Effects of bending on the performance of hole ... - OSA Publishing

Effects of bending on the performance of holeassisted single polarization fibers Xin Chen, Ming-Jun Li, Joohyun Koh, Anthony Artuso, and Daniel A. Nolan Science and Technology Division, Corning Incorporated, SP-AR-01-2, Corning, NY 14831 [email protected]

Abstract: We study the effects of bending on single polarization fiber performance through the use of finite element method in conjunction with the perfectly matched layer (PML) in cylindrical geometry. The cylindrical PML used in this paper allows us to calculate the loss associated with each polarization mode at a given wavelength, specified bending diameter, and specific orientation. We identified a series of bending characteristics of the single polarization fiber by choosing different bending diameters and different orientations. We also conducted experiments to study some aspects of the bending. Good qualitative agreement between numerical and experimental results is found, which helps to understand fiber deployment conditions and can potentially facilitate new design efforts. ©2007 Optical Society of America OCIS codes: (060.2270) Fiber characterization; (060.2420) Fibers, polarization-maintaining.

References and links 1. 2. 3. 4.

5. 6.

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8. 9. 10. 11. 12. 13. 14. 15.

M. J. Messerly, J. R. Onstott, and R. C. Mikkelson, “A broad-band single polarization optical fiber,” J. Lightwave Technol. 9, 817-820 (1991). H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Technol. Lett. 16, 182-184 (2004). J. R. Folkenberg, M. D. Nielsen, and C. Jakobsen, “Broadband single-polarization photonic crystal fiber,” Opt. Lett. 30, 1446-1448 (2005) T. Schreiber, H. Schultz, F. Röser, O. Schmidt, J. Limpert, R. Iliew, A. Petersson, C. Jacobsen, K. P. Hansen, J. Broeng, and A. Tünnermann, “Design and high power operation of a stress-induced singlepolarization single-transverse mode LMA Yb-doped photonic crystal fiber,” Proc. SPIE 6102, 61020C-161020C-9 (2006) D. A. Nolan, G. E. Berkey, M.-J. Li, X. Chen, W. A. Wood, and L. A. Zenteno, “Single-polarization fiber with a high extinction ratio,” Opt. Lett. 29, 1855-1857 (2004) D. A. Nolan, M.-J. Li, X. Chen, and J. Koh, “Single polarization fibers and applications,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference on CDROM (Optical Society of America, Washington DC, 2006) OWA1 S. Li, X. Chen, D. V. Kuksenkov, J. Koh, M.-J. Li, L. A. Zenteno, and D. A. Nolan, “Wavelength tunable stretched-pulse mode-locked all-fiber erbium ring laser with single polarization fiber,” Opt. Express 14, 6098-6102 (2006) D. Marcuse, “Influence of curvature on the losses of doubly clad fibers”, Appl. Opt. 21,4208-4213, (1982) K. Okamoto, “Single-polarization operation in highly birefringent optical fibers,” Appl. Opt. 23, 26382642 (1984) J. Olszewski, M. Szpulak, W. Urbanczyk, “Effect of coupling between fundamental and cladding modes on bending losses in photonic crystal fibers,” Opt. Express 13, 6015-6022 (2005) Y. Tsuchida, K. Saitoh, and M. Koshiba, “Design and characterization of single-mode holey fibers with low bending losses,” Opt. Express 13, 4770-4779 (2005) M.J. Li, X. Chen, D.A. Nolan, G. E. Berkey, J. Wang, W. A. Wood, and L.A. Zenteno “High Bandwidth Single Polarization Fiber With Elliptical Central Air Hole”, J. Lightwave Technol. 23, 3454-3460 (2005) W. C. Chew, and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” IEEE Photon. Tech. Lett. 7, 599-604 (1994) Jianming Jin, “The Finite Element Method in Electromagnetics,” Second Edition, John Wiley & Sons, Inc, New York 2002. P.L. Teixeira and W. C. Chew, “Systematic Derivation of Anisotropic PML Absorbing media in Cylindrical and Spherical Coordinates,” IEEE Microwave and Guid. Wave Lett. 7, 371-373 (1997)

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Received 18 Apr 2007; revised 20 Jul 2007; accepted 4 Aug 2007; published 7 Aug 2007

20 August 2007 / Vol. 15, No. 17 / OPTICS EXPRESS 10629

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Electromagnetics Module User’s Guide, COMSOL 3.2, Chapter 2, COMSOL AB (2005)

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M. Heiblum, and J. H. Harris, “Analysis of Curved Optical Waveguides by Conformal Transformation,” IEEE J. Quantum Electron. QE-11, 75-83 (1975) L. Faustini, and G. Martini, “Bending Loss in Single-Mode Fibers,” J. Lightwave Technol., 15, 671-679 (1997)

18.

1. Introduction Recently there has been an increasing interest in using single-polarization fibers in applications such as fiber lasers, fiber-optic gyroscopes, and fiber sensors to deal with unwanted and detrimental polarization effects. Different approaches in achieving single polarization operation in optical fibers have been explored [1-4]. A solid type single polarization fiber was studied [1] in early 1990s. More recent studies focused on single polarization fiber with air holes, e.g. hole-assisted fibers with a few air holes or microstructured fibers [2-6]. In the past few years, we have proposed, implemented and commercialized a holeassisted single polarization fiber [5-6]. In studying this new type of fiber, we have identified several novel properties that require special attention in order to use the fiber properly. Unlike conventional polarization maintaining (PM) fibers, such as Panda or Bowtie PM fibers, the hole-assisted single polarization fibers have a cutoff wavelength for each fundamental polarization mode. Because the fundamental mode cutoff is intrinsic to the fiber, the single polarization operation can be achieved even when the fiber is straight and in both short length (20-30cm) and long length. It also results in other significantly different behaviors than conventional PM fibers related to fiber length and bending. Understanding these effects is critical for practical applications since the fiber property could alter from the measured conditions. When properly used, some of the deployment effects can be beneficial for additional flexibility in certain applications [7]. There has been a long history of studying the bending properties of optical fibers [8], on both transmission fibers and polarization maintaining fibers. Most of these studies focused on understanding how bending affects the loss of the simple step index optical fiber in the macrobending regime as it is a critical factor for optical communications. In the context of achieving single polarization operations, Ref.[9] suggested the use of bending to force a Bowtie type highly birefringent fiber, which does not exhibit fundament cutoff, to have single polarization behavior. In order to achieve the single polarization operation the fiber must be bent at a carefully selected diameter and must be sufficiently long, which limits the applications of such fibers. Very recently the study has been extended to photonic crystal fibers to understand the coupling effect between fundamental and cladding modes on bending loss [10] and to design holey fiber with low bending losses [11]. To our knowledge, to date no literature has addressed the problem of how bending affects fibers with fundamental mode cutoff, in particular single polarization fibers. In this paper, we conduct numerical and experimental studies on bending effects on hole-assisted single polarization fibers to gain systematic understanding of this problem. We first describe numerical modeling based on a finite element method in section 2 and 3. We then present experimental results with comparison to numerical results in section 4. A brief summary is presented in section 5. 2. Finite element modeling scheme The numerical modeling conducted in this paper is based on finite element method. Note that in early studies of the fiber bending effects including those in Ref.[9], the bend loss was calculated by formula developed by Marcuse in Ref.[8] and its earlier references for fibers with step index profiles. The Marcuse formula does not account for more complex and asymmetric structures and non-step-index profile as is the case for the fibers considered in this paper. It also can not deal with fibers with fundamental mode cutoffs. More sophisticated formalism, which is capably of handling fully vectorial Maxwellian equations with #82237 - $15.00 USD

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asymmetric fiber cross section, is needed. We adopt a finite element method for fully vectorial Maxwell equation for the two dimensional fiber structure used in a previous paper [12], where the properties of single polarization fibers with central air hole were studied. In Ref.[12] the properties of the fiber were studied through analyzing the effective refractive index associated with each polarization mode. The difference in effective refractive indices at a given wavelength reveals the modal birefringence of the fiber between the two fundamental polarization modes. The cutoff wavelength for each polarization mode, which determines the key properties of the single polarization fibers, such as the single polarization bandwidth and the location of single polarization operating window, can be calculated from the cutoff wavelength of each polarization mode, i.e. the wavelength when the effective refractive index of a particular polarization mode equals that of the cladding. The finite element modeling in this paper was further enhanced from that in Ref. [12] to be able to calculate the complex effective index associated with a particular (polarization) mode. Through this enhancement, we can address the specific modeling need in this paper, where the understanding of single polarization fibers relies on a richer set of information. The imaginary part of the effective refractive index, exhibiting the loss of the fiber, increases sharply when the fiber approaches the cutoff of a polarization mode at certain wavelength. Such modeling capability was obtained through the use of the perfectly matched layer (PML) in the cylindrical coordinate system. As is widely studied in the literature, the PML is an additional domain that absorbs the incident radiation without producing reflections. The PML emulates the effect of infinite domain, which allows us to calculate the loss associated with each eigen-mode of the fiber. Although there are various interpretations of PML and its variants, Chew and Weedon [13] have interpreted the PML as coordinate stretching in the frequency domain. In such formalism, the full vectorial wave equation including the anisotropic PML still takes the conventional form, ∇ × ( μ −1∇ × E ' (r ) − k02εE ' (r ) = 0 (1) where μ is the permeability, and k0 is the free space wave number. E ' is the new electric field vector, which is linked to the original electric field vector by,

E ' (r ) = S −1E (r ) .

(2)

where

S

⎛ s x−1 ⎜ =⎜ 0 ⎜ ⎝ 0

0 s

−1 y

0

⎞ ⎟ 0⎟ ⎟ s z−1 ⎠

0

(3)

r represents a coordinate converted position vector, x y z r = [~ x , ~y , ~ z ] = [ ∫ s x ( x' )dx ', ∫ s x ( y ' )dy ' , ∫ s x ( z ' )dz '] 0

0

0

(4)

where s x , s y , and s z are complex stretching variables in Cartesian coordinate system. As a result, the permeability μ and the permittivity

ε in the PML region take the form,

μ = μ0 μr L

(5)

ε = ε 0ε r L

(6)

where

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L

⎛s s ⎜ y z ⎜ sx ⎜ =⎜ 0 ⎜ ⎜ ⎜ 0 ⎝

0

0

sz sx sy

0

0

sx s y sz

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

.

(7)

In Refs.[10, 11], the PML is a rectangular layer surrounding the fiber, in which case the symmetry of the fiber is arbitrarily broken. The rectangular PML scheme is not suitable to study the bending effect with different fiber orientations. To overcome this difficulty, we consider the cylindrical PML as described in [14]. The fiber cross section along with the PML layer is shown in Figure 1. At sufficiently large radius, the cylindrical PML plays a similar role as that of rectangle PML. In a straightforward manner, the finite element scheme can thus be implemented in the cylindrical coordinate system as that in Ref.[15]. However, for many available implementations of the finite element method, the basic coordinate system is Cartesian. To avoid the implementation of a cylindrical PML scheme as laid out in Ref.[15] from scratch, one may still use the finite element scheme implemented in Cartesian coordinate but introduce one additional transformation. The Cartesian tensor components for the cylindrical PML can be expressed by the cylindrical coordinates ( ρ ,ϕ , z ) so that the tensor matrix L becomes [16],

L

⎛ s s ⎜ s z ( ϕ cos 2 ϕ + ρ sρ sϕ ⎜ ⎜ s = ⎜ s z cos ϕ sin ϕ ( ϕ sρ ⎜ ⎜ ⎜ 0 ⎜ ⎝

sin 2 ϕ ) −

sρ ) sϕ

sϕ s ρ − ) sρ sϕ s s s z ( ϕ sin 2 ϕ + ρ cos 2 ϕ ) sρ sϕ s z cos ϕ sin ϕ (

0

0 0 sρ s ϕ sz

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(8)

s ρ , sϕ , and s z are stretching variables in the cylindrical coordinate system. By expressing the L matrix with cylindrical variables, we can take advantage of the Cartesian implementation of the finite element scheme and get the benefit of the cylindrical geometry. In our calculation, we have set the ρ PML to be the normal cladding radius of 62.5 μm , chosen the sϕ and s z to be 1 in all regions, and let s ρ take the following form,

1 ⎧ ⎪ ρ − ρ PML 2 sρ ( ρ ) = ⎨ 1 − iα ( ) ⎪ d PML ⎩

ρ ≤ ρ PML ρ > ρ PML

(9)

where α is the attenuation parameter. The thickness of the PML d PML is chosen so that the imaginary part of the effective index converges to a stable value. The main focus of this paper is to study the bending property of single polarization fibers. In the analysis of bending effects, the curved fiber is replaced by a straight one with an equivalent refractive index distribution [17],

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x neq ( x, y ) = n( x, y ) exp( ) R

(10)

where R is the bending radius. In Eq.(10) we assume the fiber is always bent along the xdirection. In order to study the effect of bending on different fiber orientation, we rotate the fiber counter-clockwise by certain degrees, for example, 45 degrees or 90 degrees while keeping the bending direction fixed. The loss of the fiber ( α B ) in a particular mode can be

Loss (dB /m )

calculated from the imaginary part of the propagation constant 20 αB = Im(β ) ≈ 8.686 Im(β ) ln(10)

0 10 20 30 40 50 60

β

so that,

(11)

Sheet: Untitled1

Cutoff 1

Cutoff 2

1500 1520 1540 1560 1580 1600 1620

Wavelength (nm) Fig. 1. Cross section view of the fiber with PML.

Fig. 2. Calculated loss as a function of wavelength for two fundamental polarization modes in a straight fiber.

3. Numerical modeling results

With the modeling tool fully established, we can study the loss of the fiber as a function of wavelength. First, we show the result of a straight fiber in Fig. 2. In the modeling, we have chosen a set of fiber parameters so that they have the single polarization operation window centered on 1550 nm. Specifically, the fiber core has refractive index 1% higher than that of the cladding. The semi-axial dimensions of the fiber core are 1.53 μm and 4.5 μm respectively. The air holes are placed right next to the fiber core with a radius of 4.5 μm. Note that the single polarization operation window is the wavelength range between the cutoff wavelengths of the two fundamental polarization modes. When the fundamental mode cutoff is reached, the loss increases sharply. The calculated curves are very similar to the transmission spectrum measured in [5] taking into account that higher loss means less transmission. We have flipped the order of the vertical axis in Fig. 2 making it more comparable to transmission spectrum in Ref. [5]. The single polarization window is around 80 nm wide while an actual fiber implemented according to the design has a bandwidth of 5560nm. The result in Figure 2 highlights that the current modeling tool can realistically describe the fiber cutoff behavior around the cutoff transition. In the next step, we study the effect of bending on the cutoffs for the fiber as shown in Fig. 1 by bending it with 5 cm, 10 cm, and 20 cm diameters. The results are shown in Fig. 3 for each polarization mode. It is clear that bending has complex effects to the loss curve. The transition to the cutoff is not necessarily smooth and sharp. In some cases, the curve appears to be slightly bumpy. However, in general, with tighter bending, the overall single polarization operating windows shifts toward a lower wavelength. At the tighter bending, the transition to cutoff is less steep, which effectively reduces the single polarization extinction ratio, defined as the power ratio in dB between two polarization modes, for some part of the

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L o s s (d B /m )

single polarization operation window. In addition, the loss curves for polarization mode 1 monotonically shift toward lower value while for the other polarization mode the order is reversed as we decrease the bending diameter. 0 10 20 30 40 50 01400

Sheet: Untitled2

Polarization Mode 1

1450

Sheet:1500 Untitled3 1550 5cm Column1

10 20 10cm 20cm 30 40 Polarization Mode 2 50 1400 1450

1500

1550

1600

1600

Wavelength (nm) Fig. 3. Calculated loss as a function of wavelength for both polarization modes at several different bending diameters with 0 degree rotation.

L o s s (d B /m )

Since the fiber cross section is asymmetric, one may wonder if the cutoff behavior varies with the fiber orientation. Using the layout in Figure 1 as a reference of having 0 degree rotation, we subsequently rotate the fiber by 45 and 90 degrees and obtain the loss as a function of wavelength. The results are shown in Figure 4 for a bending diameter of 20 cm. It can be found that fiber orientation relative to the bending direction can have significant effects on the fiber cutoff behavior. When the fiber bending direction is aligned with the direction of the two air holes (0 degree curve), the bending has less effect on the cutoff because the fiber has better bending resistance due to the higher glass-air index contrast than the other orientations. We also observe that the response to the bending is different for the two polarization modes. For the first polarization mode, the cutoff shifts towards shorter wavelength monotonically as the fiber rotation angle increases. But for the second polarization mode, the effect of bending on the 45 degree and 90 degree rotation is not significantly different as the loss curves almost overlap. 0 Polarization Mode 1 10 20 30 40 50 Sheet: Untitled4 01460 1480 1500 1520 1540 1560 1580 10 0 Degree 20 45 Degree 30 90 Degree Polarization Mode 2 40 50 1460 1480 1500 1520 1540 1560 1580

Wavelength (nm) Fig. 4. Calculated loss as a function of wavelength for both polarization modes at several orientations of 0, 45 and 90 degrees. Bending diameter is fixed at 20 cm.

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Column2 Loss (dB)

4. Experimental results

Fig5a.PDW

0 Polarization Mode 1 20cm 5 5cm 10 10cm No Bend 15 20 1000 1020 Fig5b.PDW 1040 1060 1080 1100 0 No Bend Column1 5 20cm 10 5cm 10cm Polarization Mode 2 15 20 1000 1020 1040 1060 1080 1100

Wavelength (nm) Fig. 5. Measured fundamental mode cutoff wavelength shifts at different bending diameters for 1060 nm single polarization fiber

Loss (dB)

Fig6a.PDW 0 Polarization Mode 1 20cm No Bend 5 10 15 10cm 20 Fig6b.PDW 1500 1550 1600 01450 No Bend 5 Column1 20cm 10 10cm 15 Polarization Mode 2 20 1450 1500 1550 1600

Wavelength (nm) Fig. 6. Measured fundamental mode cutoff wavelength shifts at different bending diameters for 1550 nm single polarization fiber

We conducted experimental study to see how the bending affects the cutoff behavior for the single polarization fiber. Two fibers were made according to the fiber design with single polarization operating window around 1060nm and 1550nm. For both fibers, around 2.5 meters fibers were taken for testing. We first tested a fiber with the single polarization window around 1060 nm with the results shown in Figure 5. We measured the fiber under the straight condition, which is defined as having as little bending as we can achieve, and then bent it at three bending diameters of 5 cm, 10 cm and 20 cm in one full loop. We repeated the measurement several times. It is found that the transmission curves that indicate the fiber cutoff behavior vary each time in the experiment sometimes with the ‘bumpy’ structure. The ‘bumpy’ curves over wavelength is common for bent single mode optical fibers, which is related to the reflection in the interfaces between two materials with sharply different refractive indices, for example between glass and air, resulting in coupling between the fundamental mode and the cladding modes[18]. In the case of the current fiber, the air holes introduce multiple material interfaces in an asymmetric fashion. The ‘bumpy’ structure is out of the same nature but illustrate itself in a more complex way because it varies with the bending orientation. The poor repeatability between different experiments is also #82237 - $15.00 USD

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understandable that the fiber orientation may vary each time we redeploy the fiber. However, within each experiment, with careful handling, it is reasonable to believe that the fiber orientations among several bending diameters remain approximately the same. We also tested the second fiber with the single polarization window around 1550 nm under the conditions of no bend, and with bend diameters of 10 and 20 cm in one full loop. The results are plotted in Figure 6. It can be found that the overall effects of the bending are to shift the cutoff curve toward lower wavelengths. In addition, we observe that with tighter bending, the spectrum in the cutoff transition regime is less steep. Both of the effects can be intuitively understood. When the fiber is in straight condition, the sharply increased loss toward the longer wavelength is due to reaching the intrinsic cutoff. Bending on the fiber brings additional loss to each polarization mode near the cutoff wavelength region. The increased loss at the wavelength region that is not lossy in straight condition makes the transition to the extinction of light less steep. In addition, because the cutoff is always measured at a fixed loss level, reaching the same loss earlier leads to the shift of the single polarization window to a shorter wavelength. We also note that both of the observations have been in good agreement with the numerical modeling results described above in Figure 3. The poor repeatability especially in tight bending regime is also consistent with the numerical modeling results as the bending effect depends on the orientation of asymmetric fiber structure relative to the fiber bending direction. The lack of control of bending the fiber in a known orientation is a limiting factor that prevents us from thoroughly examining the predictions from numerical modeling. 5. Conclusion

In summary, we have established a finite element modeling scheme that is capable of modeling the behavior of the single polarization fiber under bending condition. Through the introduction of cylindrical PML and the use of equivalent refractive index of a bent fiber, we can calculate the loss associated with each polarization mode at a given wavelength, bending diameter and fiber orientation. The modeling allows us to describe the realistic cutoff behavior. We uncovered several aspects of the bending effects such as the bending diameter and bending orientation on the fiber cutoff behavior. It is found that in general the bending has the effects of shifting the single polarization operating window toward a lower wavelength although the effects to each polarization mode and at different orientation can have more complex behaviors. In addition, at the tighter bending, the transition to cutoff is less steep, which effectively reduces the single polarization extinction ratio. We have also conducted experimental measurements on the bending property of hole-assisted single polarization fibers. We have found good qualitative agreement between the numerical modeling and the experimental measurements. It is expected that the modeling technique, which is used for the first time on single polarization fibers, can also be applicable to wider range of fibers such as photonic crystal fibers. The understanding of the bending behavior can also provide intriguing opportunities. Recently, the bending of single polarization fiber has enabled a wavelength tunable stretched-pulse ring laser [7].

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