Phase-controlled superimposed FBGs and their applications in spectral-phase en/decoding

doi:10.1364/OE.19.008580

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Phase-controlled superimposed FBGs and their applications in spectral-phase en/decoding

Jilin Zheng, Rong Wang, Tao Pu, Lin Lu, Tao Fang, Yang Su, Ling Li, and Xiangfei Chen »View Author Affiliations

Optics Express, Vol. 19, Issue 9, pp. 8580-8595 (2011)
http://dx.doi.org/10.1364/OE.19.008580

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▸ Article Outline
– Abstract
1. Introduction
2. Principle of phase controlling
3. Application in spectral-phase en/decoding
4. Conclusion
– References and links

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Abstract

A novel kind of superimposed fiber Bragg gratings (SI-FBGs) named SI-sampled FBGs (SI-SFBGs) is proposed to control the phase relationship among SI sub-gratings by modulating the sampling periods. The realization of such phase-controlled SI-SFBGs just needs a single uniform phase mask and sub-micrometer precision moving stage. The success of phase-controll is expected to encourage SI-SFBGs’ applications in more sophisticated fields. As a demonstration, their applications in spectral-phase en/decoding are testified by both simulation and experiment. The spectral-phase encoded (SPE) encoders with the longest code-length that FBG-based SPE encoders can achieve, i.e., 64-frequency bins, are experimentally fabricated for the first time. The results show the advantages accompanying the SI-SFBGs-based SPE encoders compared with the traditional methods.

1. Introduction

Among a number of kind of fiber Bragg gratings (FBGs), the superimposed (SI) FBGs are especially appealing since multiple FBGs are inscribed at the same location in the fiber, which will bring two main unique features. The first is that all the superimposed wavelengths are time-synchronized. The second is that these FBGs are more compact since short-length structure can accommodate more FBGs containing different wavelengths. These features lend themselves to wide applications, such as optical filters [1

R. Slavik and S. LaRochelle, “Large-band periodic filters for DWDM using multiple-superimposed fiber Bragg gratings,” IEEE Photon. Technol. Lett. 14(12), 1704–1706 (2002). [CrossRef]

Q.-J. Wang, Y. Zhang, and Y.-C. Soh, “Efficient structure for optical interleavers using superimposed chirped fiber Bragg graings,” IEEE Photon. Technol. Lett. 17(2), 387–389 (2005). [CrossRef]

], multiwavelength fiber lasers [3

Y.-G. Han, X.-Y. Dong, C.-S. Kim, M.-Y. Jeong, and J.-H. Lee, “Flexible all fiber Fabry-Perot filters based on superimposed chirped fiber Bragg gratings with continuous FSR tunability and its application to a multiwavelength fiber laser,” Opt. Express 15(6), 2921–2926 (2007). [CrossRef] [PubMed]

Y.-G. Han, J.-H. Lee, S.-B. Lee, L. Potì, and A. Bogoni, “Novel multiwavelength Erbium-doped fiber and Raman fiber ring lasers with continuous wavelength spacing tunability at room temperature,” J. Lightwave Technol. 25(8), 2219–2225 (2007). [CrossRef]

], multiplying the repetition rates of periodic pulse trains [5

J. Azaña, R. Slavík, P. Kockaert, L.-R. Chen, and S. LaRochelle, “Generation of customized ultrahigh repetition rate pulse sequences using superimposed fiber Bragg gratings,” J. Lightwave Technol. 21(6), 1490–1498 (2003). [CrossRef]

J. Magné, J. Bolger, M. Rochette, S. LaRochelle, L.-R. Chen, B.-J. Eggleton, and J. Azaña, “Generation of a 4×100 GHz pulse-train from a single-wavelength 10-GHz mode-locked laser using superimposed fiber Bragg gratings and nonlinear conversion,” J. Lightwave Technol. 24(5), 2091–2099 (2006). [CrossRef]

], frequency-amplitude en/decoding in optical code-division multiple-access (OCDMA) systems [7

D.-M. Meghavoryan and A.-V. Daryan, “Superimposed fiber Bragg grating simulation by the method of single expression for optical CDMA systems,” IEEE Photon. Technol. Lett. 15(11), 1546–1548 (2003). [CrossRef]

S. Ayotte, M. Rochette, J. Magné, L.-A. Rusch, and S. LaRochelle, “Experimental verification and capacity prediction of FE-OCDMA using superimposed FBG,” J. Lightwave Technol. 23(2), 724–731 (2005). [CrossRef]

] and so on. Conventionally, the fabrication of SI-FBGs requires changing different phase masks with different pitches or exerting a controllable strain on the fiber. However, due to the complexity of fabrication process, the precise relative phases of the SI-FBGs cannot be controlled, to the best of our knowledge. This drawback limits the applications of SI-FBGs for more sophisticated functions, since the phase control is usually needed for complex functions. For example, for spectral-amplitude OCDMA en/decoders based on SI-FBGs [7

], the relative phases of different wavelengths are not essential for such an incoherent OCDMA scheme. However, such device cannot be applied to coherent frequency domain OCDMA scheme, i.e., the spectral-phase encoded (SPE) scheme. Usually it is well known that the coherent technique is superior to incoherent technique in many aspects of the performance. Therefore, for realizing more sophisticated functions, it is necessary to control the relative phases of SI-FBGs.

Sampled FBG (SFBG) is another powerful grating which can be realized mainly by modulating the amplitude [9

F. Ouellette, P. A. Krug, T. Stephens, G. Dhosi, and B. Eggleton, “Broadband and WDM dispersion compensation using chirped sampled fibre Bragg gratings,” Electron. Lett. 31(11), 899–901 (1995). [CrossRef]

–12

Y.-T. Dai, X.-F. Chen, L. Xia, Y.-J. Zhang, and S.-Z. Xie, “Sampled Bragg grating with desired response in one channel by use of a reconstruction algorithm and equivalent chirp,” Opt. Lett. 29(12), 1333–1335 (2004). [CrossRef] [PubMed]

] or the phase [13

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann fiber Bragg gratings and phase-only sampling for high channel counts,” IEEE Photon. Technol. Lett. 14(9), 1309–1311 (2002). [CrossRef]

–16

Y. Painchaud, M. Poulin, M. Morin, and M. Guy, “Fiber Bragg grating based dispersion compensator slope-matched for LEAF fiber,” OFC2006, paper OThe2 (2006).

] in the grating. The phased-only SFBG which was developed by Rothenberg et al [13

–16

Y. Painchaud, M. Poulin, M. Morin, and M. Guy, “Fiber Bragg grating based dispersion compensator slope-matched for LEAF fiber,” OFC2006, paper OThe2 (2006).

] can achieve perfect performance in high-channel-count and chromatic dispersion compensation. However, the experimental fabrication of phased-only SBFGs usually needs specially-designed phase masks that include the sampling patterns. It will not only make a significant challenge to produce various complicated phase masks, but also brings along complexities and difficulties in practical experiments has been indicated in [15

Y.-L. Sheng, J. E. Rothenberg, H.-P. Li, Y. Wang, and J. Zweiback, “Split of phase shifts in a phase mask for fiber Bragg gratings,” IEEE Photon. Technol. Lett. 16(5), 1316–1318 (2004). [CrossRef]

,16

Y. Painchaud, M. Poulin, M. Morin, and M. Guy, “Fiber Bragg grating based dispersion compensator slope-matched for LEAF fiber,” OFC2006, paper OThe2 (2006).

]. Whereas, the amplitude SFBG, which has been especially improved by the equivalent-chirp (EC) [10

X.-F. Chen, Y. Luo, C.-C. Fan, T. Wu, and S.-Z. Xie, “Analytical expression of sampled Bragg gratings with chirp in the sampling period and its applicationin dispersion management design in a WDM system,” IEEE Photon. Technol. Lett. 12(8), 1013–1015 (2000). [CrossRef]

], equivalent phase-shift (EPS) [11

Y.-T. Dai, X.-F. Chen, D.-J. Jiang, S.-Z. Xie, and C.-C. Fan, “Equivalent phase shift in a fiber Bragg grating achieved by changing the sampling period,” IEEE Photon. Technol. Lett. 16(10), 2284–2286 (2004). [CrossRef]

] and reconstruction-equivalent-chirp (REC) [12

] technologies recently, has developed into a flexible, cost-effective and practical solution.

In this paper, we combine the SI-FBGs with the amplitude SFBGs to promote the SI-FBGs’ performance. Based on the EC and EPS technologies, the SI-FBGs with desired phase relationship are first proposed and studied, which can be realized only using a uniform-pitch phase mask and sub-micrometer precision moving stage. Being SFBGs essentially, such novel SI-FBGs are named SI-SFBGs in this paper. Then, the SI-SFBGs would be upgraded for potential sophisticated applications by controlling the phase. The application in SPE OCDMA is demonstrated and experimentally validated.

2. Principle of phase controlling

2.1 SI-SFBG

Essentially, SI-SFBG is a kind of FBG where multiple SFBGs are inscribed at the same location using the same uniform phase mask. The overall refractive index modulation is the summation of all SI gratings, which can be expressed as

δ n (z) = \sum_{i = 1}^{N} {\bar{δ n}}_{_{eff, i}} (z) {1 + \frac{1}{2} ν_{i} (z) S_{i} (z) {\exp (j \frac{2 π z}{Λ_{}} + j ϕ_{i} (z)) + c . c}}

(1)

where N represents the number of superimpositions;

{\bar{δ n}}_{_{eff, i}} (z)

is average effective index (DC index) of the i ^th SI grating;

ν_{i} (z)

is the fringe visibility of the i ^th SI grating, which also acts as the apodization function; Λ is the only grating-period of all the SI gratings, which is determined by the phase mask;

ϕ_{i} (z)

describes the spatially-varying phase of the i ^th SI grating. In theory,

ϕ_{i} (z)

is the same for all the SI gratings since only one phase mask is used and no relative displacement occurs between the fiber and mask throughout the fabrication process. So for the sake of simplicity, the value of

ϕ_{i} (z)

is set to zero for all gratings in the latter analysis. S_i (z) describes the sampling function of the i ^th SI grating. For an uniform SFBG with the sampling period of P_i , S_i (z) can be written as

S_{i} (z) = \sum_{m} F_{i, m} \exp (j \frac{2 m π}{P_{i}} z) m = 0, 1, \cdot \cdot \cdot

(2)

where F_i,m is the Fourier coefficient. Hence, the overall refractive index modulation of SI-SFBG can be rewritten as

\begin{array}{l} δ n (z) = \sum_{i = 1}^{N} {\bar{δ n}}_{_{eff, i}} (z) {1 + \frac{1}{2} ν_{i} (z) \sum_{m} F_{i, m} {\exp (j \frac{2 m π}{P_{i}} z) \exp (j \frac{2 π z}{Λ_{}}) + c . c}} \\ = \sum_{i = 1}^{N} {\bar{δ n}}_{_{eff, i}} (z) + \frac{1}{2} \sum_{i = 1}^{N} \sum_{m} {\bar{δ n}}_{_{eff, i}} (z) ν_{i} (z) F_{i, m} \exp (j (\frac{2 π z}{\underset{ghost grating period}{\underset{︸}{1 / (1 / Λ + m / P_{i})}}})) + c . c \end{array}

(3)

Equation (3) shows that every SFBG consists of a series of equivalent ghost gratings with the grating periods of

1 / (1 / Λ + m / P_{i})

, and the SI-SFBG is the linear superimposition of ghost gratings of different origins. For the −1st order ghost grating, the refractive index modulation can be written as

δ n_{- 1} (z) = \sum_{i = 1}^{N} {\bar{δ n}}_{_{eff, i}} (z) + \frac{1}{2} \sum_{i = 1}^{N} {\bar{δ n}}_{_{eff, i}} (z) ν_{i} (z) F_{i, - 1} \exp (j (\frac{2 π z}{1 / (1 / Λ - 1 / P_{i})})) + c . c

(4)

Equation (4) shows that the overall −1st order ghost grating of SI-SFBG is the linear superimposition of individual −1st order ghost grating (with the grating period of

1 / (1 / Λ - 1 / P_{i})

) of every SFBG. According to the EC technology, the grating period of ghost grating can be varied by changing the sampling period. Therefore, the SI-SFBG with multiple −1st order reflective wavelengths can be produced based on a single uniform phase mask through changing the sampling period. Figure 1 illustrates the principle of SI-SFBGs with two −1st-wavelengths.

Fig. 1 Illustration of SI-SFBGs with two −1st-wavelengths.

In Fig. 1, the sampling period of the first inscribed SFBG is P ₁, then the −1st-wavelength of λ ₁ is generated; in the second inscription, the P ₂ (P ₂<P ₁) generates the −1st-wavlength of λ ₂ (λ ₂>λ ₁). If these −1st-wavelengths are located in the desired target region, a SI-SFBG with multiple wavelengths can be fabricated based on a single uniform phase mask, which can resolve the problems existing in the traditional fabrication of SI-FBG such as high cost, poor flexibility and so on. Furthermore, the nanometer control-precision can be avoided, since the sampling period is usually in the order of hundreds of microns.

Such a SI-SFBG technology is simulated. For simplicity, only 4 superimpositions are considered. The parameters are listed in Table 1 .The simulated spectrum is shown in Fig. 2(a) . The four −1st order channels are the desired target response. The group time delay in the passband shown in Fig. 2(b) exhibits uniform synchronization, which indicates that the four channels are superimposed at the same location. Therefore, SI-SFBGs with multiple wavelengths can be designed and fabricated based on a single uniform phase mask.

Table 1 Parameters for SI-SFBGs with 4 Superimpositions and 4 Wavelengths

Superimpo-sition order	Grating period (nm)	Desired −1st order channel (nm)	Sampling period (μm)	Sampling number	Δn _eff	Apodized profile
1	Λ₁ = 531.95	λ ₁ = 1544	204.8	79	1 × 10⁻⁴	1st order Gauss
2	Λ₂ = 531.95	λ ₂ = 1544.2	195.1	83	1 × 10⁻⁴	1st order Gauss
3	Λ₃ = 531.95	λ ₃ = 1544.4	186.2	87	1 × 10⁻⁴	1st order Gauss
4	Λ₄ = 531.95	λ ₄ = 1544.6	178.1	91	1 × 10⁻⁴	1st order Gauss

Fig. 2 The simulation result: (a) Amplitude spectrum of SI-SFBGs; (b) Detailed property of the 4 −1st channels.

2.2 Phase control of SI-SFBG

The failure to control the phase relationship among SI sub-gratings severely degrades the performance of SI-FBGs. However, the SI-SFBGs, which can realize multiple-wavelengths in the −1st order channel, can control their phase relationship simply and precisely. According to the EPS principle, the change of sampling period will produce phase-shift in the nonzero reflective channel.

We introduce the modulation, i.e., chirp f_i (z), in the sampling position z, then S_i (z) can be written as

S_{i} (z) = \sum_{m} F_{i, m} \exp (j \frac{2 m π}{P_{i}} f_{i} (z)) m = 0, 1, \cdot \cdot \cdot

(5)

Here, the −1st order ghost grating can be expressed as

δ n_{- 1} (z) = \sum_{i = 1}^{N} {\bar{δ n}}_{_{eff, i}} (z) + \frac{1}{2} \sum_{i = 1}^{N} {\bar{δ n}}_{_{eff, i}} (z) ν_{i} (z) F_{i, - 1} \exp (j (\frac{2 π z}{Λ_{}} - \frac{2 π}{P_{i}} f_{i} (z))) + c . c

(6)

f_{i} (z) = z - \frac{φ_{i} (z)}{2 π} \cdot P_{i}

(The meaning of function φ_i (z) will be explained later), the −1st order ghost grating can be written as

\begin{array}{l} δ n_{- 1} (z) = \sum_{i = 1}^{N} {\bar{δ n}}_{_{eff, i}} (z) + \frac{1}{2} \sum_{i = 1}^{N} {\bar{δ n}}_{_{eff, i}} (z) ν_{i} (z) F_{i, - 1} \exp (j (\frac{2 π z}{Λ_{}} - \frac{2 π}{P_{i}} (z - \frac{φ_{i} (z)}{2 π} \cdot P_{i}))) + c . c \\ = \sum_{i = 1}^{N} {\bar{δ n}}_{_{eff, i}} (z) + \frac{1}{2} \sum_{i = 1}^{N} {\bar{δ n}}_{_{eff, i}} (z) ν_{i} (z) F_{i, - 1} \exp (j (\frac{2 π z}{1 / (1 / Λ - 1 / P_{i})} + \underset{\begin{array}{l} phase \\ modulation \end{array}}{\underset{︸}{φ_{i} (z)}})) + c . c \end{array}

(7)

The comparison between Eqs. (4) and (7) shows that, the sampling-function modulation f_i (z) has produced extra phase-modulation φ_i (z) in the −1st order ghost grating. φ_i (0) is the original phase of each SI sub-grating. Therefore, the phase relationship of SI-SFBGs can be easily controlled by modulating the sampling periods. Figure 3 reveals the principle of phase-control.

Fig. 3 Phase-controlled SI-SFBGs and corresponding reflective property.

Overall, there is little difference between Fig. 1 and Fig. 3. The only difference is that in Fig. 3, there exists a definite P ₂/2 excursion in the original position of second inscription. According to Eq. (7), such an excursion will produce a relative π phase-shift. Hence, the minus polarity of λ ₂ denotes its relative π phase-shift.

To validate the proposed principle, a phase-controlled SI-SFBG is simulated. The FBG’s parameters are similar to Table 1, and the original phases of the four −1st order reflective channels are set to be 0, π, π and 0 respectively.

In terms of reflective spectrum and group time delay, the comparison between two sets of 4-superimposed gratings is shown in Fig. 4 . Their target phases are (0,0,0,0) and (0,π,π,0) respectively. It can be found that more deeper dips occur between the phase-shifted adjacent reflective peaks, which is usually considered to be the existence of phase shift in the amplitude domain. The group time delay in the passband is synchronous. The phases in the passband of the two sets of SI-SFBGs are shown in Fig. 5(a) . The result shows that in the passband of the same phase, their phase spectrum match well. While in the passband of different designed phases, the phase spectrum exhibit dissimilarities. To search out such dissimilarities and the initial phases imposed on each wavelength, we subtract the phase spectrum of (0,0,0,0) from (0,π,π,0), which is shown in Fig. 5(b). The result clearly shows that in the designed “π” passband, their phase difference is either “π” or “-π”, otherwise zero. Therefore, the phase relationship among the sub-gratings of SI-SFBG can be controlled by modulating the sampling periods.

Fig. 4 The comparison between SI-SFBGs with the phases of (0,0,0,0) (marked with “No phase shift”) and (0,π,π,0) (marked with “Phase shift”): (a) Amplitude; (b) Group time delay.

Fig. 5 The comparison of phases in passband: (a) Phases, “A” represents the SI-SFBGs with (0,π,π,0), while “B” represents the SI-SFBGs with (0,0,0,0); (b) The phases of “A-B”.

To realize such phase-controlled SI-SFBGs just needs a single uniform phase mask and sub-micrometer transferring system, without the relative displacement between the mask and fiber. Thus, the SI-SFBGs are suitable for massive production. The realization of phase-control will promote SI-FBGs’ processing ability and encourage their applications in more complicated fields.

It should be noted the number of superimpositions has become an important concern, since the photosensitivity of the fiber mustn’t be saturated to ensure uniform reflectivity of all the SI wavelengths. Concerning such a SI-SFBG technology, there is no relative displacement between the mask and fiber, all the initial phases (

ϕ_{i} (0)

rather than

φ_{i} (0)

) of the SI sub-FBGs are the same, which means that the total number of superimposed gratings are limited to [17

A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, “Optimization of refractive index sampling for multichannel fiber Bragg gratings,” J. Quantum Electron. 39(1), 91–98 (2003). [CrossRef]

]

\sum_{i = 1}^{N} {\bar{δ n}}_{_{eff, i}} \leq Δ n_{\max}

(8)

where N is the number of superimpositions,

{\bar{δ n}}_{_{eff, i}}

is the refractive index modulation of the i ^th sub-FBG,

Δ n_{\max}

is the achievable maximal refractive index modulation of a photosensitive fiber. When all the sub-FBGs are uniform and equally modulated, Eq. (8) can be rewritten as

N \leq \frac{Δ n_{\max}}{{\bar{δ n}}_{_{eff, i}}}

(9)

Equations (8) or (9) is the basic restriction on the number of superimpositions for SI-SFBGs. The fiber used in our experiments are hydrogen-loaded photosensitive fiber, in which

Δ n_{\max}

can be up to 1.3 × 10⁻³. Experimental test shows that the total number of superimposed gratings should be no more than 16, in order to avoid saturation as well as to keep a reasonable reflectivity.

3. Application in spectral-phase en/decoding

The SPE en/decoders based on FBGs have the advantages of compact all-fiber structure, potential low cost, entire compatibility with fiber and so on. Currently, there have been reported two types of FBGs for SPE en/decoders, i.e., the step-chirped (SC) FBG-pair [18

A. Grunnet-Jepsen, A. E. Johnson, E. S. Maniloff, T. W. Mossberg, M. J. Munroe, and J. N. Sweetser, “Fiber Bragg grating based spectral encoder/ decoder for lightwave CDMA,” Electron. Lett. 35(13), 1096–1097 (1999). [CrossRef]

] and the amplitude sampled (AS) FBG-pair [19

M. Yan, M. Yao, H. Zhang, L. Xia, and Y. Zhang, “En/decoder for spectral phase-coded OCDMA system based on amplitude sampled FBG,” IEEE Photon. Technol. Lett. 20(10), 788–790 (2008). [CrossRef]

]. Both of these two structures have their own disadvantages. For the former, firstly, the fabrication of SC FBG-pair needs multiple phase masks of different pitches, and the number of required masks increases with code length; secondly, the phase control needs ~nm precision measuring and positioning equipments, and the frequent change in position between masks and fiber will produce unpredictable random phase-errors; thirdly, the grating length will increase sharply with code length, which would become a serious concern due to the high demand for complicated fabrication process and FBG package; fourthly, the FBG-pair structure will bring along, on the one hand, bulky configuration, and on the other hand, high insertion loss. For the latter, i.e., ASFBG structure, though the first two disadvantages existing in SC-FBG-pair can be overcome by EC and EPS technologies, the last two problems still exist. For example, in literature [19

M. Yan, M. Yao, H. Zhang, L. Xia, and Y. Zhang, “En/decoder for spectral phase-coded OCDMA system based on amplitude sampled FBG,” IEEE Photon. Technol. Lett. 20(10), 788–790 (2008). [CrossRef]

], 5.4cm-long FBG can only accommodate 15-frequency bins, and if the code length is doubled, the total grating length will be quadrupled. That is to say, if the 64-Walsh code is used, the FBG length will reach as long as 90cm, which is not easy to be experimentally fabricated.

However, the phase-controlled SI-FBG would provide an effective approach to resolving the last two problems mentioned above. As a matter of fact, as early as a decade ago, though, in literature [18

], the single SI-FBG was supposed to replace the SC FBG-pair, this proposal has not come into reality up to now. The fatal reason for such a dilemma lies in the difficulty of phase controlling of SI-FBGs in experiment.

In this paper, however, the phase-controlled SI-SFBGs can be used to overcome all the four main disadvantages. Fisrlty, incorporating the EC and EPS technilogies, the fabrication of SI-SFBGs based SPE encoders only needs a single uniform phase mask and sub-micrometer precision positioning devices; secondly, the SI-gratings structure will evidently reduce the required grating length and make the en/decoder’s configuration simple and power-budget-effective.

3.1 Simulation

In the former section, the principle of phase-controlling in SI-SFBG has been introduced, and theoretically, such phase-controlled SI-SFBGs would be an ideal choice for SPE encoders. In fact, however, the total number of superimposed gratings is limited. In other words, when the code length exceeds the achievable maximum number of superimposed gratings, some of the gratings should be inscribed in series, which means that all the wavelengths are not time-synchronized any longer. In this section, both the time-synchronous and –asynchronous scenarios are demonstrated.

To start with, suppose the code length is less than the achievable maximum number. In this case, there are no time-delay among different wavelengths. The encoder’s structure is just the same as Fig. 3. The Walsh codes #2 and #3 with the code-length of 16 are used, which are generated by the hadamard command in MATLAB. For comparison, the no phase-shift SI-SFBG of the same parameters is also simulated. The detailed parameters are listed in Table 2 .It should be noted the total length is about 8mm, while it would be 130mm if the traditional encoder structure was used. The reflective spectrum of the encoders with three different codes are shown in Fig. 6 . In Fig. 6(a), it is evident that deep dips occur between phase-shifted adjacent reflective peaks. Figure 6(b) shows the time-synchronization property of all the reflective peaks.

Table 2 Parameters for SI-SFBGs-Based SPE Encoders with 16-Frequency Bins and 16 Superimpositions

First wavelength (nm)	Phase bin spacing (nm)	Phase bin bandwidth (nm)	First sampling period (μm)	Last sampling period (μm)	Δn _eff	Apodized profile	Total length (mm)
1544	0.2	0.12	204.8	117	5 × 10⁻⁵	Gauss	8.192

Fig. 6 Simulation result: (a) Amplitude spectrum of three different encoders, “Code#2” and “Code#3” use the left Y-coordinate, while the “No phase shift” uses the right Y-coordinate; (b) Group time delay.

Subtracting the no phase-shifted spectrum from that of code #2 and #3, the initial phase imposed on each reflective peak is obtained, which is shown in Fig. 7 . The “0” and “π” at the top of the figures represent the employed codes. We can find that the phase values agree well with the corresponding codes. It indicates the 16 reflective channels have successfully carried code information in phase.

Fig. 7 The wavelengths’ initial phases and corresponding address codes: (a) Code#2; (b) Code#3.

Simulating the input broadband source with ~2.3ps (full width at half maximum, FWHM) Gauss pulse, the encoded waveforms are shown in Fig. 8(a) . The decoded auto-correlation and cross-correlation waveforms are shown in Fig. 8(b). The results reflect the fact that due to the nature of Walsh code [20

R. Omrani and P.-V Kumar, “Spreading sequence for asynchronous spectrally phase encoded optical CDMA,” ISIT2006, 2642–2646 (2006).

], there exists no energy in the center of cross-correlation, whereas a sharp peak in auto-correlation waveforms. Conclusively, such phase-controlled SI-SFBGs are theoretically competent for SPE en/decoding.

Fig. 8 Performance of SPE encoders with 16-frequency bins and 16 superimpositions: (a) Encoded waveforms; (b) Decoded waveforms.

It is known that large code-length can increase the spectrum-spreading gain and consequently enlarge system’s capacity. When the code-length exceeds the achievable maximum number of superimpositions, some of gratings have to be inscribed in series. If the code-length is M and the maximum number of superimpositions is N, then the maximum number of gratings inscribed in series can be figured as

S = [M / N]

(10)

where [⋅] represents the rounding upward operator.

For example, for a Walsh code with the code-length of 64, if the maximum number of superimpositions is 16, then the number of gratings inscribed in series is 4. For simplicity, the structure of SI-SFBGs with 2 superimpositions and 2 series connections is shown in Fig. 9 .

Fig. 9 Phase-controlled SI-SFBGs with 2 superimpositions and 2 series connections.

For every inscription, the method of phase controlling between adjacent wavelengths is the same as that of literature [13

]. According to the proposed SI-SFBG technology, the phase relationship between the first and second inscription is determined by the initial sampling position of λ ₃. In this figure, the initial position of λ ₃ introduces an extra offset of half its normal sampling period, which produces a relative π phase shift. Therefore, the relative phase relationship among the four wavelengths has been controlled.

Then, we simulate the spectrum of phase-controlled 4-wavelength SI-SFBGs with 2 superimpositions and 2 series connections. The detailed parameters are listed in Table 3 . Two gratings with the phases of (0,0,0,0) and (0,π, π,0) are chosen for comparison.

Table 3 Parameters for Phase-Controlled 4-Wavelength SI-SFBGs with 2 Superimpositions and 2 Series Connections

Superim-position order	Grating period (nm)	−1st order channel (nm)	Phase (radian)	Sampling period (μm)	Sampling number	Δn _eff	Apodized profile	Total length (mm)
1	Λ₁ = 531.95	λ ₁ = 1544	0/0	204.8	79	1 × 10⁻⁴	1st order Gauss	32.7
1	Λ₂ = 531.95	λ ₂ = 1544.2	0/π	195.1	83	1 × 10⁻⁴	1st order Gauss	32.7
2	Λ₃ = 531.95	λ ₃ = 1544.4	0/π	186.2	87	1 × 10⁻⁴	1st order Gauss	32.7
2	Λ₄ = 531.95	λ ₄ = 1544.6	0/0	178.1	91	1 × 10⁻⁴	1st order Gauss	32.7

The “AC” refractive index modulation (i.e.,

δ n - \sum {\bar{δ n}}_{_{eff, i}}

) of such SI-SFBGs (with the phases of (0,π,π,0)) is shown in Fig. 10 . We can see two gratings are cascaded in series, both of which are Gauss-profiled. The maximum refractive index modulation is about 2 × 10⁻⁴.

Fig. 10 The refractive index modulation of SI-SFBGs with 2 superimpositions and 2 series connections.

Figure 11 shows the simulated spectrum. In the group delay spectrum, the ~160ps steps correspond to the 1.6cm-long sub-gratings.

Fig. 11 The comparison between SI-SFBGs (2 superimpositions and 2 series connections) with the phases of (0,0,0,0) (marked with“No phase shift”) and (0,π,π,0) (marked with “Phase shift”): (a) Amplitude; (b) Group time delay.

The phase spectrum in passband is shown in Fig. 12(a) . The phase difference is shown in Fig. 12(b). It clearly shows, in the passband of target phase “π”, the phase difference is either π or -π. That is to say, the target phase relationship (0,π, π,0) has been successfully realized.

Fig. 12 The comparison of phases in passband (SI-SFBGs with 2 superimpositions and 2 series connections): (a) Phases, “A” represents the SI-SFBGs with (0,π,π,0), while “B” represents the SI-SFBGs with (0,0,0,0); (b) The phases of “A-B”.

Considering the limitation on the number of superimpositions, the 64-frequency bins are divided into 16 superimpositions and 4 series connections. The Walsh codes #5 and #6 with the code-length of 64 are used, which are generated by the hadamard command in MATLAB. For comparison, the no phase-shift 64-frequency bins SI-SFBG of the same parameters is also simulated. The detailed parameters are listed in Table 4 .Here, the total length of encoder is about 8.2cm, whereas it would be 131cm in the traditional structure, which reflects the advantage of superimposing technology in terms of shortening the required length of encoder.

Table 4 Parameters for 64-Frequency Bins SI-SFBGs-Based SPE Encoders with 16 Superimpositions and 4 Series Connections

First wavelength (nm)	Phase bin spacing (nm)	Phase bin bandwidth (nm)	First sampling period (μm)	Last sampling period (μm)	Δn _eff	Apodized profile	Total length (mm)
1544	0.08	0.08	204.8	90.6	5 × 10⁻⁵	Gauss	81.92

The simulated spectrum is shown in Fig. 13 . The four group delay steps correspond to the four sub-gratings inscribed in series arrangement. However, the encoder and decoder are dispersion-complementary.

Fig. 13 Simulated spectrum of 64-frequency bins SI-SFBGs-based SPE en/decoders with 16 superimpositions and 4 series connections.

To subtract the phase spectrum of no phase-shift SI-SFBG from that of encoder with Walsh code#5, then the initial phases imposed on the 64 wavelength channels are obtained, which is shown in Fig. 14 . The result clearly shows the phase values agree well with the corresponding code.

Fig. 14 The initial phases of SPE encoder with Walsh code#5.

In the simulation, we use ~1.3ps Gauss pulse as the broadband optical source, and the encoded waveforms and decoded waveforms are shown in Figs. 15(a) and 15(b) respectively. We can see the overall profile of encoded waveform consists of four sub-profiles. Each of the sub-profiles is about 100ps width (FWHM), which is due to the 0.08nm phase-bin spacing. The overall profile is about 600ps width due to the four sub-gratings inscribed in series. The decoded waveforms (auto- and cross-correlation) show high contrast ratio.

Fig. 15 Performance of 64-frequency bins SPE en/decoders with 16 superimpositions and 4 series connections: (a) Encoded waveforms; (b) Decoded waveforms.

Thus, the SI-SFBGs’ application in SPE en/decoding has been demonstrated by simulation. Its feasibility is validated theoretically, and its experimental validation is implemented in the following section.

3.2 Experimental validation

High code-length needs big number of superimpositions and deep refractive index modulation consequently. So the hydrogen-loaded photo-sensitive fiber is used in this experiment, in which the refractive index can be modulated up to 1.3 × 10⁻³. The pitch of uniform phase mask is 1070nm, and the ultraviolet power at 244nm is set to be 12mW. The single exposure time is no more than 1s, and because of this, the refractive index modulation at single exposure is about 6 × 10⁻⁵, and the overall refractive index modulation (16 superimpositions) is less than 1 × 10⁻³. Compared with the maximum index modulation available, a reasonable margin is reserved to ensure the uniformity of all the wavelengths’ reflectivities. In our experiments, two sets of encoders are fabricated, both of which are not anodized. The first group use the Walsh codes with code-length of 64, the total encoding bandwidth is 5.12nm. To our knowledge, this is the longest code-length that FBG-based SPE encoder can achieve. The second group use the Walsh codes with code-length of 32, the total encoding bandwidth is 3.52nm. The coherent broadband source in our lab is MLLD (TMLL1550) with the 20dB bandwidth of 4.12nm, which cannot provide sufficient spectrum for the first group. So the “measured data + simulation” method is taken to verify the performance of the encoders with 64-frequency bins, while a proof-of-principle experiment is implemented to validate the feasibility of the encoders with 32-frequency bins.

The 64-frequency bins Walsh codes are still the #5 and #6 used in the simulation. Both the phase bin spacing and width are 0.08nm. The number of superimpositions is 16, so the sub-gratings inscribed in series are 4 in number. The total length of encoder is 4.4cm. The spectrum of encoder #5, decoder #5 and encoder #6 are measured by OVA, as shown in Figs. 16(a) , 16(b) and 16(c) respectively.

Fig. 16 Measured spectrum of 64-frequency bins SPE en/decoders with 16 superimpositions and 4 series connections, the solid line represents the amplitude, the dashed line represents the group delay: (a) Encoding spectrum with Walsh code#5; (b) Decoding spectrum with Walsh code#5; (c) Encoding spectrum with Walsh code#6.

The top of each spectrum is labeled with corresponding code. The overall spectrum are relatively uniform, and the encoding spectrum and decoding spectrum are dispersion-complementary. Gauss pulse with the FWHM duration of 1.3ps is used as the input laser source. The encoded and decoded waveforms based on the measured data are computed and shown in Figs. 17(a) and 17(b) respectively. We can see the encoded waveforms extend to a duration of 440ps, which is determined by the grating’s length. The encoded waveforms exhibit rectangular profile since no apodization is used. The decoded waveforms show high contrast ratio between auto- and cross-correlation. It can be seen that there is little intensity in the center of cross-correlation whereas a sharp peak of auto-correlation. This characteristic agrees well with the simulation result.

Fig. 17 Calculated result based on the measured spectrum: (a) Encoded waveforms; (b) Decoded waveforms.

The second set of encoders employ 32-frequency bins Walsh code#2 and code#3 (generated by hadamard command in MATLAB). The phase bin spacing as well as the bin width is 0.11nm. The number of superimpositions is 4, and the subgratings inscribed in series are 8 in number. The total length of encoders is 6cm. No apodization is used. The OVA measured data of encoder#2, decoder#2 and encoder#3 are depicted in Figs. 18(a) , 18(b) and 18(c) respectively. The setup of a proof-of-principle experiment is shown in Fig. 19(a) . The 10GHz clock generated by the Pulse Pattern Generator(PPG, MP1763C) is amplified to drive the MLLD. A 10GHz pulse train with the duration of 2ps is then generated. Such a pulse train is modulated with PRBS data from the PPG by a LiNbO3 intensity modulator (LN-IM, JDSU10024180). The Optical Spectrum Analyzer (OSA, Anritsu MP9710C) can be used to monitor the spectrum adjustment among encoder, decoder and laser source. The decoded signals are sent to the Optical Sampling Oscilloscope (OSO, NRO9000, 28GHz bandwidth). The spectrum of laser source is shown in Fig. 19(b), and its 20dB bandwidth is measured to be 4.12nm that basically satisfies the requirement of encoding bandwidth. Correctly decoded waveforms and incorrectly decoded waveforms are shown in Figs. 20(a) and 20(b) respectively. We can see the auto-correlation waveforms have been successfully recovered to be narrow pulses, while the cross-correlation waveforms still remain low-intensity noise-like signals. It should be noted the 6cm-long encoder in 10Gbps system will result in as many as 11 inter-symbol-interferences (ISIs) and accompanying self beat-noise(BN), and these contribute to the variations in the peak power of the correctly decoded waveforms. Of course, to increase the number of superimpositions can be expected to shorten encoder’s length.

Fig. 18 Measured spectrum of 32-frequency bins SPE en/decoders with 4 superimpositions and 8 series connections, the solid line represents the amplitue, the dashed line represents the group delay: (a) Encoding spectrum with Walsh code#2; (b) Decoding spectrum with Walsh code#2; (c) Encoding spectrum with Walsh code#3.

Fig. 19 The proof-of-principle experiment: (a) Setup; (b) Measured spectrum of laser source.

Fig. 20 Experimental result: (a) Correctly decoded waveforms; (b) Incorrectly decoded waveforms.

Therefore, the experimental results have validated the feasibility of SI-SFBGs-based SPE encoders.

4. Conclusion

In this paper, a novel kind of SI-FBGs structure, i.e., the SI-SFBGs, is proposed and investigated. The phase relationship among SI sub-gratings is successfully controlled for the first time. The beauties of such SI-SFBGs can be summarized as: 1) To realize SI-FBGs incorporating different target wavelengths doesn’t need to change phase masks or to tune strain, which helps to ensure the stability of each FBG, especially the consistency of phase; 2) The relative phases of different SI target wavelengths can be easily and flexibly controlled by sub-micrometer precision moving stage. This phase-control would extend the application of SI-FBGs in potential more sophisticated functions. Its application in SPE OCDMA is well demonstrated by both simulation and experiment. The SPE encoders with 64-frequency bins, which is the longest code-length that FBG-based SPE encoder can achieve, is experimentally fabricated for the first time. Compared with the traditional methods, the SI-SFBGs-based SPE encoders can overcome the existing limitations and make encoders more powerful and practical.

Finally, it should be noted that, since the photosensitivity of the fiber mustn’t be saturated to ensure uniformity of all the SI wavelengths, to increase the number of superimpositions would be another challenging work to make such phase-controlled SI-SFBGs more powerful.

Acknowledgments

This work is partly supported by the National Nature Science Foundation of China under (61032005) and (60871075). The authors would like to thank the anonymous reviewers for their careful reading and helpful comments.

References and links

1.	R. Slavik and S. LaRochelle, “Large-band periodic filters for DWDM using multiple-superimposed fiber Bragg gratings,” IEEE Photon. Technol. Lett. 14(12), 1704–1706 (2002). [CrossRef]
2.	Q.-J. Wang, Y. Zhang, and Y.-C. Soh, “Efficient structure for optical interleavers using superimposed chirped fiber Bragg graings,” IEEE Photon. Technol. Lett. 17(2), 387–389 (2005). [CrossRef]
3.	Y.-G. Han, X.-Y. Dong, C.-S. Kim, M.-Y. Jeong, and J.-H. Lee, “Flexible all fiber Fabry-Perot filters based on superimposed chirped fiber Bragg gratings with continuous FSR tunability and its application to a multiwavelength fiber laser,” Opt. Express 15(6), 2921–2926 (2007). [CrossRef] [PubMed]
4.	Y.-G. Han, J.-H. Lee, S.-B. Lee, L. Potì, and A. Bogoni, “Novel multiwavelength Erbium-doped fiber and Raman fiber ring lasers with continuous wavelength spacing tunability at room temperature,” J. Lightwave Technol. 25(8), 2219–2225 (2007). [CrossRef]
5.	J. Azaña, R. Slavík, P. Kockaert, L.-R. Chen, and S. LaRochelle, “Generation of customized ultrahigh repetition rate pulse sequences using superimposed fiber Bragg gratings,” J. Lightwave Technol. 21(6), 1490–1498 (2003). [CrossRef]
6.	J. Magné, J. Bolger, M. Rochette, S. LaRochelle, L.-R. Chen, B.-J. Eggleton, and J. Azaña, “Generation of a 4×100 GHz pulse-train from a single-wavelength 10-GHz mode-locked laser using superimposed fiber Bragg gratings and nonlinear conversion,” J. Lightwave Technol. 24(5), 2091–2099 (2006). [CrossRef]
7.	D.-M. Meghavoryan and A.-V. Daryan, “Superimposed fiber Bragg grating simulation by the method of single expression for optical CDMA systems,” IEEE Photon. Technol. Lett. 15(11), 1546–1548 (2003). [CrossRef]
8.	S. Ayotte, M. Rochette, J. Magné, L.-A. Rusch, and S. LaRochelle, “Experimental verification and capacity prediction of FE-OCDMA using superimposed FBG,” J. Lightwave Technol. 23(2), 724–731 (2005). [CrossRef]
9.	F. Ouellette, P. A. Krug, T. Stephens, G. Dhosi, and B. Eggleton, “Broadband and WDM dispersion compensation using chirped sampled fibre Bragg gratings,” Electron. Lett. 31(11), 899–901 (1995). [CrossRef]
10.	X.-F. Chen, Y. Luo, C.-C. Fan, T. Wu, and S.-Z. Xie, “Analytical expression of sampled Bragg gratings with chirp in the sampling period and its applicationin dispersion management design in a WDM system,” IEEE Photon. Technol. Lett. 12(8), 1013–1015 (2000). [CrossRef]
11.	Y.-T. Dai, X.-F. Chen, D.-J. Jiang, S.-Z. Xie, and C.-C. Fan, “Equivalent phase shift in a fiber Bragg grating achieved by changing the sampling period,” IEEE Photon. Technol. Lett. 16(10), 2284–2286 (2004). [CrossRef]
12.	Y.-T. Dai, X.-F. Chen, L. Xia, Y.-J. Zhang, and S.-Z. Xie, “Sampled Bragg grating with desired response in one channel by use of a reconstruction algorithm and equivalent chirp,” Opt. Lett. 29(12), 1333–1335 (2004). [CrossRef] [PubMed]
13.	J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann fiber Bragg gratings and phase-only sampling for high channel counts,” IEEE Photon. Technol. Lett. 14(9), 1309–1311 (2002). [CrossRef]
14.	H.-P. Li, Y.-L. Sheng, Y. Li, and J. E. Rothenberg, “Phased-only sampled fiber Bragg gratings for high-channel-count chromatic dispersion compensation,” J. Lightwave Technol. 21(9), 2074–2083 (2003). [CrossRef]
15.	Y.-L. Sheng, J. E. Rothenberg, H.-P. Li, Y. Wang, and J. Zweiback, “Split of phase shifts in a phase mask for fiber Bragg gratings,” IEEE Photon. Technol. Lett. 16(5), 1316–1318 (2004). [CrossRef]
16.	Y. Painchaud, M. Poulin, M. Morin, and M. Guy, “Fiber Bragg grating based dispersion compensator slope-matched for LEAF fiber,” OFC2006, paper OThe2 (2006).
17.	A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, “Optimization of refractive index sampling for multichannel fiber Bragg gratings,” J. Quantum Electron. 39(1), 91–98 (2003). [CrossRef]
18.	A. Grunnet-Jepsen, A. E. Johnson, E. S. Maniloff, T. W. Mossberg, M. J. Munroe, and J. N. Sweetser, “Fiber Bragg grating based spectral encoder/ decoder for lightwave CDMA,” Electron. Lett. 35(13), 1096–1097 (1999). [CrossRef]
19.	M. Yan, M. Yao, H. Zhang, L. Xia, and Y. Zhang, “En/decoder for spectral phase-coded OCDMA system based on amplitude sampled FBG,” IEEE Photon. Technol. Lett. 20(10), 788–790 (2008). [CrossRef]
20.	R. Omrani and P.-V Kumar, “Spreading sequence for asynchronous spectrally phase encoded optical CDMA,” ISIT2006, 2642–2646 (2006).

OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(060.2340) Fiber optics and optical communications : Fiber optics components
(060.3735) Fiber optics and optical communications : Fiber Bragg gratings

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: March 1, 2011
Revised Manuscript: April 9, 2011
Manuscript Accepted: April 9, 2011
Published: April 18, 2011

Citation
Jilin Zheng, Rong Wang, Tao Pu, Lin Lu, Tao Fang, Yang Su, Ling Li, and Xiangfei Chen, "Phase-controlled superimposed FBGs and their applications in spectral-phase en/decoding," Opt. Express 19, 8580-8595 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-9-8580

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References

R. Slavik and S. LaRochelle, “Large-band periodic filters for DWDM using multiple-superimposed fiber Bragg gratings,” IEEE Photon. Technol. Lett. 14(12), 1704–1706 (2002). [CrossRef]
Q.-J. Wang, Y. Zhang, and Y.-C. Soh, “Efficient structure for optical interleavers using superimposed chirped fiber Bragg graings,” IEEE Photon. Technol. Lett. 17(2), 387–389 (2005). [CrossRef]
Y.-G. Han, X.-Y. Dong, C.-S. Kim, M.-Y. Jeong, and J.-H. Lee, “Flexible all fiber Fabry-Perot filters based on superimposed chirped fiber Bragg gratings with continuous FSR tunability and its application to a multiwavelength fiber laser,” Opt. Express 15(6), 2921–2926 (2007). [CrossRef] [PubMed]
Y.-G. Han, J.-H. Lee, S.-B. Lee, L. Potì, and A. Bogoni, “Novel multiwavelength Erbium-doped fiber and Raman fiber ring lasers with continuous wavelength spacing tunability at room temperature,” J. Lightwave Technol. 25(8), 2219–2225 (2007). [CrossRef]
J. Azaña, R. Slavík, P. Kockaert, L.-R. Chen, and S. LaRochelle, “Generation of customized ultrahigh repetition rate pulse sequences using superimposed fiber Bragg gratings,” J. Lightwave Technol. 21(6), 1490–1498 (2003). [CrossRef]
J. Magné, J. Bolger, M. Rochette, S. LaRochelle, L.-R. Chen, B.-J. Eggleton, and J. Azaña, “Generation of a 4×100 GHz pulse-train from a single-wavelength 10-GHz mode-locked laser using superimposed fiber Bragg gratings and nonlinear conversion,” J. Lightwave Technol. 24(5), 2091–2099 (2006). [CrossRef]
D.-M. Meghavoryan and A.-V. Daryan, “Superimposed fiber Bragg grating simulation by the method of single expression for optical CDMA systems,” IEEE Photon. Technol. Lett. 15(11), 1546–1548 (2003). [CrossRef]
S. Ayotte, M. Rochette, J. Magné, L.-A. Rusch, and S. LaRochelle, “Experimental verification and capacity prediction of FE-OCDMA using superimposed FBG,” J. Lightwave Technol. 23(2), 724–731 (2005). [CrossRef]
F. Ouellette, P. A. Krug, T. Stephens, G. Dhosi, and B. Eggleton, “Broadband and WDM dispersion compensation using chirped sampled fibre Bragg gratings,” Electron. Lett. 31(11), 899–901 (1995). [CrossRef]
X.-F. Chen, Y. Luo, C.-C. Fan, T. Wu, and S.-Z. Xie, “Analytical expression of sampled Bragg gratings with chirp in the sampling period and its applicationin dispersion management design in a WDM system,” IEEE Photon. Technol. Lett. 12(8), 1013–1015 (2000). [CrossRef]
Y.-T. Dai, X.-F. Chen, D.-J. Jiang, S.-Z. Xie, and C.-C. Fan, “Equivalent phase shift in a fiber Bragg grating achieved by changing the sampling period,” IEEE Photon. Technol. Lett. 16(10), 2284–2286 (2004). [CrossRef]
Y.-T. Dai, X.-F. Chen, L. Xia, Y.-J. Zhang, and S.-Z. Xie, “Sampled Bragg grating with desired response in one channel by use of a reconstruction algorithm and equivalent chirp,” Opt. Lett. 29(12), 1333–1335 (2004). [CrossRef] [PubMed]
J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann fiber Bragg gratings and phase-only sampling for high channel counts,” IEEE Photon. Technol. Lett. 14(9), 1309–1311 (2002). [CrossRef]
H.-P. Li, Y.-L. Sheng, Y. Li, and J. E. Rothenberg, “Phased-only sampled fiber Bragg gratings for high-channel-count chromatic dispersion compensation,” J. Lightwave Technol. 21(9), 2074–2083 (2003). [CrossRef]
Y.-L. Sheng, J. E. Rothenberg, H.-P. Li, Y. Wang, and J. Zweiback, “Split of phase shifts in a phase mask for fiber Bragg gratings,” IEEE Photon. Technol. Lett. 16(5), 1316–1318 (2004). [CrossRef]
Y. Painchaud, M. Poulin, M. Morin, and M. Guy, “Fiber Bragg grating based dispersion compensator slope-matched for LEAF fiber,” OFC2006, paper OThe2 (2006).
A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, “Optimization of refractive index sampling for multichannel fiber Bragg gratings,” J. Quantum Electron. 39(1), 91–98 (2003). [CrossRef]
A. Grunnet-Jepsen, A. E. Johnson, E. S. Maniloff, T. W. Mossberg, M. J. Munroe, and J. N. Sweetser, “Fiber Bragg grating based spectral encoder/ decoder for lightwave CDMA,” Electron. Lett. 35(13), 1096–1097 (1999). [CrossRef]
M. Yan, M. Yao, H. Zhang, L. Xia, and Y. Zhang, “En/decoder for spectral phase-coded OCDMA system based on amplitude sampled FBG,” IEEE Photon. Technol. Lett. 20(10), 788–790 (2008). [CrossRef]
R. Omrani and P.-V Kumar, “Spreading sequence for asynchronous spectrally phase encoded optical CDMA,” ISIT2006, 2642–2646 (2006).

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