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Pulse broadening in optical fiber with polarization mode dispersion and polarization dependent loss
Optics Communications 227 (2003) 83–87 www.elsevier.com/locate/optcom
Pulse broadening in optical fiber with polarization mode dispersion and polarization dependent loss Ling-wei Guo *, Ying-wu Zhou, Zu-jie Fang Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, PR China Received 19 June 2003; received in revised form 3 September 2003; accepted 5 September 2003
Abstract The pulse broadening of optical signals in a single mode fiber was studied theoretically in presence of polarization mode dispersion, polarization dependent losses, chromatic dispersion, and spectral chirping. Analytical expressions were derived for the parameters of pulse broadening characteristics without simplification assumptions with respect to the pulse shape and to the order of dispersion. Comparisons with the published theoretical work by numerical simulation showed the compatibility and capability of the expressions. Ó 2003 Elsevier B.V. All rights reserved. PACS: 48.81.Dp; 42.81.Gs Keywords: Polarization mode disperison; Polarization dependent loss
1. Introduction The effect of polarization dependent loss (PDL) is one of the key issues in the exploration of polarization mode dispersion (PMD), especially in high-speed optical communication systems because the combined effects of PMD and PDL will cause anomalous broadening [1–5]. Gisin and Hunter [1,5] considered the combined effects of PMD and PDL and derived an analytic expression of PMD. But the higher-order PMD terms and the combined effects of chromatic dispersion and chirping were neglected in their derivation. Li and Yariv [3] solved the dynamic equation, and compared theoretical results with numerical simulations. Chen et al. [2,4] gave numerical simulation results for the statistics of PMD. In this paper analytic expressions are derived in frequency domain without simplification assumptions with respect to the pulse shape and to the order of dispersion. The expressions can then be used to analyze the combined effects of PDL, PMD, chromatic dispersion and spectral chirping.
*
Corresponding author. Tel.: +862159914167; fax: +862159918183. E-mail address: [email protected] (L.-w. Guo).
0030-4018/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2003.09.019
84
L. Guo et al. / Optics Communications 227 (2003) 83–87
2. Analytical result for the amount of pulse broadening Our method is based on the so-called moment method [6]. The root-mean-square (RMS) value of the pulse broadening propagating in a fiber with arbitrarily birefringence and dispersions can be derived as follows. If a pulse Win ðxÞ ¼ Jf ðxÞ is launched into a fiber, where J is the Jones vector and f ðxÞ is the input pulse shape. Then the output pulse Wout ðxÞ can be expressed as Wout ðxÞ ¼ T ðxÞWin ¼ T ðxÞ expði/Þjf jJ :
ð1Þ
Here T ðxÞ is the transmission matrix, which is the same as GisinÕs notations [1] with the following definition: T ðxÞ ¼ AN TN ðxÞ; AN ¼ exp
a0
N X
! aj =2 ;
j¼1
TN ðxÞ ¼ exp½ði~ bN x þ ~ aN Þ ~ r=2 exp½ði~ b2 x þ ~ a2 Þ ~ r=2 exp½ði~ b1 x þ ~ a1 Þ ~ r=2: ~ aj is PDL vector of the jth trunk. Here ~ r are the Pauli matrices, bj is the PMD vector of the jth trunk, and ~ ~ bj , ~ aj and AN can be considered to be independent of optical frequency x, and ~ bj and ~ aj are parallel with qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi each other. The RMS width of the pulse defined as dt ¼ by two moments, hti ¼ w1 =w and ht2 i ¼ w2 =w, where Z 1 dWout ðxÞ dx ; w1 ¼ i Wyout ðxÞ dx 2p 1 Z 1 dWyout ðxÞ dWout ðxÞ dx w2 ¼ ; dx dx 2p 1 Z 1 dx w¼ : Wyout ðxÞWout ðxÞ 2p 1
2
ht2 i hti can be expressed in frequency domain
ð2Þ ð3Þ ð4Þ
Henceforth, we denote the derivation with respect to the optical frequency x with a prime. Substituting Eq. (1) into Eqs. (2) and (4), we obtain Z 1 dx w¼ ; ð5Þ A2N jf j2 J y TNy TN J 2p 1 Z 1 0 djf j dx 2 y y y 0 2 y y w1 ¼ i : ð6Þ AN jf j J ½TN TN ði/ Þ þ TN TN ðxÞJ þ J TN TN J jf j dx 2p 1 0
1
Using properties of the transmission matrix given in [1], we obtain Tr ðTN1 TN Þ ¼ Tr ðTN0 TN Þ ¼ 0. Based on the fact that the Pauli matrices form a basis for the 2 2 matrices of zero trace, thus there exist a three~ as that component complex vector W i ~ ~ r: ð7Þ TN1 TN0 ¼ W 2 Here a positive symbol is used in the right-hand side of Eq. (7) based on a comparison with the result of 0 ~ =2 and ~ references. In [7] one knows that J y TN1 TN J ¼ J y TNy TN0 J ¼ i~ J W J is the input Stokes vector corre~ ~ is complex. As the lasers sponding to J . W is real vector when no PDL exists. In a general case, the vector W used in telecommunications are narrow line-width sources, this implies that the Jones vector Wout ðxÞ is a constant complex scalar to first order in optical frequency. So following [1], we can also get ~ ~ ði=2ÞðW rÞWin ¼ iðk=2ÞWin , which define two input principal polarization states W , and its corresponding eigenvalue value can be written as k ¼ ðds þ igÞ:
L. Guo et al. / Optics Communications 227 (2003) 83–87
According to [6], one can set T y T a0 I þ ~ a ~ r: Then the transmission coefficient can be written as a ~ J Þ: jð~ J Þ ¼ AN ða0 þ ~
85
ð8Þ ð9Þ
The minimum and maximum transmission coefficients are thus given by jmax ¼ A2N ða0 þ aÞ; jmin ¼ A2N ða0 aÞ;
ð10Þ
where a is the norm of vector ~ a. From Eq. (10), the PDL and the attenuation for depolarized light jdepol can be deduced [8] as follows: jmax a0 þ a PDLdb ¼ 10 log10 ¼ 10 log10 ½db; ð11Þ a0 a jmin jmax jmin a ¼ ; ð12Þ C¼ jmax þ jmin a0 jmax þ jmin ¼ A2N a0 : jdepol ¼ ð13Þ 2 For the useful expression of PDL in a linear scale, we define that ~ C ¼~ a=a0 . Now substituting (7) and (8) into Eqs. (2) and (4), we obtain Z 1 dx ; ð14Þ A2N jf j2 ða0 þ ~ a ~ JÞ w¼ 2p 1 Z 1 0 i ~ djf j dx w1 ¼ i : W ðzÞ ~ r J þ ða0 þ ~ A2N jf j2 ða0 þ ~ a ~ rÞ a ~ J Þjf j a ~ J Þði/ Þ þ J y ða0 þ ~ 2 dx 2p 1 ð15Þ ~ into real and imaginary components. Then Eq. (15) can be ~ ðzÞ ~ We decompose the vector W Xin þ iK transformed into Z 1 0 1 dx ~in I ð~ ~in ~ ~Þ ~ w1 ¼ A2N jf j2 ða0 þ ~ r þ~ aX aK rJ a ~ J Þð/ Þ þ J y ½a0 X 2 2p 1 Z 1 Z 1 i dx djf j dx 2 ~in Þ ~ ~ ~ þi : ð16Þ A2 jf j J y ½ð~ aX r þ a0 K A2N ða0 þ ~ a ~ J Þjf j r þ~ a~ KJ 2 1 N 2p dx 2p 1 For w1 to be real, the last two terms of Eq. (16) must be zero for all ~ J . Then we have Z
1
djf j dx 1 dx 2p 2
Z
1
dx 2 ~in Þ ~ ~ ~ 0: ð17Þ A2N jf j J y ½ð~ aX r þ a0 K r þ~ a~ KJ 2p 1 1 ~; K ~; a0 and ~ It should be noticed that Eq. (17) uncover explicitly the interaction among jf j; X a. By using similar processing, w2 can be deduced to Z 1 2 1 ~~ ~ ðX ~~ ~ÞðX ~ ~ ~~ ~ þ 2ð~ ½X2 þ K2 þ 2ðX w2 ¼ jdepol jf j CÞ ~ J þ ðK CÞ X CÞ K CX JÞ 4 1 0 2 0 2 ~ ~ 2 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ X ðC J Þ þ 2ðC KÞðK J Þ K ðC J Þ þ ½X J þ C K þ ðK CÞ J ð/ Þ þ ½1 þ C J ð/ Þ A2N ða0 þ ~ a ~ J Þjf j
jdepol jf j
2 djf j ~ ~ ~ ~ djf j dx ~Þ ~ : ½1 þ ~ C ~ J ðK J þ C K þ ð~ CX J Þ þ jdepol dx dx 2p
ð18Þ
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L. Guo et al. / Optics Communications 227 (2003) 83–87
In the case of no PDL, we can reproduce the same as the KarlssonÕs results Eqs. (7) and (8) of [7] from our results Eqs. (16) and (18). Eqs. (16)–(18) are the main results of this paper and especially Eq. (17) gives ~; a0 and ~ an explicit expression to describe the interaction among jf j; ~ X; K a. The obvious advantage of our results is that it is not necessary to make any simplifying assumption with respect to the pulse shape and the ~in ; /0 ; f , order of PMD. In other words, this theory is valid for arbitrary frequency dependences of W ~in ðxÞ, ~ therefore it should be quite useful . For application, one can substitute measured values of W CðxÞ, jdepol ðxÞ into Eqs. (16) and (18), and performs the integration over the pulse spectrum.
3. An example and discussion To demonstrate the effectiveness of this method, we make a comparison with the calculations by Chen et al. [2], in which a PDL element was sandwiched between two wave plates with birefringence b3 ; b1 , respectively, as an example model. The same configuration as [2] is used, that is, two identical PMD sections with 20 ps DGD are arranged in parallel; and a PDL middle section is aligned 45° with respect to the PMD direction, and it was parallel with the input polarization. The transfer matrix is written as p p T ðxÞ ¼ expðjaj=2Þ U ðixb3 Þ R ð19Þ U ðaÞ R U ðixb1 Þ; 4 4 where R and U are written as follows: cosðhÞ sinðhÞ RðhÞ ¼ ; ð20Þ sinðhÞ cosðhÞ expðx=2Þ 0 : ð21Þ U¼ 0 expðx=2Þ Here the input linearly polarization electric field of the Gaussian pulse is 1 cosðhin Þ 2 2 Einput ðtÞ ¼ cosðx tÞ expðt =4T Þ : 0 1=4 sinðhin Þ ð2pÞ T 1=2
ð22Þ
By solving Eqs. (7) and (8), we get ~ ¼ ðb3 coshðaÞ b1 ; ib3 sinðxb1 Þ sinhðaÞ; ib3 cosðxb1 Þ sinhðaÞÞ; W
ð23Þ
~ ¼ ðb3 coshðaÞ b1 ; 0; 0Þ; X
ð24Þ
~ ¼ ð0; b3 sinðxb1 Þ sinhðaÞ; b3 cosðxb1 Þ sinhðaÞÞ; K
ð25Þ
a0 ¼ coshðaÞ;
ð26Þ
~ a ¼ ð0; sinhðaÞ cosðxb1 Þ; sinhðaÞ sinðxb1 ÞÞ:
According to Eqs. (24)–(26), we can find Eq. (17) is really constant equal to zero for all ~ J , and we also arrive at the same result as Eq. (17) of [1] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds ¼ b21 þ b23 þ 2b1 b3 coshðaÞ: To calculate the effect of the pulse-width broadening, we choose T ¼ 25 ps and x0 ¼ 400p (rad/ps). The pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi effect of pulse-width broadening after transmission can be calculated by s ¼ s2out s2in : Inserting Eqs. (24)–(26) in Eqs. (16) and (18), we plot the effect of the pulse-width broadening of a shown in Fig. 1. For comparison, the results reported by Chen et al. [2] are also shown in Fig. 1. According to Fig. 1, we can reproduce the result as that reported in [2] that the effective pulse-width broadening can either be larger or smaller compared with the case of a ¼ 0, but this result could not be explained by the theory of [1]. Papers about PMD theory other than [2] cannot predict these results, while the expressions of this paper have given
L. Guo et al. / Optics Communications 227 (2003) 83–87
87
Fig. 1. A comparison of the calculated effect of pulse-width broadening (solid line) with the example data (doted line) [2].
the prediction based on a generalized form rather than restricted in a simplified model. Our theory will be helpful in studying PMD and PDL combined effects without simplified assumptions respect to the pulse shape or to the order of dispersion, and is then suitable for actual transmission systems.
4. Conclusion In this work, a general theory has been developed for calculation of anomalous pulse broadening induced by the combined effects of PMD, PDL, chromatic dispersion and chirping without making any simplifying assumption with respect to the pulse shape and the order of PMD. Analyses and simulations show that it is compatible with the previous theories and more capable of dealing with pulse broadening and distortion in complicated cases, which will be key problems in high-speed optical communication networks.
References [1] [2] [3] [4] [5] [6] [7] [8]
N. Gisin, B. Huttner, Opt. Commun. 142 (1997) 119. L. Chen, J. Cameron, X. Bao, Opt. Commun. 169 (1999) 69. Y. Li, A. Yariv, J. Opt. Soc. Am. B 11 (2000) 1021. P. Lu, L. Chen, X. Bao, J. Lightwave Technol. 6 (2001) 856. B. Huttner, N. Gisin, Opt. Lett. 22 (1997) 504. J.P. Gordon, H. Kogelnik, PNAS 97 (2000) 4541. M. Karlsson, Opt. Lett. 23 (1998) 688. N. Gisin, Opt. Commun. 114 (1995) 399.
Pulse broadening in optical fiber with polarization mode dispersion and polarization dependent loss Ling-wei Guo *, Ying-wu Zhou, Zu-jie Fang Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, PR China Received 19 June 2003; received in revised form 3 September 2003; accepted 5 September 2003
Abstract The pulse broadening of optical signals in a single mode fiber was studied theoretically in presence of polarization mode dispersion, polarization dependent losses, chromatic dispersion, and spectral chirping. Analytical expressions were derived for the parameters of pulse broadening characteristics without simplification assumptions with respect to the pulse shape and to the order of dispersion. Comparisons with the published theoretical work by numerical simulation showed the compatibility and capability of the expressions. Ó 2003 Elsevier B.V. All rights reserved. PACS: 48.81.Dp; 42.81.Gs Keywords: Polarization mode disperison; Polarization dependent loss
1. Introduction The effect of polarization dependent loss (PDL) is one of the key issues in the exploration of polarization mode dispersion (PMD), especially in high-speed optical communication systems because the combined effects of PMD and PDL will cause anomalous broadening [1–5]. Gisin and Hunter [1,5] considered the combined effects of PMD and PDL and derived an analytic expression of PMD. But the higher-order PMD terms and the combined effects of chromatic dispersion and chirping were neglected in their derivation. Li and Yariv [3] solved the dynamic equation, and compared theoretical results with numerical simulations. Chen et al. [2,4] gave numerical simulation results for the statistics of PMD. In this paper analytic expressions are derived in frequency domain without simplification assumptions with respect to the pulse shape and to the order of dispersion. The expressions can then be used to analyze the combined effects of PDL, PMD, chromatic dispersion and spectral chirping.
*
Corresponding author. Tel.: +862159914167; fax: +862159918183. E-mail address: [email protected] (L.-w. Guo).
0030-4018/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2003.09.019
84
L. Guo et al. / Optics Communications 227 (2003) 83–87
2. Analytical result for the amount of pulse broadening Our method is based on the so-called moment method [6]. The root-mean-square (RMS) value of the pulse broadening propagating in a fiber with arbitrarily birefringence and dispersions can be derived as follows. If a pulse Win ðxÞ ¼ Jf ðxÞ is launched into a fiber, where J is the Jones vector and f ðxÞ is the input pulse shape. Then the output pulse Wout ðxÞ can be expressed as Wout ðxÞ ¼ T ðxÞWin ¼ T ðxÞ expði/Þjf jJ :
ð1Þ
Here T ðxÞ is the transmission matrix, which is the same as GisinÕs notations [1] with the following definition: T ðxÞ ¼ AN TN ðxÞ; AN ¼ exp
a0
N X
! aj =2 ;
j¼1
TN ðxÞ ¼ exp½ði~ bN x þ ~ aN Þ ~ r=2 exp½ði~ b2 x þ ~ a2 Þ ~ r=2 exp½ði~ b1 x þ ~ a1 Þ ~ r=2: ~ aj is PDL vector of the jth trunk. Here ~ r are the Pauli matrices, bj is the PMD vector of the jth trunk, and ~ ~ bj , ~ aj and AN can be considered to be independent of optical frequency x, and ~ bj and ~ aj are parallel with qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi each other. The RMS width of the pulse defined as dt ¼ by two moments, hti ¼ w1 =w and ht2 i ¼ w2 =w, where Z 1 dWout ðxÞ dx ; w1 ¼ i Wyout ðxÞ dx 2p 1 Z 1 dWyout ðxÞ dWout ðxÞ dx w2 ¼ ; dx dx 2p 1 Z 1 dx w¼ : Wyout ðxÞWout ðxÞ 2p 1
2
ht2 i hti can be expressed in frequency domain
ð2Þ ð3Þ ð4Þ
Henceforth, we denote the derivation with respect to the optical frequency x with a prime. Substituting Eq. (1) into Eqs. (2) and (4), we obtain Z 1 dx w¼ ; ð5Þ A2N jf j2 J y TNy TN J 2p 1 Z 1 0 djf j dx 2 y y y 0 2 y y w1 ¼ i : ð6Þ AN jf j J ½TN TN ði/ Þ þ TN TN ðxÞJ þ J TN TN J jf j dx 2p 1 0
1
Using properties of the transmission matrix given in [1], we obtain Tr ðTN1 TN Þ ¼ Tr ðTN0 TN Þ ¼ 0. Based on the fact that the Pauli matrices form a basis for the 2 2 matrices of zero trace, thus there exist a three~ as that component complex vector W i ~ ~ r: ð7Þ TN1 TN0 ¼ W 2 Here a positive symbol is used in the right-hand side of Eq. (7) based on a comparison with the result of 0 ~ =2 and ~ references. In [7] one knows that J y TN1 TN J ¼ J y TNy TN0 J ¼ i~ J W J is the input Stokes vector corre~ ~ is complex. As the lasers sponding to J . W is real vector when no PDL exists. In a general case, the vector W used in telecommunications are narrow line-width sources, this implies that the Jones vector Wout ðxÞ is a constant complex scalar to first order in optical frequency. So following [1], we can also get ~ ~ ði=2ÞðW rÞWin ¼ iðk=2ÞWin , which define two input principal polarization states W , and its corresponding eigenvalue value can be written as k ¼ ðds þ igÞ:
L. Guo et al. / Optics Communications 227 (2003) 83–87
According to [6], one can set T y T a0 I þ ~ a ~ r: Then the transmission coefficient can be written as a ~ J Þ: jð~ J Þ ¼ AN ða0 þ ~
85
ð8Þ ð9Þ
The minimum and maximum transmission coefficients are thus given by jmax ¼ A2N ða0 þ aÞ; jmin ¼ A2N ða0 aÞ;
ð10Þ
where a is the norm of vector ~ a. From Eq. (10), the PDL and the attenuation for depolarized light jdepol can be deduced [8] as follows: jmax a0 þ a PDLdb ¼ 10 log10 ¼ 10 log10 ½db; ð11Þ a0 a jmin jmax jmin a ¼ ; ð12Þ C¼ jmax þ jmin a0 jmax þ jmin ¼ A2N a0 : jdepol ¼ ð13Þ 2 For the useful expression of PDL in a linear scale, we define that ~ C ¼~ a=a0 . Now substituting (7) and (8) into Eqs. (2) and (4), we obtain Z 1 dx ; ð14Þ A2N jf j2 ða0 þ ~ a ~ JÞ w¼ 2p 1 Z 1 0 i ~ djf j dx w1 ¼ i : W ðzÞ ~ r J þ ða0 þ ~ A2N jf j2 ða0 þ ~ a ~ rÞ a ~ J Þjf j a ~ J Þði/ Þ þ J y ða0 þ ~ 2 dx 2p 1 ð15Þ ~ into real and imaginary components. Then Eq. (15) can be ~ ðzÞ ~ We decompose the vector W Xin þ iK transformed into Z 1 0 1 dx ~in I ð~ ~in ~ ~Þ ~ w1 ¼ A2N jf j2 ða0 þ ~ r þ~ aX aK rJ a ~ J Þð/ Þ þ J y ½a0 X 2 2p 1 Z 1 Z 1 i dx djf j dx 2 ~in Þ ~ ~ ~ þi : ð16Þ A2 jf j J y ½ð~ aX r þ a0 K A2N ða0 þ ~ a ~ J Þjf j r þ~ a~ KJ 2 1 N 2p dx 2p 1 For w1 to be real, the last two terms of Eq. (16) must be zero for all ~ J . Then we have Z
1
djf j dx 1 dx 2p 2
Z
1
dx 2 ~in Þ ~ ~ ~ 0: ð17Þ A2N jf j J y ½ð~ aX r þ a0 K r þ~ a~ KJ 2p 1 1 ~; K ~; a0 and ~ It should be noticed that Eq. (17) uncover explicitly the interaction among jf j; X a. By using similar processing, w2 can be deduced to Z 1 2 1 ~~ ~ ðX ~~ ~ÞðX ~ ~ ~~ ~ þ 2ð~ ½X2 þ K2 þ 2ðX w2 ¼ jdepol jf j CÞ ~ J þ ðK CÞ X CÞ K CX JÞ 4 1 0 2 0 2 ~ ~ 2 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ X ðC J Þ þ 2ðC KÞðK J Þ K ðC J Þ þ ½X J þ C K þ ðK CÞ J ð/ Þ þ ½1 þ C J ð/ Þ A2N ða0 þ ~ a ~ J Þjf j
jdepol jf j
2 djf j ~ ~ ~ ~ djf j dx ~Þ ~ : ½1 þ ~ C ~ J ðK J þ C K þ ð~ CX J Þ þ jdepol dx dx 2p
ð18Þ
86
L. Guo et al. / Optics Communications 227 (2003) 83–87
In the case of no PDL, we can reproduce the same as the KarlssonÕs results Eqs. (7) and (8) of [7] from our results Eqs. (16) and (18). Eqs. (16)–(18) are the main results of this paper and especially Eq. (17) gives ~; a0 and ~ an explicit expression to describe the interaction among jf j; ~ X; K a. The obvious advantage of our results is that it is not necessary to make any simplifying assumption with respect to the pulse shape and the ~in ; /0 ; f , order of PMD. In other words, this theory is valid for arbitrary frequency dependences of W ~in ðxÞ, ~ therefore it should be quite useful . For application, one can substitute measured values of W CðxÞ, jdepol ðxÞ into Eqs. (16) and (18), and performs the integration over the pulse spectrum.
3. An example and discussion To demonstrate the effectiveness of this method, we make a comparison with the calculations by Chen et al. [2], in which a PDL element was sandwiched between two wave plates with birefringence b3 ; b1 , respectively, as an example model. The same configuration as [2] is used, that is, two identical PMD sections with 20 ps DGD are arranged in parallel; and a PDL middle section is aligned 45° with respect to the PMD direction, and it was parallel with the input polarization. The transfer matrix is written as p p T ðxÞ ¼ expðjaj=2Þ U ðixb3 Þ R ð19Þ U ðaÞ R U ðixb1 Þ; 4 4 where R and U are written as follows: cosðhÞ sinðhÞ RðhÞ ¼ ; ð20Þ sinðhÞ cosðhÞ expðx=2Þ 0 : ð21Þ U¼ 0 expðx=2Þ Here the input linearly polarization electric field of the Gaussian pulse is 1 cosðhin Þ 2 2 Einput ðtÞ ¼ cosðx tÞ expðt =4T Þ : 0 1=4 sinðhin Þ ð2pÞ T 1=2
ð22Þ
By solving Eqs. (7) and (8), we get ~ ¼ ðb3 coshðaÞ b1 ; ib3 sinðxb1 Þ sinhðaÞ; ib3 cosðxb1 Þ sinhðaÞÞ; W
ð23Þ
~ ¼ ðb3 coshðaÞ b1 ; 0; 0Þ; X
ð24Þ
~ ¼ ð0; b3 sinðxb1 Þ sinhðaÞ; b3 cosðxb1 Þ sinhðaÞÞ; K
ð25Þ
a0 ¼ coshðaÞ;
ð26Þ
~ a ¼ ð0; sinhðaÞ cosðxb1 Þ; sinhðaÞ sinðxb1 ÞÞ:
According to Eqs. (24)–(26), we can find Eq. (17) is really constant equal to zero for all ~ J , and we also arrive at the same result as Eq. (17) of [1] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds ¼ b21 þ b23 þ 2b1 b3 coshðaÞ: To calculate the effect of the pulse-width broadening, we choose T ¼ 25 ps and x0 ¼ 400p (rad/ps). The pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi effect of pulse-width broadening after transmission can be calculated by s ¼ s2out s2in : Inserting Eqs. (24)–(26) in Eqs. (16) and (18), we plot the effect of the pulse-width broadening of a shown in Fig. 1. For comparison, the results reported by Chen et al. [2] are also shown in Fig. 1. According to Fig. 1, we can reproduce the result as that reported in [2] that the effective pulse-width broadening can either be larger or smaller compared with the case of a ¼ 0, but this result could not be explained by the theory of [1]. Papers about PMD theory other than [2] cannot predict these results, while the expressions of this paper have given
L. Guo et al. / Optics Communications 227 (2003) 83–87
87
Fig. 1. A comparison of the calculated effect of pulse-width broadening (solid line) with the example data (doted line) [2].
the prediction based on a generalized form rather than restricted in a simplified model. Our theory will be helpful in studying PMD and PDL combined effects without simplified assumptions respect to the pulse shape or to the order of dispersion, and is then suitable for actual transmission systems.
4. Conclusion In this work, a general theory has been developed for calculation of anomalous pulse broadening induced by the combined effects of PMD, PDL, chromatic dispersion and chirping without making any simplifying assumption with respect to the pulse shape and the order of PMD. Analyses and simulations show that it is compatible with the previous theories and more capable of dealing with pulse broadening and distortion in complicated cases, which will be key problems in high-speed optical communication networks.
References [1] [2] [3] [4] [5] [6] [7] [8]
N. Gisin, B. Huttner, Opt. Commun. 142 (1997) 119. L. Chen, J. Cameron, X. Bao, Opt. Commun. 169 (1999) 69. Y. Li, A. Yariv, J. Opt. Soc. Am. B 11 (2000) 1021. P. Lu, L. Chen, X. Bao, J. Lightwave Technol. 6 (2001) 856. B. Huttner, N. Gisin, Opt. Lett. 22 (1997) 504. J.P. Gordon, H. Kogelnik, PNAS 97 (2000) 4541. M. Karlsson, Opt. Lett. 23 (1998) 688. N. Gisin, Opt. Commun. 114 (1995) 399.