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The study of chromatic dispersion and chromatic dispersion slope of WI- and WII-type triple-clad single-mode fibers
ARTICLE IN PRESS
Optics & Laser Technology 37 (2005) 167–172 www.elsevier.com/locate/optlastec
The study of chromatic dispersion and chromatic dispersion slope of WI- and WII-type triple-clad single-mode fibers Xiaoping Zhang*, Xin Wang School of Information Science and Engineering, Lanzhou University, Lanzhou 730000, China Received 6 January 2004; received in revised form 4 March 2004; accepted 11 March 2004 Available online 25 May 2004
Abstract In this paper, the chromatic dispersion and chromatic dispersion slope of two kinds of triple-clad single-mode fibers with a depressed index inner cladding named WI- and WII-type were examined. A feasible approach to calculate chromatic dispersion and higher-order dispersion was established successfully, and the influences made by the optical parameters and geometric parameters on the chromatic dispersion coefficient and its slope were analyzed in detail. The calculated results show that the optical parameter R2 ; which symbolizes the third cladding effect, has a strong impact on the chromatic dispersion coefficient and the chromatic dispersion slope, and the degrees of such impact are closely related to the other parameters. r 2004 Elsevier Ltd. All rights reserved. Keywords: Triple-clad single-mode fiber; Chromatic dispersion coefficient; Chromatic dispersion slope; Zero-dispersion wavelength
1. Introduction Triple-clad single-mode fibers have been closely paid attentions because the perfect transmission properties can be achieved by adjusting the multi-parameters [1–3]. A triple-clad fiber was firstly studied by Cozens and Boucouvalas as an optical coupler for sensing [4]. Dispersion curves for a particular coaxial structure were theoretically obtained with the resonance technique [5] and later by solution of the transcendental equation [6]. A biconical-taper coaxial coupler filter was reported and later Boucouvalas reported a successive tapered coaxial coupler [7–9], again using coaxial fibers. Melo et al. [10] analyzed the transmission characteristics of four kinds of triple-clad multi-mode fibers with different refractive index profiles, and the modal dispersion and field distribution during the multi-mode propagation were studied in detail, but the corresponding properties during single-mode propagation were not discussed. Recently, the cutoff characteristics and the waveguide dispersion characteristics of the two kinds of triple-clad *Corresponding author. E-mail address: [email protected] (X. Zhang). 0030-3992/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2004.03.006
single-mode fibers have been reported [11], and the studied results indicated that the waveguide dispersion characteristics of these kinds of fibers were influenced by the optical parameter R2 ; but the corresponding chromatic dispersion and higher-order dispersion were not studied because the characteristic equations are so complex that it is difficult to examine the chromatic dispersion coefficient and chromatic dispersion slope. In this paper, we extended our work reported in Ref. [11] and focused our study on the chromatic dispersion coefficient and chromatic dispersion slope, and established successfully a feasible approach to calculate the chromatic dispersion coefficient and chromatic dispersion slope. The characteristics of the chromatic dispersion and the chromatic dispersion slope were discussed in detail. The calculated results indicate that parameter R2 ; which symbolizes the third cladding effect, has a strong impact on the chromatic dispersion coefficient and the chromatic dispersion slope and the degrees of such impact are closely related to the other parameters. Therefore, by adjusting each parameter reasonably, the chromatic dispersion coefficient and chromatic dispersion slope can easily be optimized or match with our demands.
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2. Theoretical analysis 2.1. Derivation of the characteristic equations We focus on the two structures that are mentioned in our recent work [11]. The refractive index profiles are shown in Fig. 1, which are named WI- and WII-type, respectively. The refractive index of each layer is shown as follows: 8 n1 ; 0oroa; > > > < n ; aorob; 2 nðrÞ ¼ > n 3 ; boroc; > > : n4 ; cor; where r is the radius position. According to the effective refractive index given by neff ¼ b=k0 ; where b is the longitudinal propagation constant of guided mode and k0 is the wave number in vacuum, the structures of WI- and WII-type can be divided into three and two regions, respectively, which are shown in Fig. 1. According to the wave equation, the boundary conditions of electromagnetic field and under the LP approximation, the characteristic equations in regions (I)–(III) can be obtained as follows:
where Jm ; Ym ; Im ; Km are the Bessel and modified Bessel functions. The characteristic equation of WI-type fiber is made up of (1)–(3); however, the characteristic equation of WII-type is made up of (2) and (3) for it has only two regions. The parameters used in (1)–(3) are defined as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U1 ¼ a k02 n21 b2 ; W2 ¼ a k02 n22 b2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W3 ¼ a b2 k02 n22 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 k02 n23 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W4 ¼ c b2 k02 n24 ;
U2 ¼ b
W20 ¼
P W2 ; Q
W30 ¼
U3 ¼ b
P W3 ; Q
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k02 n23 b2 ;
U20 ¼
1 U2 ; P
U30 ¼
1 U3 : P
We define the geometrical parameters as b P¼ ; c
a Q¼ ; c
the optical parameters as Dn1 n1 n3 Dn2 n2 n4 ¼ ¼ ; R2 ¼ ; Dn n3 n2 Dn n3 n2 n2 n2 n 1 n4 D ¼ 1 2 4E ; n4 2n4
R1 ¼
Jm ðU1 Þ Jm ðW2 Þ Ym ðW2 Þ 0 0 0 0 0 0 J ðW Þ Y ðW Þ J ðU Þ Y ðU Þ 0 m m m 3 m 3 2 2 0 0 0 0 0 Jm ðU3 Þ Ym ðU3 Þ Km ðW4 Þ ¼ 0; U J 0 ðU Þ W J 0 ðW Þ W Y 0 ðW Þ 0 0 0 2 m 2 2 m 2 1 m 1 0 0 0 0 0 0 0 0 0 W2 Jm ðW2 Þ W2 Ym ðW3 Þ U3 Jm ðU3 Þ U3 Ym ðU3 Þ 0 0 0 0 0 0 0 0 0 0 0 U3 Jm ðU3 Þ U3 Ym ðU3 Þ W4 K4 ðW4 Þ ðn4 oneff on2 ; WIÞ;
ð1Þ
Jm ðU1 Þ Im ðW3 Þ Km ðW3 Þ 0 0 0 0 0 0 Im ðW3 Þ Km ðW3 Þ Jm ðU3 Þ Ym ðU3 Þ 0 0 0 0 0 0 Jm ðU3 Þ Ym ðU3 Þ Km ðW4 Þ ¼ 0; U J 0 ðU Þ W I 0 ðW Þ W K 0 ðW Þ 0 0 0 3 m 3 3 m 3 1 m 1 0 0 0 0 0 0 0 0 0 W I ðW Þ W K ðW Þ U J ðU Þ U Y ðU Þ 0 3 3 3 3 3 m 3 3 m 3 m m 0 0 0 0 0 0 0 0 0 0 U3 Jm ðU3 Þ U3 Ym ðU3 Þ W4 K4 ðW4 Þ ðn2 oneff on3 ; WIÞ; ðn4 oneff on3 ; WIIÞ; Jm ðU1 Þ Im ðW3 Þ Km ðW3 Þ 0 0 0 0 I ðW Þ K ðW Þ I m m m ðU2 Þ 3 3 0 0 0 Im ðU20 Þ U J 0 ðU Þ W I 0 ðW Þ W K 0 ðW Þ 0 3 m 3 3 m 3 1 m 1 0 0 0 0 0 0 0 W3 Im ðW3 Þ W3 Km ðW3 Þ U2 Im0 ðU2 Þ 0 0 0 U20 Im0 ðU20 Þ ðn3 oneff on1 ; WI WIIÞ;
ð2Þ 0 0 Km ðU2 Þ 0 Km ðU20 Þ Km ðW4 Þ ¼ 0; 0 0 0 U2 Km ðU2 Þ 0 0 0 0 0 U2 Km ðU2 Þ W4 K4 ðW4 Þ ð3Þ
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Fig. 1. Profile of two triple-clad fibers with parameters defined: (a) WI and (b) WII.
the normalized frequency as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ¼ k0 a n21 n24 and the normalized propagation constant as 2 ðb=k0 Þ2 n24 U1 ¼1 : B¼ 2 2 V n1 n4
ð4Þ
ð5Þ
2.2. Analysis of chromatic dispersion coefficient D and dispersion slope S According to the definitions of chromatic dispersion coefficient D [12,13], the expressions of chromatic dispersion coefficient D and its slope S can be obtained as follows:
l d 2 n4 dðBV Þ N4 D d2 ðBV Þ V D¼ 1 þ D ; ð6Þ c dl2 dV dV 2 c l
l d3 n4 dðBV Þ 1 d2 n4 dðBV Þ S¼ 1þD 1þD c dl3 dV c dl2 dV ð7Þ
where N4 ¼ n4 ldn4 =dl is the group index of the outer cladding and dm n4 =dlm ðm ¼ 1; 2; 3Þ can be calculated by Sellmeier formula [14]. From Eqs. (6) and (7) we can conclude that in order to obtain D and S; dðBV Þ=dV ; V d2 ðBV Þ=dV 2 and V 2 d3 ðBV Þ=dV 3 must be calculated firstly. Deducing from Eq. (5), we obtain as follows: 2 dðBV Þ U1 U1 dU1 ¼1þ 12 ; ð8Þ dV V V dV d2 ðBV Þ dU1 U1 2 d 2 U1 ¼ 2 2U ; V 1 dV 2 dV V dV 2
d 3 U1 : ð10Þ dV 3 This means that the calculation of dðBV Þ=dV ; V d2 ðBV Þ= dV 2 and V 2 d3 ðBV Þ=dV 3 can be changed into the calculation of dU1 =dV ; d2 U1 =dV 2 and d3 U1 =dV 3 ; but it is impossible to obtain the analytical solutions of dU1 =dV ; d2 U1 =dV 2 and d3 U1 =dV 3 from determinants (1)–(3) directly. Herein, we extend the numerical analysis method established in our reported work. Setting m ¼ 0 in determinants (1)–(3), the equations change into the eigenvalue equations of fundamental mode which have the general form F ðU1 ; U2 ; U3 ; W2 ; W3 ; W4 ; V Þ ¼ 0: The first-, second- and third-order derivatives of F to V were calculated, and the partial derivatives @F =@U1 ; @2 F =@U12 ; @3 F =@U13 ; @F =@U2 ; @2 F =@U22 ; @3 F =@U23 ; @F =@U3 ; @2 F =@U32 ; @3 F =@U33 ; @F =@W2 ; @2 F =@W22 ; @3 F =@W23 ; @F =@W3 ; @2 F =@W32 ; @3 F =@W33 ; @F =@W4 ; @2 F =@W42 ; @3 F =@W43 should be analytically calculated when calculating derivatives of F to V : We found that all the resulted terms contain 18 derivatives, 12 derivatives of which have been mentioned in our reported work and new six derivatives that cannot be calculated directly are 2VU1
The value of optical parameter R2 defined above has made a distinction between WI- and WII-type fibers: R2 > 0 is corresponding to WI-type fibers, while R2 o0 is corresponding to WII-type fibers.
N4 D 2 d3 ðBV Þ N4 D d2 ðBV Þ V þ 2 V ; þ dV 3 dV 2 c l2 c l2
d3 ðBV Þ dU1 U1 2 d2 U1 dU1 U1 ¼6 6V V dV 3 dV V dV 2 dV V 2
ð9Þ
d3 U1 d3 U2 d3 U3 d3 W2 d3 W3 d3 W4 ; ; ; ; ; : dV 3 dV 3 dV 3 dV 3 dV 3 dV 3 By analyzing and calculating to the defined expressions of V ; U1 ; U2 ; U3 ; W2 ; W3 and W4 ; it has been found that the last five derivative terms can be changed into the calculation of dU1 =dV ; d2 U1 =dV 2 and d3 U1 =dV 3 : From F ðU1 ; U2 ; U3 ; W2 ; W3 ; W4 ; V Þ ¼ 0; the B–V curves can be obtained, and the U1 –V curves and the relations between dU1 =dV ; d2 U1 =dV 2 and V can be obtained from the B–V curves, respectively. So it is possible to obtain the relation between d3 U1 =dV 3 and V by using the U1 –V curves, the relations between the first- and second-order derivatives of U1 to V and the third-order derivatives of F to V : Thus, we obtain the dðBV Þ= dV V ; V d2 ðBV Þ=dV 2 V and V 2 d3 ðBV Þ=dV 3 V
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curves from Eqs. (8)–(10). And finally D–l and S–l curves can be calculated from Eqs. (6) and (7), respectively.
0.15
R2=0.6
0.14 0.13
R2=-0.8
2
0.12
R2=-1
1
0.11
2
S(ps/nm /km)
0.10
3. Calculated results and analysis 3.1. Influences of R2 and D on D and S
0.09 0.08 0.07 0.06 0.05
1:∆=3e-3 2:∆ =5e-3
0.04 0.03 0.02 0.01 0.00 1.20
λ (µm) 1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
1.75
1.70
1.75
R1=5 P=0.7(1) Q=0.5 a=2.5µm
(a) 0.15
R2=0.6
0.14
R2=-0.8
R2=-1
0.13 0.12 0.11
2
0.10
2
S(ps/nm /km)
According to the method mentioned in Section 2.2, a new numerical calculation for D and S was conducted, and the resulted D–l and S–l curves are shown in Figs. 2–6. Figs. 2 and 3 show the resulted D–l and S–l curves with varied R2 and D at R1 ¼ 5; Q ¼ 0:5; a ¼ 2:5 mm and P ¼ 0:7 (or 1 and P ¼ 1 is corresponding to R2 ¼ 1; which represents doubly clad W-type fibers), respectively. From Figs. 2 and 3, it can be seen that R2 ; which symbolizes the outer cladding effect, has a strong impact on D–l and S–l curves, and the regulars of such impact are closely related to D: In Fig. 2, there is a wavelength l0 called zero-dispersion wavelength at which the value of D is 0 and l0 is different for various
1
0.09 0.08 0.07 0.06 0.05
1:∆=5e-3 2:∆=8e-3
0.04 0.03 0.02 0.01
40
R2=0.6
35 30
R2=-0.8
R2=-1
0.00 1.20
(b)
1:∆=3e-3 2:∆=5e-3
25 20
D(ps/nm/km)
2
10
1.35
1.40
1.45
1.50
1.55
1.60
1.65
R1=5 P=0.7(1) Q=0.5 a=2.5µm
Fig. 3. S as a function of l for various R2 with R1 ¼ 5; P ¼ 0:7ð1Þ; Q ¼ 0:5; a ¼ 2:5 mm: (a) D ¼ 3 103 ; 5 103 and (b) D ¼ 5 103 ; 8 103 :
5 0 -5 -10 -15 -20 -25
λ (µm)
-30 1.20
1.25
1.30
(a)
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
1.75
1.70
1.75
R 1=5 P=0.7(1) Q=0.5 a=2.5µm 40
R2=0.6
35 30
R2 =-0.8
R2 =-1
1:∆=5e-3 2:∆=8e-3
25 20 15
D(ps/nm/km)
1.30
1
15
10 5 0 -5 -10
2
-15 -20
1
-25 -30 1.20
(b)
λ(µm) 1.25
λ (µm) 1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
R1=5 P=0.7(1) Q=0.5 a=2.5µm
Fig. 2. D as a function of l for various R2 with R1 ¼ 5; P ¼ 0:7ð1Þ; Q ¼ 0:5; a ¼ 2:5 mm: (a) D ¼ 3 103 ; 5 103 and (b) D ¼ 5 103 ; 8 103 :
R2 with defined D: It can be seen from Fig. 2(a) that l0 increases with R2 decreasing and the smaller R2 is, the bigger l0 is. But in Fig. 2(b), there is a difference that the l0 with R2 ¼ 1 is smaller than that with R2 ¼ 0:8; it means that the values of l0 can be controlled by adjusting R2 and D: It can be also seen that the range of zero-dispersion wavelength with D ¼ 5 103 is obviously wider than that with another two D; which indicates there should be an optimum value of D so that we can obtain a widest range of l0 by adjusting R2 : From Fig. 3 we can see that the dispersion slopes S of WI- and WII-type decrease meanwhile with increasing l and the influence of D on S–l curves with R2 ¼ 1 is stronger than that with R2 ¼ 0:6 and 0:8; and when D ¼ 3 103 there exist two turning-points at around l ¼ 1:35 and 1:62 mm; but, when D ¼ 5 103 there exists only one turning-point at around l ¼ 1:60 mm that leads S increase. The reason for these turningpoints appearing is that the D–l curves are not always smooth. In the spectral range of 1.20–1:75 mm; the two turning-points that lead S increase are where the chromatic dispersion coefficients vary the most slowly while the turning-point that leads S decrease is where
ARTICLE IN PRESS X. Zhang, X. Wang / Optics & Laser Technology 37 (2005) 167–172 30
Q=0.4
Q=0.5
35
Q=0.6
1:R2=0.6 2:R2=-0.8 3:R2=-1
15
D(ps/nm/km)
10
25
1
20
2
15
D(ps/nm/km)
20
3
5 0 -5
-15 -20
λ (µm) 1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
-25 1.20
1.75
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
1.75
1.80
1.70
1.75
1.80
R1=2 R2=0.6 P=0.7 Q=0.5 a=2.5µm 0.15
Q=0.4
0.14
Q=0.5
Q=0.6
0.14 0.13
0.13
0.10
0.11 0.10
S(ps/nm /km)
1
0.11
0.12
1:R2=0.6 2:R2=-0.8 3:R2=-1
0.12
0.09
2
0.08 0.07
2
0.06
3
0.05
0.09 0.08 0.07
∆ =3e-3 ∆ =5e-3 ∆ =7e-3 ∆ =1e-2
0.06 0.05
0.04
0.04
0.03
0.03
0.02
0.02 0.01
0.01 0.00 1.20
λ (µm) 1.25
(a)
R1=5 P=0.7(1) ∆=5e-3 a=2.5µm 0.15
2
0 -5 -10
(a)
S(ps/nm /km)
5
-15 -20
(b)
10
-10
-25 1.20
∆ =3e-3 ∆ =5e-3 ∆ =7e-3 ∆ =1e-2
30
25
171
1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
λ (µm)
0.00 1.20
1.75
R1=5 P=0.7(1) ∆=5e-3 a=2.5µm
Fig. 4. D and S as a function of l for various R2 and Q with R1 ¼ 5; P ¼ 0:7ð1Þ; D ¼ 5 103 ; a ¼ 2:5 mm: (a) D–l curves and (b) S–l curves.
the chromatic dispersion coefficient varies the most rapidly. 3.2. Influences of R2 and Q Fig. 4 shows the resulted D–l and S–l curves with varied R2 and Q at R1 ¼ 5; P ¼ 0:7ð1Þ; a ¼ 2:5 mm and D ¼ 5 103 ; respectively. It can be seen from Fig. 4 that Q has a more obvious impact on D–l curves with R2 ¼ 0:8 and 1 than those with R2 ¼ 0:6: The smaller R2 is, the wider the adjustable ranges of zerodispersion wavelength l0 are. From Fig. 4(b), we can see that the impact of Q on S–l curves is also obvious and strong. When the operating wavelength l lies within a longer wavelength, the values of S vary rapidly when parameter Q is adjusted. And it can also be seen that S has little changes in the spectral range of 1.525– 1:700 mm when R2 ¼ 0:8 and Q ¼ 0:4: 3.3. Influences of D on WI- and WII-type Fig. 5 shows the resulted D–l and S–l curves for WItype with varied D at R1 ¼ 2; R2 ¼ 0:6; P ¼ 0:7; Q ¼ 0:5
λ (µm) 1.25
1.30
(b)
1.35
1.40
1.45
1.50
1.55
1.60
1.65
R1=2 R2=0.6 P=0.7 Q=0.5 a=2.5µm
Fig. 5. D and S as a function of l for various D with R1 ¼ 2; R2 ¼ 0:6; P ¼ 0:7; Q ¼ 0:5; a ¼ 2:5 mm in WI-type. (a) D–l curves and (b) S–l curves.
and a ¼ 2:5 mm; respectively. It should be noted that D has minute influences on D–l and S–l curves as compared with that of Q and R2 when the else parameters are constants. Fig. 6 shows the resulted D– l and S–l curves for WII-type with varied D at R1 ¼ 2; R2 ¼ 0:8; P ¼ 0:7; Q ¼ 0:5 and a ¼ 2:5 mm; respectively. It can be seen that the influence of D on D–l curves is obvious as compared with that of WI-type, and the spectral range of zero-dispersion wavelength is much wider than that of WI-type. Although not shown here, the influences of R1 and P on D and S are also discussed and calculated results indicate that the values of R1 and P have minute influences on D and S as compared with those of R2 ; Q and D:
4. Conclusion The chromatic dispersion coefficient and its slope of two kinds of triple-clad single-mode fibers named WIand WII-type respectively are studied in this paper, and a feasible approach to calculate the chromatic dispersion
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more flexible. The studies provide an important basis for the optimization of triple-clad single-mode fibers, such as dispersion and dispersion slope compensation, and designs of new type of dispersion-shifted and dispersionflattened fibers. The method to calculate chromatic dispersion coefficient and its slope can be extended to multi-clad optical fibers and much more higher-order dispersion.
35 30
∆ =3e-3
25
∆ =5e-3 ∆ =7e-3
20
∆ =1e-2
D(ps/nm/km)
15 10 5 0 -5 -10 -15 -20
λ (µm)
-25 1.20
1.25
1.30
(a)
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
1.75
R1=2 R2=-0.8 P=0.7 Q=0.5 a=2.5µm
0.15 0.14 0.13 0.12 0.11
2
S(ps/nm /km)
0.10 0.09 0.08 0.07
∆ =3e-3 ∆ =5e-3 ∆ =7e-3
0.06 0.05 0.04
∆ =1e-2
0.03 0.02 0.01 0.00 1.20
(b)
λ (µm) 1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
1.75
References
1.80
1.80
R1=2 R2=-0.8 P=0.7 Q=0.5 a=2.5µm
Fig. 6. D and S as a function of l for various D with R1 ¼ 2; R2 ¼ 0:8; P ¼ 0:7; Q ¼ 0:5; a ¼ 2:5 mm in WII-type. (a) D–l curves and (b) S–l curves.
coefficient D and its slope S is established successfully. The calculated results indicate that optical parameter R2 ; which symbolizes the third cladding effect and makes a distinction between WI- and WII-type fibers, has a strong impact on the chromatic dispersion coefficient and its slope, and such an impact is closely related to the parameter D and Q: Thus, with properly selecting R2 ; D and Q; the chromatic dispersion and the chromatic dispersion slope can also be easily adjusted, which makes the design of the dispersion-shifted fiber
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Optics & Laser Technology 37 (2005) 167–172 www.elsevier.com/locate/optlastec
The study of chromatic dispersion and chromatic dispersion slope of WI- and WII-type triple-clad single-mode fibers Xiaoping Zhang*, Xin Wang School of Information Science and Engineering, Lanzhou University, Lanzhou 730000, China Received 6 January 2004; received in revised form 4 March 2004; accepted 11 March 2004 Available online 25 May 2004
Abstract In this paper, the chromatic dispersion and chromatic dispersion slope of two kinds of triple-clad single-mode fibers with a depressed index inner cladding named WI- and WII-type were examined. A feasible approach to calculate chromatic dispersion and higher-order dispersion was established successfully, and the influences made by the optical parameters and geometric parameters on the chromatic dispersion coefficient and its slope were analyzed in detail. The calculated results show that the optical parameter R2 ; which symbolizes the third cladding effect, has a strong impact on the chromatic dispersion coefficient and the chromatic dispersion slope, and the degrees of such impact are closely related to the other parameters. r 2004 Elsevier Ltd. All rights reserved. Keywords: Triple-clad single-mode fiber; Chromatic dispersion coefficient; Chromatic dispersion slope; Zero-dispersion wavelength
1. Introduction Triple-clad single-mode fibers have been closely paid attentions because the perfect transmission properties can be achieved by adjusting the multi-parameters [1–3]. A triple-clad fiber was firstly studied by Cozens and Boucouvalas as an optical coupler for sensing [4]. Dispersion curves for a particular coaxial structure were theoretically obtained with the resonance technique [5] and later by solution of the transcendental equation [6]. A biconical-taper coaxial coupler filter was reported and later Boucouvalas reported a successive tapered coaxial coupler [7–9], again using coaxial fibers. Melo et al. [10] analyzed the transmission characteristics of four kinds of triple-clad multi-mode fibers with different refractive index profiles, and the modal dispersion and field distribution during the multi-mode propagation were studied in detail, but the corresponding properties during single-mode propagation were not discussed. Recently, the cutoff characteristics and the waveguide dispersion characteristics of the two kinds of triple-clad *Corresponding author. E-mail address: [email protected] (X. Zhang). 0030-3992/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2004.03.006
single-mode fibers have been reported [11], and the studied results indicated that the waveguide dispersion characteristics of these kinds of fibers were influenced by the optical parameter R2 ; but the corresponding chromatic dispersion and higher-order dispersion were not studied because the characteristic equations are so complex that it is difficult to examine the chromatic dispersion coefficient and chromatic dispersion slope. In this paper, we extended our work reported in Ref. [11] and focused our study on the chromatic dispersion coefficient and chromatic dispersion slope, and established successfully a feasible approach to calculate the chromatic dispersion coefficient and chromatic dispersion slope. The characteristics of the chromatic dispersion and the chromatic dispersion slope were discussed in detail. The calculated results indicate that parameter R2 ; which symbolizes the third cladding effect, has a strong impact on the chromatic dispersion coefficient and the chromatic dispersion slope and the degrees of such impact are closely related to the other parameters. Therefore, by adjusting each parameter reasonably, the chromatic dispersion coefficient and chromatic dispersion slope can easily be optimized or match with our demands.
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2. Theoretical analysis 2.1. Derivation of the characteristic equations We focus on the two structures that are mentioned in our recent work [11]. The refractive index profiles are shown in Fig. 1, which are named WI- and WII-type, respectively. The refractive index of each layer is shown as follows: 8 n1 ; 0oroa; > > > < n ; aorob; 2 nðrÞ ¼ > n 3 ; boroc; > > : n4 ; cor; where r is the radius position. According to the effective refractive index given by neff ¼ b=k0 ; where b is the longitudinal propagation constant of guided mode and k0 is the wave number in vacuum, the structures of WI- and WII-type can be divided into three and two regions, respectively, which are shown in Fig. 1. According to the wave equation, the boundary conditions of electromagnetic field and under the LP approximation, the characteristic equations in regions (I)–(III) can be obtained as follows:
where Jm ; Ym ; Im ; Km are the Bessel and modified Bessel functions. The characteristic equation of WI-type fiber is made up of (1)–(3); however, the characteristic equation of WII-type is made up of (2) and (3) for it has only two regions. The parameters used in (1)–(3) are defined as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U1 ¼ a k02 n21 b2 ; W2 ¼ a k02 n22 b2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W3 ¼ a b2 k02 n22 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 k02 n23 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W4 ¼ c b2 k02 n24 ;
U2 ¼ b
W20 ¼
P W2 ; Q
W30 ¼
U3 ¼ b
P W3 ; Q
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k02 n23 b2 ;
U20 ¼
1 U2 ; P
U30 ¼
1 U3 : P
We define the geometrical parameters as b P¼ ; c
a Q¼ ; c
the optical parameters as Dn1 n1 n3 Dn2 n2 n4 ¼ ¼ ; R2 ¼ ; Dn n3 n2 Dn n3 n2 n2 n2 n 1 n4 D ¼ 1 2 4E ; n4 2n4
R1 ¼
Jm ðU1 Þ Jm ðW2 Þ Ym ðW2 Þ 0 0 0 0 0 0 J ðW Þ Y ðW Þ J ðU Þ Y ðU Þ 0 m m m 3 m 3 2 2 0 0 0 0 0 Jm ðU3 Þ Ym ðU3 Þ Km ðW4 Þ ¼ 0; U J 0 ðU Þ W J 0 ðW Þ W Y 0 ðW Þ 0 0 0 2 m 2 2 m 2 1 m 1 0 0 0 0 0 0 0 0 0 W2 Jm ðW2 Þ W2 Ym ðW3 Þ U3 Jm ðU3 Þ U3 Ym ðU3 Þ 0 0 0 0 0 0 0 0 0 0 0 U3 Jm ðU3 Þ U3 Ym ðU3 Þ W4 K4 ðW4 Þ ðn4 oneff on2 ; WIÞ;
ð1Þ
Jm ðU1 Þ Im ðW3 Þ Km ðW3 Þ 0 0 0 0 0 0 Im ðW3 Þ Km ðW3 Þ Jm ðU3 Þ Ym ðU3 Þ 0 0 0 0 0 0 Jm ðU3 Þ Ym ðU3 Þ Km ðW4 Þ ¼ 0; U J 0 ðU Þ W I 0 ðW Þ W K 0 ðW Þ 0 0 0 3 m 3 3 m 3 1 m 1 0 0 0 0 0 0 0 0 0 W I ðW Þ W K ðW Þ U J ðU Þ U Y ðU Þ 0 3 3 3 3 3 m 3 3 m 3 m m 0 0 0 0 0 0 0 0 0 0 U3 Jm ðU3 Þ U3 Ym ðU3 Þ W4 K4 ðW4 Þ ðn2 oneff on3 ; WIÞ; ðn4 oneff on3 ; WIIÞ; Jm ðU1 Þ Im ðW3 Þ Km ðW3 Þ 0 0 0 0 I ðW Þ K ðW Þ I m m m ðU2 Þ 3 3 0 0 0 Im ðU20 Þ U J 0 ðU Þ W I 0 ðW Þ W K 0 ðW Þ 0 3 m 3 3 m 3 1 m 1 0 0 0 0 0 0 0 W3 Im ðW3 Þ W3 Km ðW3 Þ U2 Im0 ðU2 Þ 0 0 0 U20 Im0 ðU20 Þ ðn3 oneff on1 ; WI WIIÞ;
ð2Þ 0 0 Km ðU2 Þ 0 Km ðU20 Þ Km ðW4 Þ ¼ 0; 0 0 0 U2 Km ðU2 Þ 0 0 0 0 0 U2 Km ðU2 Þ W4 K4 ðW4 Þ ð3Þ
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Fig. 1. Profile of two triple-clad fibers with parameters defined: (a) WI and (b) WII.
the normalized frequency as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ¼ k0 a n21 n24 and the normalized propagation constant as 2 ðb=k0 Þ2 n24 U1 ¼1 : B¼ 2 2 V n1 n4
ð4Þ
ð5Þ
2.2. Analysis of chromatic dispersion coefficient D and dispersion slope S According to the definitions of chromatic dispersion coefficient D [12,13], the expressions of chromatic dispersion coefficient D and its slope S can be obtained as follows:
l d 2 n4 dðBV Þ N4 D d2 ðBV Þ V D¼ 1 þ D ; ð6Þ c dl2 dV dV 2 c l
l d3 n4 dðBV Þ 1 d2 n4 dðBV Þ S¼ 1þD 1þD c dl3 dV c dl2 dV ð7Þ
where N4 ¼ n4 ldn4 =dl is the group index of the outer cladding and dm n4 =dlm ðm ¼ 1; 2; 3Þ can be calculated by Sellmeier formula [14]. From Eqs. (6) and (7) we can conclude that in order to obtain D and S; dðBV Þ=dV ; V d2 ðBV Þ=dV 2 and V 2 d3 ðBV Þ=dV 3 must be calculated firstly. Deducing from Eq. (5), we obtain as follows: 2 dðBV Þ U1 U1 dU1 ¼1þ 12 ; ð8Þ dV V V dV d2 ðBV Þ dU1 U1 2 d 2 U1 ¼ 2 2U ; V 1 dV 2 dV V dV 2
d 3 U1 : ð10Þ dV 3 This means that the calculation of dðBV Þ=dV ; V d2 ðBV Þ= dV 2 and V 2 d3 ðBV Þ=dV 3 can be changed into the calculation of dU1 =dV ; d2 U1 =dV 2 and d3 U1 =dV 3 ; but it is impossible to obtain the analytical solutions of dU1 =dV ; d2 U1 =dV 2 and d3 U1 =dV 3 from determinants (1)–(3) directly. Herein, we extend the numerical analysis method established in our reported work. Setting m ¼ 0 in determinants (1)–(3), the equations change into the eigenvalue equations of fundamental mode which have the general form F ðU1 ; U2 ; U3 ; W2 ; W3 ; W4 ; V Þ ¼ 0: The first-, second- and third-order derivatives of F to V were calculated, and the partial derivatives @F =@U1 ; @2 F =@U12 ; @3 F =@U13 ; @F =@U2 ; @2 F =@U22 ; @3 F =@U23 ; @F =@U3 ; @2 F =@U32 ; @3 F =@U33 ; @F =@W2 ; @2 F =@W22 ; @3 F =@W23 ; @F =@W3 ; @2 F =@W32 ; @3 F =@W33 ; @F =@W4 ; @2 F =@W42 ; @3 F =@W43 should be analytically calculated when calculating derivatives of F to V : We found that all the resulted terms contain 18 derivatives, 12 derivatives of which have been mentioned in our reported work and new six derivatives that cannot be calculated directly are 2VU1
The value of optical parameter R2 defined above has made a distinction between WI- and WII-type fibers: R2 > 0 is corresponding to WI-type fibers, while R2 o0 is corresponding to WII-type fibers.
N4 D 2 d3 ðBV Þ N4 D d2 ðBV Þ V þ 2 V ; þ dV 3 dV 2 c l2 c l2
d3 ðBV Þ dU1 U1 2 d2 U1 dU1 U1 ¼6 6V V dV 3 dV V dV 2 dV V 2
ð9Þ
d3 U1 d3 U2 d3 U3 d3 W2 d3 W3 d3 W4 ; ; ; ; ; : dV 3 dV 3 dV 3 dV 3 dV 3 dV 3 By analyzing and calculating to the defined expressions of V ; U1 ; U2 ; U3 ; W2 ; W3 and W4 ; it has been found that the last five derivative terms can be changed into the calculation of dU1 =dV ; d2 U1 =dV 2 and d3 U1 =dV 3 : From F ðU1 ; U2 ; U3 ; W2 ; W3 ; W4 ; V Þ ¼ 0; the B–V curves can be obtained, and the U1 –V curves and the relations between dU1 =dV ; d2 U1 =dV 2 and V can be obtained from the B–V curves, respectively. So it is possible to obtain the relation between d3 U1 =dV 3 and V by using the U1 –V curves, the relations between the first- and second-order derivatives of U1 to V and the third-order derivatives of F to V : Thus, we obtain the dðBV Þ= dV V ; V d2 ðBV Þ=dV 2 V and V 2 d3 ðBV Þ=dV 3 V
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curves from Eqs. (8)–(10). And finally D–l and S–l curves can be calculated from Eqs. (6) and (7), respectively.
0.15
R2=0.6
0.14 0.13
R2=-0.8
2
0.12
R2=-1
1
0.11
2
S(ps/nm /km)
0.10
3. Calculated results and analysis 3.1. Influences of R2 and D on D and S
0.09 0.08 0.07 0.06 0.05
1:∆=3e-3 2:∆ =5e-3
0.04 0.03 0.02 0.01 0.00 1.20
λ (µm) 1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
1.75
1.70
1.75
R1=5 P=0.7(1) Q=0.5 a=2.5µm
(a) 0.15
R2=0.6
0.14
R2=-0.8
R2=-1
0.13 0.12 0.11
2
0.10
2
S(ps/nm /km)
According to the method mentioned in Section 2.2, a new numerical calculation for D and S was conducted, and the resulted D–l and S–l curves are shown in Figs. 2–6. Figs. 2 and 3 show the resulted D–l and S–l curves with varied R2 and D at R1 ¼ 5; Q ¼ 0:5; a ¼ 2:5 mm and P ¼ 0:7 (or 1 and P ¼ 1 is corresponding to R2 ¼ 1; which represents doubly clad W-type fibers), respectively. From Figs. 2 and 3, it can be seen that R2 ; which symbolizes the outer cladding effect, has a strong impact on D–l and S–l curves, and the regulars of such impact are closely related to D: In Fig. 2, there is a wavelength l0 called zero-dispersion wavelength at which the value of D is 0 and l0 is different for various
1
0.09 0.08 0.07 0.06 0.05
1:∆=5e-3 2:∆=8e-3
0.04 0.03 0.02 0.01
40
R2=0.6
35 30
R2=-0.8
R2=-1
0.00 1.20
(b)
1:∆=3e-3 2:∆=5e-3
25 20
D(ps/nm/km)
2
10
1.35
1.40
1.45
1.50
1.55
1.60
1.65
R1=5 P=0.7(1) Q=0.5 a=2.5µm
Fig. 3. S as a function of l for various R2 with R1 ¼ 5; P ¼ 0:7ð1Þ; Q ¼ 0:5; a ¼ 2:5 mm: (a) D ¼ 3 103 ; 5 103 and (b) D ¼ 5 103 ; 8 103 :
5 0 -5 -10 -15 -20 -25
λ (µm)
-30 1.20
1.25
1.30
(a)
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
1.75
1.70
1.75
R 1=5 P=0.7(1) Q=0.5 a=2.5µm 40
R2=0.6
35 30
R2 =-0.8
R2 =-1
1:∆=5e-3 2:∆=8e-3
25 20 15
D(ps/nm/km)
1.30
1
15
10 5 0 -5 -10
2
-15 -20
1
-25 -30 1.20
(b)
λ(µm) 1.25
λ (µm) 1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
R1=5 P=0.7(1) Q=0.5 a=2.5µm
Fig. 2. D as a function of l for various R2 with R1 ¼ 5; P ¼ 0:7ð1Þ; Q ¼ 0:5; a ¼ 2:5 mm: (a) D ¼ 3 103 ; 5 103 and (b) D ¼ 5 103 ; 8 103 :
R2 with defined D: It can be seen from Fig. 2(a) that l0 increases with R2 decreasing and the smaller R2 is, the bigger l0 is. But in Fig. 2(b), there is a difference that the l0 with R2 ¼ 1 is smaller than that with R2 ¼ 0:8; it means that the values of l0 can be controlled by adjusting R2 and D: It can be also seen that the range of zero-dispersion wavelength with D ¼ 5 103 is obviously wider than that with another two D; which indicates there should be an optimum value of D so that we can obtain a widest range of l0 by adjusting R2 : From Fig. 3 we can see that the dispersion slopes S of WI- and WII-type decrease meanwhile with increasing l and the influence of D on S–l curves with R2 ¼ 1 is stronger than that with R2 ¼ 0:6 and 0:8; and when D ¼ 3 103 there exist two turning-points at around l ¼ 1:35 and 1:62 mm; but, when D ¼ 5 103 there exists only one turning-point at around l ¼ 1:60 mm that leads S increase. The reason for these turningpoints appearing is that the D–l curves are not always smooth. In the spectral range of 1.20–1:75 mm; the two turning-points that lead S increase are where the chromatic dispersion coefficients vary the most slowly while the turning-point that leads S decrease is where
ARTICLE IN PRESS X. Zhang, X. Wang / Optics & Laser Technology 37 (2005) 167–172 30
Q=0.4
Q=0.5
35
Q=0.6
1:R2=0.6 2:R2=-0.8 3:R2=-1
15
D(ps/nm/km)
10
25
1
20
2
15
D(ps/nm/km)
20
3
5 0 -5
-15 -20
λ (µm) 1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
-25 1.20
1.75
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
1.75
1.80
1.70
1.75
1.80
R1=2 R2=0.6 P=0.7 Q=0.5 a=2.5µm 0.15
Q=0.4
0.14
Q=0.5
Q=0.6
0.14 0.13
0.13
0.10
0.11 0.10
S(ps/nm /km)
1
0.11
0.12
1:R2=0.6 2:R2=-0.8 3:R2=-1
0.12
0.09
2
0.08 0.07
2
0.06
3
0.05
0.09 0.08 0.07
∆ =3e-3 ∆ =5e-3 ∆ =7e-3 ∆ =1e-2
0.06 0.05
0.04
0.04
0.03
0.03
0.02
0.02 0.01
0.01 0.00 1.20
λ (µm) 1.25
(a)
R1=5 P=0.7(1) ∆=5e-3 a=2.5µm 0.15
2
0 -5 -10
(a)
S(ps/nm /km)
5
-15 -20
(b)
10
-10
-25 1.20
∆ =3e-3 ∆ =5e-3 ∆ =7e-3 ∆ =1e-2
30
25
171
1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
λ (µm)
0.00 1.20
1.75
R1=5 P=0.7(1) ∆=5e-3 a=2.5µm
Fig. 4. D and S as a function of l for various R2 and Q with R1 ¼ 5; P ¼ 0:7ð1Þ; D ¼ 5 103 ; a ¼ 2:5 mm: (a) D–l curves and (b) S–l curves.
the chromatic dispersion coefficient varies the most rapidly. 3.2. Influences of R2 and Q Fig. 4 shows the resulted D–l and S–l curves with varied R2 and Q at R1 ¼ 5; P ¼ 0:7ð1Þ; a ¼ 2:5 mm and D ¼ 5 103 ; respectively. It can be seen from Fig. 4 that Q has a more obvious impact on D–l curves with R2 ¼ 0:8 and 1 than those with R2 ¼ 0:6: The smaller R2 is, the wider the adjustable ranges of zerodispersion wavelength l0 are. From Fig. 4(b), we can see that the impact of Q on S–l curves is also obvious and strong. When the operating wavelength l lies within a longer wavelength, the values of S vary rapidly when parameter Q is adjusted. And it can also be seen that S has little changes in the spectral range of 1.525– 1:700 mm when R2 ¼ 0:8 and Q ¼ 0:4: 3.3. Influences of D on WI- and WII-type Fig. 5 shows the resulted D–l and S–l curves for WItype with varied D at R1 ¼ 2; R2 ¼ 0:6; P ¼ 0:7; Q ¼ 0:5
λ (µm) 1.25
1.30
(b)
1.35
1.40
1.45
1.50
1.55
1.60
1.65
R1=2 R2=0.6 P=0.7 Q=0.5 a=2.5µm
Fig. 5. D and S as a function of l for various D with R1 ¼ 2; R2 ¼ 0:6; P ¼ 0:7; Q ¼ 0:5; a ¼ 2:5 mm in WI-type. (a) D–l curves and (b) S–l curves.
and a ¼ 2:5 mm; respectively. It should be noted that D has minute influences on D–l and S–l curves as compared with that of Q and R2 when the else parameters are constants. Fig. 6 shows the resulted D– l and S–l curves for WII-type with varied D at R1 ¼ 2; R2 ¼ 0:8; P ¼ 0:7; Q ¼ 0:5 and a ¼ 2:5 mm; respectively. It can be seen that the influence of D on D–l curves is obvious as compared with that of WI-type, and the spectral range of zero-dispersion wavelength is much wider than that of WI-type. Although not shown here, the influences of R1 and P on D and S are also discussed and calculated results indicate that the values of R1 and P have minute influences on D and S as compared with those of R2 ; Q and D:
4. Conclusion The chromatic dispersion coefficient and its slope of two kinds of triple-clad single-mode fibers named WIand WII-type respectively are studied in this paper, and a feasible approach to calculate the chromatic dispersion
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more flexible. The studies provide an important basis for the optimization of triple-clad single-mode fibers, such as dispersion and dispersion slope compensation, and designs of new type of dispersion-shifted and dispersionflattened fibers. The method to calculate chromatic dispersion coefficient and its slope can be extended to multi-clad optical fibers and much more higher-order dispersion.
35 30
∆ =3e-3
25
∆ =5e-3 ∆ =7e-3
20
∆ =1e-2
D(ps/nm/km)
15 10 5 0 -5 -10 -15 -20
λ (µm)
-25 1.20
1.25
1.30
(a)
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
1.75
R1=2 R2=-0.8 P=0.7 Q=0.5 a=2.5µm
0.15 0.14 0.13 0.12 0.11
2
S(ps/nm /km)
0.10 0.09 0.08 0.07
∆ =3e-3 ∆ =5e-3 ∆ =7e-3
0.06 0.05 0.04
∆ =1e-2
0.03 0.02 0.01 0.00 1.20
(b)
λ (µm) 1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
1.75
References
1.80
1.80
R1=2 R2=-0.8 P=0.7 Q=0.5 a=2.5µm
Fig. 6. D and S as a function of l for various D with R1 ¼ 2; R2 ¼ 0:8; P ¼ 0:7; Q ¼ 0:5; a ¼ 2:5 mm in WII-type. (a) D–l curves and (b) S–l curves.
coefficient D and its slope S is established successfully. The calculated results indicate that optical parameter R2 ; which symbolizes the third cladding effect and makes a distinction between WI- and WII-type fibers, has a strong impact on the chromatic dispersion coefficient and its slope, and such an impact is closely related to the parameter D and Q: Thus, with properly selecting R2 ; D and Q; the chromatic dispersion and the chromatic dispersion slope can also be easily adjusted, which makes the design of the dispersion-shifted fiber
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