provided by the large difference in refractive indexes between the core and the air part of the cladding. However, such confinement is lost for

Fig. 5. Effective mode area against the wavelength, assuming D = 1 mm, dair/D = 0.7 and several values for the core radius ri.

(10)

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Fig. 6. Effective mode area against the wavelength, assuming D = 1 mm, ri = 0.6 mm and different values for the relative air layer thickness dair/D.

longer wavelengths, as observed in the curves corresponding to ri = 0.3–0.5 mm. For the wavelength l = 1.55 mm, D = 1 mm, and dair/D = 0.7 a minimum effective mode area Aeff = 1.88 mm2 is achieved for ri = 0.6 mm. Considering n2 = 2.6  1020m2/W, we obtain a nonlinear parameter g = 56.0 W1/km. However, considering a wavelength l = 1.06 mm, a minimum effective area Aeff  0.91 mm2 is obtained for a core radius ri = 0.39 mm, which provides simultaneously zero dispersion and a nonlinear parameter g = 169.4 W1/km. Fig. 6 shows the effective mode area against the wavelength, assuming D = 1 mm, ri = 0.6 mm and

Fig. 7. Effective mode area against D at 1.55 mm, assuming ri = 0.6 mm and several values for dair/D. The circles correspond to zero dispersion.

different values for the relative air layer thickness dair/D. We observe that the mode area can be effectively reduced at any wavelength by increasing dair/D, as a result of the enhanced mode confinement provided by the increased air-filling fraction. The dependence of the effective mode area on D at 1.55 mm is illustrated in Fig. 7, assuming ri = 0.6 mm and several values for the relative air layer thickness dair/D. Figs. 4 and 7 show that a zero dispersion and an effective area Aeff = 1.505 mm2 can be achieved at l = 1.55 mm when dair/D = 0.9, D = 1.048 mm, and ri = 0.6 mm. Considering a value n2 = 2.6  1020m2/W for the Kerr coefficient, we obtain a nonlinear parameter g = 70.0 W1/km, which is, to the best of our knowledge, the highest value ever reported at that wavelength for a silica fiber, with the simultaneous occurrence of a zero dispersion. The above value must be compared with that one reported in Ref. [9] (g = 52 W1/km) as the best one achievable with conventional silica holey fibers. In fact, there are some factors which can affect the value of the nonlinear parameter, namely the value assumed for the Kerr coefficient and the expression for the effective mode area. In this paper, we have chosen to use the standard definition for the effective mode area and a commonly used value for the Kerr coefficient. The results presented above demonstrate the ability of the LS-MOF to effectively concentrate the fundamental mode field in a very small area, thus providing not only an enhanced fiber nonlinearity, but also a very low confinement loss. This property is illustrated in Fig. 8 for case dair/D = 0.9, D = 1.0 mm, and ri = 0.6 mm. We observe that the

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S.M. Rodrigues et al. / Photonics and Nanostructures – Fundamentals and Applications 11 (2013) 226–233 Table 1 Parameters of the LS-MOF providing simultaneously zero dispersion and optimum nonlinearity at l = 1.55 mm. Geometry parameters, where ri = 0.6 mm dair/D = 0.4; dair/D = 0.5; dair/D = 0.6; dair/D = 0.7; dair/D = 0.8; dair/D = 0.9; dair/D = 1.0;

D = 1.525 mm D = 1.311 mm D = 1.014 mm D = 1.010 mm D = 0.950 mm D = 1.048 mm D = 0.400 mm [JASR]

Dispersion at 1.55 mm [ps/(km nm)]

g at 1.55 mm [W1/km]

0.0 0.0 0.0 0.0 0.0 0.0 63.4

28.33 38.08 47.45 56.02 63.52 70.02 76.05

Fig. 8. Confinement loss of the LS-MOF against the wavelength for dair/D = 0.9, D = 1.0 mm, and ri = 0.6 mm. The dot represents the loss of a conventional fiber at l = 1.55 mm (0.2 dB/km).

Fig. 9. Field distributions for the two orthogonally polarized modes of the LS-MOF: (a) x-polarized mode and (b) y-polarized mode.

confinement loss of the LS-MOF is actually less than the loss of a conventional fiber at l = 1.55 mm (0.2 dB/km), achieving a value which is similar to that of a jacketed air-suspended rod with the same radius [9]. It must be noted that, due to the slight asymmetry of its core, the LS-MOF is naturally birefringent. Fig. 9 shows the field distribution for the two orthogonally polarized modes at l = 1.55 mm, when ri = 0.6 mm, D = 1 mm, and dair/D = 0.7. The effective refractive indexes of these modes are (a) neff,x = 1.309528 and (b) neff,y = 1.308259, thus confirming the fiber birefringence. Table 1 summarizes some of the results discussed above and shows the LS-MOF geometry’s parameters providing simultaneously zero dispersion and optimum nonlinear parameter at l = 1.55 mm.

5. Conclusions In this paper we have proposed a novel spiral-like design for a microstructured silica fiber. Besides the number of spiral turns, three different parameters allow the independent adjustment of the fiber parameters. Such design versatility provides a good control of the fiber dispersion properties and the possibility of achieving simultaneously zero dispersion and a high nonlinearity at specific wavelengths, as required for efficient supercontinuum generation. A record value of 70.0 W1/km for the fiber nonlinear parameter and a confinement loss below the loss of a conventional silica fiber (0.2 dB/km) were obtained at the zero dispersion wavelength of 1.55 mm, for a core radius of 0.6 mm and a cladding air-filling fraction of 90%. Moreover, a higher value g = 169.4 W1/km and zero dispersion

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have been also achieved at the important wavelength of 1.06 mm, where Yb-doped fiber lasers can be used. These results demonstrate the great potential of the proposed LS-MOF design for an efficient supercontinuum generation. Acknowledgments We acknowledge FCT (Fundac¸a˜o para a Cieˆncia e Tecnologia) for supporting through the Projects PTDC/ EEA-TEL/105254/2008, PTDC/FIS/112624/2009, and PEst-C/CTM/LA0025/2011. References [1] K. Saitoh, M. Koshiba, Numerical modeling of photonic crystal fibers, Journal of Lightwave Technology 23 (2005) 3580. [2] L. Dong, B.K. Thomas, L. Fu, Highly nonlinear silica suspended core fibers, Optics Express 16 (2008) 6423. [3] J. Dudley, J. Taylor (Eds.), Supercontinuum Generation in Optical Fibers, Cambridge University Press, Cambridge, UK, 2010. [4] P. Russel, Photonic-Crystal fibers, Journal of Lightwave Technology 24 (2006) 4729–4749. [5] J. Leong, P. Petropoulos, J. Price, H. Ebendorff-Heidepriem, S. Asimakis, R. Moore, K. Frampton, V. Finazzi, X. Feng, T. Monro, D. Richardson, High-nonlinearity dispersion-shifted lead-silicate holey fibers for efficient 1-mm pumped supercontinuum generation, Journal of Lightwave Technology 24 (2006) 183–190. [6] H. Ebendorff-Heidepriem, P. Petropoulos, S. Asimadis, V. Finazzi, R. Moore, K. Frampton, F. Koizumi, D. Richardson, T. Monro, Bismuth glass holey fibers with high nonlinearity, Optics Express 12 (2004) 5082–5087. [7] V. Ravi, K. Kumar, A. George, W. Reeves, J. Knight, P. Russell, F. Omenetto, A. Taylor, Extruded soft glass photonic crystal fiber for ultrabroad supercontinuum generation, Optics Express 10 (2002) 1520–1525.

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