- Email: [email protected]
Tunable slow and fast light in double-Lorentzian active fiber Bragg gratings
Accepted Manuscript Title: Tunable slow and fast light in double-Lorentzian active fiber Bragg gratings Author: Kai Qian Lin Gao Xu Jian Weichuang Yu Honghua Liao PII: DOI: Reference:
S0030-4026(16)31087-7 http://dx.doi.org/doi:10.1016/j.ijleo.2016.09.067 IJLEO 58210
To appear in: Received date: Revised date: Accepted date:
27-7-2016 12-9-2016 16-9-2016
Please cite this article as: Kai Qian, Lin Gao, Xu Jian, Weichuang Yu, Honghua Liao, Tunable slow and fast light in double-Lorentzian active fiber Bragg gratings, Optik - International Journal for Light and Electron Optics http://dx.doi.org/10.1016/j.ijleo.2016.09.067 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Tunable slow and fast light in double-Lorentzian active fiber Bragg gratings
Kai QIAN*, Lin GAO, Xu JIAN, Weichuang Yu, Honghua LIAO School of Information Engineering, Hubei University for Nationalities, Enshi, 445000, Hubei Province, China
*Corresponding Author: [email protected]
We proposed a new approach to tunable slow and fast light using double-Lorentzian active fiber Bragg gratings (DL-AFBGs). Group delay in DL-AFBGs is theoretically analyzed using the coupled mode theory. Unique double-Lorentzian spectral resonance of the proposed DL-AFBGs significantly modifies the dispersion of injected light near the Bragg frequency, leading to structure-based tunable slow and fast light. The Bragg frequency varies with the pump power of the FBG. In this way, the group delay of the signal can be optical controlled. Furthermore, we describe a possible realization of group delay tuning from subluminal to superluminal pulse reflection in active double-Lorentzian Bragg gratings. Tunable group delay from 242 ps to -200 ps is theoretically demonstrated. It provides a very simple approach to getting both subluminal and superluminal reflection in one component, which we might find a good future in light matter interaction enhancement, high sensitivity sensor and so on. slow and fast light, fiber Bragg gratings, active fiber, dispersion PACS number(s): 42.81.-i, 42.79.Dj, 42.79.Sz
1 Introduction With the potential applications in all-optical router communication, optical computing, nonlinearity enhancement and fiber sensor, the controllability of group delay has aroused a continuous interest in the past decades. Several schemes have been studied both theoretically and experimentally, such as electromagnetically induced transparency [1], coherent population oscillation [2-5], stimulated Raman scattering [6], coupled resonator induced transparency [7-8], fiber optical parametric amplifier [9] and hybrid approach employing wavelength conversion followed by linear dispersion [10-11]. Stimulated Brillouin scattering slow light allows for a broad tunability. However, it offers relatively narrow delay bandwidth and requires long interaction length [12-15]. It is well known that Bragg gratings can be used to generate slow and fast light as optical filter [16-17]. J. B. Khurgin has analyzed the conceptual similarity between Moiré grating and electro-magnetic-induced transparency [18-19]. S. Longhi demonstrated a class of slow light schemes based on Bragg gratings [20-25]. In particular, they experimentally observed superluminal and subluminal pulse reflection in a double-Lorentzian Bragg grating by tuning the incident pulse spectrum from the Bragg resonance to the Lorentzian peaks [23]. Although this scheme can generate both positive and negative group delay in one component, the signal wavelength requires to be precisely tuned, which will lead to unavoidable inconvenience. J. T. Mok reported tunable transmission group delay in an apodized FBG by varying the input signal intensity [26]. By introducing nonlinearity, the optical pulse can travel slowly and also remain undistorted by the formation of a soliton. In addition, B.
J. Eggleton reported tunable group velocity in an apodized FBG by varying an axial strain of the grating [27]. Recently, G. Skolianos observed 300 km/s group velocity in a deuterium-loaded FBG. It’s the slowest light reported in FBG [28]. In these approaches, fast light cannot be generated, the slow light bandwidth is quite narrow, and the group velocity tuning is inconvenient. Several optical controllable slow light schemes based on active gratings have been theoretically proposed [29-30]. According to the previous numerical analysis, the pump power variation has little effect on the group delay change [30]. However, the group delay varying with the spectrum shift was ignored. We have experimentally demonstrated optical controllable slow light in a uniform Er/Yb co-doped fiber Bragg grating [31]. This controllable slow light results from the combined effect of the dispersion of FBG and the pump-induced thermal phase shift in active fiber. Thus, the group delay varies with the pump power. It brings us a very convenient approach achieving tunable optical delay in FBGs. Controllable slow light in active linear chirped FBG and active tilted FBG were also experimentally observed [32-34]. In this paper, we present a simplified theoretical analysis of tunable group delay in active Bragg gratings by introducing complex gain coefficient. The analysis shows that the imaginary part contributes more to the tunable group delay. This simplified model can provide a new perspective to understand the active FBG slow light. Furthermore, we study the characteristic of active double-Lorentzian FBGs. Group delay tuning from subluminal to superluminal pulse reflection can be obtained by adjusting the pump power. It may bring some important applications to practical systems, such as phase array antennas and high sensitivity fiber sensors [35-37].
2 Theory The refractive index along the grating can be described by n( z) 0n {1
2h (z ) co s [ z2 /
z ( )]},
(1)
for 0 z L , where n0 is the refractive index of the grating, is the nominal period, and h( z ) ,
( z ) describe the amplitude and phase of the grating. The light field can be written as
E ( z, t ) u (z, ) exp i( t i z )
z( , ) ex ip (t i
Based on the coupled mode theory, for a weak modulation grating ( h( z ) backward amplitudes satisfy the following equations
z
)c c . . .
1 ), the forward and
(2)
du i( g )u iq z( ) , dz
(3a)
d i( g ) iq( z )u. dz
Where n0 q( z )
B c0
(3b)
is the detuning parameter, B c / (n0) is the Bragg frequency,
h( z )exp[i ( z )] is the scattering potential, and g is the gain coefficient of the active
medium. For small-signal gain condition, the pump induced complex gain coefficient can be written as
g gr igi [30]. Thus, the revised Bragg frequency varies with the pump power
B B
gi c0 . n0
(4)
The revised detuning parameter can be rewritten as n0
B c0
gi . We consider a light pulse
incident to the grating from the z 0 planes. The boundary condition is u(0, ) 1 and ( L, ) 0 . The reflection coefficient is r ( ) (0, ) / u(0, ) . The reflection spectrum R( ) and transmission spectrum T ( ) satisfy the relation R( ) T ( ) 1 , where R( ) r ( ) . The group delay for 2
light pulse reflected off of the grating
r ( gi )
d r . d
(5)
Where r arg(r ) is the phase of r . Eq. (5) shows the group delay can be represented as a function of the imaginary part of the gain. For a signal with central wavelength near the bandgap, the group delay is varying with the detuning parameter. Thus, optical controllable group delay can be achieved by modifying the pump power of the active grating.
3 Discussion Double-Lorentzian fiber Bragg grating (DL FBG) can be used to simulate the dispersion properties of a gain doublet, which is known to give rise to negative group velocities [20]. Thus, optical controllable group delay from subluminal to superluminal pulse reflection can be obtained in an active DL FBG. The reflectivity can be written as r ( ) i R0 [
1 ( B )( / 2) i
1 ( B )( / 2) i
],
(6)
Where R0 is the peak power reflectivity of each Lorentzian line, 2 is the frequency separation between the two resonances, and is the full width at half maximum (FWHM). In order to achieve optical controllability, this FBG consists of a 30-cm-long Er-Yb co-doped fiber. The apodization profile
h( z ) of the refractive index can be obtained by the Gel’fand-Levitan-Marchenko inverse scattering
method [38]. Figure 1 shows the reflection spectrum and group delay of the grating. Parameter values are n0 1.48 , B 1.21611015 rad/s, R0 0.8 , 1.02 1010 rad/s and 7.548 109 rad/s.
As is shown in Fig. 2, when the pump power is 0 mW, the central wavelength is near the right resonance peak of the DL FBG, the group delay of the signal pulse is about 242 ps. When the pump power is increased from 0 mW to 160 mW, the Bragg wavelength is assumed to shift 0.013 nm towards longer wavelength. This assumption proves to be reasonable since we have already observed the Bragg wavelength shift on the same level in previous works [31-32, 39]. The corresponding imaginary part of the gain is 50.3, the core index change n is 1.24 10-5 . In this case, the group delay decreases to -200 ps, as shown in Fig. 2.
The group delay of the reflected pulse can be expressed as a function of the imaginary part of the gain
( gi )
arg[r ( , gi )] . B
(7)
The group delay varying with the pump power is shown in Fig. 3. The Bragg wavelength shift is approximately proportional to pump power when the power is increased from 10 mW to 150 mW. Thus, optical tuning of group delay from subluminal (positive group delay) to superluminal (negative group delay) pulse reflection can be obtained.
This slow light technique is very simple and feasible to achieve continuously tunable group velocity. The key component in the proposed system is the double-Lorentzian FBG, which was written on an active fiber. Compared with other FBG slow light approaches, this approach can be used to generate controllable group delay from positive to negative in one component by adjusting the 980 nm pump power. It may have some important applications in light-matter interaction, phase array antennas and high sensitivity fiber sensors.
4 Conclusion In this work, a theoretical analysis of the active fiber Bragg grating slow light is presented. Based on the coupled mode theory, the group delay of a signal with certain central wavelength varies with the Bragg frequency, which can be modulated by controlling the pump power of the active FBG. In addition, a possible realization of tuning from subluminal to superluminal pulse reflection using an active double-Lorentzian FBG is proposed. We believe that this simple, cost-effective tunable slow and fast light approach will have a great future.
Acknowledge The authors acknowledge the support from the Hubei Provincial Department of Education (Q20151903) and Hubei University for Nationalities (4148023).
References [1] Hau L V, Harris S E, Dutton Z. Light speed reduction to 17 metres per second in an ultracold atomic gas. Nature, 1999, 397:594-598 [2] Bigelow M S, Lepeshkin N N, Boyd R W. Observation of Ultraslow Light Propagation in a Ruby Crystal at Room Temperature. Phys Rev Lett, 2003, 90:113901--113904. [3] Bigelow M S, Lepeshkin N N, Boyd R W. Superluminal and Slow Light Propagation in a Room-Temperature Solid. Science, 2003, 301:200--202 [4] Ku P C, Sedgwick F, Chang-Hasnain C J, et al. Slow light in semiconductor quantum wells. Opt. Lett, 2004, 29:2291-2293 [5] Schweinsberg A, Lepeshkin N N, Bigelow M S, et al. Observation of superluminal and slow light propagation in erbium-doped optical fiber. Europhys Lett, 2006, 15:218-224 [6] Sharping J, Okawachi Y, Gaeta A. Wide bandwidth slow light using a Raman fiber amplifier. Opt Exp, 2005, 13:6092-6098 [7] Yariv A, Xu Y, Lee R K. Coupled-resonator optical waveguide: a proposal and analysis. Opt Lett, 1999, 24: 711-713 [8] Xiao P, Hu H, Qi M. Controllable time delay using slow/fast light in active fiber ring resonators with gain manipulation. Sci. China: Phys. Mech. Astron, 2013, 56:1085-1089 [9] Dahan D, Eisenstein G. Tunable all optical delay via slow and fast light propagation in a Raman assisted fiber optical parametric amplifier: a route to all optical buffering. Opt Exp, 2005, 13:6234-6249 [10] Sharping J, Okawachi Y, Howe J, et al. All-optical, wavelength and bandwidth preserving, pulse delay based on parametric wavelength conversion and dispersion. Opt Exp, 2005, 13: 7872-7877 [11] Fok M P, Shu C. Tunable optical delay using four-wave mixing in a 35-cm highly nonlinear Bismuth-Oxide fiber and group velocity dispersion. J. Lightwave Technol. 2008, 26: 499-504 [12] Song K Y, Herraez M G, Thevenaz L. Observation of pulse delaying and Advancement in optical Fibers using stimulated Brillouin scattering. Opt Exp, 2005, 13:82-88 [13] Okawachi Y, Bigelow M S, Sharping J E, et al. Tunable all-optical delay viea Brillouin slow light in an optical fiber. Phys Rev Lett, 2005, 94:153902 [14] Thevenaz L. Slow and fast light in optical fibres. Nat Photon, 2008, 2:471-481
[15] Xing L, Zhan L, Yi L, et al. Storage capacity of slow-light tunable optical buffers based on fiber Brillouin amplifiers for real signal bit streams. Opt Exp, 2007, 16:10189 -10195 [16] Lenz G, Eggleton B J, Madsen C K, et al. Optical delay lines based on optical filters. IEEE J. Quantum Electron, 2001, 37:525-532 [17] Poladian L. Group-delay reconstruction for fiber Bragg gratings in reflection and transmission. Opt Lett, 1997, 22:1571-1573 [18] Khurgin J B. Light slowing down in Moiré fiber gratings and its implications for nonlinear optics. Phys Rev A, 2000, 62:013821 [19] Khurgin J B. Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis. J Opt Soc Am B, 2005, 22:1062-1074 [20] Longhi S. Superluminal pulse reflection in asymmetric one-dimensional photonic band gaps. Phys Rev E, 2001, 64:037601 [21] Longhi S. Superluminal optical pulse propagation at 1.5 μm in periodic fiber Bragg gratings. Phys Rev E, 2001, 64:055602 [22] Longhi S, Marano M, Laporta P. Experimental observation of superluminal pulse reflection in a double-Lorentzian photonic band gap. Phys Rev E, 2002, 65:045602 [23] Longhi S, Laporta P, Belmonte M, et al. Measurement of superluminal optical tunneling times in double-barrier photonic band gaps. Phys Rev E, 2002, 65:046610 [24] Longhi S, Marano M, Laporta P, et al. Propagation, manipulation, and control of picosecond optical pulses at 1.5μm in fiber Bragg gratings. J Opt Soc Am B, 2002, 19:2742-2757 [25] Janner D, Galzerano G, Valle G D, et al. Slow light in periodic superstructure Bragg gratings. Phys Rev E, 2005, 72:056605 [26] Mok J T, Sterke C M D, Littler I C M, et al. Dispersionless slow light using gap solitons. Nat Phys, 2006, 2:775-780 [27] Eggleton B J, Slusher R E, Sterke C M D, et al. Bragg Grating solitons. Phys Rev Lett, 1996, 76:1627 [28] Skolianos G, Arora A, Bernier M, et al. Slowing down light to 300 km/s in a deuterium-loaded fiber Bragg grating. Opt Lett, 2015, 40:1524-1527 [29] Longhi S. Group delay tuning in active fiber Bragg gratings: From superluminal to subluminal pulse reflection. Phys Rev E, 2005, 72: 056614
[30] Zhuo Z, Ham B S. Subluminal and superluminal propagation in Er3+ doped fiber Bragg grating. Eur Phys J D, 2008, 49:117-121 [31] Qian K, Zhan L, Li H G, et al. Tunable delay slow-light in an active fiber Bragg grating. Opt Exp, 2009, 17:22217-22222 [32] Shahoei H, Li M, Yao J. Continuously tunable time delay using an optically pumped linear chirped fiber Bragg grating. J Lightwave Technol, 2011, 29:1465-1472 [33] Liu W, Yao J. Ultra-wideband microwave photonic phase shifter with a 360°tunable phase shift based on an erbium-ytterbium co-doped linearly chirped FBG. Opt Lett, 2014, 39: 922-924 [34] Davis M K, Digonnet M J F, Pantell R H. Thermal Effects in Doped Fibers. J Lightwave Technol, 1998, 16:1013-1023 [35] Wen H, Matt T, Fan S, et al. Sensing with slow light in fiber Bragg gratings. IEEE Sensors J. 2012, 12:156-163 [36] Wen H, Skolianos G, Fan S, et al. Slow-light fiber-Bragg-grating strain sensor with a 280-femtostrain/√Hz resolution. J Lightwave Technol, 2014, 31:1804-1808 [37] Abbott B P, Abbott R, Abbott T D, et al. Observation of Gravitational Waves from a Binary Black Hole Merger [J]. Physical Review Letters, 2016, 116(6): 061102. [38] Song G H, Shin S Y. Design of corrugated waveguide filters by the Gel’fand-Levitan-Marchenko inverse-scattering method. J Opt Soc Am A, 1985, 2:1905-1915 [39] Shahoei H, Yao J. Tunable microwave photonic phase shifter based on slow and fast light effects in a tilted fiber Bragg grating. Opt Exp, 2012, 20:14009-14014
300 Reflectivity Delay
1.4 1.2
Reflectivity
100
0.8
0
0.6 -100 0.4
Delay (Unit: ns)
1.0
200
-200
0.2 0.0
-300 1549.96
1549.98
1550.00
1550.02
1550.04
Wavelength (Unit: nm)
Fig. 1. Reflectivity (solid line) and group delay (dash dot line) of the DL FBG.
300
Delay (Unit: ps)
200 100 0 -100 -200
signal
1549.96
1549.98
1550.00
1550.02
Wavelength (Unit: nm) Fig. 2. Delay spectrum shift with different pump power.
0 mW 160 mW
1550.04
Group delay (Unit: ps)
300 200 100 0
-100 -200 0
20
40 60 80 100 120 Pump power (Unit: mW)
Fig. 3. Group delay varying with pump power.
140
160
S0030-4026(16)31087-7 http://dx.doi.org/doi:10.1016/j.ijleo.2016.09.067 IJLEO 58210
To appear in: Received date: Revised date: Accepted date:
27-7-2016 12-9-2016 16-9-2016
Please cite this article as: Kai Qian, Lin Gao, Xu Jian, Weichuang Yu, Honghua Liao, Tunable slow and fast light in double-Lorentzian active fiber Bragg gratings, Optik - International Journal for Light and Electron Optics http://dx.doi.org/10.1016/j.ijleo.2016.09.067 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Tunable slow and fast light in double-Lorentzian active fiber Bragg gratings
Kai QIAN*, Lin GAO, Xu JIAN, Weichuang Yu, Honghua LIAO School of Information Engineering, Hubei University for Nationalities, Enshi, 445000, Hubei Province, China
*Corresponding Author: [email protected]
We proposed a new approach to tunable slow and fast light using double-Lorentzian active fiber Bragg gratings (DL-AFBGs). Group delay in DL-AFBGs is theoretically analyzed using the coupled mode theory. Unique double-Lorentzian spectral resonance of the proposed DL-AFBGs significantly modifies the dispersion of injected light near the Bragg frequency, leading to structure-based tunable slow and fast light. The Bragg frequency varies with the pump power of the FBG. In this way, the group delay of the signal can be optical controlled. Furthermore, we describe a possible realization of group delay tuning from subluminal to superluminal pulse reflection in active double-Lorentzian Bragg gratings. Tunable group delay from 242 ps to -200 ps is theoretically demonstrated. It provides a very simple approach to getting both subluminal and superluminal reflection in one component, which we might find a good future in light matter interaction enhancement, high sensitivity sensor and so on. slow and fast light, fiber Bragg gratings, active fiber, dispersion PACS number(s): 42.81.-i, 42.79.Dj, 42.79.Sz
1 Introduction With the potential applications in all-optical router communication, optical computing, nonlinearity enhancement and fiber sensor, the controllability of group delay has aroused a continuous interest in the past decades. Several schemes have been studied both theoretically and experimentally, such as electromagnetically induced transparency [1], coherent population oscillation [2-5], stimulated Raman scattering [6], coupled resonator induced transparency [7-8], fiber optical parametric amplifier [9] and hybrid approach employing wavelength conversion followed by linear dispersion [10-11]. Stimulated Brillouin scattering slow light allows for a broad tunability. However, it offers relatively narrow delay bandwidth and requires long interaction length [12-15]. It is well known that Bragg gratings can be used to generate slow and fast light as optical filter [16-17]. J. B. Khurgin has analyzed the conceptual similarity between Moiré grating and electro-magnetic-induced transparency [18-19]. S. Longhi demonstrated a class of slow light schemes based on Bragg gratings [20-25]. In particular, they experimentally observed superluminal and subluminal pulse reflection in a double-Lorentzian Bragg grating by tuning the incident pulse spectrum from the Bragg resonance to the Lorentzian peaks [23]. Although this scheme can generate both positive and negative group delay in one component, the signal wavelength requires to be precisely tuned, which will lead to unavoidable inconvenience. J. T. Mok reported tunable transmission group delay in an apodized FBG by varying the input signal intensity [26]. By introducing nonlinearity, the optical pulse can travel slowly and also remain undistorted by the formation of a soliton. In addition, B.
J. Eggleton reported tunable group velocity in an apodized FBG by varying an axial strain of the grating [27]. Recently, G. Skolianos observed 300 km/s group velocity in a deuterium-loaded FBG. It’s the slowest light reported in FBG [28]. In these approaches, fast light cannot be generated, the slow light bandwidth is quite narrow, and the group velocity tuning is inconvenient. Several optical controllable slow light schemes based on active gratings have been theoretically proposed [29-30]. According to the previous numerical analysis, the pump power variation has little effect on the group delay change [30]. However, the group delay varying with the spectrum shift was ignored. We have experimentally demonstrated optical controllable slow light in a uniform Er/Yb co-doped fiber Bragg grating [31]. This controllable slow light results from the combined effect of the dispersion of FBG and the pump-induced thermal phase shift in active fiber. Thus, the group delay varies with the pump power. It brings us a very convenient approach achieving tunable optical delay in FBGs. Controllable slow light in active linear chirped FBG and active tilted FBG were also experimentally observed [32-34]. In this paper, we present a simplified theoretical analysis of tunable group delay in active Bragg gratings by introducing complex gain coefficient. The analysis shows that the imaginary part contributes more to the tunable group delay. This simplified model can provide a new perspective to understand the active FBG slow light. Furthermore, we study the characteristic of active double-Lorentzian FBGs. Group delay tuning from subluminal to superluminal pulse reflection can be obtained by adjusting the pump power. It may bring some important applications to practical systems, such as phase array antennas and high sensitivity fiber sensors [35-37].
2 Theory The refractive index along the grating can be described by n( z) 0n {1
2h (z ) co s [ z2 /
z ( )]},
(1)
for 0 z L , where n0 is the refractive index of the grating, is the nominal period, and h( z ) ,
( z ) describe the amplitude and phase of the grating. The light field can be written as
E ( z, t ) u (z, ) exp i( t i z )
z( , ) ex ip (t i
Based on the coupled mode theory, for a weak modulation grating ( h( z ) backward amplitudes satisfy the following equations
z
)c c . . .
1 ), the forward and
(2)
du i( g )u iq z( ) , dz
(3a)
d i( g ) iq( z )u. dz
Where n0 q( z )
B c0
(3b)
is the detuning parameter, B c / (n0) is the Bragg frequency,
h( z )exp[i ( z )] is the scattering potential, and g is the gain coefficient of the active
medium. For small-signal gain condition, the pump induced complex gain coefficient can be written as
g gr igi [30]. Thus, the revised Bragg frequency varies with the pump power
B B
gi c0 . n0
(4)
The revised detuning parameter can be rewritten as n0
B c0
gi . We consider a light pulse
incident to the grating from the z 0 planes. The boundary condition is u(0, ) 1 and ( L, ) 0 . The reflection coefficient is r ( ) (0, ) / u(0, ) . The reflection spectrum R( ) and transmission spectrum T ( ) satisfy the relation R( ) T ( ) 1 , where R( ) r ( ) . The group delay for 2
light pulse reflected off of the grating
r ( gi )
d r . d
(5)
Where r arg(r ) is the phase of r . Eq. (5) shows the group delay can be represented as a function of the imaginary part of the gain. For a signal with central wavelength near the bandgap, the group delay is varying with the detuning parameter. Thus, optical controllable group delay can be achieved by modifying the pump power of the active grating.
3 Discussion Double-Lorentzian fiber Bragg grating (DL FBG) can be used to simulate the dispersion properties of a gain doublet, which is known to give rise to negative group velocities [20]. Thus, optical controllable group delay from subluminal to superluminal pulse reflection can be obtained in an active DL FBG. The reflectivity can be written as r ( ) i R0 [
1 ( B )( / 2) i
1 ( B )( / 2) i
],
(6)
Where R0 is the peak power reflectivity of each Lorentzian line, 2 is the frequency separation between the two resonances, and is the full width at half maximum (FWHM). In order to achieve optical controllability, this FBG consists of a 30-cm-long Er-Yb co-doped fiber. The apodization profile
h( z ) of the refractive index can be obtained by the Gel’fand-Levitan-Marchenko inverse scattering
method [38]. Figure 1 shows the reflection spectrum and group delay of the grating. Parameter values are n0 1.48 , B 1.21611015 rad/s, R0 0.8 , 1.02 1010 rad/s and 7.548 109 rad/s.
As is shown in Fig. 2, when the pump power is 0 mW, the central wavelength is near the right resonance peak of the DL FBG, the group delay of the signal pulse is about 242 ps. When the pump power is increased from 0 mW to 160 mW, the Bragg wavelength is assumed to shift 0.013 nm towards longer wavelength. This assumption proves to be reasonable since we have already observed the Bragg wavelength shift on the same level in previous works [31-32, 39]. The corresponding imaginary part of the gain is 50.3, the core index change n is 1.24 10-5 . In this case, the group delay decreases to -200 ps, as shown in Fig. 2.
The group delay of the reflected pulse can be expressed as a function of the imaginary part of the gain
( gi )
arg[r ( , gi )] . B
(7)
The group delay varying with the pump power is shown in Fig. 3. The Bragg wavelength shift is approximately proportional to pump power when the power is increased from 10 mW to 150 mW. Thus, optical tuning of group delay from subluminal (positive group delay) to superluminal (negative group delay) pulse reflection can be obtained.
This slow light technique is very simple and feasible to achieve continuously tunable group velocity. The key component in the proposed system is the double-Lorentzian FBG, which was written on an active fiber. Compared with other FBG slow light approaches, this approach can be used to generate controllable group delay from positive to negative in one component by adjusting the 980 nm pump power. It may have some important applications in light-matter interaction, phase array antennas and high sensitivity fiber sensors.
4 Conclusion In this work, a theoretical analysis of the active fiber Bragg grating slow light is presented. Based on the coupled mode theory, the group delay of a signal with certain central wavelength varies with the Bragg frequency, which can be modulated by controlling the pump power of the active FBG. In addition, a possible realization of tuning from subluminal to superluminal pulse reflection using an active double-Lorentzian FBG is proposed. We believe that this simple, cost-effective tunable slow and fast light approach will have a great future.
Acknowledge The authors acknowledge the support from the Hubei Provincial Department of Education (Q20151903) and Hubei University for Nationalities (4148023).
References [1] Hau L V, Harris S E, Dutton Z. Light speed reduction to 17 metres per second in an ultracold atomic gas. Nature, 1999, 397:594-598 [2] Bigelow M S, Lepeshkin N N, Boyd R W. Observation of Ultraslow Light Propagation in a Ruby Crystal at Room Temperature. Phys Rev Lett, 2003, 90:113901--113904. [3] Bigelow M S, Lepeshkin N N, Boyd R W. Superluminal and Slow Light Propagation in a Room-Temperature Solid. Science, 2003, 301:200--202 [4] Ku P C, Sedgwick F, Chang-Hasnain C J, et al. Slow light in semiconductor quantum wells. Opt. Lett, 2004, 29:2291-2293 [5] Schweinsberg A, Lepeshkin N N, Bigelow M S, et al. Observation of superluminal and slow light propagation in erbium-doped optical fiber. Europhys Lett, 2006, 15:218-224 [6] Sharping J, Okawachi Y, Gaeta A. Wide bandwidth slow light using a Raman fiber amplifier. Opt Exp, 2005, 13:6092-6098 [7] Yariv A, Xu Y, Lee R K. Coupled-resonator optical waveguide: a proposal and analysis. Opt Lett, 1999, 24: 711-713 [8] Xiao P, Hu H, Qi M. Controllable time delay using slow/fast light in active fiber ring resonators with gain manipulation. Sci. China: Phys. Mech. Astron, 2013, 56:1085-1089 [9] Dahan D, Eisenstein G. Tunable all optical delay via slow and fast light propagation in a Raman assisted fiber optical parametric amplifier: a route to all optical buffering. Opt Exp, 2005, 13:6234-6249 [10] Sharping J, Okawachi Y, Howe J, et al. All-optical, wavelength and bandwidth preserving, pulse delay based on parametric wavelength conversion and dispersion. Opt Exp, 2005, 13: 7872-7877 [11] Fok M P, Shu C. Tunable optical delay using four-wave mixing in a 35-cm highly nonlinear Bismuth-Oxide fiber and group velocity dispersion. J. Lightwave Technol. 2008, 26: 499-504 [12] Song K Y, Herraez M G, Thevenaz L. Observation of pulse delaying and Advancement in optical Fibers using stimulated Brillouin scattering. Opt Exp, 2005, 13:82-88 [13] Okawachi Y, Bigelow M S, Sharping J E, et al. Tunable all-optical delay viea Brillouin slow light in an optical fiber. Phys Rev Lett, 2005, 94:153902 [14] Thevenaz L. Slow and fast light in optical fibres. Nat Photon, 2008, 2:471-481
[15] Xing L, Zhan L, Yi L, et al. Storage capacity of slow-light tunable optical buffers based on fiber Brillouin amplifiers for real signal bit streams. Opt Exp, 2007, 16:10189 -10195 [16] Lenz G, Eggleton B J, Madsen C K, et al. Optical delay lines based on optical filters. IEEE J. Quantum Electron, 2001, 37:525-532 [17] Poladian L. Group-delay reconstruction for fiber Bragg gratings in reflection and transmission. Opt Lett, 1997, 22:1571-1573 [18] Khurgin J B. Light slowing down in Moiré fiber gratings and its implications for nonlinear optics. Phys Rev A, 2000, 62:013821 [19] Khurgin J B. Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis. J Opt Soc Am B, 2005, 22:1062-1074 [20] Longhi S. Superluminal pulse reflection in asymmetric one-dimensional photonic band gaps. Phys Rev E, 2001, 64:037601 [21] Longhi S. Superluminal optical pulse propagation at 1.5 μm in periodic fiber Bragg gratings. Phys Rev E, 2001, 64:055602 [22] Longhi S, Marano M, Laporta P. Experimental observation of superluminal pulse reflection in a double-Lorentzian photonic band gap. Phys Rev E, 2002, 65:045602 [23] Longhi S, Laporta P, Belmonte M, et al. Measurement of superluminal optical tunneling times in double-barrier photonic band gaps. Phys Rev E, 2002, 65:046610 [24] Longhi S, Marano M, Laporta P, et al. Propagation, manipulation, and control of picosecond optical pulses at 1.5μm in fiber Bragg gratings. J Opt Soc Am B, 2002, 19:2742-2757 [25] Janner D, Galzerano G, Valle G D, et al. Slow light in periodic superstructure Bragg gratings. Phys Rev E, 2005, 72:056605 [26] Mok J T, Sterke C M D, Littler I C M, et al. Dispersionless slow light using gap solitons. Nat Phys, 2006, 2:775-780 [27] Eggleton B J, Slusher R E, Sterke C M D, et al. Bragg Grating solitons. Phys Rev Lett, 1996, 76:1627 [28] Skolianos G, Arora A, Bernier M, et al. Slowing down light to 300 km/s in a deuterium-loaded fiber Bragg grating. Opt Lett, 2015, 40:1524-1527 [29] Longhi S. Group delay tuning in active fiber Bragg gratings: From superluminal to subluminal pulse reflection. Phys Rev E, 2005, 72: 056614
[30] Zhuo Z, Ham B S. Subluminal and superluminal propagation in Er3+ doped fiber Bragg grating. Eur Phys J D, 2008, 49:117-121 [31] Qian K, Zhan L, Li H G, et al. Tunable delay slow-light in an active fiber Bragg grating. Opt Exp, 2009, 17:22217-22222 [32] Shahoei H, Li M, Yao J. Continuously tunable time delay using an optically pumped linear chirped fiber Bragg grating. J Lightwave Technol, 2011, 29:1465-1472 [33] Liu W, Yao J. Ultra-wideband microwave photonic phase shifter with a 360°tunable phase shift based on an erbium-ytterbium co-doped linearly chirped FBG. Opt Lett, 2014, 39: 922-924 [34] Davis M K, Digonnet M J F, Pantell R H. Thermal Effects in Doped Fibers. J Lightwave Technol, 1998, 16:1013-1023 [35] Wen H, Matt T, Fan S, et al. Sensing with slow light in fiber Bragg gratings. IEEE Sensors J. 2012, 12:156-163 [36] Wen H, Skolianos G, Fan S, et al. Slow-light fiber-Bragg-grating strain sensor with a 280-femtostrain/√Hz resolution. J Lightwave Technol, 2014, 31:1804-1808 [37] Abbott B P, Abbott R, Abbott T D, et al. Observation of Gravitational Waves from a Binary Black Hole Merger [J]. Physical Review Letters, 2016, 116(6): 061102. [38] Song G H, Shin S Y. Design of corrugated waveguide filters by the Gel’fand-Levitan-Marchenko inverse-scattering method. J Opt Soc Am A, 1985, 2:1905-1915 [39] Shahoei H, Yao J. Tunable microwave photonic phase shifter based on slow and fast light effects in a tilted fiber Bragg grating. Opt Exp, 2012, 20:14009-14014
300 Reflectivity Delay
1.4 1.2
Reflectivity
100
0.8
0
0.6 -100 0.4
Delay (Unit: ns)
1.0
200
-200
0.2 0.0
-300 1549.96
1549.98
1550.00
1550.02
1550.04
Wavelength (Unit: nm)
Fig. 1. Reflectivity (solid line) and group delay (dash dot line) of the DL FBG.
300
Delay (Unit: ps)
200 100 0 -100 -200
signal
1549.96
1549.98
1550.00
1550.02
Wavelength (Unit: nm) Fig. 2. Delay spectrum shift with different pump power.
0 mW 160 mW
1550.04
Group delay (Unit: ps)
300 200 100 0
-100 -200 0
20
40 60 80 100 120 Pump power (Unit: mW)
Fig. 3. Group delay varying with pump power.
140
160