In a uniform fiber Bragg grating, if the input signal is a Gaussian pulse the dispersion is zero near center wavelength and becomes appreciable only near the band edges and side lobes of the reflection spectrum. However for chirped Gaussian pulses, group velocity dispersion and the reflected light must become asymmetric. Here the chirped Gaussian pulses can be treated as a symmetric but nonuniform input signal. The present paper describes that for the case of symmetric Gaussian pulse, the group velocity dispersion and pulse distortion remain symmetric however strong the grating may be. On the other hand both tend to be more asymmetric for the case of strong grating while the input signal is symmetric with nonuniform shape

Raghuwanshi, Sanjeev Kumar

Talabattula, Srinivas

Open Access Repository of IISc Research Publications

Asymmetric group velocity dispersion and pulse distortion in a uniform fiber Bragg grating

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IACS Institutional Repository

Indian J. Phys. 82(12) 1681-1687(2008) mp Asymmetric group velocity dispersion and pulse distortion in a uniform fiber Bragg grating Sanjeev Kumar Raghuwanshi* and Snnivas Talabattula Applied Photonics Laboratory, Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore-560 012, India E-mail sanjeevkrfa ece use ernet in Received 2 March 2008, accepted 4 Ju/y 2008 Abstract : In a uniform fiber Bragg grating, if the input signal is a Gaussian pulse the dispersion is zero near center wavelength and becomes appreciable only near the band edges and side lobes of the reflection spectrum However for chirped Gaussian pulses, group velocity dispersion and the reflected light must become asymmetric Here the chirped Gaussian pulses can be treated as a symmetric but nonuniform input signal The present paper describes that for the case of symmetric Gaussian pulse, the group velocity dispersion and pulse distortion remain symmetric however strong the grating may be On the other hand both tend to be more asymmetric for the case of strong grating while the input signal is symmetric with nonuniform shape Keywords : Coupled mode theory of fiber Bragg gratings, group velocity dispersion effect, chirped Gaussian pulses PACS Nos. : 42.79.Dj, 42 79 -e 1. Introduction Recently, there has been growing interest in the dispersive properties of fiber Bragg gratings for applications such as dispersion compensation, pulse shaping and fiber and semiconductor laser components [1]. Although many of these rely on the ability to tailor the dispersion in the nonuniform gratings, here we introduce the basis for determining delay and dispersion from the known (complex) reflectivity of a uniform Bragg grating. The amplitude and power reflection coefficients in a uniform fiber Bragg grating for the case of non-phase-matched contradirectional modes coupling are given by [2-14] : -K s\nh(f2L) ^sinh(/?L) + /rtcosh(rtL) v ' #n .. A ,. © 2008 IACS Corresponding Author 1682 Sanjeev Kumar Raghuwanshi and SrinivasTalabattula \pf~ n2 + 2 sinhz(f?L) Q2cost\2(f2L) + lEi) 2 sinh2(flL) (2) Here the parameters ru k and Q represent the mismatch between the momentum of the mode with the refractive index perturbation, mode coupling coefficient and the phase detuning parameter, respectively. Turan Erdogan [2] presents a derivation of the above equations. The group delay and dispersion of the reflected light can be determined from the phase of the amplitude reflection coefficient [2]. If we denote 0p = phase{p) then, the delay time rv for light reflected off a grating is 0„ T" du 2nC dX (3) 1MB S 1M0 14606 1881 Mnm) Figure 1 . Calculated reflection spectra and corresponding group delay for an uniform Bragg grating with (a), (b) KL = 2 and (c), (d) KL = 8. I xAA/\i" 1&82 " 0 6 18488 18S0 18502 15804 18808 lV . t f X(nm\ 4rdM IJ —^Vr i v . — i •TO 2 18*04 18488 184*8 1M 18802 18804 18808 1»0« Figure 2. Calculated group velocity dispersion for the same case as Figure 1 with (a) KL = 2 and (b) *L = 8. Figure 1 shows the delay rp calculated for the two examples of the grating. We see that for unchirped uniform grating both the reflectivity and delay are symmetric about the center wavelength. Since the dispersion dp (in ps/km) is the rate of change of delay with wavelength, we find " d\ 2T„ A2 d2e 2nC d\2 Asymmetric group velocity dispersion and pulse distortion in a uniform fiber Bragg grating 1683 ~2nC\ 2 ^ dcv2 (4) In a uniform grating, the dispersion is zero near center wavelength as shown in Figure 2 and becomes appreciable only near the band edges and side lobes of the reflection spectrum, where it tends to vary rapidly with wavelength. 2. Asymmetric pulse distortion Gaussian pulses can be considered as a symmetric signal, on the other hand chirped Gaussian pulses can be presumed to be nonuniform but symmetric signal. Figure 3 demonstrates various kinds of Gaussian pulses with different values of chirp parameter [3]. Now if we feed these Gaussian pulses to a uniform Bragg grating, we can visualize the pulse distortion as well as reflected mode field due to these chirp factors. We consider only the positive chirped Gaussian pulses into analysis because the negative chirp factor will only reverse the whole effect. First we present the results of chirp free pulses. Figure 4 shows the fundamental reflected mode field and corresponding peak power reflection spectrum. 5 o °[ ~-\/\}^J "S r* C l l X » .• T5S0 we* 1S«' 1»'0 1650 5 DM Figure 3. Gaussian pulses (left side) and corresponding phase spectrum (right side) for various values of chirp factor. Figure 4. Fundamental reflected mode field and corresponding peak power reflection spectrum while C=0and /,L = 2 Figures 5 and 6, show the reflected light for the two cases, one for weak grating (KL = 2) and another for strong grating (*L = 6), while chirping factor is kept at C = 0 and C = 2, respectively. One can see from Figures 5 and 6 that for chirp free pulses the reflected power is symmetric in both the cases, while for C = 2, they are considerably asymmetric. Reflected pulse would be more asymmetric for strong grating. 1684 Sanjeev Kumar Raghuwanshi and SnnivasTalabattula r I M S 154115 1550 15501 1351 1549 1549 5 1550 1350 5 1551 »(nm) x (nm) Figure 5. Reflected light at z = 0, while (a) C = 0, (b) C = 0 a n c U i . = 2 1649 5 1610 1550 1 x<nm) 1549 1S48 5 ia<j0 l&fc)-* JL(nm) Figure 6. Reflected light at z = 0, while (a) C = 0 (b) C = 2and hL = 8 3. Asymmetric group velocity dispersion Dispersion plots and corresponding contour maps for various values of chirp factors are shown in Figures 7, 8 and 9, respectively for KL = 2. For the case of chirp free pulse 1560 0 15604 1550? 1550 1549 B 1549 4 — ™ _ . = - , -j n y — u i ^ m *-" 11"LM' <- r~~*- r -*» ~ - --1550 41 I 1S8° Ccrtotr Map 0-(paftm) ft48 154J 6 5 1550 1550 5 1561 U - ^ - i ColoirMaplXpftAvn) Figure 7. Dispersion and corresponding contour map for C = 0 and ^ L = 2) Figure 8. Dispersion and corresponding contour map for C = 2 (/vL = 2) the dispersion is zero at center wavelength where the peak reflection occurs; on the other hand, for considerably chirped pulses the dispersion curve no longer remains symmetric but picks up a considerable curvature. For highly chirped Gaussian pulses (C = 4), the dispersion curve picks up larger curvature at all the wavelength of interest This feature can be considered as an asymmetric response of uniform fiber Bragg grating even for weak grating case. Figures 10 and 14 show the fundamental reflected mode field and corresponding peak power reflection spectrum for the case of moderately strong {KL = 4) and strong grating {KL = 6). The reflection bandwidth of the spectrums are 0.4 nm (Figure 10) and 0.6 nm (Figure 14), respectively. Dispersion and contonr maps for different values of C and KL are displayed in Figures 11-13. Comparison of Asymmetric group velocity dispersion and pulse distortion in a uniform fiber Bragg grating 1685 2 1 0 1 ? Figure 9. Dispersion and corresponding contour map for C = 4 (*X = 2) 2 1 0 Figure 11. Dispersion and corresponding contour map for C = 0 (KL = 4) M 1649 6 I S M 15803 1661 2 1 0 1 2 Figure 13. Dispersion and corresponding contour map for C = 4(«£. = 4) | 0 6 {., 1M9 1M9 1 V S 1 *< Figure 10. Fundamental reflected mode field and corresponding peak power reflection spectrum while C = 0 and KL = 4 r t 1550 8^ 1550 4 L ) 1550 2 ?«60l i54«ef 1546 5 15A0 1550 5 1561 X<nm> Contot* Map tXo»*mm Figure 12. Dispersion and corresponding contour map for C«2( /vL = 4) Joe 1&4SS 1560 15605 1511 Figure 14. Fundamental reflected mode field and corresponding peak power reflection spectrum while C=0andKL = 6 1686 SanjeevKumarRaghuwanshiand Snnivas Talabattula 4 Ajjl Tnv** -J t 1 | 1549 8 1 19496| 1^49 15496 1550 15605 1451 2 1 0 1 2 Cortour Map CMpMnn) Figure 15. Dispersion and corresponding contour map for C = 0 (* L = 6) 80 60 40 | 20 r ° 20 -40 -ao r _r" "4" ^549 1549 5 1540 -•1 \ jri 1550 5 15* 15506 1550 4 1580 2 1590 1549 8 1S4»6 1549 4 f '*" * ' / / -r - T ___ _ _ _ _ - ™ _ / ' ( | \ / - - ~ r „. •-^ I I I 2 1 0 1 2 Coriow Map O^psmn) Figure 16. Dispersion and corresponding contour map forC = 2 ( * L = 6) -4 ^ _x ^ " i _ A 1550 6 1550 4 1550 2 1 1550 1549 fit 1549 6 V 1549 4 f 1549 1549 5 1550 15505 1551 X(rwn) 2 1 0 1 2 Contour Map CHps/nm) Figure 17. Dispersion and corresponding contour map for C = 4 (*L = 6) Figures 7, 11 and 15 suggests that for the case of symmetric Gaussian pulse the zero dispersion bandwidth is more pronounced for the case of strong grating (B.W. « 0.6 nm) [15] while for chirp free pulses dispersion accumulates more for some singular points in the case of a strong grating (±80 ps/nm). The dispersion graph becomes significantly asymmetric for strong grating for considerably chirped Gaussian pulses. It is quite symmetric for weak grating (±5 ps/nm) case even for the same simulation parameters. 4. Conclusion We have found that for the case of strong grating the dispersion is large at some particular point of bands which may be due to rapid phase variations. Infact, the weak Asymmetric group velocity dispersion and pulse distortion in a uniform fiber Bragg grating grating shows lesser dispersion but the dispersion is more spread out over the band as compared to strong grating The dispersion graph tends to be more asymmetric for the strongly chirped pulses. Simulation also shows that the dispersion is high due to strong grating at some singular point of wavelength There is no effect of chirp factor on absolute value of dispersion as such The only advantage of using strong grating for chirp free pulse is that we can achieve a large band of zero dispersion at some center wavelength where the peak reflection occurs References [1] F Ouellette Optics Letters 16 847 (1987) [2] T Erdogan IEEE J Lightwave Technol 15 1277 (1997) [3] G P Agarwal Nonlinear Fiber Optics (New York Academic) (2001) [4] A Othonos and K Kalli Fiber Bragg Gratings Fundamentals and Applications in Telecommunications and Sensing (London Artech House) (1999) [5] K 0 Hill IEEE J Lightwave Technol 15 1263 (1997) [6] L R Chen, S D Benjamin, P W E Smith and J E Sipe IEEE J Lightwave Technol 15 1503 (1997) [7] C R Giles IEEE J Lightwave Technol 15 1391 (1997) [8] J E Sipe, B J Eggleton and T A Strasser Opt Commun 15 269 (1998) [9] A Carballar, M A Muriel and J Azana IEEE Photon Technol Lett 11 694 (1999) [10] S K Liaw, K P Ho, Y K Chen and C C Lee Optical and Quantum Electronics 31 77 (1999) [11] Stefano Longhi Phys Rev E64 037601 (2001) [12] S Longhi, M Marano, P Laporta, O Svelto and M Belmonte J Opt Soc Am B19 2742 (2002) [13] Jaepong and Byoungho Lee IEEE Photon Technol Lett 17 408 (2005) [14] Stefano Longhi Phys Rev E72 056614 (2005) [15] S K Raghuwanshi and S Talabattula J Instrum Soc India 37 297 (2007) 