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Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 20 — Oct. 6, 2003
  • pp: 2634–2640
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High-birefringence linearly chirped grating based optical device for PMD compensation

Xiaoke Yi, Chao Lu, Xiufeng Yang, Wen-De Zhong, Fang Wei, and Yixin Wang  »View Author Affiliations


Optics Express, Vol. 11, Issue 20, pp. 2634-2640 (2003)
http://dx.doi.org/10.1364/OE.11.002634


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Abstract

We propose a novel delay element for polarization mode dispersion (PMD) compensation by employing a tunable high-birefringence linearly chirped grating. The device can adjust differential group delay in a linearly continuous way and its performance is demonstrated by compensating 10-Gb/s signal with the first-order PMD. The tradeoff between PMD compensation capability of the device and the power penalty caused by the chromatic dispersion of the grating has also been studied.

© 2003 Optical Society of America

1. Introduction

Polarization mode dispersion (PMD) represents a major impairment for high-bit-rate systems, producing pulse broadening and distortion, thus leading to performance degradation. A number of all-optical compensation techniques have been presented to overcome PMD limitations [1

1. S. Lee, R. Khosravani, J. Peng, A. E. Willner, V. Grubsky, D. S. Starodubov, and J. Feinberg, “High-birefringence nonlinearly-chirped fiber Bragg grating for tunable compensation of polarization mode dispersion,” Conference on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1999) pp.. 272–274.

] [2

2. J. Kim, H. Yong, N. Park, and B. Lee., “Polarization-mode-dispersion compensator using a polarization beam splitter and quarter-wave plates,” Appl. Opt. 40, 4473–4475 (2001). [CrossRef]

] [3

3. H. Rosenfeldt, C. Knothe, and E. Brinkmeyer, “Component for optical PMD-compensation in a WDM environment,” European Conference on Optical Communication, pp.135–136 (2000).

]. The heart of an optical PMD compensator is a delay element that can introduce variable differential group delay (DGD) between the two orthogonal polarization axes [4

4. E. Brinkmeyer, “PMD compensation,” European Conference on Optical Communication, No.9.3.1 (2002)

]. Typically, the delay line is achieved by using free space optics [2

2. J. Kim, H. Yong, N. Park, and B. Lee., “Polarization-mode-dispersion compensator using a polarization beam splitter and quarter-wave plates,” Appl. Opt. 40, 4473–4475 (2001). [CrossRef]

] or normal linearly chirped grating [3

3. H. Rosenfeldt, C. Knothe, and E. Brinkmeyer, “Component for optical PMD-compensation in a WDM environment,” European Conference on Optical Communication, pp.135–136 (2000).

]. The former tends to be bulky and its performance may be easily affected by operational environment, and the later has a potential disadvantage in terms of the birefringence induced polarization dependence of the chirped grating [3

3. H. Rosenfeldt, C. Knothe, and E. Brinkmeyer, “Component for optical PMD-compensation in a WDM environment,” European Conference on Optical Communication, pp.135–136 (2000).

][5

5. M. Schiano and G. Zaffiro, “Polarization mode dispersion in chirped fiber gratings,” European Conference on Optical Communication (Optical Society of America, Washington, D.C., 1998) pp.403–404.

]. Another optical PMD compensation technique is the use of a piece of high-birefringence (Hi-Bi) fiber and single or multiple polarization controllers, which has the limitations in tunability and flexibility [6

6. T. Takahashi, T. Imai, and M. Aiki, “Automatic compensation technique for timewise fluctuating polarization mode dispersion in in-line amplifier systems,” Electron. Lett. 30, 348–349 (1994). [CrossRef]

][7

7. T. Ozeki, M. Yoshimura, T. Kudo, and H. Ibe, “Polarization mode dispersion equalization experiment using a variable equalizing optical circuit controlled by a pulse-waveform-comparison technique,” Conference on Optical Fiber Communication, (Optical Society of America, Washington, D.C., 1994) TuN4.

]. Recently, a Hi-Bi grating has been introduced for PMD compensation, where a nonlinearly chirped grating written on a Hi-Bi fiber was used to provide a differential time delay for different states of polarization [1

1. S. Lee, R. Khosravani, J. Peng, A. E. Willner, V. Grubsky, D. S. Starodubov, and J. Feinberg, “High-birefringence nonlinearly-chirped fiber Bragg grating for tunable compensation of polarization mode dispersion,” Conference on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1999) pp.. 272–274.

][8

8. Z. Pan, Y. Xie, S. lee, and A. E. Willner, “Tuanble compensation for polarization-mode dispersion using a birefringent nonlinearly-chirped Bragg grating in a dual-pass configuration,” U.S. Patent 6,400,869 B2, 2002.

]. However, the design and fabrication of the nonlinearly chirped grating increases the scheme’s complexity, and the nonlinear characteristic of the DGD makes adjustment difficult to meet the required DGD values.

In this paper, we propose a novel delay element for adjustable PMD compensation based on a tunable Hi-Bi linearly chirped fiber Bragg grating (LCFBG). It provides a linearly adjustable DGD within the operating bandwidth, without affecting wavelengths outside the bandwidth. Furthermore, the LCFBG can be easily fabricated, compared to the nonlinearly chirped grating [1

1. S. Lee, R. Khosravani, J. Peng, A. E. Willner, V. Grubsky, D. S. Starodubov, and J. Feinberg, “High-birefringence nonlinearly-chirped fiber Bragg grating for tunable compensation of polarization mode dispersion,” Conference on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1999) pp.. 272–274.

], which makes our design more economically attractive. Additionally, the group delay change of the grating is doubled in the DGD curve, which increases the maximum achievable DGD value, and improves the tunable PMD compensation capability of the proposed device. The PMD compensations of a 10-Gb/s signal with 87 ps and 106 ps DGD are used as examples to demonstrate the performance of the device.

2. Principle of operation

Fig. 1. System diagram (a) Schematic diagram of the group delay and amplitude response of a Hi-Bi LCFBG (b) Configuration of the proposed delay element. The incoming signal has polarization components along both the fast (Pf) and slow (Ps) axis. The device generates the relative delay between the two polarizations for the Bragg reflected signal (λi), while it does not affect the signal (λ 0) outside the grating bandwith.

Figure 1(a) shows the schematic diagram of the LCFBG written into a fiber that has a large refractive index difference between its fast and slow axis. The high birefringence gives two almost identical gratings for the orthogonal polarization directions. The position difference of the gratings is Δλ, which is determined by the refractive index difference between the two polarizations [1

1. S. Lee, R. Khosravani, J. Peng, A. E. Willner, V. Grubsky, D. S. Starodubov, and J. Feinberg, “High-birefringence nonlinearly-chirped fiber Bragg grating for tunable compensation of polarization mode dispersion,” Conference on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1999) pp.. 272–274.

]. Due to the fact that the group delay and chromatic dispersion depends on the incident port of the fiber grating, when the fast axis polarization signal at input wavelength λi enters the Hi-Bi LCFBG from the longer wavelength port, it will be reflected and experiences a time delay t1; whereas, the slow axis polarization signal will be reflected and has time delay t2 when it enters from the shorter wavelength port. The time delay difference, Δt=t1-t2, is the DGD between the two axes of polarizations. The key feature of our grating is that the linear variation of the DGD is a function of wavelengths. Furthermore, since the group delay of the grating will be shifted to longer or shorter wavelength by stretching or compressing the grating, the DGD between the fast and slow axes will be increased or decreased for a given wavelength. The linear variation makes it easy for the adjustment of the DGD to any given value.

As shown in Fig. 1(b), the proposed delay element consists of a four-port polarization beam splitter (PBS) and a Hi-Bi LCFBG. The PBS splits the incoming optical signal into two orthogonal polarizations: the fast axis polarization signal (Pf) enters the Hi-Bi LCFBG from the longer wavelength port and the slow one (Ps) from the shorter wavelength port. The polarization states of the signal (λi) within the bandwidth of the grating will be reflected and differently delayed by the grating, as illustrated in Fig. 1(a). By properly tuning the LCFBG, we can adaptively generate a required DGD (Δt) for the PMD compensation. The two orthogonally polarized optical signals are then combined without interference and directed to output 1 of the PBS, where an optical circulator can be used to separate the input signal and the PMD compensated optical signal. All light (λ0) outside the reflection bandwidth of the grating will not be affected and directed to output 2. In addition, when the grating is tuned, the change in DGD (Δτ) will be twice of the change of the grating’s group delay. This increases the maximum achievable DGD value and improves the tunable PMD compensation capability of the proposed delay element.

3. Experimental results and discussions

3.1 Device characterizations

Fig. 2. Relative time delay for two polarization states.

The 5 cm length Hi-Bi LCFBG was written by exposing a hydrogen-loaded Hi-Bi fiber to 244 nm light through a linearly chirped phase mask. To avoid the power loss due to the leakage of the grating, the reflectivity was greater than 99% within the grating bandwidth. The time delay curves of the reflected signals from the unstressed Hi-Bi LCFBG were measured as a function of wavelengths [9

9. Y. Horiuchi, Y. Namihira, and H. Wakabayashi, “Chromatic dispersion measurement in 1.55 um narrow-band region using a tunable external-cavity laser,” IEEE Photon.Technol. Lett. 1, 458–460 (1989). [CrossRef]

] for each polarization inputting from the longer wavelength port and shorter wavelength port respectively. As shown in Fig. 2, the group delay of the grating reveals good linearity, and the delay curves for the two polarization axes are shifted by 0.38 nm in wavelength relative to each other due to the high birefringence of the fiber.

We mounted the grating on a cantilever beam tuning structure [10

10. Z. Qin, Q. Zeng, X. Yang, D. Feng, L. Ding, G. Kai, Z. Liu, S. Yuan, X. Dong, and N. Liu, “Bidirectional grating wavelength shifter with a broad-range tunablility by using a beam of uniform strength,” IEEE Photon. Technol. Lett. 13, 326–328 (2001). [CrossRef]

], which provides ±2.6 nm linear tuning capability through stretching and compression. Fig. 3(a) shows the shape of the reflection spectrum for each polarization, which does not change over the tuning ranges. The state of polarization of the signal reflected from grating was also measured using a polarimeter. As shown in Fig. 3(b), the two orthogonally polarized light after grating reflection appear on the two opposite ends of the Poincare sphere, and no significant change is observed due to the mechanical stressing.

Fig. 3. (a) Reflection spectrum of two polarization states. Wavelength tuning shifts the passband of the grating to longer or shorter wavelength regime for light polarized along the slow (dashed line) and fast (solid line) polarizations without changing in shape of spectrum (b) Measured polarization states.
Fig. 4. DGD as a function of wavelength tuning of the grating.

For PMD compensation or emulation, the device was operated at a single wavelength, 1532 nm in the experiment, where no grating induced DGD exists for the unstressed grating, as shown in Fig. 2. Fig. 4 shows the relationship between the DGD and the wavelength tuning of the grating. It can be seen that when the grating is tuned 2 nm to the longer or shorter wavelength, the DGD between the two polarization states continuously increases or decreases by approximate 110 ps, respectively. The data fitting result shows that the slope of the DGD curve is 55.158 ps/nm, which means that 1 nm wavelength tuning gives approximately 55.158 ps change in DGD. The R-square value of the data fitting is higher than 0.99, and the maximum deviation from the linear fit is ±5 ps, which shows a good linearity of the time delay characteristics and indicates that our proposed device can adjust DGD in a linearly continuous way.

3.2 PMD compensation results and discussions

To demonstrate its capability, a first-order PMD compensation experiment was carried out. A 1532.0 nm DFB laser was externally modulated at 10-Gb/s using nonreturn-to-zero signal. Two polarization maintaining fibers of length 49 m and 60 m with a mean beat length of Λ=3 mm at 1550 nm were used to emulate the first order PMD, which provide about 87 ps and 106 ps DGD, respectively. The top diagram in Fig. 5 shows the baseline eye diagram after the intensity modulator. The middle diagrams in Fig. 5 show the eye diagrams with PMD emulation. It is noted that the eyes are completely closed since the time delays of 87 ps and 106 ps are around one bit period. The bottom diagrams in Fig. 5 show that the eye diagrams are entirely recovered after PMD was compensated by 1.6 nm and 1.9 nm wavelength tuning of the grating respectively, which confirms the effectiveness of our proposed device.

Fig. 5. Eye diagram measurement.

The PMD compensation capability of the device is mainly determined by the characteristics of the Hi-Bi LCFBG, and improved capability can be achieved by adopting higher chirp ratio phase mask to fabricate the Hi-Bi grating. However, this performance improvement will be limited by the induced distortion due to the existence of chromatic dispersion at the fast and slow axes of the compensator. The chromatic dispersion induced pulse broadening will decrease the receiver sensitivity due to the increase in the inter-symbol interference (ISI) and the reduction in the signal-to-noise ratio (SNR) [11

11. G. P. Agrawal, Fiber-optic communications systems, Third Edition, (John Wiley & Sons Inc, 2002), chap.5. [CrossRef]

]. We investigate the impact of chromatic dispersion of the delay element on the bit error rate (BER) performance of the signal, by analyzing the data modulated optical Gaussian RZ pulse transmission at the bit rate of 10-Gb/s.

The electric field for the time slot of bit n (the index begins with 1) can be expressed as:

Pin(t)=ε̂aM=llbn+mG(t(κ+n1M)Tb)
(1)

where t∊[(n-1)Tb, nTb],ε^a is the input Jones polarization vector, G(t) is taken as G(t)=P0e2ln2(tT0)2ejω0t , Po is the peak power of the signal, To is the full-width at the half-maximum (FWHM) of the optical power, ω 0 is the central optical frequency, bn is the n th bit value (“0” or “1”) of the modulating pseudorandom binary sequence (PRBS), Tb=1/rb is the bit period and rb is the bit rate, κ is the relative position of the pulse peak with respect to bit frame. The summation in Eq. (1) includes a pulse of the bit (n) as well as the energy from neighboring pulses. The number of neighboring bits from each side is given by l and l=1 is used in the simulation.

The pulse first propagates through a first-order PMD emulator which induces a DGD (τ 0), and launched into the designed delay element via a polarization controller. Taking into account the existing chromatic dispersion for principal states of polarization (PSPs) of the proposed compensator and the frequency independent PSPs of the designed delay element as indicated in Fig. 3(b), the electrical field at the compensator output can be simply expressed as [12

12. C. D. Poole and C. R. Giles, “Polarization-dependent pulse compression and broadening due to polarization dispersion in dispersion-shifted fiber,” Opt. Lett. 13, 155–157 (1988). [CrossRef] [PubMed]

][13

13. J. P. Gorden and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. National Academy of Sciences 97, 4541–4550 (2000). [CrossRef]

]:

Eout=a+E++aE
(2a)
E±=12π+Ea(ω)ejωtε̂±ej(ϕ±±Δτ2(ωω0)+12ψ±(ωω0)2)dω
(2b)

where ε^ ± is the output PSP [14

14. C. D. Poole and R. E. Wagner, “Phenomenological approach to polarization dispersion in long single-mode fibers,” Electron. Lett. 22, 1029–1030 (1986). [CrossRef]

], a ± are the projections of the input polarization onto the principal states, ϕ ± denotes phase terms and Ea(ω) is the spectrum obtained by taking the Fourier transform of Eq. (1). ψ±=λ2d±2πc , d + and d - are the chromatic dispersion of the Hi-Bi LCFBG at the fast axis (from long-λ port) and slow axis (from short-λ port) respectively, which has the same absolute value with opposite sign (d +=-d -). Δτ=τ 0-τd, τd is the DGD of the compensator and Δτ=0 for the first order PMD compensation implemented by adjusting the LCFBG.

An integrate-and-dump receiver (the integration time is equal to Tb) with optimal decision threshold is considered, which has a sensitivity of -19.25 dBm for a BER of 10-9. The BER is estimated by finding the maximum value of 0-bit and the minimum value of 1-bit and the corresponding noise terms for each signal. Thermal noise in the receiver is assumed as the dominant noise source in the simulation.

Fig. 6. BER curves for different input powers at 50 ps FWHM pulsewidth.

Figure 6 shows the calculated BER values for different input powers, where T 0=50 ps, κ=0.5, input wavelength λ=1532 nm, and |a +|2=|a -|2=0.5. The power penalty caused by the first-order PMD of τ 0=50 ps is about 1.1 dB at the BER of 10-9. As shown in Fig. 6, the power penalty becomes almost zero after the PMD compensation using the proposed delay element (d -=27.58 ps/nm). When the value of d - is increased to 340 ps/nm and 510 ps/nm, the power penalty after the PMD compensation increases to 0.16 dB and 0.39 dB, respectively. The chromatic dispersion induced distortion will be more significant with the increasing of the transmission speed [11

11. G. P. Agrawal, Fiber-optic communications systems, Third Edition, (John Wiley & Sons Inc, 2002), chap.5. [CrossRef]

], which restricts the system performance improvement. It is noted that although the PMD compensation capability of the device can be improved through increasing the chromatic dispersion value of the Hi-Bi LCFBG, the results in Fig. 6 demonstrate that it is at the price of increasing the power penalty and thus restricting the transmission bit rate.

4. Conclusions

We have proposed and demonstrated a novel delay element for adjustable PMD compensation. The device utilizes a Hi-Bi linearly chirped grating, which can be tuned mechanically to provide linearly continuous DGD. The performance of our proposed device has been demonstrated experimentally by compensating a 10-Gb/s signal with 87 ps or 106 ps DGD. The impact of the chromatic dispersion of the LCFBG on the BER performance has been discussed. We have shown that there is a tradeoff between the PMD compensation capability of the device and the grating dispersion induced power penalty.

Acknowledgments

The authors would like to acknowledge the helpful discussions with Dr. Hui Dong, North Jiaotong University, China. The authors also want to thank the anonymous reviewers for their helpful comments.

References and links

1.

S. Lee, R. Khosravani, J. Peng, A. E. Willner, V. Grubsky, D. S. Starodubov, and J. Feinberg, “High-birefringence nonlinearly-chirped fiber Bragg grating for tunable compensation of polarization mode dispersion,” Conference on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1999) pp.. 272–274.

2.

J. Kim, H. Yong, N. Park, and B. Lee., “Polarization-mode-dispersion compensator using a polarization beam splitter and quarter-wave plates,” Appl. Opt. 40, 4473–4475 (2001). [CrossRef]

3.

H. Rosenfeldt, C. Knothe, and E. Brinkmeyer, “Component for optical PMD-compensation in a WDM environment,” European Conference on Optical Communication, pp.135–136 (2000).

4.

E. Brinkmeyer, “PMD compensation,” European Conference on Optical Communication, No.9.3.1 (2002)

5.

M. Schiano and G. Zaffiro, “Polarization mode dispersion in chirped fiber gratings,” European Conference on Optical Communication (Optical Society of America, Washington, D.C., 1998) pp.403–404.

6.

T. Takahashi, T. Imai, and M. Aiki, “Automatic compensation technique for timewise fluctuating polarization mode dispersion in in-line amplifier systems,” Electron. Lett. 30, 348–349 (1994). [CrossRef]

7.

T. Ozeki, M. Yoshimura, T. Kudo, and H. Ibe, “Polarization mode dispersion equalization experiment using a variable equalizing optical circuit controlled by a pulse-waveform-comparison technique,” Conference on Optical Fiber Communication, (Optical Society of America, Washington, D.C., 1994) TuN4.

8.

Z. Pan, Y. Xie, S. lee, and A. E. Willner, “Tuanble compensation for polarization-mode dispersion using a birefringent nonlinearly-chirped Bragg grating in a dual-pass configuration,” U.S. Patent 6,400,869 B2, 2002.

9.

Y. Horiuchi, Y. Namihira, and H. Wakabayashi, “Chromatic dispersion measurement in 1.55 um narrow-band region using a tunable external-cavity laser,” IEEE Photon.Technol. Lett. 1, 458–460 (1989). [CrossRef]

10.

Z. Qin, Q. Zeng, X. Yang, D. Feng, L. Ding, G. Kai, Z. Liu, S. Yuan, X. Dong, and N. Liu, “Bidirectional grating wavelength shifter with a broad-range tunablility by using a beam of uniform strength,” IEEE Photon. Technol. Lett. 13, 326–328 (2001). [CrossRef]

11.

G. P. Agrawal, Fiber-optic communications systems, Third Edition, (John Wiley & Sons Inc, 2002), chap.5. [CrossRef]

12.

C. D. Poole and C. R. Giles, “Polarization-dependent pulse compression and broadening due to polarization dispersion in dispersion-shifted fiber,” Opt. Lett. 13, 155–157 (1988). [CrossRef] [PubMed]

13.

J. P. Gorden and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. National Academy of Sciences 97, 4541–4550 (2000). [CrossRef]

14.

C. D. Poole and R. E. Wagner, “Phenomenological approach to polarization dispersion in long single-mode fibers,” Electron. Lett. 22, 1029–1030 (1986). [CrossRef]

OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(230.5440) Optical devices : Polarization-selective devices
(260.2030) Physical optics : Dispersion
(350.2770) Other areas of optics : Gratings

ToC Category:
Research Papers

History
Original Manuscript: August 4, 2003
Revised Manuscript: September 22, 2003
Published: October 6, 2003

Citation
Xiaoke Yi, Chao Lu, Xiufeng Yang, Wen-De Zhong, Fang Wei, and Yixin Wang, "High-birefringence linearly chirped grating based optical device for PMD compensation," Opt. Express 11, 2634-2640 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-20-2634


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References

  1. S. Lee, R. Khosravani, J. Peng, A. E. Willner, V. Grubsky, D. S. Starodubov, and J. Feinberg, �??Highbirefringence nonlinearly-chirped fiber Bragg grating for tunable compensation of polarization mode dispersion,�?? Conference on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1999) pp. 272-274.
  2. J. Kim, H. Yong, N. Park, and B. Lee., �??Polarization-mode-dispersion compensator using a polarization beam splitter and quarter-wave plates,�?? Appl. Opt. 40, 4473-4475 (2001). [CrossRef]
  3. H. Rosenfeldt, C. Knothe, and E. Brinkmeyer, �??Component for optical PMD-compensation in a WDM environment,�?? European Conference on Optical Communication, pp.135-136 (2000).
  4. E. Brinkmeyer, �??PMD compensation,�?? European Conference on Optical Communication, No.9.3.1 ( 2002)
  5. M. Schiano, and G. Zaffiro, �??Polarization mode dispersion in chirped fiber gratings,�?? European Conference on Optical Communication (Optical Society of America, Washington, D.C., 1998) pp. 403-404.
  6. T. Takahashi, T. Imai and M. Aiki, �??Automatic compensation technique for timewise fluctuating polarization mode dispersion in in-line amplifier systems,�?? Electron. Lett. 30, 348-349 (1994). [CrossRef]
  7. T. Ozeki, M. Yoshimura, T. Kudo, and H. Ibe, �??Polarization mode dispersion equalization experiment using a variable equalizing optical circuit controlled by a pulse-waveform-comparison technique,�?? Conference on Optical Fiber Communication, (Optical Society of America, Washington, D.C., 1994) TuN4.
  8. Z. Pan, Y. Xie, S. lee, A. E. Willner, �??Tuanble compensation for polarization-mode dispersion using a birefringent nonlinearly-chirped Bragg grating in a dual-pass configuration,�?? U.S. Patent 6,400,869 B2, 2002.
  9. Y. Horiuchi, Y. Namihira, H. Wakabayashi, �??Chromatic dispersion measurement in 1.55 um narrow-band region using a tunable external-cavity laser,�?? IEEE Photon.Technol. Lett. 1, 458-460 (1989). [CrossRef]
  10. Z. Qin, Q. Zeng, X. Yang, D. Feng, L. Ding, G. Kai, Z. Liu, S. Yuan, X. Dong, N. Liu, �??Bidirectional grating wavelength shifter with a broad-range tunablility by using a beam of uniform strength,�?? IEEE Photon. Technol. Lett. 13, 326-328 (2001). [CrossRef]
  11. G. P. Agrawal, Fiber-optic communications systems, Third Edition, (John Wiley & Sons Inc, 2002), chap.5. [CrossRef]
  12. C. D. Poole and C. R. Giles, �??Polarization-dependent pulse compression and broadening due to polarization dispersion in dispersion-shifted fiber,�?? Opt. Lett. 13, 155-157 (1988). [CrossRef] [PubMed]
  13. J. P. Gorden, and H. Kogelnik, �??PMD fundamentals: Polarization mode dispersion in optical fibers,�?? Proc. National Academy of Sciences 97, 4541-4550 (2000). [CrossRef]
  14. C. D. Poole, R. E. Wagner, �??Phenomenological approach to polarization dispersion in long single-mode fibers,�?? Electron. Lett. 22, 1029-1030 (1986). [CrossRef]

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