Hybrid technique based on chirp effect and phase shifts for spectral Talbot effect in sampled fiber Bragg gratings (FBGs)

Transcription

1 Optica Applicata, Vol. XLI, No. 1, 011 Hybrid technique baed on chirp effect and phae hift for pectral Talbot effect in apled fiber Bragg grating (FBG) GUO DENG *, WEI PAN Center for Inforation Photonic and Counication, School of Inforation Science and Technology, Southwet Jiaotong Univerity, Chengdu , Sichuan, China * Correponding author: cipc.wjtu@gail.co To ipleent the pectral Talbot effect in apled fiber Bragg grating (SFBG), a hybrid technique baed on a chirp effect and phae hift i propoed. Firtly, the general phae condition i derived a the principle of thi hybrid technique, and it alo can be ued to deontrate other reported technique, including the linear chirp, the periodical chirp, and the ultiple phae hift technique. According to the general phae condition and the equivalent chirp coefficient in the Talbot effect, ulti-channel diperion copenator are deigned with different arrangeent of chirp coefficient and phae hift. Moreover, the diperion value can be tuned in thee device by dynaically adjuting the phae hift of the hybrid technique. Nuerical iulation are carried out to confir the perforance of uch device realized by uing the hybrid technique. Keyword: fiber Bragg grating (FBG), Talbot effect, diperion, optical fiber counication. 1. Introduction A a tructured type of fiber Bragg grating, the apled fiber Bragg grating (SFBG) ha aroued coniderable interet in a wide range of application. The ot iportant feature of SFBG i it ulti-channel characteritic, which provide the SFBG with treendou advantage in optical counication yte and ening yte. Therefore, lot of novel device baed on SFBG are propoed recently, uch a ulti-channel cob filter [1 7], ulti-channel diperion copenator [7 1], ulti-channel add/drop ultiplexer [13], OCDMA en/decoder [14], and environental paraeter enor [15], etc. Aong thee application, the pectral Talbot effect (i.e., the elf-iaging effect) play an iportant role in contructing the dene wavelength pacing device. The Talbot phenoena can be realized by the chirp effect, including the linear chirp [, 3, 16], the nonlinear chirp [17], and the periodical chirp [6]. The phae hift technique i alo ued to obtain the Talbot phenoena, uch a linear phae hift, ultiple phae hift (MPS) technique [18]. Meanwhile, the Talbot phenoena can be oberved by the cobination of chirp effect and phae hift [4, 1].

2 6 GUO DENG, WEI PAN In thi work, we focu on the ipleentation of the Talbot effect with a hybrid technique baed on the chirp effect and phae hift. It i noted that our technique i a general ethod copared with the previou invetigation (chirp effect, phae hift technique, or the cobination technique). On the bai of thi hybrid technique, variou SFBG with different channel pacing, different channel grid, or different diperion are obtained. Such SFBG can operate a ulti-channel filter or ulti-channel diperion copenator. Furtherore, tunable characteritic are offered by tuning the phae hift in SFBG. The reainder of thi paper i organized a follow. In Section, according to the phae condition required by the Talbot effect under the linear chirp, the general phae condition for linear chirp and linear phae hift technique i derived firtly. Under thi general phae condition, the hybrid technique for the Talbot effect i propoed due to the equivalence between the chirp and the phae hift. With thi hybrid technique, different channel pacing can be realized by adjuting the chirp coefficient or the phae hift. Moreover, on the bai of the general phae condition and the hybrid technique, we are convenient to deontrate the linear chirp, the periodical chirp, the linear phae hift technique, the MPS technique, and the approach developed by DAI et al. [4]. Dai approach i conidered a an expanion of the hybrid technique by inerting contant phae hift into each apling period. Extenively, a non-zero contant phae i introduced to tune the wavelength channel grid. Then, in Section 3, we dicu the ipleentation of ulti-channel diperion copenator by the hybrid technique. A variation in either the chirp coefficient or the phae hift lead to a iilar influence on the diperion. Therefore, we can get the expected diperion characteritic by changing the chirp coefficient or the phae hift. Cobined with the tunable channel grid indicated in Section, diperion copenator with the tunable channel grid and tunable diperion are preented. Finally, we give concluding reark about the hybrid technique in Section 4.. Hybrid technique for pectral Talbot effect.1. General phae condition and hybrid technique Since the hybrid technique need a general phae condition to upport it principle, we firtly derive the general condition. Conidering a apled fiber Bragg grating (SFBG), the linear chirp (Λ(z) = Cz) along the z-axi i adopted, where and C refer to the center grating period and the chirp coefficient, repectively. According to the expreion (0) in [16], to realize the Talbot effect, the required general phae induced by the chirp at apling point hould be derived a the following equation: Φ ( z = kp) = k π, k = 0, 1,,... (1) where P repreent the apling period, and are arbitrary poitive integer with that (/) i a non-integer and irreducible nuber. If Eq. (1) i provided, the SFBG

3 Hybrid technique baed on chirp effect... 7 pectru preent a narrower channel pacing by tie and a iilar diperion characteritic, a copared with that of the unifor SFBG. It i called a the integer or fractional Talbot effect. The product ( ) can be an even integer or an odd integer, which reult in the direct Talbot effect or the invere Talbot effect [16]. To contruct the phae condition (1), variou technique are propoed and invetigated. The linear chirp effect and the phae hift technique are the two ot popular technique. In our hybrid technique, the adopted chirp i linear and the adopted phae i alo linear. In detail, the phae hift θ k are proportional to the order of apling period: θ k = (k 1)α, k = 1,, 3,... () where α =(/)π i a fixed phae value. Equation () give rie to an accuulated quadratic phae condition, which i inditinguihable fro the linear chirp. Therefore, Eq. (1) can be regarded a a general phae condition for the hybrid technique, ore than a general condition only for the linear chirp. Then, we propoe our hybrid technique (i.e., a cobination of the chirp effect and the phae hift technique) to ipleent the Talbot effect. For a given chirp coefficient C and a erie of phae hift θ k = (k 1)α, when the relationhip between the i etablihed a CP π + α = π (3) the condition in direct analogy to (1) i alo obtained for the hybrid technique: Φ ( z = kp) CP = π + α k = k π (4) Equation (4) i the principle bae of the hybrid technique. It predict that the oberved phenoena are analogou to the phenoena in [16]. Note that the general phae condition (1) and (4) are the principle bae of the hybrid technique. Furtherore, (1) and (4) are general condition for other technique, uch a the chirp effect, the phae hift technique, and approach reported by DAI et al. [4]. We will orderly deontrate thee technique by the general condition and the hybrid technique in the following ubection... Linear chirp effect With the ae chirp effect indicated in the hybrid technique, the phae induced by the linear chirp effect i: Φ ( z) = πc z (5)

4 8 GUO DENG, WEI PAN If the chirp coefficient C i pecified a: C = P Equation (1) i derived by ubtituting (6) into (5) a z = kp. In other word, the linear chirp i a pecial type of the hybrid technique if we pecify α = 0 in (4). Thi i the ot popular approach invetigated for Talbot effect induced by chirp. In particular, the periodically chirped SFBG (PC-SFBG), propoed in our previou work [6], i an expanded exaple of the linear chirp. With a phae ditribution: Φ ( z) = N 1 k = 0 π ( z kmp), P the phae of the grating aple located at z = kp are: (6) 0 z kmp MP (7) Φ ( z = kp) π (8) π π =,,, , , A pointed out in [6], (8) i equivalent to the general phae condition (1) when i an even integer. Conequently, the fractional or integer Talbot phenoena are alo oberved in PC-SFBG..3. Phae hift technique For the linear phae hift technique, a erie of phae hift decribed in () are inerted into the SFBG to derive the condition (9). The phae condition for thi technique i a follow: Φ ( z = kp) = θ k = k π (9) Evidently, thi linear phae hift technique i a pecial exaple of the hybrid technique with C = 0 in (4). Recently, YUSUKE NASU et al. have propoed the ultiple phae hift (MPS) technique [18], which can be conidered a a pecial branch of the hybrid technique too. If we introduce the θ' k (according to the MPS) into the SFBG, (1) can be rewritten a Φ ( z = kp) = θ' k = k π ± Kπ (10) where K i a variable-integer aociated with the order of apling period k and detailed MPS paraeter. In addition, DAI et al. [4, 1] have propoed a novel approach to deign ulti- -channel filter and ulti-channel copenator, which alo can be explained in

5 Hybrid technique baed on chirp effect... 9 a b Fig. 1. Coparion of Talbot phenoena between our technique (olid line) and Dai approach (dotted line): =1, =3, α =0.5π, C = Λ /(1P 0 ). Reflection characteritic (a), and group delay characteritic (b). the way of the hybrid technique. For Dai approach, the phae hift i θ k =kα, not θ k =(k 1)α a in the hybrid technique. Obviouly, after a contant phae hift α i inerted into each apling period in the hybrid technique, Dai approach i fored. Thi proce i eaier to undertand a copared with the coplex derivation in [4]. A illutrated in [18], the contant phae hift contribute to the hift of wavelength grid, without deviation in the reflection or diperion characteritic. Figure 1 how the correponding hift in channel grid, including the reflection characteritic and the diperion characteritic. 3. Hybrid technique for ulti-channel diperion copenator In Section, we illutrate and dicu the hybrid technique which refer to ulti- -channel filter and tunable ulti-channel filter. Meanwhile, the hybrid technique can be ued to deign ulti-channel diperion copenator. According to [1], we conclude the equivalent chirp coefficient C 0 with a ore general expreion: k π CP + α k C 0 P = π π (11)

6 30 GUO DENG, WEI PAN A long a (11) i offered, the SFBG realized by the hybrid technique ha iilar diperion (D = 1/(cC 0 ) derived in Appendix) to that of the SFBG with the equivalent chirp coefficient C 0 (a uch le chirp coefficient). Alo, we can ee that the chirp effect and the phae hift exert the ae contribution on C 0 in (11). Naely, the variation on the chirp coefficient C or the phae hift α deterine the diperion of the SFBG equivalently. For intance, the paraeter for the Talbot phenoena are C = Λ /(1P 0 ) and α =0.5π. To obtain the linear group delay characteritic, we adopt the inc-like profile [1] intead of the Hanning function. For one SFBG with α =0.54π (C i fixed), it reflection characteritic and diperion characteritic are hown in Fig. a; for another SFBG with C = (1/ ) Λ /P 0 (α i fixed), the correponding reflection characteritic and diperion characteritic are hown in Fig. b. Evidently, the two SFBG do have the ae diperion value and iilar reflection peak during the channel. A diperion of 78 p/n, a channel pacing of n, and a 3-dB bandwidth of 0.5 n are oberved. Therefore, different diperion value can be obtained by adjuting the phae hift in SFBG to the fixed chirp coefficient. Thi i analogou to the ethod that different diperion copenator can be fabricated through the ae phae ak [1]. a b Fig.. Reflection and diperion characteritic of SFBG with different chirp coefficient or phae hift: α =0.54π, C = (1/1) Λ /P (a); α =0.5π, C =(1/1+0.04) /P 0 (b).

7 Hybrid technique baed on chirp effect a b c d Fig. 3. Diperion characteritic of SFBG with different phae hift: α =0.5π (a), α =0.5π (b), α =0.54π (c), α =0.56π (d) when a fixed chirp coefficient C = Λ /(1P 0 ) i ued. Moreover, tunable diperion copenator can be achieved by thi hybrid technique. For an operating SFBG, variation on the phae hift α reult in different diperion value. Figure 3 illutrate the diperion characteritic of thi novel tunable diperion copenator. The abolute value of diperion (the lope of group delay) decreae a the deviation of α increae (i.e., the increae of the equivalent chirp coefficient C 0 ), excepting that the diperion i zero a C 0 = 0. In detail, we can get the fitting value of diperion fro Fig. 3, a hown in Fig. 4. Thee diperion value are cloe to the theoretical value D = 1/cC 0. Siilar to the tunable ulti-channel filter entioned in Section, if α and α' can be adjuted iultaneouly and inde- Fig. 4. Fitting value of diperion correponding to Fig. 3.

8 3 GUO DENG, WEI PAN pendently, both the wavelength grid and the diperion of SFBG can be tuned individually. Conequently, ulti-channel tunable copenator (tunable channel grid and tunable diperion) are realized by a phae ak (the ae chirp coefficient). Additionally, ince the dynaic adjutent in α and α' hould be controlled accurately, a uitable tuning approach with high accuracy i required for tunable characteritic. In thi work, we are ainly devoted to the ipleentation of the hybrid technique in theory and the tuning approach i beyond the dicuion of thi paper. A long a uch an approach i etablihed, thee SFBG deigned through the hybrid technique are excellent choice for optical counication yte. 4. Concluion In thi work, we have propoed a hybrid technique to oberve the Talbot phenoena in SFBG. The principle bae of thi hybrid technique i a general phae condition realized by the linear chirp and the phae hift. With the ue of the hybrid technique, ulti-channel diperion copenator are deigned. Furtherore, the hybrid technique i characterized by everal outtanding feature. Firtly, it i able to act a a general technique for other iilar technique, uch a the linear chirp and ultiple phae hift. Secondly, becaue of the equivalence between the chirp coefficient and the phae hift in the general phae condition, different diperion are realized by different arrangeent between the chirp coefficient and the phae hift. Finally, tunable characteritic on the diperion are introduced into thee device by dynaically tuning the phae hift α and α'. Appendix In a chirped grating, the group delay τ g reult fro the twice optical path length fro the input of the grating to the reflection point: τ g n L z L L =, z c (A1) where c i the velocity of light in vacuu, L i the length of grating. In particular, for the linear chirp Λ(z) = C 0 z, there i a relationhip between the reflected wavelength and the reflection point z: z Λ = = C 0 λ λ nc 0 (A) where λ =nλ and λ 0 =n are the reflected wavelength and the Bragg wavelength, repectively. With (A1) and (A), we can get the diperion D a follow: D dτ g d nlc ( λ λ 0 ) = = = dλ dλ cc cc 0 (A3)

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