Dispersion characteristics of complex refractive-index planar slab optical waveguide by using finite element method

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Dispersion characteristics of complex refractive-index planar slab optical waveguide by using finite element method Sanjeev Kumar Raghuwanshi a , Santosh Kumar b,∗ , Ajay Kumar a a b

Photonics Laboratory, Department of Electronics Engineering, Indian School of Mines, Dhanbad 826004, Jharkhand, India Department of Electronics & Communication Engineering, DIT University, Dehra Dun 248 009, Uttarakhand, India

a r t i c l e

i n f o

Article history: Received 25 October 2013 Accepted 19 May 2014 Available online xxx Keywords: Chirp-type’s refractive index profile Waveguide dispersion Group delay

a b s t r a c t In this paper, finite element method (FEM) mode analyses of planar slab optical waveguide having complicated refractive index profile are presented. We try to estimate the dispersion graph, mode cut-off condition, group delay and waveguide dispersion for the case of ␣-power and chirped-type refractive index profile. In order to obtain the more accurate result, we have derived the higher-order polynomial, which establishes the suitable relationship between b and V for different profile of optical waveguide. On the basis of the derived polynomials, the waveguide dispersion is analyzed for different type of refractive index profile waveguide. Our study shows that the waveguide dispersion can be substantially reduced when we deployed the optical waveguide having linearly chirped-type refractive index profile. Earlier too, the arbitrary refractive index profile has been analyzed but to the best of our knowledge chirped-type refractive index profile has not been analyzed till date for the case of planar slab optical waveguide. © 2014 Elsevier GmbH. All rights reserved.

1. Introduction In the variational method, the boundary-value problem is transformed into the equivalent variational problem and is solved by applying the variational principle [1–4]. In classical analytical procedures without subdivision processes, the system is modeled using analytical functions defined over the whole region of interest, and therefore these procedures generally are applicable only to simple geometries and materials. One of the simplest methods that employ the discretization procedure is the finite-difference method (FDM) [5–8]. In FDM, the domain is discretized into small lattice regions using regular rectangular grids. However, a rectangular grid is not suitable for curved boundaries or interfaces, because they intersect gridlines obliquely at points other than the nodes. Moreover, a regular grid is not suitable for problems with very steep variations of fields. In the finite-element method (FEM), the domain of the problem is discretized into smaller elements [9–12]. The solution of the problem is approximated in each element and it is connected at the nodal points to form the solution model in the entire analysis domain. In FEM, a simple form of the function is adopted to approximate the field in each element. So, FEM is suitable for the mode analysis of optical waveguides

∗ Corresponding author. Tel.: +91 7060194847. E-mail address: [email protected] (S. Kumar).

having arbitrary refractive-index profiles and complicated waveguide structures. Needless to say, FEM is also applicable to the stress analysis of optical waveguides. In this paper, an FEM waveguide analysis for slab waveguides is described. Waveguide analysis and stress analysis are totally different problems. However, the discretization procedures and the formulation of the functional are rather similar in both problems. Such versatility in the mathematical procedures is a great advantage of FEM.

2. Finite-element method analyses of inhomogeneous planar slab waveguides In this section, the formulation of FEM is described, taking the TE mode as an example in the inhomogeneous slab waveguide. As shown in Fig. 1, the transverse region 0 ≤ x ≤ A, which has an inhomogeneous refractive-index profile, is denoted as the core, and the refractive indices in the cladding and the substrate are assumed to be constant. The maximum refractive index in the core is n1 , and the refractive indices in the cladding and the substrate are n0 and ns , respectively. Here we assume ns ≥ n0 . The wave equation for the TE mode is given by [6–8] d2 Ey dx2

+ {k2 n(x)2 − ˇ2 }Ey = 0.

(1)

http://dx.doi.org/10.1016/j.ijleo.2014.05.049 0030-4026/© 2014 Elsevier GmbH. All rights reserved.

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Fig. 1. Refractive-index distribution of the asymmetric inhomogeneous slab optical waveguide.

Fig. 3. Cutoff normalized frequency c of slab waveguides with ␣-power refractiveindex profiles.

The wave equation and boundary condition are then rewritten as d2 R + [v2 q() − w2 ]R = 0, d2

(2)

dR() are continuous at  = 0 and  = D. d

R() and

(3)

Here, normalized transverse wave number w, normalized frequency  and normalized refractive-index distribution q() are defined by



w=a

 = ka

shows the schematic of the ␣-power refractive-index profiles given by [11,12,16]

ˇ2 − k2 n2s ,



Fig. 4. Normalized propagation constant (b) of slab waveguides with ␣-power refractive-index profiles.

n21 − n2s ,

(4)

2

n(x) − n2s n21 − n2s

q() =

n2 (x) =

The solution of the wave equation (2) under the constraints of the boundary condition equation (3) is obtained as the solution of the variational problem that satisfies the stationary condition of the functional.

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

n2s ,

x < 0,



˛

x

n21 − (n21 − n2s ) − 1 , 0 ≤ x ≤ 2a, a n2s ,

(6)

x > 2a.

When the eigenvalue ˇ of matrix C is obtained, as shown in Appendix A, the corresponding eigenvector R0 , R1 , . . ., RN is calculated by the matrix operation. Actually, however, the relative value of Ri (i = 1 − N) with respect, for example, to R0 ( = / 0) is obtained and R0 is still unknown. Here R0 can be determined when we normalized the total optical power P carried by the mode to 1. Fig. 2

Next, we show the results of FEM analyses for TE modes in the ␣-power refractive-index profiles given by Eq. (6). Here 2a(= A) is a full core width and the number of divisions in the core is N = 100. The step-index slab waveguide is also analyzed by setting ˛ =∞ in Eq. (6). Figs. 3–5 show the cutoff normalized frequency c , the normalized propagation constant b and the propagation constant in terms of ˇ/k for the TE0 mode of slab waveguides with ␣-power refractive-index profiles, respectively. Here we assumed n1 = 3.5 and ns = 3.17. It is known through literature that the cutoff normalized frequency for the TE1 mode in a stepindex slab waveguide is given by c0 = /2 [17–19]. Fig. 6 shows the dependencies of the numerical error (c − c0 )/c0 × 100 (%) for the cutoff normalized frequency with respect to the number of core divisions N in the FEM analysis. It is apparent that the numerical error becomes less than 2% for N > 30 [13–14]. When the propagation constant ˇ is obtained, we can calculate the waveguide dispersion characteristics for a waveguide having

Fig. 2. Schematic of ␣-power refractive index profiles.

Fig. 5. Propagation constant in terms of ˇ/k as a function of V for slab waveguides with ␣-power refractive-index profiles.

I[R] = −

∞ 

−∞

dR d

2

∞ d +

[v2 q() − w2 ]R2 d

(5)

−∞

3. Dispersion characteristics of graded/linearly chirp types refractive-index profile of planar slab waveguides

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Fig. 6. Percentage error for the cut-off normalized frequency with respect to the number of core divisions N in FEM analysis.

3

2

Fig. 8. Normalized waveguide dispersion parameter v(d (vb)/dv2 ) of slab waveguide with ␣-power refractive index profiles.

calculate the waveguide dispersion by using the following expression [15]: an arbitrary refractive index profile. When we obtain d(vb)/dv and v(d2 (vb)/dv2 ) for an arbitrary refractive index profiles, we can

DW = −

n1 − ns d2 (vb) v c dv2

(7)

Figs. 7 and 8 show the normalized delay d(vb)/dv and the normalized waveguide dispersion v(d2 (vb)/dv2 ), respectively. Now we define a generalized linear chirp-type refractive index profile which is more complicated than previous case and is defined as [19]

n (x) =

⎧



  x  x x 2 





⎪ , ⎪ ⎨ n1 − (n1 − ns ), 1 + a − 1 − 1 exp −˛ a − 1 cos 2Nc a − 1 ⎪ ⎪ ⎩ n1 (1 − ),

Fig. 7. Normalized delay d(vb)/dv of slab waveguide with ␣-power refractive index profiles.

r ≤ a,

(8)

r > a, where n1 is the refractive index at the center of the waveguide at x = a, ˛ controls the decay or growth of the profile envelope, Nc is the number of cycles in a core radius, a is the core radius and ns is the cladding or substrate refractive index. It can be noted that the parameters, which define the index profile, can be divided in two parts. One, the fiber parameters like a, n1 , ns and other, and the profile parameters like ˛ and Nc1 . By varying these parameters (a, ns , n1 , Nc , ˛), one can generate profiles from simple-step index-type to complex multiple cladding type as shown in Fig. 9. For example, the profile parameters are Nc = 0 and ˛ = 0, respectively, for the step index profile. Figs. 10–12 show the normalized propagation constant b, propagation constant in terms of ˇ/k and normalized group delay d(vb)/dv for the TE01 mode of slab waveguides with linear chirp

Fig. 9. A variety of linearly chirp-type refractive index profiles generated by varying refractive-index profile parameters.

Please cite this article in press as: S.K. Raghuwanshi, et al., Dispersion characteristics of complex refractive-index planar slab optical waveguide by using finite element method, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.05.049

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ARTICLE IN PRESS S.K. Raghuwanshi et al. / Optik xxx (2014) xxx–xxx Table 1 The polynomial equation derived for Fig. 4. The value of ˛ 1

Fig. 10. Normalized propagation constant (b) of slab waveguides with linear chirp types of refractive index profiles.

2

4

8

Fig. 11. Propagation constant in terms of ˇ/k as a function of V for slab waveguides with linear chirp types of refractive index profiles.

types of refractive-index profiles. The waveguide dispersion can be computed straight forward from Fig. 12 and Eq. (7). It is apparent from the above figures that the waveguide dispersion can be substantially reduced when we deployed the optical waveguide having complicated refractive index profile. However, in order to obtain the more accurate results, we have derived the higher-order polynomial equations, which relates the b and V. Using the described polynomial equations, we can get the similar result and more accurate response of the waveguide dispersion. The polynomial equations, which describe the relation between b and V represented in Fig. 4, can be shown by Table 1. On the basis of the polynomial, we can obtain the same result represented in Figs. 4, 7 and 8 as shown in Fig. 13. Fig. 13 shows that we can able to obtain the similar results using the derived polynomials represented in Table 2. Fig. 13a shows the b–V relation similar to Fig. 4. On the basis of this, we can calculate the response for d(bV)/dV, V(d2 (bV)/dV2 ) represented in Fig. 13b and c, respectively. Finally, we can calculate the waveguide dispersion, represented in Fig. 13d. Hence, this shows that we can consider the derived polynomial as an appropriate mathematical equation for the calculation of the waveguide dispersion. In the same manner, we can show the higher-order polynomial equations, which shows the similar result equivalent to Figs. 10 and 12. Table 2 shows the derived polynomial equation.

100

(n1 −ns ) c

2

v dd(vv2b)

b = P1 V10 + P2 V9 + P3 V8 + P4 V7 + P5 V6 + P6 V5 + P7 V4 + P8 V3 + P9 V2 + P10 V + P11 , where P1 = −4.9542 × 10−5 , P2 = 0.0010788, P3 = −0.010093, P4 = 0.052969, P5 = −0.16987, P6 = 0.33443 P7 = −0.36011, P8 = 0.073745, P9 = 0.26295 P10 = −0.012483, P11 = 0.00096925 b = P1 V10 + P2 V9 + P3 V8 + P4 V7 + P5 V6 + P6 V5 + P7 V4 + P8 V3 + P9 V2 + P10 V + P11 , where P1 = −6.5927 × 10−5 , P2 = 0.0014693, P3 = −0.014014 P4 = 0.07431, P5 = −0.23624, P6 = 0.44139 P7 = −0.38911, P8 = −0.11718, P9 = 0.53152 P10 = −0.02195, P11 = 0.0010291 b = P1 V10 + P2 V9 + P3 V8 + P4 V7 + P5 V6 + P6 V5 + P7 V4 + P8 V3 2 + P9 V + P10 V + P11 , where P1 = −0.00015784, P2 = 0.0032833, P3 = −0.029176 P4 = 0.14353, P5 = −0.4196, P6 = 0.70598 P7 = −0.51238, P8 = −0.30209, P9 = 0.78515 P10 = −0.02195, P11 = 0.0010127 b = P1 V10 + P2 V9 + P3 V8 + P4 V7 + P5 V6 + P6 V5 + P7 V4 + P8 V3 + P9 V2 + P10 V + P11 , where P1 = −0.00021655, P2 = 0.0044599, P3 = −0.039111 P4 = 0.18889, P5 = −0.5374, P6 = 0.86311 P7 = −0.54764, P8 = −0.4817, P9 = 0.97593 P10 = −0.028173, P11 = 0.0010389 b = P1 V10 + P2 V9 + P3 V8 + P4 V7 + P5 V6 + P6 V5 + P7 V4 + P8 V3 + P9 V2 + P10 V + P11 , where P1 = −0.00021682, P2 = 0.0043925, P3 = −0.037526 P4 = 0.17353, P5 = −0.45511, P6 = 0.59589 P7 = −0.011264, P8 = −1.1172, P9 = 1.3425 P10 = −0.048549, P11 = 0.001316

Table 2 The polynomial equation derived for the Fig. 11. The Parameter value (Nc , ˛) Nc = 2.5, ˛=0

Nc = 2.5, ˛ = 2.8

Nc = 2.5, ˛ = 1.5

Nc = 2.5, ˛ = 0.8

Fig. 12. Normalized delay d(vb)/dv of slab waveguide with linear chirp types of refractive index profiles.

Derived polynomial (b Vs V), waveguide dispersion (DW ) = −

Derived Polynomial (b Vs V) waveguide dispersion (DW ) = −

n1 −ns c

2

v dd(vv2b)

b = P1 V10 + P2 V9 + P3 V8 + P4 V7 + P5 V6 + P6 V5 + P7 V4 + P8 V3 + P9 V2 + P10 V + P11 , where P1 = 1.093 × 10−9 , P2 = −7.1591 × 10−8 P3 = 1.9731 × 10−6 , P4 = −2.945 × 10−5 P5 = 0.00025341, P6 = −0.0012195 P7 = 0.0029216, P8 = −0.0049003 P9 = 0.02782, P10 = −0.0046656, P11 = 0.00081357 b = P1 V10 + P2 V9 + P3 V8 + P4 V7 + P5 V6 + P6 V5 + P7 V4 + P8 V3 2 + P9 V + P10 V + P11 , where P1 = 9.3895 × 10−11 , P2 = −7.1824 × 10−9 P3 = 2.1904 × 10−7 , P4 = −3.3851 × 10−6 P5 = 2.6863 × 10−5 , P6 = −8.7342 × 10−5 P7 = 2.4894 × 10−5 , P8 = −0.0019229 P9 = 0.022921, P10 = −0.0046392 P11 = 0.00083409 b = P1 V10 + P2 V9 + P3 V8 + P4 V7 + P5 V6 + P6 V5 + P7 V4 + P8 V3 + P9 V2 + P10 V + P11 , where P1 = 1.5045 × 10−9 , P2 = −9.39 × 10−8 P3 = 2.4107 × 10−6 , P4 = −3.2135 × 10−5 P5 = 0.00022406, P6 = −0.00061224 P7 = −0.0010989, P8 = 0.0054249 P9 = 0.026597, P10 = 0.0012403 P11 = 0.00016156 b = P1 V10 + P2 V9 + P3 V8 + P4 V7 + P5 V6 + P6 V5 + P7 V4 + P8 V3 + P9 V2 + P10 V + P11 , where P1 = 5.7037 × 10−10 , P2 = −3.5936 × 10−8 P3 = 9.3515 × 10−7 , P4 = −1.2701 × 10−5 P5 = 9.1019 × 10−5 , P6 = −0.00026699 P7 = −0.00026845, P8 = 0.00030811 P9 = 0.024313, P10 = −0.0028788 P11 = 0.0006529

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Fig. 13. Simulation result obtained by the derived polynomial represented in Table 1: (a) b–V relation; (b) d(bV)/dV–V relation; (c) V(d2 (bV)/dV2 )–V; (d) waveguide dispersion for different value of ˛.

Fig. 14. Simulation result obtained by the derived polynomial represented in Table 2: (a) b–V relation; (b) d(bV)/dV–V relation; (c) V(d2 (bV)/dV2 )–V; (d) waveguide dispersion for different value of Nc , ˛.

On the basis of the derived polynomial equations, the waveguide dispersion plot can be represented as shown in Fig. 14d. It is apparent while comparing Fig. 13d with Fig. 14d that in the case of complicated chirped-type profile case, the waveguide dispersion

is negative. Earlier also, the profile as shown in Fig. 9c has been analyzed in greater detail for the case of circular core fiber [19]. In literature [19], it has been shown, ±2 ps/km-nm waveguide dispersion over the band, but in our case the waveguide dispersion is

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much smaller as shown in Fig. 14d. From Fig. 14d, it is also apparent that the optimum parameter for lower waveguide dispersion is Nc = 2.5, ˛ = 1.5, corresponds to Fig. 9c.

−

1 D ∂I = RN−1 2 N ∂RN

4. Conclusion

−1 − (qN−1 + qN )

+ RN

In the conclusion of the paper, we have characterized the dispersion property of linearly chirp types of refractiveindex profile. The accuracy of FEM has been tested with respect to the number of core division. We have achieved very good agreement with the previously published results. It has been demonstrated that the numerical error becomes less than 2% for the number of the core divisions in FEM analyses ≥30. This study will be useful in optical communication systems where low dispersion link has to be deployed.

+

w2 3

1 − (qN−1 + 3qN )

 D 2 N

+w

D N

2

 D 2

12

N

+

w2 6

12



N

= 0.

R () =

⎪ ⎪ ⎪ ⎩

N 

Ri ∅i (),

(A1)

0 ≤  ≤ D,

RN exp[−w( − D)],

 dR 2 d

2 [2 q() − w2 ]R2 d − wRN .

0

(A2)

0

The stationary condition of the functional (A2) is given, by partial differentiation with respect to Ri (i = 0 − N) as 1 ∂I 0= = −w0 R0 ıi.0 − 2 ∂Ri

D

dR d∅i d + 2 d d

D

0

q()R∅i d

R∅i d − wRN ıi.N ,

i = 0, . . ., N.

(A3)

Noticing that the sampling function ∅i () is zero outside the region i−1 ≤  ≤ i+1 , Eq. (A2) can be rewritten as 1 D ∂I = R0 2 N ∂R0

1 − (3q0 + q1 )

2

 D 2

12

N

+R1

−1 − (q0 + q1 )

  2 D 2 N

12

w2 3

+

 D 2 N

w2 6

+

+ w0

 D 2  N

D N



= 0, (A3)

−

1 D ∂I = Ri−1 2 N ∂Ri

−1 − (qi−1 + qi )

2

 D 2

12

N

+ Ri

2 − (qi−1 + 6qi + qi+1 )

+ Ri+1

−1 − (qi + qi+1 )

i = 1, . . ., N − 1,

ı2 +

ı2 +

2 12

2w2 2 ı , 3

ı2 +

2 12

w2 2 ı + w0 ı, 3

w2 2 ı , 6

ı2 +

i = 1, . . ., N − 1,

(A7) i = 0, . . ., N − 1,

w2 2 ı + wı, 3

where discretization step ı is given by ı=

D N

(A8)

Eq. (A6) is a dispersion equation (eigenvalue equation) for the TE modes in the arbitrary refractive-index profiles. When the refractive-index distribution q() of the waveguide and the normalized frequency  are given, the propagation constant ˇ (implicitly contained in w and w0 ) is calculated from Eqs. (A6) and (A7). When we set w = 0 (ˇ = kns ) in Eq. (A7), the solution of Eq. (A6) gives the cutoff -value of the waveguide. The extension of the results for the TM mode is straightforward. References

0

−

12

0

D −w

12

ci,i+1 = ci+1,i = −1 − (qi + qi+1 )

 > D,

D d +

2

2

cN,N = 1 − (qN−1 + 3qN )

where R0 − RN are field values at the sampling points. In order to deal with the waveguides having arbitrary inhomogeneous refractive-index profiles. Substituting Eq. (A1) into Eq. (5), we obtain the functional

D 

c0.0 = 1 − (3q0 + q1 ) ci,i = 2 − (qi−1 + 6qi + qi+1 )

 < 0,

i=0

2

(A6)

The matrix elements of C are given by

R0 exp(w0 ),

I = −w0 R02 −

(A5)

Eqs. (A3)–(A5) are a set of (N + 1)th-order simultaneous equations having R0 − RN as the unknown values. In order that Eqs. (A3)–(A5) have nontrival solutions expect for R0 = R1 = · · · = RN = 0, the determinant of the matrix C should be

In order to simplify calculation of the functional, the field profile in the core is discretized and expressed as [9,10]

⎧ ⎪ ⎪ ⎪ ⎨

N

  2 D 2

det(C) = 0 Appendix A.

 D 2 

 D 2 

w2 6

+

N

2

 D 2

12

N

2

 D 2

12

N

+

2w2 + 3

 D 2  N

   w2 D 2 6

N

= 0, (A4)

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