Supermode analysis of multicore photonic crystal fibers

Optics Communications 283 (2010) 2686–2689

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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m

Supermode analysis of multicore photonic crystal fibers Chunying Guan ⁎, Libo Yuan, Jinhui Shi Photonics Research Center, College of Science, Harbin Engineering University, Harbin 150001, China

a r t i c l e

i n f o

Article history: Received 18 October 2009 Received in revised form 4 March 2010 Accepted 4 March 2010 Keywords: Photonic crystal fiber Multicore fiber Supermode Optical fiber

a b s t r a c t We reported on the supermode selection of total-internal-reflection photonic crystal fibers (PCFs) with linearly and circularly distributed multicores. The supermode characteristics were investigated using a fullvector finite-element method (FEM). The near and far-field patterns of supermodes supported by the multicore photonic crystal fiber were presented. A Talbot cavity was employed to select the wanted supermode. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Fiber lasers have attracted a tremendous amount of attention because of their ability to provide high-efficiency and excellent beam quality even at high-power levels compared with conventional laser systems. However, the scalability of output power of single-mode optical fiber laser is limited by mode area and nonlinear effects such as stimulated Raman scattering and stimulated Brillouin scattering. To overcome these limitations, multicore fibers (MCF) are introduced, and many theoretical and experimental results have been reported [1–3]. Multicore fiber lasers have an advantage of large mode areas, resulting in higher power thresholds for nonlinear processes [2,4]. Because of distributed nature of the cores, thermomechanical effects are mitigated compared with those of single-core lasers [5]. Optical fibers with multiple cores in one cladding will be of great importance in the future application of high-density optical transmission lines. In the past few years, photonic crystal fibers (PCFs) have attracted considerable interests as an alternative to conventional optical fibers in many applications [6–9]. PCFs can confine light into a guiding core based on total-internal-reflection or the photonic band-gap (PBG) effect. Total-internal-reflection occurs when the refractive index in the core is higher than that in the air-filled microstructured region. In a PCF, the number of holes and their sizes, shapes and arrangements can provide a fertile ground for achieving great flexibility and unique properties, which are not available in conventional optical fibers. Thus, PCFs could serve as a fiber host for developing a wide range of fiber devices, especially for optical communications, optical sensors and integrated optics. The recent developments revealed that the

⁎ Corresponding author. Tel./fax.: +86 451 82519850. E-mail address: [email protected] (C. Guan). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.03.007

multicore PCFs were drawn easily by introducing many defects along the entire fiber length [10,11]. An 18-core PCF laser has recently been studied [12]. In the present paper, the supermode selections of linearly and circularly distributed multicore total-internal-reflection PCFs are investigated using a full-vector finite-element method (FEM). The near and far-field patterns of supermodes supported by the multicore fiber are presented. A Talbot cavity is employed to select the wanted supermode. 2. Structure of multicore PCFs We focus on PCFs with triangular lattices because the defect core in this PCF has good axial symmetry. The cross-sections of a linear-corearray PCF and a circular-core-array PCF are illustrated in Fig. 1. Multiple cores are arranged in a linear array or a circular array in the PC cladding, where the cores are pure silica with refractive index n = 1.444, the air-hole pitch is Λ = 2.3 μm, the space between the two cores is 2Λ, the diameter of air holes is d, the ratio of the air-hole diameter dpto pitch Λ is 0.45 and the air-filling ratio is  ffiffiffi the air-hole 2 f = π = 2 3 × ðd =ΛÞ = 0:18. For the circular-core-array PCF, the number of fiber cores is easily controlled by the radius R of circularcore-array. Each core of PCFs with these parameters only supports a single mode to propagate in the optical communication band. The multicore PCFs can be fabricated using the stack-and-draw technique [13]. 3. Mode analysis In this paper, the full-vector FEM, using perfectly matched absorbing boundary layers, is successfully applied to supermode analysis of the MCFs. The FEM solver is a particularly powerful numerical method to resolve the behavior of PCFs because of its ability

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Fig. 1. Cross-sections of multicore PCFs.

to simulate complex geometries. The FEM allows the cross section of the waveguide to be divided into a patchwork of triangular elements which can be of different sizes, shapes, refractive indices and anisotropies. The mode field distribution and effective propagation constant can be obtained simply by using FEM.

the array performs in a nearly pure sinusoid, which implies that the far-field mainly consists of two vertical line-shaped spots with an angular spread in the horizontal direction. The phases of other supermodes have mixed symmetry. Supposing that the near field is known, the far-field can be calculated using diffraction equation

3.1. Linear-core-array PCF 2 −jk e−jkz ðx−x0 Þ ∫ ∫ aðx0 ; y0 ; z0 Þe 2z ½ jλz −∞−∞

+∞ +∞

In multicore PCFs, each core designed is aimed to only support a single mode. However, even when this triangular symmetry is broken in a multicore PCF, the splitting between the polarization states is so small that we can neglect it. Therefore, the array holds the same number of supermodes as that of its core. The propagation constants of the supermodes are slightly different from the individual modes, similar to conventional multicore optical fibers. 4-core and 5-core cases for linear-core-array PCFs are discussed in details and the supermodes of 4-core and 5-core PCFs are shown in Figs. 2 and 3, respectively. Here, the optical wavelength is 1.55 μm. The modes are numbered to be an integer n, i.e., decreasing order of their propagation constants. The first mode (n = 1) that possesses the largest propagation constant is the in-phase one, in which all cores have a constant phase over the entire array. Contrary to the in-phase mode, the last mode (antiphase mode, n = 4 or n = 5) has the form of phase reversal between two adjacent cores and the near field along

aðx; y; zÞ =

+ ðy−y0 Þ

2

 dx dy 0 0

ð1Þ

where x0 and y0 are the coordinate positions at the end of the fiber, while x and y are the coordinate positions in propagating distance z of light field. Since integrating Eq. (1) directly is quite difficult, here we resort to FFT method for calculating the far-field distribution [14]. Figs. 4 and 5 show the far-field patterns of supermodes in the 4-core PCF and 5-core PCF at z = 0.5 mm, respectively. We recognize that the far-field pattern of the in-phase mode renders extremely close to a linear field distribution and the far-field pattern of the last mode (n = 4 or n = 5) seems to be nearly two linear fields. The beam power coming from a multicore fiber can be transmitted into two far-field beams with specified spatial angle, indicating the importance in splitting laser beams. Our numerical simulations are coincident with our prediction above. Apart from in-phase mode and antiphase mode,

Fig. 2. Normalized electrical field intensity profiles of supermodes in a 4-core PCF.

Fig. 3. Normalized electrical field intensity profiles of supermodes in a 5-core PCF.

Fig. 4. Far-field distribution of a 4-core PCF.

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Fig. 6. Normalized electrical field intensity profiles of supermodes in a 6-core PCF.

far-field intensity patterns of other modes are also divided into two identical speckles, however, each speckle is surrounded by obvious side-lobes. These results are similar to conventional linear-array multicore fibers, while far-field distributions of multicore PCF reveal an additional different feature that obvious side-lobes appear on the edge of y-directional far-field. 3.2. Circular-core-array PCF For the circular-core-array PCF, we analyze a 6-core PCF with the same parameters in the previous section. The near and far-field patterns are illustrated in Fig. 6 and Fig. 7, respectively, where n = 1 and n = 6 are in-phase and antiphase modes. The circular-core-array PCF exhibits an advantage of obtaining high power because the farfield of the in-phase mode shows a Gauss-like profile and the power mostly concentrates in the center of the field. 4. Mode selection Among all supermodes, only the in-phase mode has the preferable far-field intensity profile. Since all supermodes in MCFs compete with each other by sharing the population inversion, it is very important to select the in-phase mode and suppress the other modes. A Talbot cavity between the fiber end and the feedback mirror was used to select the wanted supermode [15]. In Talbot resonators one makes use of the fact that the self reproductive properties of a coherent periodic wave field propagating through free space. In the one-dimensional case, a coherent wave field with a periodic structure and an infinite extension reproduces its own field distribution after a certain distance of propagation, called the Talbot distance. If one assumes mutual coherence among individual emitters of the multicore PCF, a feedback mirror, which is placed at half Talbot distance, results in self-imaging of the emitting structure on multicore PCF's end face. The resonator would suffer strong losses lack of phase locking. The amplitude coupling coefficients of the supermodes were adjusted by varying

distance between the fiber end and the mirror. The amplitude coupling from the jth mode to the ith mode was defined as [16]

ηij ðzÞ =

j

+∞ +∞

j

∫ ∫ a ðx; y; z= 0Þ Ta ðx; y; zÞdxdy

−∞− ∞

i

j

ð2Þ

where ai(x, y, z = 0) and aj(x, y, z) represent the emitted field and the reflected field, respectively. Here, only self-coupling coefficients are calculated. The calculated self-coupling coefficients of all supermodes in linear 4-core fibers and circular 6-core fibers at λ = 1.55 μm are shown in Fig. 8 and the distance is normalized with respect to the airhole pitch Λ. The supermodes of n = 2(4) and n = 3(5) produce identical self-coupling coefficients for the circular-core-array PCF owing to the circular symmetry. It is clear that self-coupling coefficient decreases faster for the high-order mode than the loworder mode at the large distance, and importantly the in-phase mode has highest self-coupling coefficient, whereas the antiphase supermode has the lowest self-coupling coefficient. The in-phase supermode is completely distinguished from the other supermodes to achieving nearly perfect phase locking by placing a mirror at the large distance. The supermode selection of the circular-core-array PCF is superior to that of the linear-core-array PCF. For smaller distance, the self-coupling coefficient undergoes oscillations owing to the Talbot reimaging effect [17]. 5. Conclusion In summary, the multicore fiber has attracted considerable attention in the fiber laser field because of the opportunity of achieving extremely high power and high quality beam. We have studied the near-field and far-field distribution properties of the supermode in linearly and circularly distributed multicore totalinternal-reflection PCFs based on the full-vector finite-element method. Because of the linear superposition of supermodes, the

Fig. 7. Far-field distribution of a 6-core PCF.

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Fig. 8. Amplitude self-coupling coefficient of each supermode for the 4-core PCF(a) and 6-core PCF(b).

multicore PCF allows one to achieve high powers and good beam quality in fiber lasers. The supermode selection of multicore PCF laser has been analyzed by employing the Talbot cavity. The supermode selection of the circular-core-array PCF is superior to that of the linear-core-array PCF. The potential properties are beneficial to further development in novel fiber-based components and fiber laser application. The high gain of the in-phase supermode is of key importance for making the fiber laser. The gain factor of the in-phase supermodes and other supermodes in the multicore PCFs will be discussed in future work.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

[11]

Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant 60877046, and in part by the Special Foundation for Harbin Young Scientists under Grant 2008RFQXG031.

[12] [13] [14] [15] [16] [17]

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