Numerical analyses of splice losses of photonic crystal fibers

Optics Communications 282 (2009) 4527–4531

Contents lists available at ScienceDirect

Optics Communications journal homepage: www.elsevier.com/locate/optcom

Numerical analyses of splice losses of photonic crystal fibers Zhongnan Xu a,b, Kailiang Duan b,*, Zejin Liu a, Yishan Wang b, Wei Zhao b a b

College of Optoelectronic Science and Engineering, National University of Defense Technology, Changsha, Hunan 410073, China State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an 710119, China

a r t i c l e

i n f o

Article history: Received 19 June 2009 Received in revised form 19 August 2009 Accepted 19 August 2009

PACS: 02.70.Dh 42.25.Bs 42.81.Bm 42.81.Dp 42.81.Gs

a b s t r a c t Splice losses between a photonic crystal fiber (PCF) and a single mode fiber (SMF) or a PCF are numerically investigated by using finite element method (FEM) with the circular perfectly matched layer (PML). Results show that the splice loss between a SMF and a PCF with air holes completely collapsed can reach many times of that between a SMF and a PCF without air-hole collapse. We calculate the rotation losses between two identical PCFs of three kinds: large mode area, polarization maintaining and grapefruit. It is shown that for the large mode area PCF and the grapefruit PCF, the rotation losses are sensitive to the wavelength when the rotation angle is larger than zero degree. The non-circular mode field distribution is the main source of the rotation loss. Ó 2009 Elsevier B.V. All rights reserved.

Keywords: Splice loss Rotation loss PCF FEM PML

1. Introduction In recent years PCFs, also called microstructure fibers or holey fibers, have attracted much attention due to their unique waveguide properties, such as endless single-mode guiding [1], tailorable group-velocity dispersion [2], high nonlinearity [3] and flexible designing, etc. PCFs are typically classified into two types: index-guiding PCFs [4], which guide light by total internal reflection mechanism as the conventional fibers, and photonic bandgap fibers [5], which confine light in the low-index core by using photonic bandgap effects. PCFs have been used for fiber-optic devices and fiber sensing applications that are difficult to be realized by conventional fibers. In many experiments and applications, PCFs are often needed to be spliced with conventional fibers, the splice loss of which mainly comes from fundamental mode mismatch, angular misalignment and core offset [6]. Air-hole collapse of PCFs is another source of splice loss, but it also can be used to reduce splice loss when the * Corresponding author. Address: State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an 710119, China. Tel.: +86 029 88887617 (K. Duan). E-mail address: [email protected] (K. Duan). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.08.026

two fibers have mode mismatch [7]. Many methods have been proposed for low-loss splicing of conventional fibers with PCFs by using a fusion splicer [8,9] or a CO2 laser [10,11], and the theory used for two conventional fibers is applied to analyze the splice loss between PCFs and conventional fibers [10,12]. The splicing of two PCFs is also confronted in the applications of PCFs, the experiment of which has been done by Bruno Bourliaguet et al. [13] and the results showed that the splice loss is more sensitive than that of a conventional fiber with a PCF. In this paper, splice losses resulting from the fundamental mode mismatch between a PCF and a SMF or a PCF are investigated by using FEM with the circular PML [14,15]. By ignoring other factors, we first calculate the splice loss between a PCF and a SMF with airholes collapsed. The results show that when the PCF’s air holes are completely collapsed the splice loss can be many times of that when the PCF without air-hole collapse. Second, we focus our study on the rotation losses of two identical PCFs of three types of configurations, i.e., large mode area, polarization maintaining and grapefruit, without air-hole collapse. From the calculation results, we find that the rotation losses are sensitive to wavelength for the large mode area PCF and grapefruit PCF when the rotation angle is larger than zero, and increase with increasing of the rotation angles.

4528

Z. Xu et al. / Optics Communications 282 (2009) 4527–4531

2. Theoretical formulation

3. Numerical results

FEM has been one of the most successful numerical methods in solving waveguide problems owing to its high flexibility and precision. By properly choosing finite triangular elements across the profile of PCFs and using the continuous conditions on the boundary of each element, FEM has been widely used to solve electromagnetic problems in PCFs. For a better description of the mode field of PCFs, the complex areas are often chosen to have finer elements than the other parts. In addition, an anisotropic material is assumed in the calculation as a computational window border (i.e., PML) which will absorb the entire incident field without any reflection. PML surrounding the profile of the PCF are often chosen to have eight square regions. Recently, Pierre Viale proposed a kind of circular PML [15] which was shown to be more simple and efficient in the study of electromagnetic problems in PCFs. By using FEM we can calculate the fundamental transverse mode distributions of PCFs and SMFs. Then the effective areas of the fundamental modes of the fibers can be given by [16]

3.1. Splicing a PCF to a SMF

R R þ1 ð jEðx; yÞj2 dxdyÞ2 Aeff ¼ R R1 þ1 jEðx; yÞj4 dxdy 1

ð1Þ

where E(x, y) is the transverse electric field distribution of the fundamental mode of the fiber. From Eq. (1), the effective radius xeff = (Aeff/p)1/2 can be obtained. Under the Gaussian profile approximation of the fundamental modes, the splice loss aps between a PCF and a SMF can be approximately expressed by [9]

aps ¼ 20 log



2xPCF xSMF x2PCF þ x2SMF

 ð2Þ

where xPCF and xSMF are the fundamental mode radii of the PCF and the SMF, respectively. During the splicing process, air-hole collapse is often inevitable for PCFs. For the PCFs with a few air holes, such as grapefruit PCF, the effect of air-hole collapse has been experimentally studied [17]. For the PCFs with a large number of air holes in the cladding, the relation between the air-hole radius d and pitch K (the distance between nearest neighbor air holes) is approximately given by [18,19]



K K0

2

pffiffi 3 2

 p4

3 2

 p4

¼ pffiffi

 2 d0 K0

 2

ð3Þ

d0

K

where d0 and K0 are the initial air-hole radius and the pitch, respectively. Generally, the splice loss between PCFs and conventional fibers can be calculated by using Eqs. (1) and (2). However, the contribution of the microstructure to the mode fields makes the splice loss calculated by Eq. (2) inaccurate for splicing two PCFs. The rigorous analysis of the splice ap between two PCFs is given by the mode overlap integral [20]:

0 1 2   RR   Ep  Es dxdy B C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ap ¼ 10 log @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A RR R R   jEp j2 dxdy jEs j2 dxdy  

As examples, two representative fibers, a large mode area PCF LMA-10 and a single mode fiber SMF-28, are considered to analyze the splice loss between a PCF and a SMF by using Eqs. (1) and (2). For LMA-10, due to its large number of air holes in the cladding, we use Eq. (3) to calculate the new air-hole radius and the new pitch when the air holes are collapsed. The schematic illustration for LMA-10 is shown in Fig. 1. LMA-10 comprises of seven layers of air holes with one central air hole missed. Its cladding diameter is 125 lm and its pitch is K = 7.14 lm with normalized air-hole diameter being d/K = 0.46. The refractive index n of the pure silica background is given by the Sellmeier equation [21] as follows:

n2 ¼ 1 þ þ

0:6961663k2 2

2

k  0:00467914826 0:8974794k2

þ

0:4079426k2 k  0:01351206312 2

k2  97:93400252

where k is the wavelength. To calculate the fundamental mode distributions of LMA-10 one needs only to consider one quarter of the cross section due to their symmetry [22]. Our calculation of the fundamental mode field diameter (FMFD) of LMA-10 is 9.1 lm for k = 1.55 lm, which agrees with Ref. [9]. In the range of d/k = 0–0.46, the confinement losses are less than 104 dB/m, so it can be neglected in the simulation. We consider a full adiabatic process [23], so that the fundamental mode would not couple to any other modes while propagating through the splice region. We only calculate the splice loss caused by the mode mismatch between LMA-10 and SMF-28 without considering angular misalignment, core offset and Fresnel reflection at the interface. Fig. 2 shows the effective index (solid line) and the fundamental mode radius (dash line) versus the air-hole radius of LMA-10 for k = 1.55 lm. From Fig. 2 it can be seen that the effective index and fundamental mode radius decreases with the increasing of air-hole radius. The maximum refractive index is 1.444, which is close to the background material, and the maximum mode radius is 41 lm, which is close to the silica cylinder radius, when the air holes are completely collapsed. Fig. 3 shows the splice loss between LMA-10 and SMF-28 (FMFD = 10.4 lm for k = 1.55 lm [8]) versus the air-hole radius. It can be seen that the splice loss decreases with increasing air-hole radius, but slightly increases when the air-hole radius is larger than d = 1.148 lm, as shown in the insert of Fig. 3. Zero splice loss is found at d = 1.148 lm and the maximum value is 12.05 dB when d = 0, which is about 150 times of the splice loss when without air-hole collapse. The FMFD increases with decreasing air-hole radius because smaller air holes have lower restriction on the mode field as shown in Fig. 2. However, even if air holes are completely collapsed, the

ð4Þ

where Ep and Es are the transverse electric field distributions of the fundamental modes of PCFs. This equation reveals that the splice loss ap is closely related with the angle which comes from the inner product of Ep and Es, so we define this type of splice loss as rotation loss to distinguish it from the conventional splice loss.

ð5Þ

Fig. 1. Microstructure scheme of LMA-10.

Z. Xu et al. / Optics Communications 282 (2009) 4527–4531

Fig. 2. Effective index (solid line) and fundamental mode radius (dash line) of LMA10 for k = 1.55 lm.

Fig. 3. Splice loss between LMA-10 and SMF-28 for k = 1.55 lm, and the insert shows the zero splice loss point.

cladding which is formed by the infinite air also can strictly bind the fundamental mode in the pure silica area. Due to the significant different mode radii between the infinite air cladding fiber and SMF-28, the splice loss can reach 12 dB for k = 1.55 lm. This is more than the measured loss in Ref. [9] because Eq. (3) is an approximate relationship and based on the assumption of purely transverse silica flow when the air holes are collapsed. Furthermore, the air-hole collapse also can reduce splice loss as shown in the insert of Fig. 3.

3.2. Splicing a PCF to a PCF Eq. (2) is widely applied to deal with the splice loss of a PCF and a SMF. However, this equation assumes that the mode field distri-

4529

bution has circular symmetry. The splice loss will be zero if Eq. (2) is used for two identical PCFs. In fact, due to their non-circular mode distributions, the splice loss can be very large if the configurations of the two identical PCFS are not matched. To correctly calculate the splice of two PCFs, Eq. (4) should be used. In what follows, we focus our study only on the rotation loss of two identical PCFs without considering air-hole collapse. The rotation loss depends only on the rotation angles, which is defined as the angle between the corresponding symmetry axes of the two identical PCFs. As examples, we study only three types of PCFs, one of which has been shown in Fig. 1 and the other two are shown in Fig. 4. Fig. 4a shows the cross section of a grapefruit PCF whose core diameter is 28.4 lm and cladding diameter is 125 lm. To simulate the actual structure of the grapefruit PCF, the model of the air hole is chosen to be formed by a section of large circle (R = 20 lm) and two tangents of three symmetrically distributed inscribed circles (r/ R = 0.5) in the large one as shown in Fig. 4b. Fig. 4c shows the cross section of a PMPCF which comprises of four layers of air holes with one pair of air holes enlarged. The pitch is K = 7.16 lm and the normalized air holes for the small holes and larger holes are d/K = 0.46 and d2/K = 0.92, respectively. The cladding diameter is 64 lm. Their background refractive indexes are also given by Eq. (5). The maximum rotation angle h for LMA-10 is 30° as shown in Fig. 1, and the maximum rotation angles h for the grapefruit PCF and the PMPCF are 30° and 90°, respectively, as shown in Fig. 4a and c. Fig. 5 shows the rotation losses of LMA-10 and the grapefruit PCF versus the rotation angles for k = 1 lm, 1.55 lm, 2 lm and 2.5 lm, respectively. It can be seen that the rotation loss of LMA10, as shown in Fig. 5a, increases with increasing rotation angle and for the same rotation angle the rotation loss increases with increasing wavelength. The maximum rotation losses of LMA-10 are 0.182 dB, 0.19 dB, 0.197 dB and 0.205 dB for k = 1 lm, 1.55 lm, 2 lm and 2.5 lm, respectively. The rotation loss for k = 1.55 lm is less than the loss in Ref. [13] because the structure of the PCF in Ref. [13] has smaller d/K than that of LMA-10 and its outer layers of air holes have been destroyed. Fig. 5b shows the rotation loss of the grapefruit PCF, from which it can be seen that the rotation loss increases with increasing rotation angle, but decreases with increasing wavelength. The maximum rotation losses are 0.083 dB, 0.08 dB, 0.078 dB and 0.075 dB for k = 1 lm, 1.55 lm, 2 lm and 2.5 lm, respectively. Compared with LMA-10, the grapefruit PCF has lower splice loss due to its large core and tiny gap, although it has sixfold symmetry. The fundamental modes of the PMPCF can be classified into X polarized mode and Y polarized mode, and the two polarized modes have different propagation constants, so they are non-degenerated. In the simulation, when one of the two identical spliced PMPCFs is rotated, the polarization direction of the polarized modes will change. However, we focus on the rotation loss between the polarized modes with the same polarization direction relative to the

Fig. 4. Microstructure schemes for (a) the grapefruit PCF and (c) the PMPCF, (b) illustrates the geometric model of one air hole in (a).

4530

Z. Xu et al. / Optics Communications 282 (2009) 4527–4531

Fig. 5. Rotation losses of (a) LMA-10 and (b) the grapefruit PCF for k = 1 lm, 1.55 lm, 2 lm, and 2.5 lm, respectively.

Fig. 6. (a) The rotation loss of X polarized mode of the PMPCF for k = 1 lm, (b) the rotation loss differences of X polarized mode between the rotation losses for k = 1.55 lm, 2 lm and 2.5 lm and that for k = 1 lm, respectively, (c) the rotation loss differences between Y polarized mode and X polarized mode for k = 1 lm, 1.55 lm, 2 lm, and 2.5 lm, respectively.

microstructure. The rotation losses of both of the polarized modes are calculated by using Eq. (4), and the results are shown in Fig. 6 for the PMPCF of Fig. 4c. From Fig. 6a, it can be seen that the rotation loss of X polarized mode increases quickly with increasing rotation angle for k = 1 lm and is infinite when rotation angle is 90°. Fig. 6b shows the rotation loss differences of X polarized mode between the rotation losses for k = 1.55 lm, 2 lm and 2.5 lm and the rotation loss for k = 1 lm, respectively. It can be seen that the differences decrease and then increase with increasing rotation angle for the same wavelength. The maximum absolute value of the differences is about 0.116 dB at h = 66° for k = 2.5 lm, where the rotation loss is about 8 dB for k = 1 lm. The rotation loss differences of Y polarized mode and X polarized mode are calculated for k = 1 lm, 1.55 lm, 2 lm, and 2.5 lm, respectively, as shown in Fig. 6c. It can be seen that the maximum absolute value can reach 0.12 dB at h = 69° for k = 1 lm, where the rotation loss is about 10 dB for k = 1 lm. From Fig. 6 we can find the rotation losses of X polarized mode and Y polarized mode are not sensitive to the wavelength and the relative rotation loss differences of X polarized mode and Y polarized mode are comparably very small.

sults show that the splice loss between a SMF and a PCF with air holes completely collapsed can reach many times that of a SMF and a PCF without air-hole collapse. Even for the splice between two identical PCFs, the rotation loss is non-zero due to their noncircular mode field distributions. The splice loss increases with increasing wavelength for the large mode area PCF and decreases with increasing wavelength for the grapefruit PCF at the same rotation angle, but for the PMPCF, the rotation losses of X polarized modes and Y polarized modes are not sensitive to wavelength, and the relative rotation loss differences of the two polarized modes are comparably very small. The results in this paper are useful for analyzing the splice loss of fibers. Acknowledgement This work is supported by State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences (CAS). References

4. Conclusion The splice losses between a PCF and a SMF or a PCF have been numerically investigated by using FEM with the circular PML. Re-

[1] T.A. Birks, J.C. Knight, P. St, J. Russell, Opt. Lett. 22 (1997) 961. [2] J.C. Knight, J. Arriaga, T.A. Birks, A. Ortigosa-Blanch, W.J. Wadsworth, P. St, J. Russell, IEEE Photon. Technol. Lett. 12 (2000) 807. [3] N.G.R. Broderick, T.M. Monro, P.J. Bennett, D.J. Richardson, Opt. Lett. 24 (1999) 1395.

Z. Xu et al. / Optics Communications 282 (2009) 4527–4531 [4] J.C. Knight, T.A. Birks, R.F. Cregan, P.S.J. Russell, J.P. Sandro, Opt. Mater. 11 (1999) 143. [5] R.F. Cregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.S.J. Russell, P.J. Roberts, D.C. Allan, Science 285 (1999) 1537. [6] Andrew D. Yablon, Optical Fiber Fusion Splicing, Springer Series in Optical Sciences, vol. 103, Springer, Berlin, 2005, p. 112 (Chapter 4). [7] C. Kerbage, A. Hale, A. Yablon, R.S. Windeler, B.J. Eggleton, Appl. Phys. Lett. 79 (2001) 3191. [8] P.J. Bennett, T.M. Monro, D.J. Richardson, Opt. Lett. 24 (1999) 1203. [9] Limin Xiao, M.S. Demokan, Wei Jin, Yiping Wang, Chun-Liu Zhao, J. Lightwave Technol. 25 (2007) 3563. [10] J.H. Chong, M.K. Rao, Opt. Express 11 (2003) 1365. [11] J.H. Chong, M.K. Rao, Y. Zhu, P. Shum, IEEE Photon. Technol. Lett. 15 (2003) 942. [12] R. Thapa, K. Knabe, K.L. Corwin, B.R. Washburn, Opt. Express 14 (2006) 9576. [13] Bruno Bourliaguet, Claude Paré, Frédéric Émond, André Croteau, Antoine Proulx, Réal Vallée, Opt. Express 11 (2003) 3412.

4531

[14] U. Pekel, R. Mittra, IEEE Microwave Guided Wave Lett. 5 (1995) 258. [15] Pierre Viale, Sebastien Fevrier, Frederie Gerome, Herve Vilard, in: Proceedings of the COMSOL Multiphysics User’s Conference, 15 November, Paris, 2005. [16] Govind P. Agrawal, Nonlinear Fiber Optics, Optics and Photonics, third ed., Academic, San Diego, California, 2001 (Chapters 2 and 44). [17] E. Magi, H. Nguyen, B.J. Eggleton, Opt. Express 13 (2005) 453. [18] Jesper Lagsgaard, Anders Bjarklev, Opt. Commun. 237 (2004) 431. [19] Boris T. Kuhlmey, Hong C. Nguyen, J. Opt. Soc. Am. B 23 (2006) 1965. [20] Yeuk Lai Hoo, Wei Jin, Jian Ju, Hoi Lut Ho, Microwave Opt. Technol. Lett. 40 (2004) 378. [21] C.V.I. Melles Griot, Dispersion Equations Technical Notes, Optical Components and Assemblies, 427. [22] T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, L.C. Botton, J. Opt. Soc. Am. B 19 (2002) 2322. [23] J.K. Chandalia, B.J. Eggleton, R.S. Windeler, S.G. Kosinski, X. Liu, C. Xu, IEEE Photon. Technol. Lett. 13 (2001) 52.