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Chromatic dispersion behavior of Si-NC–Er doped optical fiber
Optics Communications 281 (2008) 4530–4535
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Chromatic dispersion behavior of Si-NC–Er doped optical fiber A. Salman Ogli *, A. Rostami Photonics and Nanocrystal Research Laboratory (PNRL), Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz 51664, Iran
a r t i c l e
i n f o
Article history: Received 15 January 2008 Received in revised form 29 April 2008 Accepted 29 April 2008
Keywords: Chromatic dispersion Si-NC (Silicon nanocrystal) Optical amplifier
a b s t r a c t The chromatic dispersion for conventional and Er-doped fibers using the refractive index approximation is examined. A first, analytical method for investigation of dispersion in step index triple clad optical fiber is used. To design of zero-dispersion shifted fiber for optical communication purpose manipulation of the refractive index and radius of the core are considered. We show that in presence of the Si-NC–Er ions, zero-dispersion wavelength is displaced and the dispersion quantity is increased. In this work, we try to optimize system parameters to obtain minimum dispersion and dispersion shifted fiber with control of the doping levels of Er ions and Si-NC as well as doping profiles. For especial case, we assumed the Gaussian inhomogeneous core refractive index for zero-dispersion wavelength and dispersion managements. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction High-speed data communication and processing are basic industrial and scientific demand recently. Optical method is one of best alternatives for doing these tasks. Main physical medium for optical communication and processing is optical fiber [1–5]. Transmission bit rate in optical communication is higher than other data transmission methods. But, this is still far from the fundamental limit to the information transfer rate, and future systems are expected to reach data transfer rates of several terabits per second in a single mode optical fiber for transmission over large distance the optical signal needs to be amplified at regular intervals in order to maintain sufficient light intensity. For loss compensation in optical communications, we need optical amplifiers which there are some alternatives such as semiconductor optical amplifiers (SOA) and erbium doped fiber amplifier (EDFA) [6–9]. Another parameter for investigation of quality of the fiber optic is chromatics dispersion [10]. In this work, we focused on the chromatic dispersion in conventional (step index and triple clad (W profile index)) and Si-NC–Er doped optical fibers (homogenous and inhomogeneous). The characteristics of the chromatic dispersion are discussed in detail in this paper. Since by adjusting fiber parameters the chromatic dispersion in traditional fiber (triple clad step index) can easily be optimized, but in Si-NC–Er doped fiber amplifier (inhomogeneous distribution) by varying the essential fiber parameters, manipulation of the dispersion characteristics are hard. Finally, we analyze the optical amplification process and dispersion characteristics of Si-NC–Er doped fiber.
The organization of the paper is as follows: In Sections 2 and 3 theoretical and mathematical backgrounds for description of modal behavior of the fiber and chromatic dispersion are presented. Simulation results are presented and discussed in Section 4. Finally the paper ends with a short conclusion in Section 5. 2. Theoretical background First, we suppose the refractive index profile for traditional fiber is shown in Fig. 1 and the refractive index of each layer are given as follows:
nðrÞ ¼
n1 ; 0 < r < Rcore ; n2 ; Rcore < r < Rclad ;
where n1, n2 and r are refractive index for core and cladding and the radius of fiber, respectively. According to the wave equation and the boundary condition of electromagnetic field and under the LP approximation the characteristics equations in one region can be obtained as follows:
det
jm ðu1 Þ
0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.04.073
km ðu2 Þ;
u1 jm ðu1 Þ u2 km ðu2 Þ;
¼0
ð2Þ
where Jm and Km are the Bessel and modified Bessel functions, respectively. The parameters used in determinant are defined as follows:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk02 n21 Þ b2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 ¼ Rcore b2 ðk02 n22 Þ;
u1 ¼ Rcore * Corresponding author. Tel./fax: +98 411 339 3724. E-mail address: [email protected] (A. Salman Ogli).
ð1Þ
4531
A. Salman Ogli, A. Rostami / Optics Communications 281 (2008) 4530–4535
Fig. 1. Refractive index profile for conventional fiber.
Fig. 2. The Gaussian refractive index approximation of Si-NC–Er doped fiber.
where b is the longitudinal propagation constant of guided mode and k0 is the wave number in the vacuum. Finally, when Si-Nc and Er ions are used as dopants, the refractive index profile is changed and shown in Fig. 2. In this work, we suppose that the Er ions and Si-NCs have the Gaussian distribution. In this strategy, we approximate the Gaussian profile with step index format. Then the approximated refractive index in presence of the wave equation is solved analytically or numerically (see Fig. 3). The refractive index of each layer is shown as follows:
8 n10 ; > > > < n11 ; nðrÞ ¼ > n > 1; > : n2 ;
0 < r < R10 ; R10 < r < R20 ;
ð3Þ
R20 < r < Rcore ; Rcore < r < Rclad ;
According to the wave equation and the boundary condition of the electromagnetic fields and under the LP approximation the characteristic equation in three regions can be obtain as follows:
0
n11 < neff < n10
B0 B B B0 det B B u J ðu Þ B 1m 1 B @0
km ðw2 Þ
0
0
0
im ðw22 Þ
km ðw22 Þ
im ðw3 Þ
km ðw3 Þ
0
0
0
im ðw33 Þ
km ðw33 Þ
km ðw4 Þ
0
0
w2 Im ðw2 Þ w2 K m ðw2 Þ 0 w22 Im ðw22 Þ w22 K m ðw22 Þ
0 jm ðu1 Þ
0 jm ðu2 Þ
0
0 jm ðu2 Þ
0 ym ðu2 Þ
w3 Im ðw3 Þ w3 K m ðw3 Þ 0
1 C C C C C¼0 C C C A
0
w33 Im ðw33 Þ w33 K m ðw33 Þ w4 K m ðw4 Þ 1 0 0 0 C B0 jm ðu22 Þ ym ðu22 Þ jm ðu3 Þ ym ðu3 Þ 0 C B C B C B0 0 0 jm ðu33 Þ ym ðu33 Þ km ðw4 Þ C ¼ 0; B det B C 0 0 C B u1 J m ðu1 Þ u2 J m ðu2 Þ u2 Y m ðu2 Þ 0 C B A @0 u22 J m ðu22 Þ u22 Y m ðu22 Þ u3 J m ðu3 Þ u3 Y m ðu3 Þ 0 0
n2 < neff < n1
0 jm ðu1 Þ
im ðw2 Þ
w33 Im ðw33 Þ w33 K m ðw33 Þ w4 K m ðw4 Þ 1 0 0 C B0 jm ðu22 Þ ym ðu22 Þ im ðw3 Þ km ðw3 Þ 0 C B C B C B0 0 0 im ðw33 Þ km ðw33 Þ km ðw4 Þ C¼0 det B C B u J ðu Þ u J ðu Þ u Y ðu Þ 0 0 0 2 m 2 2 m 2 C B 1m 1 C B A @0 u22 J m ðu22 Þ u22 Y m ðu22 Þ w3 Im ðw3 Þ w3 K m ðw3 Þ 0 0
n1 < neff < n11
jm ðu1 Þ
Fig. 3. Approximation of the Gaussian profile of Si-NC–Er doped fiber.
0
0 ym ðu2 Þ
0
u33 J m ðu33 Þ u33 Y m ðu33 Þ w4 K m ðw4 Þ
ð4Þ
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A. Salman Ogli, A. Rostami / Optics Communications 281 (2008) 4530–4535
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n210 n22 ;
where jm, ym, km, Im are the Bessel and modified Bessel functions. The parameters used in determinant are defined as follows:
v ¼ k0 Rcore
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk02 n210 Þ b2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 ¼ R10 ðk02 n211 Þ b2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w2 ¼ R10 b2 ðk02 n211 Þ; u22 ¼ R20 ðk02 n211 Þ b2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w22 ¼ R20 b2 ðk02 n211 Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u3 ¼ R20 ðk02 n21 Þ b2 ; w3 ¼ R20 b2 ðk02 n21 Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u33 ¼ Rcore ðk02 n21 Þ b2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w33 ¼ Rcore b2 ðk02 n21 Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w4 ¼ Rcore b2 ðk02 n22 Þ;
and the normalized propagation constant is given as
u1 ¼ R10
2
B¼
ðb2 =k0 Þ n22 : n210 n22
The optical parameters also, defined as
D¼
ð5Þ
In this stage, we suppose that the refractive index profile for W-type fiber is shown in Fig. 4. Also, Fig. 5 shows homogeneously Si-Nc and Er ions doped fiber refractive index profile. With calculation of neff for different region of fiber, we can obtain the characteristic equation and dispersion coefficient. In this simulation the normalized frequency is defined as
n210 n22 : 2n22
In this section Si-NC–Er doped fibers from refractive index distribution was modeled and using the boundary conditions modal analysis including Eigen values and vectors were done. In the next section dispersion properties of the mentioned fibers are extracted. Effect of different parameters on dispersion property is considered and we try to optimize optical and geometrical parameters to obtain a given dispersion profile. 3. Mathematical analysis of chromatic dispersion According to standard definition and relation of chromatic dispersion, in the following chromatic dispersion coefficient is calculated. 2
N2 D d ðBvÞ ; v dv2 c k 2 k d n2 dðBvÞ ð1 þ D Þ; Material dispersion ¼ c dk2 dv
Waveguide dispersion ¼
ð6Þ ð7Þ
In the following for simplicity some relations are given for calculation of mentioned above equations.
u dðBvÞ u1 dðu1 Þ 1 ¼1þ ; 12 dv v v dv 2 2 2 d ðBvÞ dðu1 Þ u1 d ðu1 Þ ¼ 2u1 ; v 2 dv dv dv2 v
ð8Þ ð9Þ
2 In above statement N 2 ¼ n2 k dn is the group index of the outer dk cladding layer [10]. In the next section some simulated results are given and discussed.
4. Results and discussion Fig. 4. Triple clad profile traditional fiber.
In this section the simulated results are categorized to four groups. In each group effect of especial parameters on dispersion coefficient are considered. 4.1. Influences of DRcore,Dn on dispersion coefficient in conventional step index fiber
Fig. 5. Profile of homogenously doped of Si-Nc and Er ions.
According to numerical calculation of dispersion coefficient, the obtained D–k curve is shown in Figs. 6–8. In Fig. 6, it is shown that zero-dispersion wavelength can obtain for specified parameters of the fiber (Rcore = 2.5 lm, n1 = 1.5, n2 = 1.4953). Considering Fig. 7, it is observed that with increasing differential refractive index between core and clad the dispersion coefficient is increased. It is shown that with increasing of differential refractive index, the waveguide dispersion coefficient is increased because of the increasing of difference between core and clad refractive index. So, total dispersion is increased but flattened dispersion curve can be obtained. Effect of DRcore (varying of core radius) on the dispersion coefficient is illustrated in Fig. 8. With varying of core radius the amount of dispersion quantity is increased but with increase of core radius, slope of dispersion coefficient is decreased too. For explaining of
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40
20 Material Dispersion
30
Waveguide Dispersion Total Dispersion
10
20 0
Dispersion
Dispersion
10 0 -10
Rcore>2.5 Micrometer
-10
-20
Rcore=2.5 Micrometer
-20 -30
-30 -40 1.1
Rcore<2.5 Micrometer
1.2
1.3
1.4
1.5
1.6
1.7
1.8
-40 1.1
Wavelength
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Wavelength Fig. 6. Dispersion vs. wavelength (lm) Rcore = 2.5 lm, n1 = 1.5, n2 = 1.4953. Fig. 8. Dispersion vs. wavelength (lm) n1 = 1.5, n2 = 1.4953.
20
10 5
10
Delta r=0.65 Delta r=0.5 Delta r=0.2 Delta r=0.02
0
0
Dispersion
Dispersion
-5 -10 -15
-10
-20
-20
-30
-25 Dn1=4.69e-3 Dn2=5.5e-3 Dn3=6.49e-3 Dn1=8.49e-3
-30 -35 1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Wavelength Fig. 7. Dispersion vs. wavelength (lm) Rcore = 2.5 lm, Dn = n1 n2.
-40
-50 1.2
1.3
1.4
Wavelength
1.5
1.6
1.7 -6
x 10
Fig. 9. Dispersion vs. wavelength (lm) n1 = 1.5, n2 = 1.4953, Delta r = r2 r1.
this result, it should be noted that, in core with large radius internal reflection is occurred absolutely. 4.2. Influence of Delta r(r = r2 r1), Delta n(n = n2 n1) on dispersion coefficient for conventional triple clad fiber (W-type)
4.3. Influences of NSi-Nc,RSi-Nc Concentration of Si-Nc and Er doped and R10(Radius of Si-Nc and Er doped) on dispersion coefficient at homogenous doped Si-NC–Er at traditional profile
Effect of Delta r (different between core and clad radius) on the dispersion coefficient is illustrated in Fig. 9. With decrease of this parameter the total dispersion is increased, but slope of dispersion coefficient is decreased. In Fig. 10 the effect of differential refractive index between core and clad (Delta n = n2 n1) on dispersion quantity is investigated and it is shown that with increase of Delta n, the dispersion coefficient is decreased. The result of this effect is observed in pulse width propagating through optical fiber.
The effects of Si-NC doped radius and concentration of Si-NC on dispersion characteristic are investigated in Figs. 11–13. The effect of Si-NC and Er doping is illustrated in Fig. 11. We show that with increasing of Si-NC and Er ions concentrations, dispersion coefficient is increased. We have shown that effect of concentration of Si-NC and Er ions on dispersion coefficient are investigated in Figs. 12 and 13. It can be seen that with increasing of the Si-NC at core of the fiber because of the inhomogeneous distribution the dispersion in this state is increased. Moreover with regulation the amount
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A. Salman Ogli, A. Rostami / Optics Communications 281 (2008) 4530–4535
20
10
20 Delta n=5.96e-4 Delta n=5.47e-4 Delta n=4.96e-4 Delta n=4.46e-4
10
0
Dispersion
Dispersion
0
-10
-20
-10
-20
-30
-30
-40
-40
-50
-50
1.2
1.3
1.4
1.5
1.6
1.2
1.7
1.4
1.5
1.6
1.7 -6
Wavelength
x 10
Fig. 10. Dispersion vs. wavelength (lm) n1 = 1.5, n2 = 1.4953, Delta n = n2 n1.
x 10
Fig. 12. Dispersion vs. landa (lm) n11 = 1.50034, n1 = 1.4936035, n3 = 1.4942, n4 = 1.4940 R10: radius of Si-Nc and Er doped (lm).
5
30
1.3
-6
Wavelength
40
R10=0.1 R10=0.9 R10=1.2 R10=1.8
Material Dispersion waveguide Dispersion Total Dispersion
0
Nsi-nc=1e22 Nsi-nc=7e21 Nsi-nc=4e21 Nsi-nc=1e21
20 -5
Dispersion
Dispersion
10 0 -10
-10
-15
-20 -20
-30 -40 -50 1.1
-25
1.2
1.3
1.4
1.5
1.6
1.7
1.4
1.8
1.45
1.5
Wavelength
Wavelength
1.55 -6
x 10
Fig. 11. Dispersion vs. wavelength (lm) n11 = 1.50034, n1 = 1.4936035, n3 = 1.4942, n4 = 1.4940.
Fig. 13. Dispersion vs. wavelength (lm) n11 = 1.50034, n1 = 1.4936035, n3 = 1.4942, n4 = 1.4940 NSi-nc: concentration of Si-Nc and Er doped (1/cm3).
of Si-Nc and radius of Si-NC, we can attain zero dispersion at 1.55 lm.
NC in core is Gaussian, the effect of waveguide dispersion is weak and the total dispersion is strongly increased that result comes back to inhomogeneous distribution. In this section simulation results including investigation of the dispersion properties of the Si-NC Er doped fibers have been presented. It was shown that dispersion manipulation in this fiber is so hard and dispersion coefficient is high.
4.4. Influences of Delta n(n10 n1) on dispersion coefficient at Gaussian profile of Si-NC refractive index In this section, we assume Si-Nc and Er ions at core to be Gaussian profile. Then, the increase of the dispersion coefficient in this section is severe because of inhomogeneous distribution of Si-NC and Er ions at center of the fiber. In Figs. 14 and 15, effects of the Gaussian distribution and Delta n (n11 n1) in dispersion coefficient are investigated. It is shown that when distribution of Si-
5. Conclusion The chromatic dispersion coefficient of two kinds of single mode fiber (traditional and triple clad (W form structure)) and
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A. Salman Ogli, A. Rostami / Optics Communications 281 (2008) 4530–4535
35
Delta n=.00014 Delta n=.0002 Delta n=.0003 Delta n=.00001
25.4
25
25.2
20
25
Dispersion
Dispersion
30
Material Dispesion Waveguide Dispesion Hole Dispesion
15 10 5
24.8 24.6 24.4
0
24.2 -5
24 -10
23.8 -15 1.1
1.54 1.2
1.3
1.4
1.5
1.6
1.7
1.8
Si-NC Er-doped optical fiber (homogenous and Gaussian profile) were studied in this paper. We have been shown that the zero-dispersion wavelength can be shifted such as traditional fibers. It was shown that with control of dr and dn wavelength shifting can be done. Also, we have shown that by variation of the parameters the dispersion coefficient of the fiber strongly is changed. In the case of Si-Nc Er-doped in the core of the fiber the refractive index profile is varied. Because of this variation, position of the zero-dispersion wavelength is changed. Moreover, for homogeneous Si-Nc distribution in the core the zero-dispersion wavelength was tuned to 1.55 lm using density and radius of the nanocrystal. But in the case of the Gaussian distribution of Si-Nc and Er ions the dispersion coefficient is increased and capability of zero-dispersion wavelength tuning was so hard. It was shown that by changing the Gaussian parameters the amount of dispersion coefficient can be controlled.
1.55
Wavelength
Wavelength Fig. 14. Dispersion vs. wavelength (lm) n11 = 1.50034, n10 = 1.50004, n1 = 1.4936035, n3 = 1.4942, n4 = 1.4940.
1.545
1.555
1.56 -6
x 10
Fig. 15. Dispersion vs. wavelength (lm) n11 = 1.50034, n10 = 1.50004, n1 = 1.4936035, n3 = 1.4942, n4 = 1.4940 Delta n (n = n10 n1).
References [1] N. Daldosso, D. Navarro, M. Melchiorri, C. Gristina, P. Pellegrino, B. Garrido, C. Sada, G. Battaglin, F. Gourbilleau, R. Rizik, L. Pavaesi, IEEE – J. Quantum Confinement (2005) (November). [2] D. Pacifici, G. Franzo, F. Priolo, F. Lacona, L.D. Negro, Phys. Rev. B 67 67 (2003) 45301. [3] P.G. Kik, A. Polman, J. Appl. Phys. 91 (2002) 1. [4] F. Lucarz, A.J. Kenyon, Appl. Phys. Lett. (2003). [5] P.G. Kik, A. Polman, J. Appl. Phys. 88 (2000) 4. [6] C.E. Chryssou, A.J. Kenyon, C.W. Pitt, Mater. Sci. Eng. B81 (2001). [7] P.G. Kik, A. Polman, F.O.M. Institute (2002). [8] V.Yu. Timoshenko, D.M. Zhigunov, P.K. Kashkarov, O.A. Shaligina, S.A. Tetrukov, R.J. Zhang, M. Fujii, Sh. Hayashi, J. Nanocryst. Solid 352 (2006). [9] N. Daldosso, M. Melchioori, L. Pavesi, G. Pucker, F. Gourbilleau, S. Chausserie, A. Belarousi, X. Potier, C. Dufour, J. Lumin. 121 (2006). [10] Xiaoping Zhang, Xin Wang, Opt. Laser Technol. 37 (2005) 167.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Chromatic dispersion behavior of Si-NC–Er doped optical fiber A. Salman Ogli *, A. Rostami Photonics and Nanocrystal Research Laboratory (PNRL), Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz 51664, Iran
a r t i c l e
i n f o
Article history: Received 15 January 2008 Received in revised form 29 April 2008 Accepted 29 April 2008
Keywords: Chromatic dispersion Si-NC (Silicon nanocrystal) Optical amplifier
a b s t r a c t The chromatic dispersion for conventional and Er-doped fibers using the refractive index approximation is examined. A first, analytical method for investigation of dispersion in step index triple clad optical fiber is used. To design of zero-dispersion shifted fiber for optical communication purpose manipulation of the refractive index and radius of the core are considered. We show that in presence of the Si-NC–Er ions, zero-dispersion wavelength is displaced and the dispersion quantity is increased. In this work, we try to optimize system parameters to obtain minimum dispersion and dispersion shifted fiber with control of the doping levels of Er ions and Si-NC as well as doping profiles. For especial case, we assumed the Gaussian inhomogeneous core refractive index for zero-dispersion wavelength and dispersion managements. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction High-speed data communication and processing are basic industrial and scientific demand recently. Optical method is one of best alternatives for doing these tasks. Main physical medium for optical communication and processing is optical fiber [1–5]. Transmission bit rate in optical communication is higher than other data transmission methods. But, this is still far from the fundamental limit to the information transfer rate, and future systems are expected to reach data transfer rates of several terabits per second in a single mode optical fiber for transmission over large distance the optical signal needs to be amplified at regular intervals in order to maintain sufficient light intensity. For loss compensation in optical communications, we need optical amplifiers which there are some alternatives such as semiconductor optical amplifiers (SOA) and erbium doped fiber amplifier (EDFA) [6–9]. Another parameter for investigation of quality of the fiber optic is chromatics dispersion [10]. In this work, we focused on the chromatic dispersion in conventional (step index and triple clad (W profile index)) and Si-NC–Er doped optical fibers (homogenous and inhomogeneous). The characteristics of the chromatic dispersion are discussed in detail in this paper. Since by adjusting fiber parameters the chromatic dispersion in traditional fiber (triple clad step index) can easily be optimized, but in Si-NC–Er doped fiber amplifier (inhomogeneous distribution) by varying the essential fiber parameters, manipulation of the dispersion characteristics are hard. Finally, we analyze the optical amplification process and dispersion characteristics of Si-NC–Er doped fiber.
The organization of the paper is as follows: In Sections 2 and 3 theoretical and mathematical backgrounds for description of modal behavior of the fiber and chromatic dispersion are presented. Simulation results are presented and discussed in Section 4. Finally the paper ends with a short conclusion in Section 5. 2. Theoretical background First, we suppose the refractive index profile for traditional fiber is shown in Fig. 1 and the refractive index of each layer are given as follows:
nðrÞ ¼
n1 ; 0 < r < Rcore ; n2 ; Rcore < r < Rclad ;
where n1, n2 and r are refractive index for core and cladding and the radius of fiber, respectively. According to the wave equation and the boundary condition of electromagnetic field and under the LP approximation the characteristics equations in one region can be obtained as follows:
det
jm ðu1 Þ
0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.04.073
km ðu2 Þ;
u1 jm ðu1 Þ u2 km ðu2 Þ;
¼0
ð2Þ
where Jm and Km are the Bessel and modified Bessel functions, respectively. The parameters used in determinant are defined as follows:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk02 n21 Þ b2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 ¼ Rcore b2 ðk02 n22 Þ;
u1 ¼ Rcore * Corresponding author. Tel./fax: +98 411 339 3724. E-mail address: [email protected] (A. Salman Ogli).
ð1Þ
4531
A. Salman Ogli, A. Rostami / Optics Communications 281 (2008) 4530–4535
Fig. 1. Refractive index profile for conventional fiber.
Fig. 2. The Gaussian refractive index approximation of Si-NC–Er doped fiber.
where b is the longitudinal propagation constant of guided mode and k0 is the wave number in the vacuum. Finally, when Si-Nc and Er ions are used as dopants, the refractive index profile is changed and shown in Fig. 2. In this work, we suppose that the Er ions and Si-NCs have the Gaussian distribution. In this strategy, we approximate the Gaussian profile with step index format. Then the approximated refractive index in presence of the wave equation is solved analytically or numerically (see Fig. 3). The refractive index of each layer is shown as follows:
8 n10 ; > > > < n11 ; nðrÞ ¼ > n > 1; > : n2 ;
0 < r < R10 ; R10 < r < R20 ;
ð3Þ
R20 < r < Rcore ; Rcore < r < Rclad ;
According to the wave equation and the boundary condition of the electromagnetic fields and under the LP approximation the characteristic equation in three regions can be obtain as follows:
0
n11 < neff < n10
B0 B B B0 det B B u J ðu Þ B 1m 1 B @0
km ðw2 Þ
0
0
0
im ðw22 Þ
km ðw22 Þ
im ðw3 Þ
km ðw3 Þ
0
0
0
im ðw33 Þ
km ðw33 Þ
km ðw4 Þ
0
0
w2 Im ðw2 Þ w2 K m ðw2 Þ 0 w22 Im ðw22 Þ w22 K m ðw22 Þ
0 jm ðu1 Þ
0 jm ðu2 Þ
0
0 jm ðu2 Þ
0 ym ðu2 Þ
w3 Im ðw3 Þ w3 K m ðw3 Þ 0
1 C C C C C¼0 C C C A
0
w33 Im ðw33 Þ w33 K m ðw33 Þ w4 K m ðw4 Þ 1 0 0 0 C B0 jm ðu22 Þ ym ðu22 Þ jm ðu3 Þ ym ðu3 Þ 0 C B C B C B0 0 0 jm ðu33 Þ ym ðu33 Þ km ðw4 Þ C ¼ 0; B det B C 0 0 C B u1 J m ðu1 Þ u2 J m ðu2 Þ u2 Y m ðu2 Þ 0 C B A @0 u22 J m ðu22 Þ u22 Y m ðu22 Þ u3 J m ðu3 Þ u3 Y m ðu3 Þ 0 0
n2 < neff < n1
0 jm ðu1 Þ
im ðw2 Þ
w33 Im ðw33 Þ w33 K m ðw33 Þ w4 K m ðw4 Þ 1 0 0 C B0 jm ðu22 Þ ym ðu22 Þ im ðw3 Þ km ðw3 Þ 0 C B C B C B0 0 0 im ðw33 Þ km ðw33 Þ km ðw4 Þ C¼0 det B C B u J ðu Þ u J ðu Þ u Y ðu Þ 0 0 0 2 m 2 2 m 2 C B 1m 1 C B A @0 u22 J m ðu22 Þ u22 Y m ðu22 Þ w3 Im ðw3 Þ w3 K m ðw3 Þ 0 0
n1 < neff < n11
jm ðu1 Þ
Fig. 3. Approximation of the Gaussian profile of Si-NC–Er doped fiber.
0
0 ym ðu2 Þ
0
u33 J m ðu33 Þ u33 Y m ðu33 Þ w4 K m ðw4 Þ
ð4Þ
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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n210 n22 ;
where jm, ym, km, Im are the Bessel and modified Bessel functions. The parameters used in determinant are defined as follows:
v ¼ k0 Rcore
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk02 n210 Þ b2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 ¼ R10 ðk02 n211 Þ b2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w2 ¼ R10 b2 ðk02 n211 Þ; u22 ¼ R20 ðk02 n211 Þ b2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w22 ¼ R20 b2 ðk02 n211 Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u3 ¼ R20 ðk02 n21 Þ b2 ; w3 ¼ R20 b2 ðk02 n21 Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u33 ¼ Rcore ðk02 n21 Þ b2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w33 ¼ Rcore b2 ðk02 n21 Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w4 ¼ Rcore b2 ðk02 n22 Þ;
and the normalized propagation constant is given as
u1 ¼ R10
2
B¼
ðb2 =k0 Þ n22 : n210 n22
The optical parameters also, defined as
D¼
ð5Þ
In this stage, we suppose that the refractive index profile for W-type fiber is shown in Fig. 4. Also, Fig. 5 shows homogeneously Si-Nc and Er ions doped fiber refractive index profile. With calculation of neff for different region of fiber, we can obtain the characteristic equation and dispersion coefficient. In this simulation the normalized frequency is defined as
n210 n22 : 2n22
In this section Si-NC–Er doped fibers from refractive index distribution was modeled and using the boundary conditions modal analysis including Eigen values and vectors were done. In the next section dispersion properties of the mentioned fibers are extracted. Effect of different parameters on dispersion property is considered and we try to optimize optical and geometrical parameters to obtain a given dispersion profile. 3. Mathematical analysis of chromatic dispersion According to standard definition and relation of chromatic dispersion, in the following chromatic dispersion coefficient is calculated. 2
N2 D d ðBvÞ ; v dv2 c k 2 k d n2 dðBvÞ ð1 þ D Þ; Material dispersion ¼ c dk2 dv
Waveguide dispersion ¼
ð6Þ ð7Þ
In the following for simplicity some relations are given for calculation of mentioned above equations.
u dðBvÞ u1 dðu1 Þ 1 ¼1þ ; 12 dv v v dv 2 2 2 d ðBvÞ dðu1 Þ u1 d ðu1 Þ ¼ 2u1 ; v 2 dv dv dv2 v
ð8Þ ð9Þ
2 In above statement N 2 ¼ n2 k dn is the group index of the outer dk cladding layer [10]. In the next section some simulated results are given and discussed.
4. Results and discussion Fig. 4. Triple clad profile traditional fiber.
In this section the simulated results are categorized to four groups. In each group effect of especial parameters on dispersion coefficient are considered. 4.1. Influences of DRcore,Dn on dispersion coefficient in conventional step index fiber
Fig. 5. Profile of homogenously doped of Si-Nc and Er ions.
According to numerical calculation of dispersion coefficient, the obtained D–k curve is shown in Figs. 6–8. In Fig. 6, it is shown that zero-dispersion wavelength can obtain for specified parameters of the fiber (Rcore = 2.5 lm, n1 = 1.5, n2 = 1.4953). Considering Fig. 7, it is observed that with increasing differential refractive index between core and clad the dispersion coefficient is increased. It is shown that with increasing of differential refractive index, the waveguide dispersion coefficient is increased because of the increasing of difference between core and clad refractive index. So, total dispersion is increased but flattened dispersion curve can be obtained. Effect of DRcore (varying of core radius) on the dispersion coefficient is illustrated in Fig. 8. With varying of core radius the amount of dispersion quantity is increased but with increase of core radius, slope of dispersion coefficient is decreased too. For explaining of
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40
20 Material Dispersion
30
Waveguide Dispersion Total Dispersion
10
20 0
Dispersion
Dispersion
10 0 -10
Rcore>2.5 Micrometer
-10
-20
Rcore=2.5 Micrometer
-20 -30
-30 -40 1.1
Rcore<2.5 Micrometer
1.2
1.3
1.4
1.5
1.6
1.7
1.8
-40 1.1
Wavelength
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Wavelength Fig. 6. Dispersion vs. wavelength (lm) Rcore = 2.5 lm, n1 = 1.5, n2 = 1.4953. Fig. 8. Dispersion vs. wavelength (lm) n1 = 1.5, n2 = 1.4953.
20
10 5
10
Delta r=0.65 Delta r=0.5 Delta r=0.2 Delta r=0.02
0
0
Dispersion
Dispersion
-5 -10 -15
-10
-20
-20
-30
-25 Dn1=4.69e-3 Dn2=5.5e-3 Dn3=6.49e-3 Dn1=8.49e-3
-30 -35 1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Wavelength Fig. 7. Dispersion vs. wavelength (lm) Rcore = 2.5 lm, Dn = n1 n2.
-40
-50 1.2
1.3
1.4
Wavelength
1.5
1.6
1.7 -6
x 10
Fig. 9. Dispersion vs. wavelength (lm) n1 = 1.5, n2 = 1.4953, Delta r = r2 r1.
this result, it should be noted that, in core with large radius internal reflection is occurred absolutely. 4.2. Influence of Delta r(r = r2 r1), Delta n(n = n2 n1) on dispersion coefficient for conventional triple clad fiber (W-type)
4.3. Influences of NSi-Nc,RSi-Nc Concentration of Si-Nc and Er doped and R10(Radius of Si-Nc and Er doped) on dispersion coefficient at homogenous doped Si-NC–Er at traditional profile
Effect of Delta r (different between core and clad radius) on the dispersion coefficient is illustrated in Fig. 9. With decrease of this parameter the total dispersion is increased, but slope of dispersion coefficient is decreased. In Fig. 10 the effect of differential refractive index between core and clad (Delta n = n2 n1) on dispersion quantity is investigated and it is shown that with increase of Delta n, the dispersion coefficient is decreased. The result of this effect is observed in pulse width propagating through optical fiber.
The effects of Si-NC doped radius and concentration of Si-NC on dispersion characteristic are investigated in Figs. 11–13. The effect of Si-NC and Er doping is illustrated in Fig. 11. We show that with increasing of Si-NC and Er ions concentrations, dispersion coefficient is increased. We have shown that effect of concentration of Si-NC and Er ions on dispersion coefficient are investigated in Figs. 12 and 13. It can be seen that with increasing of the Si-NC at core of the fiber because of the inhomogeneous distribution the dispersion in this state is increased. Moreover with regulation the amount
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20
10
20 Delta n=5.96e-4 Delta n=5.47e-4 Delta n=4.96e-4 Delta n=4.46e-4
10
0
Dispersion
Dispersion
0
-10
-20
-10
-20
-30
-30
-40
-40
-50
-50
1.2
1.3
1.4
1.5
1.6
1.2
1.7
1.4
1.5
1.6
1.7 -6
Wavelength
x 10
Fig. 10. Dispersion vs. wavelength (lm) n1 = 1.5, n2 = 1.4953, Delta n = n2 n1.
x 10
Fig. 12. Dispersion vs. landa (lm) n11 = 1.50034, n1 = 1.4936035, n3 = 1.4942, n4 = 1.4940 R10: radius of Si-Nc and Er doped (lm).
5
30
1.3
-6
Wavelength
40
R10=0.1 R10=0.9 R10=1.2 R10=1.8
Material Dispersion waveguide Dispersion Total Dispersion
0
Nsi-nc=1e22 Nsi-nc=7e21 Nsi-nc=4e21 Nsi-nc=1e21
20 -5
Dispersion
Dispersion
10 0 -10
-10
-15
-20 -20
-30 -40 -50 1.1
-25
1.2
1.3
1.4
1.5
1.6
1.7
1.4
1.8
1.45
1.5
Wavelength
Wavelength
1.55 -6
x 10
Fig. 11. Dispersion vs. wavelength (lm) n11 = 1.50034, n1 = 1.4936035, n3 = 1.4942, n4 = 1.4940.
Fig. 13. Dispersion vs. wavelength (lm) n11 = 1.50034, n1 = 1.4936035, n3 = 1.4942, n4 = 1.4940 NSi-nc: concentration of Si-Nc and Er doped (1/cm3).
of Si-Nc and radius of Si-NC, we can attain zero dispersion at 1.55 lm.
NC in core is Gaussian, the effect of waveguide dispersion is weak and the total dispersion is strongly increased that result comes back to inhomogeneous distribution. In this section simulation results including investigation of the dispersion properties of the Si-NC Er doped fibers have been presented. It was shown that dispersion manipulation in this fiber is so hard and dispersion coefficient is high.
4.4. Influences of Delta n(n10 n1) on dispersion coefficient at Gaussian profile of Si-NC refractive index In this section, we assume Si-Nc and Er ions at core to be Gaussian profile. Then, the increase of the dispersion coefficient in this section is severe because of inhomogeneous distribution of Si-NC and Er ions at center of the fiber. In Figs. 14 and 15, effects of the Gaussian distribution and Delta n (n11 n1) in dispersion coefficient are investigated. It is shown that when distribution of Si-
5. Conclusion The chromatic dispersion coefficient of two kinds of single mode fiber (traditional and triple clad (W form structure)) and
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35
Delta n=.00014 Delta n=.0002 Delta n=.0003 Delta n=.00001
25.4
25
25.2
20
25
Dispersion
Dispersion
30
Material Dispesion Waveguide Dispesion Hole Dispesion
15 10 5
24.8 24.6 24.4
0
24.2 -5
24 -10
23.8 -15 1.1
1.54 1.2
1.3
1.4
1.5
1.6
1.7
1.8
Si-NC Er-doped optical fiber (homogenous and Gaussian profile) were studied in this paper. We have been shown that the zero-dispersion wavelength can be shifted such as traditional fibers. It was shown that with control of dr and dn wavelength shifting can be done. Also, we have shown that by variation of the parameters the dispersion coefficient of the fiber strongly is changed. In the case of Si-Nc Er-doped in the core of the fiber the refractive index profile is varied. Because of this variation, position of the zero-dispersion wavelength is changed. Moreover, for homogeneous Si-Nc distribution in the core the zero-dispersion wavelength was tuned to 1.55 lm using density and radius of the nanocrystal. But in the case of the Gaussian distribution of Si-Nc and Er ions the dispersion coefficient is increased and capability of zero-dispersion wavelength tuning was so hard. It was shown that by changing the Gaussian parameters the amount of dispersion coefficient can be controlled.
1.55
Wavelength
Wavelength Fig. 14. Dispersion vs. wavelength (lm) n11 = 1.50034, n10 = 1.50004, n1 = 1.4936035, n3 = 1.4942, n4 = 1.4940.
1.545
1.555
1.56 -6
x 10
Fig. 15. Dispersion vs. wavelength (lm) n11 = 1.50034, n10 = 1.50004, n1 = 1.4936035, n3 = 1.4942, n4 = 1.4940 Delta n (n = n10 n1).
References [1] N. Daldosso, D. Navarro, M. Melchiorri, C. Gristina, P. Pellegrino, B. Garrido, C. Sada, G. Battaglin, F. Gourbilleau, R. Rizik, L. Pavaesi, IEEE – J. Quantum Confinement (2005) (November). [2] D. Pacifici, G. Franzo, F. Priolo, F. Lacona, L.D. Negro, Phys. Rev. B 67 67 (2003) 45301. [3] P.G. Kik, A. Polman, J. Appl. Phys. 91 (2002) 1. [4] F. Lucarz, A.J. Kenyon, Appl. Phys. Lett. (2003). [5] P.G. Kik, A. Polman, J. Appl. Phys. 88 (2000) 4. [6] C.E. Chryssou, A.J. Kenyon, C.W. Pitt, Mater. Sci. Eng. B81 (2001). [7] P.G. Kik, A. Polman, F.O.M. Institute (2002). [8] V.Yu. Timoshenko, D.M. Zhigunov, P.K. Kashkarov, O.A. Shaligina, S.A. Tetrukov, R.J. Zhang, M. Fujii, Sh. Hayashi, J. Nanocryst. Solid 352 (2006). [9] N. Daldosso, M. Melchioori, L. Pavesi, G. Pucker, F. Gourbilleau, S. Chausserie, A. Belarousi, X. Potier, C. Dufour, J. Lumin. 121 (2006). [10] Xiaoping Zhang, Xin Wang, Opt. Laser Technol. 37 (2005) 167.