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Mode-locked performance of hybrid soliton pulse source utilizing fiber grating external cavity lasers
Optics Communications 260 (2006) 227–231 www.elsevier.com/locate/optcom
Mode-locked performance of hybrid soliton pulse source utilizing fiber grating external cavity lasers Nuran Dogru
*
Department of Electrical and Electronics, University of Gaziantep, 27310 Gaziantep, Turkey Received 24 May 2005; received in revised form 2 October 2005; accepted 6 October 2005
Abstract Mode-locking characteristic of hybrid soliton pulse source (HSPS) utilizing linearly chirped raised-cosine flat top apodized fiber Bragg grating (FBG) is investigated by using coupled-mode equations. It is found that the fundamental repetition frequency range of HSPS is significantly extended by using linearly chirped raised-cosine flat top apodized FBG instead of linearly chirped Gaussian apodized FBG. The range of repetition frequencies over which proper mode-locking is obtained is 2–3.3 GHz with linearly chirped raised-cosine flat top apodized grating whereas this range is 2.1–2.95 GHz with linearly chirped Gaussian apodized grating. 2005 Elsevier B.V. All rights reserved. Keywords: Pulse generation; Hybrid soliton pulse source; Active mode-locking; Semiconductor lasers
1. Introduction Mode-locked semiconductor lasers can be used as optical pulse source in many applications, including telecommunication systems, measurement systems, and as clock sources for optoelectronic processing systems. Their versatility and reliability is matched by the potential for production of relatively cheap and reliable devices. Hybrid soliton pulse source (HSPS) is one such device, developed as a pulse source for soliton transmission system. This device has demonstrated an extremely wide-mode-locking frequency range due to a novel wavelength self-tuning mechanism [1], which makes it very useful when precise control over the operating frequency is required. The device has also proved to be robust in operation, providing a consistent output over a large range of driving parameters. This attributes allowed the source to be used in a soliton transmission experiment, where information was transmitted at 10 Gb/s over 27,000 km [2]. Several examples of short optical pulse sources using active mode-locking have been pro*
Tel.: +90 342 3601200; fax: +90 342 3601103. E-mail address: [email protected].
0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.10.018
posed. This method provides low timing jitter picosecond pulses, locked to an external electrical reference frequency, for high-speed optical communications systems. Active mode-locking with external cavity gratings is one technique used to generate such pulses from semiconductor lasers by applying an rf drive current at frequency matching the roundtrip of the laser cavity. Fibers grating external cavity lasers have been experimentally demonstrated in the modelocking regime at 2.5 GHz [1] and 10 GHz [3]. The HSPS described in this paper is being developed at AT&T Bell Laboratories as a possible source for ultra-long distance soliton communication systems. The HSPS integrates a semiconductor laser gain section and fiber Bragg reflector in a hybrid device, to capitalize on the inherent advantages each possesses. The semiconductor section allows simple and efficient pumping of the device, and the fiber Bragg reflector provides excellent wavelength and optical bandwidth control. Together these components provide a source which is both simple and inexpensive, whilst satisfying the exacting performance requirements for ultra-long distance soliton transmission systems. The Bragg reflectors are manufactured using holographic techniques, which allows great control over the
228
N. Dogru / Optics Communications 260 (2006) 227–231
Bragg wavelength and intensity profile. The reflector is produced by exposing an optical fiber to a two beam interference pattern, at a wavelength near the peak of a defect absorption band at 242 nm [4]. The absorbed UV light permanently modifies the index of refraction in the core, transferring the interference pattern to the core index profile, and therefore, forming a grating, or Bragg reflector. The grating intensity profile follows that of the laser beams, and is therefore approximately Gaussian. This provides for a very clean reflection spectrum free from any significant side-lobes. HSPS utilizing a uniform grating Bragg reflector integrated into a fiber external cavity has been shown to operate close to the required frequency with transform-limited pulses and very high output powers [5]. However, this device worked well only under specific operating conditions, and showed spectral instabilities when operating parameters were changed. Utilizing with a linearly chirped grating has overcome the spectral instability problem of the earlier device. An interesting feature of these devices is the extremely wide operating repetition frequency range, which can be enhanced by using chirped gratings, as reported in [1,6]. In [6], calculated and measured results showed that transform-limited pulses are generated over a repetition frequency range of 850 MHz using linearly chirped Gaussian apodized fiber Bragg grating (FBG). However, in this paper, it is shown that the repetition frequency range of proper mode-locking where transform-limited pulses is obtained is extended to 1.3 GHz by utilizing linearly chirped raised-cosine flat top FBG. 2. Model The structure of the HSPS utilizing linearly chirped raised-cosine flat top apodized FBG, considered in our analysis is shown in Fig. 1. The geometry is same as the one reported in [7,8]. The HSPS system is made up of a multi-quantum well (MQW) semiconductor laser, a fiber and a FBG. One facet of diode is high reflectivity (HR) coated and the other antireflection (AR) coated. The light from the AR coated facet is coupled to the FBG reflector. The field in this system travels between the HR coated laser end and effective cavity length of the grating. The output power is taken through the grating. The mode-locked HSPS model is based on time-domain solution of the coupled-mode equations. Assume that the longitudinal, effective-refractive index variation of the Bragg grating is given as
dc+rf AR
Fiber Bragg grating
HR
Light output MQW Laser
Fiber Fig. 1. Schematic of HSPS.
2p nðzÞ ¼ nco þ Dn 1 þ m cos z ; KðzÞ
ð1Þ
where nco is the unperturbed effective index of the fiber, Dn represents the dc index change spatially averaged over a grating period, m is the modulation index of the grating. The z-dependent grating period K(z) is linearly chirped and is taken as KðzÞ ¼ Ko þ
1 dko z; 2nco dz
ð2Þ
where Ko = ko/2nco is the pitch of the unchirped Bragg grating at the operating wavelength ko. The grating pitch is accepted to be linearly chirped such that the operating wavelength ko corresponds to the center of the grating and the wavelength is chirped by C = dko/dz (cm/cm) around the center. Using couple mode theory [9], the inter-coupling relations between forward-propagation field E+(z, t) and reverse-propagation field E(z, t) can be written as dEþ 2j 1 d/ þ ¼ j d þ ð3Þ E jjE ; m 2 dz dz dE 2j 1 d/ ¼j dþ ð4Þ E þ jjEþ ; m 2 dz dz where j is the coupling coefficient between forward and reverse waves, d is the deviation from real part of propagation constant b, and u is the grating chirp. The loss in fiber and grating can be neglected since it is a few centimeters long. If the term inside the parenthesis in Eqs. (3) and (4) is called the dc ‘‘self-coupling’’ coefficient, as in [10] r¼dþ
2j 1 du . m 2 dz
ð5Þ
The coupling coefficient is assumed raised cosine flat-top [10] as 1 pz jðzÞ ¼ jp 1 þ cos ; ð6Þ 2 FWHMj where jp is the peak value of the Gaussian variation and FWHMj is the full-width half-maximum (FWHM) of this variation and is taken simply as Lg(grating length)/2 throughout calculations [10]. The coupled-mode equations are solved using a piecewise-uniform approach. First, the equations are solved analytically and then the grating is divided into M sections each having an equal length Dz. Assuming the boundary conditions at z = L/2 as E+(L/2) = 1 and E(L/2) = 0, the calculations are carried out back to forth (from z = L/2 to L/2), the parameters of each section are calculated and these parameters are put into 2 · 2 propagation matrix Ti. The fields at the ith section can be calculated from the known fields of the previous section such that þ þ Ei Ei1 ¼ T . ð7Þ i E E i i1
N. Dogru / Optics Communications 260 (2006) 227–231
" Ti ¼
coshðci DzÞ j jcii
j rcii
sinhðci DzÞ
sinhðci DzÞ
j jcii
#
sinhðci DzÞ
coshðci DzÞ þ j rcii sinhðci DzÞ
;
ð8Þ
where the coupling coefficients are related as c2i ¼ j2i r2i .
ð9Þ
Note that the values of ji, ri and hence ci are different for each section and must be calculated individually before putting into the propagation matrix. The number of sections needed for the calculations is determined by the required accuracy. For most cases, M 100 is sufficient [10]. M may not be arbitrarily large, since the approximations that lead to the coupled-mode equations are not valid when a uniform grating section is only a few grating periods long. Thus, we require Dz K which means M 2ncoL/ko. For the modeling of a multi-section system like HSPS, coupled-mode equations must be modified in such a way that they include gain, and loss in the laser. In each laser section, the carrier density is calculated from the rate equation dN ðz; tÞ IðtÞ N ðz; tÞ ao ðN ðz; tÞ N o Þ ¼ mg Sðz; tÞ; dt eV sn 1 þ eSðz; tÞ
ð10Þ
where I(t) is the injection current, V is the active layer volume, e is the electronic charge, N(z, t) is the carrier density, sn is the carrier lifetime, S(z, t) is the photon density and it is proportional to jE+j2 + jEj2, No is the carrier density at transparency, e is the gain saturation parameter and ao is the differential gain. The laser sections also include the coupling between carrier density and the refractive index through the linewidth enhancement factor and it is given as Dn ¼
ko Caao DN ðz; tÞ; 4p
ð11Þ
where C is the confinement factor, a is the linewidth enhancement factor, and DN(z, t) is the change in the carrier density. For each time step the new field values are calculated and boundary conditions applied. This process is repeated for a sufficient number of modulation periods to obtain stable mode-locked pulses.
longer grating. The group delay curve with respect to the wavelength becomes linear for longer gratings. With help of this feature, HSPS can be mode-locked for a wider repetition frequency range, since the whole reflection spectrum can be used [6]. Even if the length of the grating is constant, its reflection spectrum can be made wider or narrower with the application of a proper amount of chirp. As the magnitude of the chirp is increased, the spectrum becomes wider and the side lobe disappears. Although the group delay curve for the unchirped reflector is nonlinear, it becomes linear as the rate of chirp is increased. Therefore, the higher chirp values will result in good gratings for HSPS applications. Although higher modulation index (m) values result in a more symmetric reflection spectrum, there is no big difference between the group delay characteristics for different m values. All these parameters are used to tailor the response of the grating and their optimum configuration can be found. In our simulation, a grating length of 4 cm is used with maximum power reflectivity of 0.5 at 1.55 lm. Linear chirp ˚ /cm and 0.8. The fundavalue and m are taken as 1.9 A mental mode locking frequency is chosen as 2.5 GHz and step size is 0.6875 ps. The other MQW laser diode parameters are given in Table 1. If the period of the sinusoidal variation of refractive ˚ /cm) through the grating index is reduced linearly (1.9 A length, then the reflectivity and group delay response of the Gaussian apodized FBG will be as in Fig. 2(a). As seen in figure, the reflection spectrum covers a wide range of Table 1 Laser diode parameters 2 · 1017 cm3 10 · 1016 cm2 5 · 105 0.8 0.01 0.9 0.1 0.8 · 109 s 25 cm1
Gain saturation parameter Differential gain Spontaneous coupling factor Field coupling factor AR coating field reflectivity HR coating field reflectivity Optical confinement factor Carrier lifetime Internal loss
e ao bsp g r3 r1 C sn aint
0.6
400 350
3. Results One of the important features of the HSPS is its very wide operating frequency range [1], due to the spatially distributed nature of the grating. The length of the grating can be changed in order to adjust the spectral width of the reflection spectrum. As the length of the grating is increase, the full-width at half-maximum (FWHM) of the reflection spectrum increases as well. Since the wider the FWHM of the reflection spectrum, the better the wavelength self-tuning for mode-locked HSPS [6], it is preferable to have a
reflectivity
0.5
a
0.4
300 250
b 0.3
200 150
0.2
100 0.1 0 1549.2
group delay, ps
The propagation matrix for the ith mode can be written as
229
50 1549.6
1550
1550.4
0 1550.8
wavelenght, nm Fig. 2. Reflectivity and group delay characteristics for linearly chirped Gaussian apodized and linearly chirped raised cosine flat top FBG.
230
N. Dogru / Optics Communications 260 (2006) 227–231
˚ . If grating is raisedwavelengths giving a FWHM 3.78 A cosine flat top then the resulting reflection spectrum and group delay will be shown in Fig. 2(b). The reflection spec˚ ) compared to that of trum is wider (FWHM = 6.21 A Gaussian apodized grating. Also, peak reflectivity occurs at the operating wavelength giving a symmetric reflection spectrum for linearly chirped raised-cosine flat top FBG as seen in Fig. 2(b). This is because of zero dc index change of these gratings. In this case, grating behaves as if it has only ac index variation. In other words, the grating shows a performance similar to the unapodized gratings. These results reveal that linearly chirped raised-cosine flat top
3.3 GHz TBP=0.420
3.2 GHz TBP=0.500
3.1 GHz TBP=0.420
3 GHz TBP=0.500
2.9 GHz TBP=0.414
2.8 GHz TBP=0.390
field spectra
2.7 GHz TBP=0.408
2.6 GHz TBP=0.392
2.5 GHz TBP=0.394
FBG is superior to the linearly chirped Gaussian apodized FBG because of wider and symmetric reflection spectrum. By applying a linear chirp across the reflector, the effective cavity length becomes wavelength dependent. When the mode-locking repetition frequency is changed, the device self-tunes its wavelength to give the correct cavity length to keep on resonance with the modulation frequency. When the mode-locking repetition frequency is increased the effective cavity length will decrease, and the output will move to longer wavelengths. This feature of the HSPS is shown in Fig. 3, where the field spectra of the pulses are shown for different mode-locking repetition frequencies ranging from 2 to 3.3 GHz. A constant dc bias of 6 mA and an rf current 20 mA are used over the whole repetition frequency range. The shift of the entire spectrum to lower or higher frequencies is obvious when the modelocking repetition frequency is changed. The change of 1.3 GHz in mode-locking repetition frequency gives a shift of 94 GHz in the field spectrum of the Bragg reflector. This shift was found for 29 GHz in [6]. Calculated pulsewidths as a function of rf current is shown in Fig. 4 for different values of dc bias and modulation repetition frequency is 2.5 GHz. This plot includes entire range of operation for which the device produces clean, stable, single pulses. Longer pulses occur for small drive levels. The dc bias also has a small effect on the pulsewidth producing slightly shorter pulses at lower dc drives levels. The peak power rises as both the dc and rf drive levels are increased, as expected. Obtained pulses are shorter than that of linearly chirped Gaussian apodized FBG [6]. The calculated results for time–bandwidth product and pulsewidth versus rf mode-locking repetition frequency are shown in Figs. 5 and 6 for the range of dc and rf values producing clean pulses. These results show the broad operating range of this source with pulsewidths of 40–80 ps and time–bandwidth products of 0.3–0.5 over rf repetition frequencies from 2 to 3.3 GHz. Mode-locking range was found for 850 MHz (2.1–2.95 GHz) in [6] although
2.4 GHz TBP=0.397
90 80
2.3 GHz TBP=0.400
pulse width,ps
70 2.2 GHz TBP=0.424
2.1 GHz TBP=0.432
60 50 40 30
2 GHz TBP=0.443
20 -80
-60
-40
-20
0
20
40
60
80
Frequency offset, GHz Fig. 3. Field spectra of HSPS for different mode-locking repetition frequencies.
0
10
20
30
40
50
60
rf current,mA Fig. 4. Pulsewidth as a function of rf current for all dc current with a mode-locking repetition frequency of 2.5 GHz.
N. Dogru / Optics Communications 260 (2006) 227–231
close to 0.4 are achieved for all bias conditions, with pulsewidths from 40 to 80 ps. In conclusion, HSPS utilizing linearly chirped raisedcosine flat top FBG works over a large modulation repetition frequency range of 1.3 GHz giving a time–bandwidth product close to 0.4.
time-bandwidth product
0.55 0.5 0.45 0.4 0.35
4. Conclusion
0.3 0.25 0.2
1.6
1.9
2.2
2.5
2.8
3.1
3.4
mode-locking repetition frequency, GHz Fig. 5. Time–bandwidth product as a function of mode-locking repetition frequency for all rf and dc current.
100 90 80
pulsewidth, ps
231
It has been demonstrated with extremely wide modelocking repetition frequency range and stable operation by the use of a linearly chirped Bragg reflector integrated in a fiber external cavity. The device produces a clean stable output of near transform limited pulses over a large range of drive parameters, making a good source for use in many practical systems. However, this repetition frequency range where transform-limited pulses are generated is significantly extended by using linearly chirped raised-cosine flat top FBG giving a range of 1.3 GHz instead of linearly chirped Gaussian apodized FBG having a range of 850 MHz. References
70 60 50 40 30 20 1.6
1.9
2.2
2.5
2.8
3.1
3.4
mode-locking repetition frequency, GHz Fig. 6. Pulsewidth as a function of mode-locking repetition frequency for all rf and dc current.
time–bandwidth product changes between 0.3 and 0.7. At the center of the repetition frequency range where the device is designed to operate, time–bandwidth products
[1] P.A. Morton, V. Mizrahi, P.A. Andrekson, T. Tanbun-Ek, R.A. Logan, P. Lemaire, D.L. Coblentz, A.M. Sergent, K.W. Wecht, P.F. Sciortino Jr., IEEE Photon. Technol. Lett. 5 (1993) 28. [2] P.A. Morton, V. Mizrahi, G. Harvey, L. Mollenauer, T. Tanbun-Ek, R.A. Logan, H.M. Presby, T. Erdogan, P. Lemaire, A.M. Sergent, K.W. Wecht, IEEE Photon. Technol. Lett. 7 (1995) 111. [3] R. Paoletti, D. Bertone, R. Fang, G. Magnetti, M. Meliga, G. Meneghi, G. Morello, G. Rossi, L. Tallone, M. Scofet, IEEE Photon. Technol. Lett. 12 (2000) 245. [4] V. Mizrahi, J.E. Sipe, J. Lightwave Technol. 11 (1993) 1513. [5] P.A. Morton, V. Mizrahi, S.G. Kosinski, L.F. Mollenauer, T. Tanbun-Ek, R. Logan, D.L. Coblentz, A.M. Sergent, K.W. Wecht, Electron. Lett. 28 (1992) 561. [6] M.S. Ozyazici, P.A. Morton, L.M. Zhang, V. Mizrahi, IEEE Photon. Technol. Lett. 7 (1995) 1142. [7] N. Dogru, M.S. Ozyazici, Opt. Commun. 220 (2003) 97. [8] N. Dogru, M.S. Ozyazici, IEE Proc. J. Optoelectron. 151 (2004) 166. [9] H. Kogelnik, C.V. Shank, J. Appl. Phys. 43 (1972) 2327. [10] T. Erdogan, J. Lightwave Technol. 15 (1997) 1277.
Mode-locked performance of hybrid soliton pulse source utilizing fiber grating external cavity lasers Nuran Dogru
*
Department of Electrical and Electronics, University of Gaziantep, 27310 Gaziantep, Turkey Received 24 May 2005; received in revised form 2 October 2005; accepted 6 October 2005
Abstract Mode-locking characteristic of hybrid soliton pulse source (HSPS) utilizing linearly chirped raised-cosine flat top apodized fiber Bragg grating (FBG) is investigated by using coupled-mode equations. It is found that the fundamental repetition frequency range of HSPS is significantly extended by using linearly chirped raised-cosine flat top apodized FBG instead of linearly chirped Gaussian apodized FBG. The range of repetition frequencies over which proper mode-locking is obtained is 2–3.3 GHz with linearly chirped raised-cosine flat top apodized grating whereas this range is 2.1–2.95 GHz with linearly chirped Gaussian apodized grating. 2005 Elsevier B.V. All rights reserved. Keywords: Pulse generation; Hybrid soliton pulse source; Active mode-locking; Semiconductor lasers
1. Introduction Mode-locked semiconductor lasers can be used as optical pulse source in many applications, including telecommunication systems, measurement systems, and as clock sources for optoelectronic processing systems. Their versatility and reliability is matched by the potential for production of relatively cheap and reliable devices. Hybrid soliton pulse source (HSPS) is one such device, developed as a pulse source for soliton transmission system. This device has demonstrated an extremely wide-mode-locking frequency range due to a novel wavelength self-tuning mechanism [1], which makes it very useful when precise control over the operating frequency is required. The device has also proved to be robust in operation, providing a consistent output over a large range of driving parameters. This attributes allowed the source to be used in a soliton transmission experiment, where information was transmitted at 10 Gb/s over 27,000 km [2]. Several examples of short optical pulse sources using active mode-locking have been pro*
Tel.: +90 342 3601200; fax: +90 342 3601103. E-mail address: [email protected].
0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.10.018
posed. This method provides low timing jitter picosecond pulses, locked to an external electrical reference frequency, for high-speed optical communications systems. Active mode-locking with external cavity gratings is one technique used to generate such pulses from semiconductor lasers by applying an rf drive current at frequency matching the roundtrip of the laser cavity. Fibers grating external cavity lasers have been experimentally demonstrated in the modelocking regime at 2.5 GHz [1] and 10 GHz [3]. The HSPS described in this paper is being developed at AT&T Bell Laboratories as a possible source for ultra-long distance soliton communication systems. The HSPS integrates a semiconductor laser gain section and fiber Bragg reflector in a hybrid device, to capitalize on the inherent advantages each possesses. The semiconductor section allows simple and efficient pumping of the device, and the fiber Bragg reflector provides excellent wavelength and optical bandwidth control. Together these components provide a source which is both simple and inexpensive, whilst satisfying the exacting performance requirements for ultra-long distance soliton transmission systems. The Bragg reflectors are manufactured using holographic techniques, which allows great control over the
228
N. Dogru / Optics Communications 260 (2006) 227–231
Bragg wavelength and intensity profile. The reflector is produced by exposing an optical fiber to a two beam interference pattern, at a wavelength near the peak of a defect absorption band at 242 nm [4]. The absorbed UV light permanently modifies the index of refraction in the core, transferring the interference pattern to the core index profile, and therefore, forming a grating, or Bragg reflector. The grating intensity profile follows that of the laser beams, and is therefore approximately Gaussian. This provides for a very clean reflection spectrum free from any significant side-lobes. HSPS utilizing a uniform grating Bragg reflector integrated into a fiber external cavity has been shown to operate close to the required frequency with transform-limited pulses and very high output powers [5]. However, this device worked well only under specific operating conditions, and showed spectral instabilities when operating parameters were changed. Utilizing with a linearly chirped grating has overcome the spectral instability problem of the earlier device. An interesting feature of these devices is the extremely wide operating repetition frequency range, which can be enhanced by using chirped gratings, as reported in [1,6]. In [6], calculated and measured results showed that transform-limited pulses are generated over a repetition frequency range of 850 MHz using linearly chirped Gaussian apodized fiber Bragg grating (FBG). However, in this paper, it is shown that the repetition frequency range of proper mode-locking where transform-limited pulses is obtained is extended to 1.3 GHz by utilizing linearly chirped raised-cosine flat top FBG. 2. Model The structure of the HSPS utilizing linearly chirped raised-cosine flat top apodized FBG, considered in our analysis is shown in Fig. 1. The geometry is same as the one reported in [7,8]. The HSPS system is made up of a multi-quantum well (MQW) semiconductor laser, a fiber and a FBG. One facet of diode is high reflectivity (HR) coated and the other antireflection (AR) coated. The light from the AR coated facet is coupled to the FBG reflector. The field in this system travels between the HR coated laser end and effective cavity length of the grating. The output power is taken through the grating. The mode-locked HSPS model is based on time-domain solution of the coupled-mode equations. Assume that the longitudinal, effective-refractive index variation of the Bragg grating is given as
dc+rf AR
Fiber Bragg grating
HR
Light output MQW Laser
Fiber Fig. 1. Schematic of HSPS.
2p nðzÞ ¼ nco þ Dn 1 þ m cos z ; KðzÞ
ð1Þ
where nco is the unperturbed effective index of the fiber, Dn represents the dc index change spatially averaged over a grating period, m is the modulation index of the grating. The z-dependent grating period K(z) is linearly chirped and is taken as KðzÞ ¼ Ko þ
1 dko z; 2nco dz
ð2Þ
where Ko = ko/2nco is the pitch of the unchirped Bragg grating at the operating wavelength ko. The grating pitch is accepted to be linearly chirped such that the operating wavelength ko corresponds to the center of the grating and the wavelength is chirped by C = dko/dz (cm/cm) around the center. Using couple mode theory [9], the inter-coupling relations between forward-propagation field E+(z, t) and reverse-propagation field E(z, t) can be written as dEþ 2j 1 d/ þ ¼ j d þ ð3Þ E jjE ; m 2 dz dz dE 2j 1 d/ ¼j dþ ð4Þ E þ jjEþ ; m 2 dz dz where j is the coupling coefficient between forward and reverse waves, d is the deviation from real part of propagation constant b, and u is the grating chirp. The loss in fiber and grating can be neglected since it is a few centimeters long. If the term inside the parenthesis in Eqs. (3) and (4) is called the dc ‘‘self-coupling’’ coefficient, as in [10] r¼dþ
2j 1 du . m 2 dz
ð5Þ
The coupling coefficient is assumed raised cosine flat-top [10] as 1 pz jðzÞ ¼ jp 1 þ cos ; ð6Þ 2 FWHMj where jp is the peak value of the Gaussian variation and FWHMj is the full-width half-maximum (FWHM) of this variation and is taken simply as Lg(grating length)/2 throughout calculations [10]. The coupled-mode equations are solved using a piecewise-uniform approach. First, the equations are solved analytically and then the grating is divided into M sections each having an equal length Dz. Assuming the boundary conditions at z = L/2 as E+(L/2) = 1 and E(L/2) = 0, the calculations are carried out back to forth (from z = L/2 to L/2), the parameters of each section are calculated and these parameters are put into 2 · 2 propagation matrix Ti. The fields at the ith section can be calculated from the known fields of the previous section such that þ þ Ei Ei1 ¼ T . ð7Þ i E E i i1
N. Dogru / Optics Communications 260 (2006) 227–231
" Ti ¼
coshðci DzÞ j jcii
j rcii
sinhðci DzÞ
sinhðci DzÞ
j jcii
#
sinhðci DzÞ
coshðci DzÞ þ j rcii sinhðci DzÞ
;
ð8Þ
where the coupling coefficients are related as c2i ¼ j2i r2i .
ð9Þ
Note that the values of ji, ri and hence ci are different for each section and must be calculated individually before putting into the propagation matrix. The number of sections needed for the calculations is determined by the required accuracy. For most cases, M 100 is sufficient [10]. M may not be arbitrarily large, since the approximations that lead to the coupled-mode equations are not valid when a uniform grating section is only a few grating periods long. Thus, we require Dz K which means M 2ncoL/ko. For the modeling of a multi-section system like HSPS, coupled-mode equations must be modified in such a way that they include gain, and loss in the laser. In each laser section, the carrier density is calculated from the rate equation dN ðz; tÞ IðtÞ N ðz; tÞ ao ðN ðz; tÞ N o Þ ¼ mg Sðz; tÞ; dt eV sn 1 þ eSðz; tÞ
ð10Þ
where I(t) is the injection current, V is the active layer volume, e is the electronic charge, N(z, t) is the carrier density, sn is the carrier lifetime, S(z, t) is the photon density and it is proportional to jE+j2 + jEj2, No is the carrier density at transparency, e is the gain saturation parameter and ao is the differential gain. The laser sections also include the coupling between carrier density and the refractive index through the linewidth enhancement factor and it is given as Dn ¼
ko Caao DN ðz; tÞ; 4p
ð11Þ
where C is the confinement factor, a is the linewidth enhancement factor, and DN(z, t) is the change in the carrier density. For each time step the new field values are calculated and boundary conditions applied. This process is repeated for a sufficient number of modulation periods to obtain stable mode-locked pulses.
longer grating. The group delay curve with respect to the wavelength becomes linear for longer gratings. With help of this feature, HSPS can be mode-locked for a wider repetition frequency range, since the whole reflection spectrum can be used [6]. Even if the length of the grating is constant, its reflection spectrum can be made wider or narrower with the application of a proper amount of chirp. As the magnitude of the chirp is increased, the spectrum becomes wider and the side lobe disappears. Although the group delay curve for the unchirped reflector is nonlinear, it becomes linear as the rate of chirp is increased. Therefore, the higher chirp values will result in good gratings for HSPS applications. Although higher modulation index (m) values result in a more symmetric reflection spectrum, there is no big difference between the group delay characteristics for different m values. All these parameters are used to tailor the response of the grating and their optimum configuration can be found. In our simulation, a grating length of 4 cm is used with maximum power reflectivity of 0.5 at 1.55 lm. Linear chirp ˚ /cm and 0.8. The fundavalue and m are taken as 1.9 A mental mode locking frequency is chosen as 2.5 GHz and step size is 0.6875 ps. The other MQW laser diode parameters are given in Table 1. If the period of the sinusoidal variation of refractive ˚ /cm) through the grating index is reduced linearly (1.9 A length, then the reflectivity and group delay response of the Gaussian apodized FBG will be as in Fig. 2(a). As seen in figure, the reflection spectrum covers a wide range of Table 1 Laser diode parameters 2 · 1017 cm3 10 · 1016 cm2 5 · 105 0.8 0.01 0.9 0.1 0.8 · 109 s 25 cm1
Gain saturation parameter Differential gain Spontaneous coupling factor Field coupling factor AR coating field reflectivity HR coating field reflectivity Optical confinement factor Carrier lifetime Internal loss
e ao bsp g r3 r1 C sn aint
0.6
400 350
3. Results One of the important features of the HSPS is its very wide operating frequency range [1], due to the spatially distributed nature of the grating. The length of the grating can be changed in order to adjust the spectral width of the reflection spectrum. As the length of the grating is increase, the full-width at half-maximum (FWHM) of the reflection spectrum increases as well. Since the wider the FWHM of the reflection spectrum, the better the wavelength self-tuning for mode-locked HSPS [6], it is preferable to have a
reflectivity
0.5
a
0.4
300 250
b 0.3
200 150
0.2
100 0.1 0 1549.2
group delay, ps
The propagation matrix for the ith mode can be written as
229
50 1549.6
1550
1550.4
0 1550.8
wavelenght, nm Fig. 2. Reflectivity and group delay characteristics for linearly chirped Gaussian apodized and linearly chirped raised cosine flat top FBG.
230
N. Dogru / Optics Communications 260 (2006) 227–231
˚ . If grating is raisedwavelengths giving a FWHM 3.78 A cosine flat top then the resulting reflection spectrum and group delay will be shown in Fig. 2(b). The reflection spec˚ ) compared to that of trum is wider (FWHM = 6.21 A Gaussian apodized grating. Also, peak reflectivity occurs at the operating wavelength giving a symmetric reflection spectrum for linearly chirped raised-cosine flat top FBG as seen in Fig. 2(b). This is because of zero dc index change of these gratings. In this case, grating behaves as if it has only ac index variation. In other words, the grating shows a performance similar to the unapodized gratings. These results reveal that linearly chirped raised-cosine flat top
3.3 GHz TBP=0.420
3.2 GHz TBP=0.500
3.1 GHz TBP=0.420
3 GHz TBP=0.500
2.9 GHz TBP=0.414
2.8 GHz TBP=0.390
field spectra
2.7 GHz TBP=0.408
2.6 GHz TBP=0.392
2.5 GHz TBP=0.394
FBG is superior to the linearly chirped Gaussian apodized FBG because of wider and symmetric reflection spectrum. By applying a linear chirp across the reflector, the effective cavity length becomes wavelength dependent. When the mode-locking repetition frequency is changed, the device self-tunes its wavelength to give the correct cavity length to keep on resonance with the modulation frequency. When the mode-locking repetition frequency is increased the effective cavity length will decrease, and the output will move to longer wavelengths. This feature of the HSPS is shown in Fig. 3, where the field spectra of the pulses are shown for different mode-locking repetition frequencies ranging from 2 to 3.3 GHz. A constant dc bias of 6 mA and an rf current 20 mA are used over the whole repetition frequency range. The shift of the entire spectrum to lower or higher frequencies is obvious when the modelocking repetition frequency is changed. The change of 1.3 GHz in mode-locking repetition frequency gives a shift of 94 GHz in the field spectrum of the Bragg reflector. This shift was found for 29 GHz in [6]. Calculated pulsewidths as a function of rf current is shown in Fig. 4 for different values of dc bias and modulation repetition frequency is 2.5 GHz. This plot includes entire range of operation for which the device produces clean, stable, single pulses. Longer pulses occur for small drive levels. The dc bias also has a small effect on the pulsewidth producing slightly shorter pulses at lower dc drives levels. The peak power rises as both the dc and rf drive levels are increased, as expected. Obtained pulses are shorter than that of linearly chirped Gaussian apodized FBG [6]. The calculated results for time–bandwidth product and pulsewidth versus rf mode-locking repetition frequency are shown in Figs. 5 and 6 for the range of dc and rf values producing clean pulses. These results show the broad operating range of this source with pulsewidths of 40–80 ps and time–bandwidth products of 0.3–0.5 over rf repetition frequencies from 2 to 3.3 GHz. Mode-locking range was found for 850 MHz (2.1–2.95 GHz) in [6] although
2.4 GHz TBP=0.397
90 80
2.3 GHz TBP=0.400
pulse width,ps
70 2.2 GHz TBP=0.424
2.1 GHz TBP=0.432
60 50 40 30
2 GHz TBP=0.443
20 -80
-60
-40
-20
0
20
40
60
80
Frequency offset, GHz Fig. 3. Field spectra of HSPS for different mode-locking repetition frequencies.
0
10
20
30
40
50
60
rf current,mA Fig. 4. Pulsewidth as a function of rf current for all dc current with a mode-locking repetition frequency of 2.5 GHz.
N. Dogru / Optics Communications 260 (2006) 227–231
close to 0.4 are achieved for all bias conditions, with pulsewidths from 40 to 80 ps. In conclusion, HSPS utilizing linearly chirped raisedcosine flat top FBG works over a large modulation repetition frequency range of 1.3 GHz giving a time–bandwidth product close to 0.4.
time-bandwidth product
0.55 0.5 0.45 0.4 0.35
4. Conclusion
0.3 0.25 0.2
1.6
1.9
2.2
2.5
2.8
3.1
3.4
mode-locking repetition frequency, GHz Fig. 5. Time–bandwidth product as a function of mode-locking repetition frequency for all rf and dc current.
100 90 80
pulsewidth, ps
231
It has been demonstrated with extremely wide modelocking repetition frequency range and stable operation by the use of a linearly chirped Bragg reflector integrated in a fiber external cavity. The device produces a clean stable output of near transform limited pulses over a large range of drive parameters, making a good source for use in many practical systems. However, this repetition frequency range where transform-limited pulses are generated is significantly extended by using linearly chirped raised-cosine flat top FBG giving a range of 1.3 GHz instead of linearly chirped Gaussian apodized FBG having a range of 850 MHz. References
70 60 50 40 30 20 1.6
1.9
2.2
2.5
2.8
3.1
3.4
mode-locking repetition frequency, GHz Fig. 6. Pulsewidth as a function of mode-locking repetition frequency for all rf and dc current.
time–bandwidth product changes between 0.3 and 0.7. At the center of the repetition frequency range where the device is designed to operate, time–bandwidth products
[1] P.A. Morton, V. Mizrahi, P.A. Andrekson, T. Tanbun-Ek, R.A. Logan, P. Lemaire, D.L. Coblentz, A.M. Sergent, K.W. Wecht, P.F. Sciortino Jr., IEEE Photon. Technol. Lett. 5 (1993) 28. [2] P.A. Morton, V. Mizrahi, G. Harvey, L. Mollenauer, T. Tanbun-Ek, R.A. Logan, H.M. Presby, T. Erdogan, P. Lemaire, A.M. Sergent, K.W. Wecht, IEEE Photon. Technol. Lett. 7 (1995) 111. [3] R. Paoletti, D. Bertone, R. Fang, G. Magnetti, M. Meliga, G. Meneghi, G. Morello, G. Rossi, L. Tallone, M. Scofet, IEEE Photon. Technol. Lett. 12 (2000) 245. [4] V. Mizrahi, J.E. Sipe, J. Lightwave Technol. 11 (1993) 1513. [5] P.A. Morton, V. Mizrahi, S.G. Kosinski, L.F. Mollenauer, T. Tanbun-Ek, R. Logan, D.L. Coblentz, A.M. Sergent, K.W. Wecht, Electron. Lett. 28 (1992) 561. [6] M.S. Ozyazici, P.A. Morton, L.M. Zhang, V. Mizrahi, IEEE Photon. Technol. Lett. 7 (1995) 1142. [7] N. Dogru, M.S. Ozyazici, Opt. Commun. 220 (2003) 97. [8] N. Dogru, M.S. Ozyazici, IEE Proc. J. Optoelectron. 151 (2004) 166. [9] H. Kogelnik, C.V. Shank, J. Appl. Phys. 43 (1972) 2327. [10] T. Erdogan, J. Lightwave Technol. 15 (1997) 1277.