New mode solver for anisotropic fibers with crystalline jackets

New mode solver for anisotropic fibers with crystalline jackets E. I. Golant, P. I. Lazarev∗ , Ya. B. Martynov, A. B. Pashkovskii, M. V. Paukshto and

V. B. Sulimov

OPTIVA, Inc., USA

ABSTRACT

We developed algorithms and full-vector mode solver as a tool for new optical components design that allows simulation of components with anisotropic optical characteristics. Algorithms are developed without assumption of weak guidance, which allows applications of the method and software in simulation of components with strong optical anisotropy. Method and software were tested by comparison with semi-analytical solutions for isotropic and anisotropic fibers. We observed reasonably good agreement (up to five non-zero digits in the effective indices) between calculated and analytical data for several low order modes. In this paper we present results of calculations for single polarization D-shaped and cylindrical fibers with anisotropic coating. Keywords: Finite-difference, beam propagation method, electromagnetic mode interaction, anisotropic thin films,

birefringence, D-shaped fiber, polarization maintaining fiber.

1. INTRODUCTION

In this paper we present algorithms and software based on vector finite-difference beam propagation method (FD BPM) [1]. The software allows fast and accurate calculation of the full sequence of vector modes, starting with the lowest effective index of refraction (propagation constant) mode for different optic waveguide cross-sections: circular, D-shape, rectangular or any other forms. Algorithms and solver allow for anisotropy of refraction indices as well as anisotropy of optical losses. We tested software on anisotropic step-index circular fibers, which allowed closed form analytical solutions, satisfactory accuracy of five significant digits for higher mode calculations were obtained.

New software Mode Solver for Anisotropic Waveguides – (MSAW), allows to calculate Maxwell equation eigenvectors and eigenvalues in the fibers with complex geometry (multi-layer circular fiber for example) made of different anisotropic materials.

2. MODE SOLVER FOR ANISOTROPIC WAVEGUIDES

Finite-difference algorithms that are used to simulate step-index fibers usually have poor convergence in the places of sharp refractive index variations. MSAW is free from this problem because of conservative finite-difference equations that were produced by using integro- interpolating method (IIM). IIM is based on the idea that Maxwell equations can form a set of equations on the finite-difference grid being integrated over a set of small adjacent pieces of surfaces and volumes. This procedure leads to the set of finite-difference equations having one remarkable quality: each conservation law, which is valid for the differential equations, is also strictly valid for the finite-difference equations on the grid. Hence, we can use rough difference grids, reduce computation time, and, minimize computation errors. The numerical solution obtained with IIM scheme rapidly converges to the exact solution of the differential equations with grid step reduction.

MSAW can calculate higher propagating fiber modes as well. This was achieved by the development of the special algorithm for the reference index correction. In addition, MSAW can use the solution obtained on the rough grid as the

∗ [email protected]; phone 1 650 616-7600; fax 1 650 616-7602; http://www.optivainc.com; OPTIVA, Inc., 377 Oyster Point Blvd. # 13, South San Francisco, CA 94080, USA

Optical Design and Analysis Software II, Richard C. Juergens, Editor,Proceedings of SPIE Vol. 4769 (2002) © 2002 SPIE · 0277-786X/02/$15.00162

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initial condition for the solution on denser grid, which makes computation time substantially shorter. These features were used in resonance effect investigations.

MSAW code has been tested on various fiber cross-section geometries and material parameters and has shown good accuracy in propagation constant computations. Up to five correct decimal digits can be obtained depending on the number of grid points and the size of computation window. For example, the comparison of solutions produced by MSAW with analytical expressions of Dai et al [1] is shown below in Table 1, which lists the results of effective index calculations for nine modes of two-layer fiber with 4µm core radius.

Table 1. Effective indices obtained with MSAW solver (no index) vs. those from dispersion equation [1] (index 0)

3. RESONANCE EFFECT SIMULATION IN CIRCULAR FIBERS COATED WITH ANISOTROPIC JACKET

3.1 Modal fields simulations

First we studied the resonance between modes in a tube-like waveguide formed by the jacket and one of the fundamental mode polarizations of the circular fiber lying inside the tube. Note that the refractive index of the surrounding fibers media equals to unit.

To affect only one polarization in the circular fiber one must create a single polarization state tube-like fiber. This can be achieved with coating jacket materials with strong optical anisotropy. If the anisotropic jacket with transversely oriented optical axis is used and one of the diagonal refractive index tensor components say nYY is less than or equal to the cladding refractive index (nclad) and another component say nXX larger than nclad, then all modes in the tube-like fiber will be mainly X-polarized. The resonance condition must be fulfilled when the effective refractive index of one of the tube-like fiber mode equals to the effective index of the fundamental mode of the inner circular fiber. For this design the interaction between two fibers is strong and the perturbation theory is not applicable. We used MSAW software to investigate this case. A step index isotropic fiber was coated with the uniaxial jacket with the transversely oriented optical axis. The parameters of the fiber are as follows: core radius is 2.45µm, core i sotropic index is 1.456, and clad radius is 30 µm with isotropic index 1.450. The surrounding coating is in the form of the coaxial cylinder with external

radius of 36 µm and diagonal refractive index tensor components of n YY =1.3 nZZ = 1.450 at Y and Z directions while 1.4528< nXX<1.4538 at X direction. Z-axis is supposed to be oriented along the fiber whereas X and Y-axes are in the fiber cross-section. Light wavelength was 1550 nm.

Calculations showed that effective index and field distribution of the Y-polarized main mode of the combined fiber remains undisturbed while nXX varies in the above mentioned interval (Fig. 1).

Mode neff V U neff 0

HE11 2.2859913 6.06022921 2.102053790 2.285987010

EH11 2.283910 6.06022921 4.402393929 2.283910071

HE12 2.283502 6.06022921 4.735455926 2.283487375

TE01 2.285121 6.06022921 3.264610250 2.285121196

TE02 2.282025 6.06022921 5.768150396 2.281980238

HE21 2.285061 6.06022921 3.335785465 2.285055989

EH21 2.282518 6.06022921 5.428344395 2.282508986

HE22 2.28188 6.06022921 5.871056814 2.281813756

HE31 2.283895 6.06022921 4.452613454 2.283848314

Proc. SPIE Vol. 4769 163



Figure 1: Electric field EY component of the Y-polarized circular fiber main mode

X-polarization of the main mode forms several hybrid modes with some tube-like fiber modes. The “portraits” and effective indices of these hybrid modes depend greatly on nXX. They all are mainly X-polarized, but can be separated in two groups: with non-zero and zero mean X- component (Fig.2).

(a) (b)

Proc. SPIE Vol. 4769164


(c) (d)

(e) (f)

(g) (h)

Figure 2: Portraits of the first eight tube- like fiber modes: “non zero” T1 - (a); T2 - (b); T6 – (c); T12 – (d) and “zero” T3 – (e); T4 – (f); T5 – (g); T7 – (h).



Simulations show that X- polarized main mode can form hybrid mode only either with the whole “non zero” tube-like fiber mode or with asymmetric combinations of the fragments of the “zero” modes. In resonance condition (equivalence of elementary fibers modes effective indices) X- polarized main mode and “non zero” tube- like fiber mode strongly affect each other and the interaction with “zero” tube-like fiber mode is absent. Fig. 3 shows the hybrid modes evolution with nXX variation. Two resonance points are present on Fig. 3.

n XXOM

1.4524 1.4526 1.452 8 1.4530 1.4532 1.45 34 1.4536

n e ff

1 .4504

1.4505

1.4506

1.4507

1.4508

1.4509

1.4510

1.4511

1.4512

1.4513

n XXO M

1.45300 1.45305 1.45310 1.45315 1.45320

n eff

1 .45092

1.45093

1.45094

1.45095

T 1

T 2

HE 11 Y -pola rized

HE 11+T 1

HE 11-T 1-->HE 11+T 2

HE 11-T 2

Figure 3: Hybrid modes evolution with nXX variation

First resonance take place at nXX=1.453125 where effective index of the first “non- zero” tube- like fiber mode (T1) is equal to effective index of the main circular fiber mode (HE11). Evolution of the interacting modes effective indices in the vicinity of this resonance point is shown on insight to Fig. 3. The following interesting phenomenon was discovered. To the left from the resonance point, fields of all combined fiber hybrid modes, except one, appear to be the differences between HE11 and existing “non-zero” tube-like fiber modes (we refer to them further as HE11-Ti, i = 1,2,6,12,..) (Fig. 4).



(a) (b)

Figure 4: Portraits of HE11-T1 (a) and HE11-T2 (b).

In these hybrids tube-like fiber modes dominate: the portion of HE11 is small and effective index of the hybrid almost equals to that of the corresponding tube-like fiber mode. But there is one hybrid mode, which appears to be the sum of T1 and HE11 (we shall mark this mode further as HE11+T1) (Fig. 5).

(a) (b)

Figure 5: Portrait of the whole hybrid HE11+T1 (a) and close insight on its tube- like fiber portion T1 (b). We see T1 on (a) as a weak circle.

Here HE11 dominates and the effective index of the hybrid nearly equals to that of HE11. In this region of nXX the birefringence between two modes- Y-polarized HE11 and X-polarized HE11+T1 (b+) is very small (b+~10-6) but it slowly increases as nXX approaches the resonance point. And b+ increases greatly after the resonance. Near the first resonance point, two modes: HE11+T1 and HE11-T1 transform with the biggest magnitude. As nXX increases, portion of T1 mode in hybrid HE11+T1 increases and in hybrid HE11-T1 decreases. Just after the resonance T1 dominates in the first hybrid and HE11 in the second (Fig. 6).



(a) (b)

(c) (d)



(e) (f)

Figure 6: Electric field EX component of HE11+T1 (a), (c), (e) and HE11-T1 (b), (d), (f) modes transformation as nXX pass the resonance point: nXX=1.4529 (a), (b); nXX=1.453125 (c), (d); nXX=1.4532 (e), (f).

Here the birefringence between Y-polarized HE11 and X-polarized HE11-T1 is also very small (~10-6), and it increases greatly while passing the resonance from right to left. So to the left from the resonance point we have HE11+T1 and to the right HE11-T1 hybrid modes, which are mainly X-polarized HE11 with a tiny portion of T1. The birefringence here is very small as it should be, as we try to affect the circular fiber modal field through its evanescent tail on the outer fiber surface. Only in the resonance point the birefringence between both Y-polarized HE11 - HE11+T1 and Y-polarized HE11 – HE11-T1 is comparatively large (~5 10-6). Here both hybrids differ drastically from their parent modes Fig. 6 (c), (d).

With the increase of nXX in the interval between the first and the second resonance points the hybrid mode HE11-T1 transforms gradually to HE11+T2 with dominant HE11 and tiny second “non zero” tube-like fiber mode T2. Near the second resonance point everything repeats with HE11+T2 and HE11-T2 playing first fiddle.

The increase of nXX in the region after the second resonance leads to the slow transformation of HE11-T2 to HE11+T4+ and

then to HE11+T6, where T4+ is the “non zero” mode constructed from the parts of the “zero” T4 tube-like fiber mode

(Fig.7), T6 is the third “non zero” tube-like fiber mode and HE11 the dominant.



(a) (b)

Figure 7: Portrait of the whole hybrid HE11+T4+ (a) and close insight on its tube- like fiber portion T4

+ (b). T4+ consists from the T4

fragments (fig.10 (f)) having the same sign. T4 consists from the same fragments but with opposite signs.

Third resonance between HE11+T6 and HE11-T6 is very weak with poor increase in modal birefringence and hence hardly has any practical use.

4. RESONANCE EFFECTS IN ANISOTROPIC D-SHAPED FIBERS

4.1 Verification

We simulated coating on the polished side of a D-shaped fiber with a highly anisotropic material. This type coating might improve the fiber’s polarization maintaining properties. It is especially promising to make one of the two orthogonal polarization modes radiative (leaky) and keep another mode propagating.

MSAW was tested on the isotropic D-shaped fiber numerically calculated in [2] by two different methods. We produced this calculation in order to test the ability of MSAW code to simulate modal birefringence caused not only by the material but also by the geometry of the fiber.

The refractive indices of the core and the cladding were taken as nco =1.46 and ncl =1.456, respectively. The radius was 2.0µm and the wavelength was 0.6328µm. The circularly symmetric (not D-shaped) fiber was calculated first. The calculated effective index was neff = 1.4578433575. Compared with the exact effective index neff = 1.4578423497, the difference is on the order of 10-6.

The second example deals with the same fiber but with one side polished, producing D-shaped cladding. Refractive index out of the D-side is supposed to be 1.0, and distance between core and D-interface denoted by d was chosen as d = 0.0, and d = 1.0 µm .The results of these calculations are listed in Table 2.

Table 2. Calculations of the simulated coating on the polished side of a D-shaped fiber with a highly anisotropic material and comparison with prior art

d(µm) Dominant field Electric BPM [2] SIMP[2] Our BPM Solver

0 0 1 1

Ex Ey Ex Ey

1.4574934 1.4575080 1.4577754 1.4577755

1.4574877 1.4575152 1.4577679 1.4577745

1.4574578 1.4574823 1.4577580 1.4577602



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It is seen from Table 2, that polarization sensitivity of our solver, determined by the differences in effective refractive indexes of X and Y– dominant field modes, is no less that of [2]. Differences in the absolute values may be attributed to different numerical approximations involved.

4.2 D-shaped fiber with weak core–coating coupling

Polishing of the cladding of a fiber is a usual procedure to get more access to the evanescent fields of the propagating modes in order to control them. Birefringent crystal placed on the polished surface of the fiber was proposed in [3] to fabricate fiber polarizers. Alternatively, it is now possible to coat the polished surface of the D-shaped fiber with highly birefringent and at the same time highly dichroic thin films [4].

To investigate modal birefringence in that type of fiber design, simulations were performed for a fiber having a core radius of 2.45µm and index of 1.456, a cladding radius of 50µm and index of 1.449; the wavelength of light was 1.55

µm. The polished surface of the fiber is supposed to be perpendicular to Y -axes. The distance from the core of the fiber to the polished surface was varied from zero to 10µm, so as to observe the decrease of the effective refractive index from

its maximum down to its minimum point. Two examples were simulated: the first when the polished surface coated with the anisotropic film [4] of 1.62µm thickness, and the second with fully isotropic coating of the same thickness. In the

anisotropic coating, indices were chosen as: nX = 1.449, nZ = 1.449 while nY = 1.465. Note that nY is greater than the core index of the fiber. In the isotropic coating all indices were equal to 1.449. The modal birefringence B, defined as the difference between effective indices for orthogonal line-polarization modes, was calculated for these two examples as a function of distance d from the core to the polished surface. The results are shown in Fig. 8.

Figure 8: Modal birefringence B versus distance d from the core to the polished surface of the D-shape fiber with isotropic (solid line) and anisotropic (dashed line) coatings of polished surface.

It may be seen from Fig. 8 that despite the low thickness, the anisotropic coating makes possible nearly two times greater modal birefringence than isotropic. In addition, the simulations have shown that in the anisotropic film the major component of the X-polarized mode (X-component) is about two times less than Y-component of the Y-polarized mode. That opens the way of gaining extinction rate by the use of anisotropic losses reported in the films [5]. To investigate this possibility, calculations of absorption length as a function of distance d from the core to the polished surface were carried out and the results are shown in Fig. 9. The imaginary parts of anisotropic indices of the film were taken from [5] as kX = 0.029, kY = 0.172 and kZ = 0.286. In our configuration, it is kY that corresponds to power absorption at the direction perpendicular to the polished surface of the fiber. Simulations showed that kY most strongly affects absorption in that design.

d mm

0 2 4 6 8 10

Mod

al b

iref

ringe

nce

B

0e+0

1e-4

2e-4

3e-4

4e-4

5e-4

6e-4



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Figure 9: Absorption length versus distance d from the core to the polished surface of the D-shape fiber with anisotropic coating: for Y-polarized mode (solid line) and X-polarized mode (dashed line).

These results demonstrate that combination of anisotropic refractive indices with anisotropic losses produces polarization sensitive devices.

4.4. D-shape fiber with strong core–coating coupling

In a previous section, a D-shape fiber with a relatively weak core–coating coupling was investigated. In this section, calculations were aimed at the search for highly birefringent design in fibers with strong core–coating coupling. The simulations have led us to a fiber having a core radius of 4.45µm and index of 1.456, a cladding radius of 50µm and

index of 1.449; the wavelength of light was 1.55 µm. The polished surface of the fiber is supposed to be perpendicular toY-axes and have a coating of 3.24µm thickness. The distance from the core of the fiber to the polished surface was equal

to zero, so as to observe the maximum geometrical birefringence. In the coating, indices were chosen as: nX = 1.449, nZ = 1.449 while nY was varied from 1.449 (isotropic case) to 1.470. The results are shown in Fig. 10.

It may be seen from Fig.10 that the anisotropic coating makes possible enormously high modal birefringence up to 1.4 10-3. In addition, the simulations have shown that there is a critical point at nY = 1.4655 where birefringence reaches its maximum and where X- polarized mode is gone. The horizontal line in Fig.10, marked with rectangles, characterizes the X-polarized mode propagating in the core and being, because of its polarization, practically independent of nY. The curve marked with circles represents the orthogonal Y-polarized mode that concentrates mainly in the core, though it expands to the coating, as nY increases. In the latter case, the curve becomes asymptotic to the line marked with triangles that represents the lowest mode propagating in rectangular waveguide formed by thin film coating. It is seen from Fig.10, that as nY increases all but one of the modes become strongly coupled to the coating to leave the only X-polarized mode uncoupled and that makes the design very attractive for many polarization-sensitive applications

d , mm

0 2 4 6 8 10

Abs

orp

tion

len

gth,

m

0 .00

0 .02

0 .04

0 .06

0 .08

0 .10

0 .12

0 .14

0 .16



nY

1.445 1.450 1.455 1.460 1.465 1.470 1.475 1.480

nef

f

1.452

1.453

1.454

1.455

1.456

1.457

1.458

1.459

1.460

Y-polarization

X-polarization

Figure 10: Effective refractive index neff versus tensor refractive index component nY of the anisotropic coating perpendicular the to

the polished surface of the D-shape fiber.

5. CONCLUSIONS

MSAW was tested by comparison with analytical solution. Test results showed 5 digits accuracy of MSAW calculations. The software might be successfully used for higher mode simulation. MSAW is not limited to any particular geometry of waveguide and can be used for wide variety of applications.

6. REFERENCES

1. J.D.Dai and C.K.Jen, “Analysis of cladded uniaxial single-crystal fibers,” J.Opt.Soc.Am.A, 8, pp. 2021-2025, 1991. 2. Y.Hsueh, M.Yang, and H.Chang “Three-Dimensional Noniterative Full-Vectorial Beam Propagating Method Based

on the Alternating Direction Implicit Method”, J. of Light Wave Tech. 17, pp. 2389-2397, 1999. 3. R.A.Bergh, H.C.Lefevre, and H.J.Shaw “Single-mode fiber-optic polarizer,” Optics Letters. 5, pp. 479-481, 1980. 4. Y.Bobrov, C.Cobb, P.Lazarev, P.Bos, D.Bryand and H. Wonderrly, “Lyotropic Thin Film Polarizers,” Proc. SID,

Int. Symp. Digest of Technical Papers, Long Beach, California, May 16-18, XXXI, pp. 1102-1107, 2000. 5. Y. Bobrov and V.Novak, “Determination of anisotropic complex refractive indices of thin film E-polarizers,” Mol.

Materials 14, pp. 21-31, 2001.







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New mode solver for anisotropic fibers with crystalline jackets

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