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Influence of uniaxial crystal material cladding on reflectivity and dispersion of uniform fiber Bragg grating
Optics Communications 271 (2007) 109–115 www.elsevier.com/locate/optcom
Influence of uniaxial crystal material cladding on reflectivity and dispersion of uniform fiber Bragg grating Shanglin Hou a
a,b,* ,
Chunlian Hu b, Xiaomin Ren a, Xia Zhang a, Yongqing Huang
a
Key Laboratory of Optical Communication and Lightwave Technologies of Ministry of Education, Beijing University of Posts and Telecommunications, P.O. Box 66, BUPT, Beijing 100876, China b School of Science, Lanzhou University of Technology, Lanzhou 730050, China Received 18 October 2005; received in revised form 28 August 2006; accepted 10 October 2006
Abstract The fiber Bragg grating with cladding made of uniaxial crystal material whose optical axis is parallel to the axis of fiber Bragg grating proposed in the paper published in 2003 was investigated again and an error was corrected in the calculation, its effective index, reflectivity and dispersion were examined using coupled-mode theory and numeric solution. The calculated results indicate that no low frequency cutoff phenomenon exists in the HE11 mode, more power is transmitted by the core of the fiber with cladding made of isotropic material, the reflectivity of the fiber Bragg grating with cladding made of uniaxial crystal material is much higher than that with cladding made of isotropic material, the parameter Kcl, i.e., the ratio of the extraordinary to the ordinary ray refractive index, has a stronger impact on the reflectivity, Bragg wavelength and the dispersion of this kind of fiber Bragg grating when it varied from 1.00 to 1.01 than in other regions. This means that the characteristics of the fiber Bragg grating with uniaxial crystal cladding can be changed through adjusting Kcl while keeping its length, periodicity and the other parameters as constants. 2006 Elsevier B.V. All rights reserved. Keywords: Uniaxial crystal materials; Fiber Bragg grating; Reflectivity; Dispersion
1. Introduction Fiber gratings have developed rapidly since the optical sensitivity was described in 1978 [1]. Numerous applications have been demonstrated that utilize fiber grating as dispersion compensator, density wavelength division multiplexer (DWDM), fiber laser, fiber sensors and so on [2–5]. Stevenson and Cozens et al. analyzed an optical fiber with a single-crystal core for the first time in 1974 [6,7]. A doubly-clad W-type optical fiber with an inner cladding made of uniaxial crystal material has been studied in 2002 [8], and the studied results indicated that the adjustable range *
Corresponding author. Address: Key Laboratory of Optical Communication and Lightwave Technologies of Ministry of Education, Beijing University of Posts and Telecommunications, P.O. Box 66, BUPT, Beijing 100876, China. Tel.: +86 10 6228400486; fax: +86 931 2756544. E-mail address: [email protected] (S. Hou). 0030-4018/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.10.009
of zero dispersion wavelength of this kind fiber is wilder than that of common doubly-clad optical fiber. The fiberoptic polarizer with cladding made of uniaxial crystal material was also described [9]. Electro-optic effect and elasto-optic effect in a chirped fiber grating with cladding made of uniaxial crystal material are theoretically analyzed and the results indicate that the reflective spectra of the chirped grating can be changed by the electric field and the strain applied to the fiber grating cladding along z-axis [10]. The paper published in 2003 [11] predicted the characteristics of a new type of fiber Bragg grating with cladding made of uniaxial crystal material whose optical axis, i.e., z-axis, is parallel to the axis of fiber Bragg grating, and the calculated results indicate that parameter Kcl, i.e., the ratio of the extraordinary ray refractive index to the ordinary ray index, has a strong impact on the reflectivity, Bragg wavelength, the bandwidth of the reflected wave and the dispersion.
110
S. Hou et al. / Optics Communications 271 (2007) 109–115
In this work, the reflectivity and the dispersion of the fiber Bragg grating with cladding made of uniaxial crystal material were calculated again and an error was corrected in calculating the effective index. The calculated results which are different from Ref. [11] were described in detail, which indicate that no low frequency cutoff phenomenon or no cutoff wavelength kC exists in the HE11 mode, more power is confined in the core of the fiber with cladding made of isotropic material and the power transmitted by the cladding is more easily coupled into the backwardpropagating guided fundamental mode. The parameter Kcl has a strong impact on the reflectivity and Bragg wavelength of this kind of fiber Bragg grating, especially when Kcl is varied from 1 to 1.01. The reflectivity of the fiber Bragg grating with cladding made of uniaxial crystal material is much higher than that with cladding made of isotropic material. The dispersion is very similar as described in Ref. [11] except when Kcl = 1. Therefore, the characteristics of the fiber Bragg grating with uniaxial crystal cladding can be optimized through adjusting Kcl while keeping its length, periodicity and the other parameters as constants.
8 < ðr2 þ k 2 n2 n2z2 b2 Þez ¼ 0 t z nt : ðr2 þ k 2 n2 b2 Þh ¼ 0 t
t
ðr > aÞ;
where nt = nx = ny and n0 > nt. k ¼ 2p is the wave number in k vacuum and b is the propagation constant. If b2 < k 2 n20 , the parameters are defined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nz n0 nt ; U ¼ a k 2 n20 b2 ; K cl ¼ ; D ¼ nt n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 W ¼ a b k nt ; V ¼ ak n20 n2t ; where Kcl is the ratio of the extraordinary to the ordinary ray refractive index and V is normalized frequency. Through matching the relationship between axial and tangential field components, and using the boundary conditions of electromagnetic field, the characteristic equation of the ideal normal mode, i.e., the mode in an ideal waveguide without grating perturbation, can be obtained as follows: 2 2 n J 0m ðU Þ K 0 ðK cl W Þ m2 Q n0 1 F 20 þ m þ ; ¼ K cl n2t U 2 W 2 nt K cl UJ m ðU Þ WK m ðK cl W Þ
2. Theoretical analysis
ð3Þ
2.1. The characteristic equation of the ideal normal mode The sketch of the fiber Bragg grating with cladding made of uniaxial crystal material was shown in Fig. 1. It has a core radius a and a core refractive index n0. The cladding with an infinite diameter is made of uniaxial crystal material whose optical axis is taken to be parallel to the axis of the fiber, i.e. z-axis, and its principal axis indices are nx, ny and nz, respectively, which satisfy: nx = ny 5 nz. For the optical fiber without the grating perturbation, the axial electric and magnetic field components satisfy wave equation as follows [12]: ( ðr2t þ k 2 n20 b2 Þez ¼ 0 ðr < aÞ; ð1Þ ðr2t þ k 2 n20 b2 Þhz ¼ 0
X
cladding O
ð2Þ
z
core
Z
2a
Λ L Fig. 1. The sketch of the fiber Bragg grating with cladding made of uniaxial crystal material.
where Jm, Km are the Bessel and modified Bessel functions respectively. The parameters F and Q are defined as follows: F ¼
J 0m ðU Þ K 0 ðW Þ þ m ; UJ m ðU Þ WK m ðW Þ
Q¼
1 1 þ : U2 W 2
2.2. The grating coupling coefficient We studied an unchirped uniform grating only and assumed that a perturbation to the core refractive index of the grating can be described as follows: 2p z ; ð4Þ dnco ðzÞ ¼ dnco 1 þ v cos K where dnco is ‘‘dc’’ index change spatially averaged over a grating period and its value is generally 1.0 · 104. v is the fringe visibility of the index change and its value is 1 in this paper. K is the grating period, L is the grating length, N is the total numbers of the grating periods and K = L/N as shown in Fig. 1. According to the coupled-mode theory, the traverse coupling coefficient between the forward-propagating guided fundamental mode and the backward-propagating guided fundamental mode can be described as follows [13,14]: Z Z xe0 K¼ ½n2 ðzÞ n20 ~ et ~ et dx dy; ð5Þ 4P core where P is the power of the incident ray and normalized P = 1, ~ et is traverse components of electric field of the HE11 mode, x is circle frequency, e0 is the permittivity in free space, n(z) is the core index of the fiber Bragg grating and it can be expressed as
S. Hou et al. / Optics Communications 271 (2007) 109–115
nðzÞ ¼ n0 þ dnco ðzÞ: We define Z Z ^ ¼ xe0 ~ et dx dy: et ~ K 4P core
ð6Þ
k 2 bðQ2 F 2 Þ : 2ðb2 Q2 k 2 n20 F 2 Þ
Then, the coupling coefficient can be described as 2p K ¼ r þ 2j cos z ; K
ð7Þ
ð8Þ
ð9Þ
^ co is called the ‘‘dc’’ (period-average) where r ¼ 2n0 Kdn ^ co is called the ‘‘ac’’ coupling coefficient and j ¼ n0 vKdn coupling coefficient. 2.3. Reflectivity c According to coupled-mode theory and the boundary condition of the fiber Bragg grating, the amplitude reflection coefficient can be shown as follows [15]: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^2 L j sinh j2 r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q¼ ^ sinh ^2 L þ i j2 r ^2 cosh ^2 L r j2 r j2 r ð10Þ and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^2 L sinh2 j2 r 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ¼ jqj ¼ ; 2 ^2 L rj^2 cosh2 j2 r
ð11Þ
^ ¼ d þ r is a general ‘‘dc’’ self-coupling coefficient. where r The detuning d is defined as p 1 1 d ¼ b ¼ 2pneff : ð12Þ K k kB The Bragg wavelength kB is expressed as kB 2neff K:
3. Calculated results and analysis 3.1. The effective index of the HE11 mode
Inserting the traverse component of electric field derived in Section 2.1 to Eq. (7), we can obtain ^¼ K
111
The parameters were set exactly as same as those in Ref. [11], such as m = 1, n0 = 1.46, D = 0.02, a = 5 lm, L = 1.06 cm, N = 20,000. A cluster of curves of the effective index as function of the normalized frequency V, i.e., (b/k)–V curves of the HE11 mode can be obtained with various Kcl from Eq. (3), as shown in Fig. 2. It can be seen that the effective indices increase with increasing V, and the shape of the curves is very similar to that with Kcl = 1. But the effective indices also increase with the increasing Kcl for a given value of V. For both Kcl > 1 and Kcl < 1, V can be arbitrarily small so that the wavelength can be very large due to correcting an error in calculating the characteristics equation, i.e., Eqs. (3) or (5) in Ref. [11], that is to say, no low frequency cut-off phenomenon and no cutoff wavelength kC exist when Kcl < 1 in the HE11 mode, contrary to the findings described in Ref. [11]. This means that when Kcl < 1, the operating wavelength k of the single-mode fiber is not limited when the single mode condition is satisfied. 3.2. Influence of Kcl on reflectivity c and Bragg wavelength kB The curves of the reflection coefficient c as functions of the wavelength k, i.e., c–k curves with varied Kcl can been obtained from Eq. (11) shown in Fig. 3(a–c) for L = 1.06 cm, 0.106 cm and 10.6 cm, respectively and Kcl = 0.2, 0.75, 0.8, 1.0, 1.2 and 1.8, the other parameters are shown in the figure caption. From Fig. 3(a), it can be seen that Kcl has a strong impact on the c–k curves. When Kcl < 1, the resulted curves moved to the left of the curve with Kcl = 1, but when Kcl > 1, the resulted curves moved to the right of the curve with Kcl = 1. The larger the Kcl deviated from 1, the larger the resulted curves deviated from the curve with Kcl = 1 and the degrees of the deviation β/k
ð13Þ
1.4594 1.4592 1.4590
2.4. Dispersion dq
Kcl=1.0
1.4588
Kcl=0.1
1.4586
The phase of Eq. (10) is expressed as hq. A delay time sq of the grating is defined as follows [3]:
Kcl=1.2
1.4584
Kcl=0.3
1.4582
Kcl=0.5
1.4580
sq ¼
dhq k2 dhq ¼ : dx 2pc dk
ð14Þ
The dispersion dq (in ps/nm) is the first-order derivatives of sq to k, therefore,
Kcl=0.7
1.4578 1.4576
Kcl=0.9
1.4574 1.4572 0.5
dsq 2sq k2 d 2 hq 2pc d2 hq ¼ dq ¼ ¼ : dk k 2pc dk2 k2 dx2
ð15Þ
1.0
1.5
2.0
2.5
3.0
3.5
4.0
V
Fig. 2. Effective index of HE11 mode as a function of V with various Kcl.
112
S. Hou et al. / Optics Communications 271 (2007) 109–115 ×100 % 1.0
Kcl=0.2
0.8
Kcl=0.8 Kcl=1.2 Kcl=1.8
Kcl=1.0
0.6
0.4
0.2
0.0 1548.50
1548.75
1549.00
1549.25
1549.50
1549.75
1550.00
1550.25
(nm) ×100% 0.05
Kcl=0.75
Kcl=1.8
0.04
0.03
0.02
Kcl=1.0 0.01
0.00 1548.0
1548.5
1549.0
1549.5
1550.0
1550.5
1551.0
(nm)
×100 % 1.0
K cl =0.75 0.8
[11] was added to show that no low frequency cutoff phenomenon exists. The curve with Kcl = 1 is consistent with that reported in the previous study [3], but its maximum value of the reflection coefficient c which is 68.3% is much lower than the others and its Bragg wavelength shifts to the side of Kcl < 1. It indicates that the variation of Kcl also results in the variation of the maximum value of the reflection coefficient c. When Kcl are varied values and L = 0.106 cm, the c–k curves are shown in Fig. 3(b). From Fig. 3(b), the maximum values of reflection coefficient c with Kcl = 0.75 or 1.8 are much greater than that with Kcl = 1. Fig. 3(c) shows the c–k curve with Kcl = 0.75 and L = 10.6 cm. In order to further study the influence of Kcl on reflectivity, the curves of reflectivity versus wavelength with Kcl varied from 1.000 to 1.009 by steps of 0.001 are calculated and the resulted c–k curves are shown in Figs. 4 and 5(a–c) with (a) Kcl = 1.007, (b) Kcl = 1.008, (c) Kcl = 1.009 and L = 1.06 cm which indicate that the reflection coefficient c varied greatly. From Fig. 4, it can be seen that the maximum value of the reflection coefficient c decreases with increasing Kcl until Kcl = 1.004, that is to say, for Kcl = 1.004, the maximum value of the reflectivity coefficient is the lowest, only 7.7%, but for Kcl = 1.006, it suddenly increases to 85.1%. From Fig. 5, it can be seen that the reflectivity c is asymmetric on the two sides of the maximum reflectivity. Fig. 5(a) shows that the maximum values of reflection coefficient c reach 100% and the maximum has a greater width of 0.26 nm; when Kcl = 1.008, there are many peak values which reach 100% when k < 1548.97 nm, and the widest bandwidth of 0.1785 nm exists near 1548.3 nm as shown in Fig. 5(b) for wavelength between 1540 nm to 1550 nm. From Fig. 5(c), there are three peak values which reach 100% and the bandwidth of the left one is larger than that of the two others, the peak values on the right side of the widest one are much higher than the corresponding ones γ ×100% 0.9 0.8
0.6
0.7 0.6
0.4
0.5 0.4
0.2
0.3
Fig. 3. Reflection spectra versus wavelength for various Kcl and L with n0 = 1:46, D = 0.02, a = 5 lm. (a) N = 20,000, L = 1.06 cm; (b) N = 2000, L = 0.106 cm and (c) N = 200,000, L = 10.6 cm.
are remarkably different when Kcl < 1 or Kcl > 1. These results are similar to those described in Ref. [11], but the curve with Kcl = 0.2 that cannot exist according to Ref.
0.1 0.0 1549.00 1549.25
1.006 1.004 1549.50
λ(
nm )
1.002 1549.75 1550.00
cl
(nm )
0.2
K
0.0 1549.20 1549.22 1549.24 1549.26 1549.28 1549.30 1549.32 1549.34
1.000
Fig. 4. Reflection spectra versus wavelength with Kcl from 1.000 to 1.006 and n0 = 1.46, D = 0.02, a = 5 lm, N = 20,000, L = 1.06 cm.
S. Hou et al. / Optics Communications 271 (2007) 109–115
γ ×100%
×100% 1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.0 1548.50
0.2
0.0 1548.5 1548.75
1549.00
1549.25
1549.50
1549.75
1550.00
1.020 1.018
1549.0
(nm)
λ(
1549.5
nm
γ ×100%
)
1.016 1.012 1550.5
K cl
1.010
Fig. 6. Reflection spectra versus wavelength with Kcl from 1.01 to 1.02 and n0 = 1.46, D = 0.02, a = 5 lm, N = 20,000, L = 1.06 cm.
0.8
0.6
0.4
0.2
1542
1546
1544
1548
1550
λ (nm)
γ ×100% 1.0
0.8
0.6
while its maximum value varies slightly from 100% to 98.9%. Fig. 7 shows the curve of the maximum value of reflection coefficient as function of Kcl, i.e., cmax–Kcl curve with N = 20,000 and L = 1.06 cm. It indicates that the maximum reflectivity varies greatly near to Kcl = 1. From left to Kcl = 1.004, it decreases rapidly and reaches the lowest point when Kcl = 1.004, then increase rapidly to 100% when Kcl changes from 1.007 to 1.009, at last it decreases to 94.5% and remains at this value. Fig. 8 shows that the curves of kB versus Kcl with N = 20,000 and L = 1.06 cm. It can be seen that when Kcl < 1, kB varies more rapidly than that when Kcl > 1, but the degree is not as strong as described in Ref. [11]. When Kcl near to 1.0, kB decreases rapidly and reaches the lowest γ ×100%
0.4
1.0
0.2
0.8
0.0 1549.0
1.014
1550.0
1.0
0.0 1540
113
0.6
1549.5
1550.0
1550.5
1551.0
1551.5
1552.0
λ (nm)
Fig. 5. Reflection spectra versus wavelength with (a) Kcl = 1.007, (b) Kcl = 1.008, (c) Kcl = 1.009 and n0 = 1.46, D = 0.02, a = 5 lm, N = 20,000, L = 1.06 cm.
on the left side. Fig. 6 shows the reflection coefficient c versus wavelength k with Kcl varied from 1.01 to 1.02 by steps of 0.002, it can be seen that the two sides of the peak are symmetric, but its top angle varies from right angle to smooth and its bandwidth decreases with Kcl increasing,
0.4
0.2
0.0 0.0
0.5
1.0
1.5
2.0
2.5
Kcl Fig. 7. The maximum reflectivity as a function of Kcl with n0 = 1.46, D = 0.02, a = 5 lm.
114
S. Hou et al. / Optics Communications 271 (2007) 109–115 λ B (nm)
η × 100%
1550.0
1.0
1549.9
kcl=1.0
1549.8 1549.7
0.8
kcl=0.2
1549.6 1549.5
0.6
1549.4
kcl=1.2
1549.3
kcl=0.8
0.4
1549.2 1549.1 1549.0
0.2
1548.9 1548.8
0.0
1548.7 0.0
0.5
1.0
1.5
2.0
0.5
2.5
1.0
1.5
point when Kcl is between 1.00 and 1.01. These results are consist with results shown in the above Figs. 1–4. In order to interpret the pronounced minimum in the reflectivity curve close to a refractive index ratio of unity of the cladding material, the power distribution between core and cladding of this kind of fiber was predicted as following part. 3.3. Influence of Kcl on power distribution It is clear that a substantial amount of light power is transported by the cladding, while most of the power is confined in the core. Thus, the total power flow is the sum of power carried by both the core and by the cladding in the HE11 mode. The formula expressing this statement is Ptotal = Pcore + Pclad, where Pcore and Pclad is the integration of the longitudinal Poynting vector ~ S z with the corresponding cross-sectional area by inserting the traverse component of electric field derived in Section 2.1. We define g = Pcore/Ptotal expressing the power distribution. Thus, the curves of g versus V with various Kcl are calculated and the resulted g–V curves are shown in Fig. 9, while the other parameters are shown in the figure caption. From Fig. 9, it can be seen that more power is confined in the core when Kcl near to 1 for a given value of V, i.e., V = 2.0. It indicates that less power is confined in the core of the fiber with cladding made of uniaxial crystal material than that with cladding made of isotropic material while keeping the other parameters of the fiber as constant. In order to further study the influence of Kcl on the power distribution, the curve of g versus Kcl with V = 2.0 is calculated and the resulted g–Kcl curve is shown in Fig. 10. It can be seen that g increases with Kcl increasing and it is 76% when Kcl = 1, maximum of 100% when Kcl = 1.004 and then drop to zero when Kcl = 1.007. It indicates that more power is transmitted by the core of the fiber with cladding made of isotropic material.
2.5
3.0
3.5
4.0
V
K cl Fig. 8. kB as a function of Kcl with n0 = 1.46, D = 0.02 and a = 5 lm.
2.0
Fig. 9. g as a function of V with various Kcl and n0 = 1.46, D = 0.02, a = 5 lm.
η × 100%
1.0
V=2.0
0.8
0.6
0.4
0.2
0.0 0.0
0.5
1.0
1.5
2.0
Kcl
Fig. 10. g as a function of Kcl with n0 = 1.46, D = 0.02, a = 5 lm.
It follows from the above discussion that more power is confined in the core of the fiber with cladding made of isotropic material. For fiber Bragg grating, the power transmitted by the cladding is more easily coupled into the backward-propagating guided fundamental mode, thus, the more power carried by the cladding, the higher reflectivity of the fiber Bragg grating. 3.4. Influence of Kcl on dispersion The dispersion curves calculated from Eq. (15) with Kcl = 0.2, 1.0 and 1.8 are shown in Fig. 11, where the corresponding curves of the reflectivity are also given, in order to show the influence of Kcl on the dispersion. It can be seen that the dispersion curves also shifted when Kcl < 1 or Kcl > 1 and its shifting rule is very similar to that of the reflectivity curves as described in Ref. [11]. When k = kB, i.e., at the maximum reflectivity, the dispersion is
S. Hou et al. / Optics Communications 271 (2007) 109–115
γ × 100% 1.0
dispersion
Kcl=0.2
0.8
reflectivity
Kcl=1.8
dρ(in ps/nm)
Kcl=1.0
8
6
0.6 4 0.4
115
mode, the parameter Kcl has a stronger impact on the reflectivity and Bragg wavelength between 1.00 and 1.01 than in others regions. The reflectivity of the fiber Bragg grating with cladding made of uniaxial crystal material is much higher than that with cladding made of isotropic material. The dispersion is much similar as described in Ref. [11] except for Kcl = 1. Therefore, the characteristics of the fiber Bragg grating with uniaxial crystal cladding can be optimized through adjusting Kcl while keeping its length, periodicity and the other parameters as constants.
2 0.2
0.0 1548.5
Acknowledgements 0 1549.5
1549.0
1550.0
λ(nm)
Fig. 11. Dispersion and reflectivity for various Kcl with n0 = 1.46, D = 0.02, a = 5 lm, N = 20,000, L = 1.06 cm.
minimum and the peak value of the dispersion appears on the right side of the peak value of reflectivity. But when Kcl = 1.0, because of its lower reflectivity, the distance between the two peak values of the dispersion and of reflectivity is greater than that with other Kcl. 4. Conclusion The reflectivity and the dispersion of the fiber Bragg grating with cladding made of uniaxial crystal material were calculated more precisely and the calculated results which are different from the paper published in 2003 [11] were described in detail, which indicate that no low frequency cutoff phenomenon or no cutoff wavelength kC exists in the HE11 mode, more power is confined in the core of the fiber with cladding made of isotropic material and the power transmitted by the cladding is more easily coupled into the backward-propagating guided fundamental
This work was supported by National Basic Research Program of China (2003CB314906), National 863 High Technology Project of China (2003AA311010), the Key grant Project of Chinese Ministry of Education (No. 104046) and Natural Science Fund of Gansu Province (ZS032-B25-006), China. References [1] K.O. Hill, Y. Fujii, D.C. Johnson, et al., Appl. Phys. Lett. 32 (1978) 647. [2] K.O. Hill, G. Meltz, J. Lightwave Technol. 15 (1997) 1263. [3] Turan Erdogan, J. Lightwave Technol. 15 (1997) 1277. [4] G.A. Ball, W.H. Glenn, J. Lightwave Technol. 10 (1992) 1338. [5] A.D. Kersey, Optic Fiber Technol. 2 (1996) 291. [6] J.L. Stevenson, R.B. Dyott, Electron. Lett. 10 (1974) 449. [7] J. Cozens, Electron. Lett. 12 (1974) 41. [8] X.P. Zhang, Z.H. Tan, Opt. Commun. 204 (2002) 127. [9] M. Wakaki, Y. Komachi, H. Machida, Appl. Opt. 35 (1996) 2591. [10] X.P. Zhang, D.C. Lv, Opt. Quant. Electron. 36 (2004) 469. [11] X.P. Zhang, S.L. Hou, Optics Commun. 219 (2003) 193. [12] A.W. Snydan, J.D. Love, Optical Waveguide Theory, Chapman and Hall, London, 1983. [13] P.D. Ye, Fiber Optic Theory, Knowledge Press, Beijing, 1985. [14] A. Yariv, IEEE J. Quantum Electron. 9 (1973) 919. [15] H. Kogelnik, in: T. Tamir (Ed.), Guided-wave Optoelectronics, Springer-Verlag, New York, 1990.
Influence of uniaxial crystal material cladding on reflectivity and dispersion of uniform fiber Bragg grating Shanglin Hou a
a,b,* ,
Chunlian Hu b, Xiaomin Ren a, Xia Zhang a, Yongqing Huang
a
Key Laboratory of Optical Communication and Lightwave Technologies of Ministry of Education, Beijing University of Posts and Telecommunications, P.O. Box 66, BUPT, Beijing 100876, China b School of Science, Lanzhou University of Technology, Lanzhou 730050, China Received 18 October 2005; received in revised form 28 August 2006; accepted 10 October 2006
Abstract The fiber Bragg grating with cladding made of uniaxial crystal material whose optical axis is parallel to the axis of fiber Bragg grating proposed in the paper published in 2003 was investigated again and an error was corrected in the calculation, its effective index, reflectivity and dispersion were examined using coupled-mode theory and numeric solution. The calculated results indicate that no low frequency cutoff phenomenon exists in the HE11 mode, more power is transmitted by the core of the fiber with cladding made of isotropic material, the reflectivity of the fiber Bragg grating with cladding made of uniaxial crystal material is much higher than that with cladding made of isotropic material, the parameter Kcl, i.e., the ratio of the extraordinary to the ordinary ray refractive index, has a stronger impact on the reflectivity, Bragg wavelength and the dispersion of this kind of fiber Bragg grating when it varied from 1.00 to 1.01 than in other regions. This means that the characteristics of the fiber Bragg grating with uniaxial crystal cladding can be changed through adjusting Kcl while keeping its length, periodicity and the other parameters as constants. 2006 Elsevier B.V. All rights reserved. Keywords: Uniaxial crystal materials; Fiber Bragg grating; Reflectivity; Dispersion
1. Introduction Fiber gratings have developed rapidly since the optical sensitivity was described in 1978 [1]. Numerous applications have been demonstrated that utilize fiber grating as dispersion compensator, density wavelength division multiplexer (DWDM), fiber laser, fiber sensors and so on [2–5]. Stevenson and Cozens et al. analyzed an optical fiber with a single-crystal core for the first time in 1974 [6,7]. A doubly-clad W-type optical fiber with an inner cladding made of uniaxial crystal material has been studied in 2002 [8], and the studied results indicated that the adjustable range *
Corresponding author. Address: Key Laboratory of Optical Communication and Lightwave Technologies of Ministry of Education, Beijing University of Posts and Telecommunications, P.O. Box 66, BUPT, Beijing 100876, China. Tel.: +86 10 6228400486; fax: +86 931 2756544. E-mail address: [email protected] (S. Hou). 0030-4018/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.10.009
of zero dispersion wavelength of this kind fiber is wilder than that of common doubly-clad optical fiber. The fiberoptic polarizer with cladding made of uniaxial crystal material was also described [9]. Electro-optic effect and elasto-optic effect in a chirped fiber grating with cladding made of uniaxial crystal material are theoretically analyzed and the results indicate that the reflective spectra of the chirped grating can be changed by the electric field and the strain applied to the fiber grating cladding along z-axis [10]. The paper published in 2003 [11] predicted the characteristics of a new type of fiber Bragg grating with cladding made of uniaxial crystal material whose optical axis, i.e., z-axis, is parallel to the axis of fiber Bragg grating, and the calculated results indicate that parameter Kcl, i.e., the ratio of the extraordinary ray refractive index to the ordinary ray index, has a strong impact on the reflectivity, Bragg wavelength, the bandwidth of the reflected wave and the dispersion.
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S. Hou et al. / Optics Communications 271 (2007) 109–115
In this work, the reflectivity and the dispersion of the fiber Bragg grating with cladding made of uniaxial crystal material were calculated again and an error was corrected in calculating the effective index. The calculated results which are different from Ref. [11] were described in detail, which indicate that no low frequency cutoff phenomenon or no cutoff wavelength kC exists in the HE11 mode, more power is confined in the core of the fiber with cladding made of isotropic material and the power transmitted by the cladding is more easily coupled into the backwardpropagating guided fundamental mode. The parameter Kcl has a strong impact on the reflectivity and Bragg wavelength of this kind of fiber Bragg grating, especially when Kcl is varied from 1 to 1.01. The reflectivity of the fiber Bragg grating with cladding made of uniaxial crystal material is much higher than that with cladding made of isotropic material. The dispersion is very similar as described in Ref. [11] except when Kcl = 1. Therefore, the characteristics of the fiber Bragg grating with uniaxial crystal cladding can be optimized through adjusting Kcl while keeping its length, periodicity and the other parameters as constants.
8 < ðr2 þ k 2 n2 n2z2 b2 Þez ¼ 0 t z nt : ðr2 þ k 2 n2 b2 Þh ¼ 0 t
t
ðr > aÞ;
where nt = nx = ny and n0 > nt. k ¼ 2p is the wave number in k vacuum and b is the propagation constant. If b2 < k 2 n20 , the parameters are defined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nz n0 nt ; U ¼ a k 2 n20 b2 ; K cl ¼ ; D ¼ nt n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 W ¼ a b k nt ; V ¼ ak n20 n2t ; where Kcl is the ratio of the extraordinary to the ordinary ray refractive index and V is normalized frequency. Through matching the relationship between axial and tangential field components, and using the boundary conditions of electromagnetic field, the characteristic equation of the ideal normal mode, i.e., the mode in an ideal waveguide without grating perturbation, can be obtained as follows: 2 2 n J 0m ðU Þ K 0 ðK cl W Þ m2 Q n0 1 F 20 þ m þ ; ¼ K cl n2t U 2 W 2 nt K cl UJ m ðU Þ WK m ðK cl W Þ
2. Theoretical analysis
ð3Þ
2.1. The characteristic equation of the ideal normal mode The sketch of the fiber Bragg grating with cladding made of uniaxial crystal material was shown in Fig. 1. It has a core radius a and a core refractive index n0. The cladding with an infinite diameter is made of uniaxial crystal material whose optical axis is taken to be parallel to the axis of the fiber, i.e. z-axis, and its principal axis indices are nx, ny and nz, respectively, which satisfy: nx = ny 5 nz. For the optical fiber without the grating perturbation, the axial electric and magnetic field components satisfy wave equation as follows [12]: ( ðr2t þ k 2 n20 b2 Þez ¼ 0 ðr < aÞ; ð1Þ ðr2t þ k 2 n20 b2 Þhz ¼ 0
X
cladding O
ð2Þ
z
core
Z
2a
Λ L Fig. 1. The sketch of the fiber Bragg grating with cladding made of uniaxial crystal material.
where Jm, Km are the Bessel and modified Bessel functions respectively. The parameters F and Q are defined as follows: F ¼
J 0m ðU Þ K 0 ðW Þ þ m ; UJ m ðU Þ WK m ðW Þ
Q¼
1 1 þ : U2 W 2
2.2. The grating coupling coefficient We studied an unchirped uniform grating only and assumed that a perturbation to the core refractive index of the grating can be described as follows: 2p z ; ð4Þ dnco ðzÞ ¼ dnco 1 þ v cos K where dnco is ‘‘dc’’ index change spatially averaged over a grating period and its value is generally 1.0 · 104. v is the fringe visibility of the index change and its value is 1 in this paper. K is the grating period, L is the grating length, N is the total numbers of the grating periods and K = L/N as shown in Fig. 1. According to the coupled-mode theory, the traverse coupling coefficient between the forward-propagating guided fundamental mode and the backward-propagating guided fundamental mode can be described as follows [13,14]: Z Z xe0 K¼ ½n2 ðzÞ n20 ~ et ~ et dx dy; ð5Þ 4P core where P is the power of the incident ray and normalized P = 1, ~ et is traverse components of electric field of the HE11 mode, x is circle frequency, e0 is the permittivity in free space, n(z) is the core index of the fiber Bragg grating and it can be expressed as
S. Hou et al. / Optics Communications 271 (2007) 109–115
nðzÞ ¼ n0 þ dnco ðzÞ: We define Z Z ^ ¼ xe0 ~ et dx dy: et ~ K 4P core
ð6Þ
k 2 bðQ2 F 2 Þ : 2ðb2 Q2 k 2 n20 F 2 Þ
Then, the coupling coefficient can be described as 2p K ¼ r þ 2j cos z ; K
ð7Þ
ð8Þ
ð9Þ
^ co is called the ‘‘dc’’ (period-average) where r ¼ 2n0 Kdn ^ co is called the ‘‘ac’’ coupling coefficient and j ¼ n0 vKdn coupling coefficient. 2.3. Reflectivity c According to coupled-mode theory and the boundary condition of the fiber Bragg grating, the amplitude reflection coefficient can be shown as follows [15]: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^2 L j sinh j2 r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q¼ ^ sinh ^2 L þ i j2 r ^2 cosh ^2 L r j2 r j2 r ð10Þ and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^2 L sinh2 j2 r 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ¼ jqj ¼ ; 2 ^2 L rj^2 cosh2 j2 r
ð11Þ
^ ¼ d þ r is a general ‘‘dc’’ self-coupling coefficient. where r The detuning d is defined as p 1 1 d ¼ b ¼ 2pneff : ð12Þ K k kB The Bragg wavelength kB is expressed as kB 2neff K:
3. Calculated results and analysis 3.1. The effective index of the HE11 mode
Inserting the traverse component of electric field derived in Section 2.1 to Eq. (7), we can obtain ^¼ K
111
The parameters were set exactly as same as those in Ref. [11], such as m = 1, n0 = 1.46, D = 0.02, a = 5 lm, L = 1.06 cm, N = 20,000. A cluster of curves of the effective index as function of the normalized frequency V, i.e., (b/k)–V curves of the HE11 mode can be obtained with various Kcl from Eq. (3), as shown in Fig. 2. It can be seen that the effective indices increase with increasing V, and the shape of the curves is very similar to that with Kcl = 1. But the effective indices also increase with the increasing Kcl for a given value of V. For both Kcl > 1 and Kcl < 1, V can be arbitrarily small so that the wavelength can be very large due to correcting an error in calculating the characteristics equation, i.e., Eqs. (3) or (5) in Ref. [11], that is to say, no low frequency cut-off phenomenon and no cutoff wavelength kC exist when Kcl < 1 in the HE11 mode, contrary to the findings described in Ref. [11]. This means that when Kcl < 1, the operating wavelength k of the single-mode fiber is not limited when the single mode condition is satisfied. 3.2. Influence of Kcl on reflectivity c and Bragg wavelength kB The curves of the reflection coefficient c as functions of the wavelength k, i.e., c–k curves with varied Kcl can been obtained from Eq. (11) shown in Fig. 3(a–c) for L = 1.06 cm, 0.106 cm and 10.6 cm, respectively and Kcl = 0.2, 0.75, 0.8, 1.0, 1.2 and 1.8, the other parameters are shown in the figure caption. From Fig. 3(a), it can be seen that Kcl has a strong impact on the c–k curves. When Kcl < 1, the resulted curves moved to the left of the curve with Kcl = 1, but when Kcl > 1, the resulted curves moved to the right of the curve with Kcl = 1. The larger the Kcl deviated from 1, the larger the resulted curves deviated from the curve with Kcl = 1 and the degrees of the deviation β/k
ð13Þ
1.4594 1.4592 1.4590
2.4. Dispersion dq
Kcl=1.0
1.4588
Kcl=0.1
1.4586
The phase of Eq. (10) is expressed as hq. A delay time sq of the grating is defined as follows [3]:
Kcl=1.2
1.4584
Kcl=0.3
1.4582
Kcl=0.5
1.4580
sq ¼
dhq k2 dhq ¼ : dx 2pc dk
ð14Þ
The dispersion dq (in ps/nm) is the first-order derivatives of sq to k, therefore,
Kcl=0.7
1.4578 1.4576
Kcl=0.9
1.4574 1.4572 0.5
dsq 2sq k2 d 2 hq 2pc d2 hq ¼ dq ¼ ¼ : dk k 2pc dk2 k2 dx2
ð15Þ
1.0
1.5
2.0
2.5
3.0
3.5
4.0
V
Fig. 2. Effective index of HE11 mode as a function of V with various Kcl.
112
S. Hou et al. / Optics Communications 271 (2007) 109–115 ×100 % 1.0
Kcl=0.2
0.8
Kcl=0.8 Kcl=1.2 Kcl=1.8
Kcl=1.0
0.6
0.4
0.2
0.0 1548.50
1548.75
1549.00
1549.25
1549.50
1549.75
1550.00
1550.25
(nm) ×100% 0.05
Kcl=0.75
Kcl=1.8
0.04
0.03
0.02
Kcl=1.0 0.01
0.00 1548.0
1548.5
1549.0
1549.5
1550.0
1550.5
1551.0
(nm)
×100 % 1.0
K cl =0.75 0.8
[11] was added to show that no low frequency cutoff phenomenon exists. The curve with Kcl = 1 is consistent with that reported in the previous study [3], but its maximum value of the reflection coefficient c which is 68.3% is much lower than the others and its Bragg wavelength shifts to the side of Kcl < 1. It indicates that the variation of Kcl also results in the variation of the maximum value of the reflection coefficient c. When Kcl are varied values and L = 0.106 cm, the c–k curves are shown in Fig. 3(b). From Fig. 3(b), the maximum values of reflection coefficient c with Kcl = 0.75 or 1.8 are much greater than that with Kcl = 1. Fig. 3(c) shows the c–k curve with Kcl = 0.75 and L = 10.6 cm. In order to further study the influence of Kcl on reflectivity, the curves of reflectivity versus wavelength with Kcl varied from 1.000 to 1.009 by steps of 0.001 are calculated and the resulted c–k curves are shown in Figs. 4 and 5(a–c) with (a) Kcl = 1.007, (b) Kcl = 1.008, (c) Kcl = 1.009 and L = 1.06 cm which indicate that the reflection coefficient c varied greatly. From Fig. 4, it can be seen that the maximum value of the reflection coefficient c decreases with increasing Kcl until Kcl = 1.004, that is to say, for Kcl = 1.004, the maximum value of the reflectivity coefficient is the lowest, only 7.7%, but for Kcl = 1.006, it suddenly increases to 85.1%. From Fig. 5, it can be seen that the reflectivity c is asymmetric on the two sides of the maximum reflectivity. Fig. 5(a) shows that the maximum values of reflection coefficient c reach 100% and the maximum has a greater width of 0.26 nm; when Kcl = 1.008, there are many peak values which reach 100% when k < 1548.97 nm, and the widest bandwidth of 0.1785 nm exists near 1548.3 nm as shown in Fig. 5(b) for wavelength between 1540 nm to 1550 nm. From Fig. 5(c), there are three peak values which reach 100% and the bandwidth of the left one is larger than that of the two others, the peak values on the right side of the widest one are much higher than the corresponding ones γ ×100% 0.9 0.8
0.6
0.7 0.6
0.4
0.5 0.4
0.2
0.3
Fig. 3. Reflection spectra versus wavelength for various Kcl and L with n0 = 1:46, D = 0.02, a = 5 lm. (a) N = 20,000, L = 1.06 cm; (b) N = 2000, L = 0.106 cm and (c) N = 200,000, L = 10.6 cm.
are remarkably different when Kcl < 1 or Kcl > 1. These results are similar to those described in Ref. [11], but the curve with Kcl = 0.2 that cannot exist according to Ref.
0.1 0.0 1549.00 1549.25
1.006 1.004 1549.50
λ(
nm )
1.002 1549.75 1550.00
cl
(nm )
0.2
K
0.0 1549.20 1549.22 1549.24 1549.26 1549.28 1549.30 1549.32 1549.34
1.000
Fig. 4. Reflection spectra versus wavelength with Kcl from 1.000 to 1.006 and n0 = 1.46, D = 0.02, a = 5 lm, N = 20,000, L = 1.06 cm.
S. Hou et al. / Optics Communications 271 (2007) 109–115
γ ×100%
×100% 1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.0 1548.50
0.2
0.0 1548.5 1548.75
1549.00
1549.25
1549.50
1549.75
1550.00
1.020 1.018
1549.0
(nm)
λ(
1549.5
nm
γ ×100%
)
1.016 1.012 1550.5
K cl
1.010
Fig. 6. Reflection spectra versus wavelength with Kcl from 1.01 to 1.02 and n0 = 1.46, D = 0.02, a = 5 lm, N = 20,000, L = 1.06 cm.
0.8
0.6
0.4
0.2
1542
1546
1544
1548
1550
λ (nm)
γ ×100% 1.0
0.8
0.6
while its maximum value varies slightly from 100% to 98.9%. Fig. 7 shows the curve of the maximum value of reflection coefficient as function of Kcl, i.e., cmax–Kcl curve with N = 20,000 and L = 1.06 cm. It indicates that the maximum reflectivity varies greatly near to Kcl = 1. From left to Kcl = 1.004, it decreases rapidly and reaches the lowest point when Kcl = 1.004, then increase rapidly to 100% when Kcl changes from 1.007 to 1.009, at last it decreases to 94.5% and remains at this value. Fig. 8 shows that the curves of kB versus Kcl with N = 20,000 and L = 1.06 cm. It can be seen that when Kcl < 1, kB varies more rapidly than that when Kcl > 1, but the degree is not as strong as described in Ref. [11]. When Kcl near to 1.0, kB decreases rapidly and reaches the lowest γ ×100%
0.4
1.0
0.2
0.8
0.0 1549.0
1.014
1550.0
1.0
0.0 1540
113
0.6
1549.5
1550.0
1550.5
1551.0
1551.5
1552.0
λ (nm)
Fig. 5. Reflection spectra versus wavelength with (a) Kcl = 1.007, (b) Kcl = 1.008, (c) Kcl = 1.009 and n0 = 1.46, D = 0.02, a = 5 lm, N = 20,000, L = 1.06 cm.
on the left side. Fig. 6 shows the reflection coefficient c versus wavelength k with Kcl varied from 1.01 to 1.02 by steps of 0.002, it can be seen that the two sides of the peak are symmetric, but its top angle varies from right angle to smooth and its bandwidth decreases with Kcl increasing,
0.4
0.2
0.0 0.0
0.5
1.0
1.5
2.0
2.5
Kcl Fig. 7. The maximum reflectivity as a function of Kcl with n0 = 1.46, D = 0.02, a = 5 lm.
114
S. Hou et al. / Optics Communications 271 (2007) 109–115 λ B (nm)
η × 100%
1550.0
1.0
1549.9
kcl=1.0
1549.8 1549.7
0.8
kcl=0.2
1549.6 1549.5
0.6
1549.4
kcl=1.2
1549.3
kcl=0.8
0.4
1549.2 1549.1 1549.0
0.2
1548.9 1548.8
0.0
1548.7 0.0
0.5
1.0
1.5
2.0
0.5
2.5
1.0
1.5
point when Kcl is between 1.00 and 1.01. These results are consist with results shown in the above Figs. 1–4. In order to interpret the pronounced minimum in the reflectivity curve close to a refractive index ratio of unity of the cladding material, the power distribution between core and cladding of this kind of fiber was predicted as following part. 3.3. Influence of Kcl on power distribution It is clear that a substantial amount of light power is transported by the cladding, while most of the power is confined in the core. Thus, the total power flow is the sum of power carried by both the core and by the cladding in the HE11 mode. The formula expressing this statement is Ptotal = Pcore + Pclad, where Pcore and Pclad is the integration of the longitudinal Poynting vector ~ S z with the corresponding cross-sectional area by inserting the traverse component of electric field derived in Section 2.1. We define g = Pcore/Ptotal expressing the power distribution. Thus, the curves of g versus V with various Kcl are calculated and the resulted g–V curves are shown in Fig. 9, while the other parameters are shown in the figure caption. From Fig. 9, it can be seen that more power is confined in the core when Kcl near to 1 for a given value of V, i.e., V = 2.0. It indicates that less power is confined in the core of the fiber with cladding made of uniaxial crystal material than that with cladding made of isotropic material while keeping the other parameters of the fiber as constant. In order to further study the influence of Kcl on the power distribution, the curve of g versus Kcl with V = 2.0 is calculated and the resulted g–Kcl curve is shown in Fig. 10. It can be seen that g increases with Kcl increasing and it is 76% when Kcl = 1, maximum of 100% when Kcl = 1.004 and then drop to zero when Kcl = 1.007. It indicates that more power is transmitted by the core of the fiber with cladding made of isotropic material.
2.5
3.0
3.5
4.0
V
K cl Fig. 8. kB as a function of Kcl with n0 = 1.46, D = 0.02 and a = 5 lm.
2.0
Fig. 9. g as a function of V with various Kcl and n0 = 1.46, D = 0.02, a = 5 lm.
η × 100%
1.0
V=2.0
0.8
0.6
0.4
0.2
0.0 0.0
0.5
1.0
1.5
2.0
Kcl
Fig. 10. g as a function of Kcl with n0 = 1.46, D = 0.02, a = 5 lm.
It follows from the above discussion that more power is confined in the core of the fiber with cladding made of isotropic material. For fiber Bragg grating, the power transmitted by the cladding is more easily coupled into the backward-propagating guided fundamental mode, thus, the more power carried by the cladding, the higher reflectivity of the fiber Bragg grating. 3.4. Influence of Kcl on dispersion The dispersion curves calculated from Eq. (15) with Kcl = 0.2, 1.0 and 1.8 are shown in Fig. 11, where the corresponding curves of the reflectivity are also given, in order to show the influence of Kcl on the dispersion. It can be seen that the dispersion curves also shifted when Kcl < 1 or Kcl > 1 and its shifting rule is very similar to that of the reflectivity curves as described in Ref. [11]. When k = kB, i.e., at the maximum reflectivity, the dispersion is
S. Hou et al. / Optics Communications 271 (2007) 109–115
γ × 100% 1.0
dispersion
Kcl=0.2
0.8
reflectivity
Kcl=1.8
dρ(in ps/nm)
Kcl=1.0
8
6
0.6 4 0.4
115
mode, the parameter Kcl has a stronger impact on the reflectivity and Bragg wavelength between 1.00 and 1.01 than in others regions. The reflectivity of the fiber Bragg grating with cladding made of uniaxial crystal material is much higher than that with cladding made of isotropic material. The dispersion is much similar as described in Ref. [11] except for Kcl = 1. Therefore, the characteristics of the fiber Bragg grating with uniaxial crystal cladding can be optimized through adjusting Kcl while keeping its length, periodicity and the other parameters as constants.
2 0.2
0.0 1548.5
Acknowledgements 0 1549.5
1549.0
1550.0
λ(nm)
Fig. 11. Dispersion and reflectivity for various Kcl with n0 = 1.46, D = 0.02, a = 5 lm, N = 20,000, L = 1.06 cm.
minimum and the peak value of the dispersion appears on the right side of the peak value of reflectivity. But when Kcl = 1.0, because of its lower reflectivity, the distance between the two peak values of the dispersion and of reflectivity is greater than that with other Kcl. 4. Conclusion The reflectivity and the dispersion of the fiber Bragg grating with cladding made of uniaxial crystal material were calculated more precisely and the calculated results which are different from the paper published in 2003 [11] were described in detail, which indicate that no low frequency cutoff phenomenon or no cutoff wavelength kC exists in the HE11 mode, more power is confined in the core of the fiber with cladding made of isotropic material and the power transmitted by the cladding is more easily coupled into the backward-propagating guided fundamental
This work was supported by National Basic Research Program of China (2003CB314906), National 863 High Technology Project of China (2003AA311010), the Key grant Project of Chinese Ministry of Education (No. 104046) and Natural Science Fund of Gansu Province (ZS032-B25-006), China. References [1] K.O. Hill, Y. Fujii, D.C. Johnson, et al., Appl. Phys. Lett. 32 (1978) 647. [2] K.O. Hill, G. Meltz, J. Lightwave Technol. 15 (1997) 1263. [3] Turan Erdogan, J. Lightwave Technol. 15 (1997) 1277. [4] G.A. Ball, W.H. Glenn, J. Lightwave Technol. 10 (1992) 1338. [5] A.D. Kersey, Optic Fiber Technol. 2 (1996) 291. [6] J.L. Stevenson, R.B. Dyott, Electron. Lett. 10 (1974) 449. [7] J. Cozens, Electron. Lett. 12 (1974) 41. [8] X.P. Zhang, Z.H. Tan, Opt. Commun. 204 (2002) 127. [9] M. Wakaki, Y. Komachi, H. Machida, Appl. Opt. 35 (1996) 2591. [10] X.P. Zhang, D.C. Lv, Opt. Quant. Electron. 36 (2004) 469. [11] X.P. Zhang, S.L. Hou, Optics Commun. 219 (2003) 193. [12] A.W. Snydan, J.D. Love, Optical Waveguide Theory, Chapman and Hall, London, 1983. [13] P.D. Ye, Fiber Optic Theory, Knowledge Press, Beijing, 1985. [14] A. Yariv, IEEE J. Quantum Electron. 9 (1973) 919. [15] H. Kogelnik, in: T. Tamir (Ed.), Guided-wave Optoelectronics, Springer-Verlag, New York, 1990.