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Raman competition from ultra-broadband modulation instability gain in the As2Se3 glass photonic crystal fiber filled with argon gas
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Raman competition from ultra-broadband modulation instability gain in the As2Se3 glass photonic crystal fiber filled with argon gas Helin Wanga, , Aijun Yangb, Bin Wua, Zhan Zhenga, Kan Lia ⁎
a b
Centre for Optics and Optoelectronics Research, Zhejiang University of Technology, Hangzhou, DC 310023, China College of Science, Zhejiang University of Technology, Hangzhou, DC 310023, China
HIGHLIGHTS
photonic crystal fiber filled with argon gas was designed. • APressure-dependent modulation instability gain was obtained. • Raman competition ultra-broadband from ultra-broadband modulation instability gain was found. • Provide a feasible way to get a bandwidth-tunable gain spectrum. • ABSTRACT
Based on the As2Se3 glass photonic crystal fiber (PCF) filled with Argon gas, pressure- and power-dependent modulation instability(MI) gain properties are studied. Two narrow MI gain bandwidths in the Stokes or anti-Stokes region expand gradually with the increasing light power, and merge into an ultra-broadband gain profile eventually by increasing the pump power from 1 W to 9000 W at a fixed gas pressure. Further increasing pump power will result in the reduction of the gain bandwidth and the formation of a constant narrow-band gain peak, which is a novel phenomenon and observed in our work for the first time. When the argon gas pressure is changed from 1 P0 to 400 P0, a similar gain characteristic is also found. However, their maximum gain bandwidths are reduced from 5065 nm to 3833 nm with the increase of the gas pressure, and their pump power required to generate the narrow-band gain spectrum are reduced from 8450 W to 3060 W. These results indicate that the formation of the ultra-broadband MI gain mainly depends on some nonlinear effects rather than stimulated Raman scattering (SRS) effect, while the single gain peak with a narrow bandwidth mainly results from the SRS effect. And they also show that, it is feasible to control the MI gain characteristic in the midinfrared region by changing the Argon gas pressure in the As2Se3 PCF, and this scheme can also be used in optical communication and spectral shaping in the future.
1. Introduction Modulation instability (MI) phenomenon, as a special four-wave mixing process, has attracted much attention, and it has been observed in some fields such as fluid dynamics, nonlinear optics and plasma physics [1–3]. As far as the physical mechanism is concerned, MI results from the interplay between the nonlinear and dispersive effect, and it can not only appear in the anomalous group velocity dispersion (GVD) regime [4], but also occur in the normal GVD regime in some cases, such as in the presence of higher even-order linear dispersions [5] and the co-propagation of two or more optical fields in an optical fiber [6]. Although MI may bring some negative effects in some systems such as optical communication, it is useful in wavelength conversion and optical amplification, and has been applied to produce optical pulse trains [7] and optical solitons [8], make MI laser [9], and generate supercontinuum (SC) [10] etc. For the fiber, our previous research results indicate that MI process can be used to generate a broadband gain, and ⁎
it depends on the fiber dispersion, the fiber nonlinearity and the stimulate Raman scattering (SRS) besides the pump light parameters [11–14]. Among them, the SRS effect is decided by the fiber substrate material (for example, As2Se3), which can’t be changed once the fiber is made. Thus, in order to manipulate MI gain effectively in experiment, it is convenient to adjust the fiber dispersion and nonlinearity by changing its effective mode refractive index and mode area. Considering that the photonic crystal fiber (PCF) can be designed flexibly [15], PCF is usually used to obtain a tunable MI gain by changing its geometry structure or choosing the different frequency light based on its zero dispersion wavelength (ZDW). These methods are feasible in the practical applications, but they may waste too much time and money on the experiment. Usually, it is hard to find the commercial light source satisfying the experiment requirement in practice. Moreover, the ZDW of PCF is invariable once it is made, which can’t be used to obtain a tunable MI gain. Thus, to avoid these difficulties, a PCF fiber filled with argon gas is considered in this work, which can be used to
Corresponding author. E-mail address: [email protected] (H. Wang).
https://doi.org/10.1016/j.optlastec.2019.105935 Received 10 November 2018; Received in revised form 2 July 2019; Accepted 28 October 2019 0030-3992/ © 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Helin Wang, et al., Optics and Laser Technology, https://doi.org/10.1016/j.optlastec.2019.105935
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get a tunable MI gain by changing the gas pressure rather than its structure or pump parameters. Besides, the substrate material of PCF plays an essential role on MI gain properties. Usually, the fiber material with a high nonlinearity and a high Raman response can make the MI gain observed easily, so the chalcogenide glass photonic crystal fibers such as As2Se3 PCF with the high Raman response and the high nonlinearity (almost 1000 times higher than the standard silica PCF) have attracted much attention in recent years [16–18]. Thus, in order to get a high, tunable and ultrabroadband MI gain, the effect of high Raman scattering and high nonlinearity on the MI gain in the As2Se3 glass PCF filled with Argon gas is discussed in detail, and some new MI properties including the bandwidth, amplitude and spectral peak position are found in this paper.
n0 =
+
For studying the MI gain characteristic of PCF fiber filled with Argon gas, the scalar nonlinear Schrödinger equation (NLSE) including all nonlinear phenomena such as fiber dispersion (FD), self-phase modulations (SPM), self-steepening (SS), stimulated Raman scattering (SRS) and four wave mixing (FWM) is adopted, and it can be expressed generally as follows [11]:
in 1 n n!
= i T U (z , t )
R ( )|U (z, t
nU
(
n02 1 pT0 n02 + 2 p0 T
1 2
1
,
n 02 1 pT0 n02 + 2 p0 T
+
1+
5.15 × 105 2 0
+
4.19 × 1011 4 0
,
4.32 × 10 23 8 0
) uu ((zz)) s
= iM 2 × 2
as
us (z ) , uas (z )
(2)
2×2
where M designates the stability matrix of the system. Suppose that us(z), u* as(z) ∝ exp(-iηz) is a solution of Eq. (2), then the MI phenomenon appears when η possesses a nonzero imaginary part. The perturbation parameter η depends on the pump wavelength, the pump power and the fiber parameters (including the fiber dispersion and its nonlinear coefficient), and it is determined by the eigenvalues of the matrix M2×2. Generally, the MI gain coefficient, defined as gMI(Ω) = 2|Im(η)|, reads:
(1)
where U(z, t) is the electric field amplitude, z is the longitudinal coordinate along the fiber. T = [1 + (i/ω)(∂/∂t)], α is the fiber loss. t is the time in a reference frame travelling. βn = ∂nβ(ω)/∂ωn|ω = ωp is the nth-order frequency-dependent dispersion coefficient of PCF filled with Argon gas at the pump frequency ωp. γ = n2ωp/cAeff is the nonlinear coefficient, n2 is the nonlinear refractive index of As2Se3 material, and Aeff is the effective mode area of the PCF fiber. It is worth noting that, the fiber nonlinearity γ and the dispersion βn depend on the refractive index of the fiber core and the gas hole in the fiber clad. Thus, if the gas hole of PCF fiber is filled with argon gas, the effective mode refractive index of PCF can be controlled by changing the pressure of argon gas filled into the clad. In the fiber, the refractive index of argon gas can be written as [19]:
nAr = 1 + 2
6 0
a a d us (z ) = i a11 a12 21 22 dz uas (z )
(z , t ) tn
)| d
4.09 × 1017
4
where nAr is the refractive index of Argon gas in the fiber clad at the gas pressure p and the temperature T, while n0 is the refractive index of Argon gas at the normal conditions (T0 = 273.15 K, p0 = 1 atm). λ0 is the pump wavelength. Usually, by restricting our analysis to the undepleted-dump approximation, MI in the NLSE is revealed by a line stability analysis of its continuous wave (CW) pumped with a power Pp, and the steady solution of Eq. (1) can be expressed by Ust(z) = (Pp)1/2exp(iγPpz). MI gain can be discussed by adding small perturbations into this steady solution. Now assume that the perturbation signals are us(z) and uas(z), then the real solution of Eq. (1) can be expressed by U(z) = Ust(z) + [us(z) exp(iΩt) + uas(z)exp(−iΩt)]exp(iγPpz), where Ω = ωp − ωs = ωas − ωp(Ω > 0) is the detuning frequency, ωs and ωas are the Stokes and Anti-Stokes frequency, respectively. After inserting U(z) into Eq. (1), two coupled linear ordinary differential equations on the perturbation signal us(z) and uas(z) are attained by picking out the Stokes term and the complex conjugate of the anti-Stokes term, and making them linearized. They can be written as:
2. MI gain of As2Se3 photonic crystal fiber
U (z , t ) + U (z , t ) + z 2
1 + 5.547 × 10
gMI ( ) = 2 Im{ PP R ( ) ± [D 2 ( ) + 2 PP R ( ) D ( ) +
2P 2 2R2 ( P
)]1 2 } (3)
, 2n
where Pp is the pump power, D(Ω)=∑β2nΩ /(2n)! (n ≥ 1) is the sum of the even order dispersion coefficients. For the As2Se3 chalcogenide glass, the nonlinear index can be estimated by the explicit KramersKronig transformation equation [20,21]. ξ = Ω/ωp. R(Ω) is the Fourier transform of Raman response function R(t) of As2Se3 PCFs [22,23]. The calculated Raman susceptibility and Raman gain spectrum are shown in Fig. 1. It indicates that, the detuning frequency of its Raman gain peak is 7 THz (the corresponding detuning angle frequency is 2π × 7 = 43.96 THz), and its Raman gain bandwidths is about 10 THz
1 2
,
Fig. 1. The Raman susceptibility and normalized Raman gain spectral of As2Se3 fiber material. 2
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Fig. 2. Optical parameters of the designed As2Se3 PCF filled with Argon gas. (a) The structure of PCF; (b) The nonlinear coefficient and effective mode area of PCF; (c) The dispersion of PCF; (d) The loss of PCF.
(the corresponding detuning 2π × 10 = 62.8 THz).
angle
frequency
bandwidth
is
fiber. Therefore, the pressure-dependent MI gain will be discussed in detail in the following sections when the pump wavelength is 3768 nm, which is close to the fiber ZDW and lies in the anomalous dispersion region of the fiber.
3. Characteristic for As2Se3 PCF filled with Argon gas To study the pressure-dependent tunable MI gain in a As2Se3 total internal reflection photonic crystal fiber (TIR-PCF) with a hexagonal structure, a TIR-PCF fiber filled with argon gas in the air hole is firstly designed with the finite-difference frequency-domain techniques. The fiber pitch Λ = 4 μm and its clad hole diameter d = 1.2 μm (see Fig. 2(a)). The fiber core diameter is 2Λ = 8 μm and its effective fiber core diameter is about 6.93 μm. During the simulation, the total dispersion only includes the material dispersion and waveguide dispersion due to the fundamental mode transmission and the non-birefringence characteristic of the designed fiber. When the Argon gas pressure filled into the PCF fiber increases from 1 P0 to 400 P0, the fiber dispersion decreases gradually (see Fig. 2(c)) at the same wavelength while its nonlinearity and loss increase gradually (see Fig. 2(b) and (d)). It is worth pointing out that, although the fiber loss increases with the gas pressure, it is very low when the pump wavelength λp < 4500 nm (see Fig. 2(d)), so it can be neglected in the calculation if the pump wavelength is set between 3700 nm and 3800 nm. Moreover, the zero dispersion wavelength (ZDW) of As2Se3 PCF fiber shifts from 3724 nm to 3766 nm (see Fig. 2(c)) with the increase of gas pressure, while MI gain depends on the fiber dispersion and its nonlinearity besides the Raman scattering (see Eq. (3)). Thus, the variation of the gas pressure can generate an important influence on the MI gain by changing the fiber dispersion and its nonlinearity. In other words, it is more convenient to obtain a tunable MI gain by controlling the gas pressure in the fiber than by changing the pump wavelength or the geometry structure of the
4. Pressure-dependent tunable MI gain properties In order to clarity the pressure-dependent tunable MI gain characteristic, the influence of gas pressure variation on MI gain in the case of the different pump power is analyzed in detail. Fig. 3(a)–(d) show that the MI gain varies with the sideband wavelength when the pump power Pp is changed from 1 W to 9000 W and the Argon gas pressure p = 1 p0, 100 p0, 200 p0, 300 p0, respectively. To better express these power-dependent MI gains, the color bar is used to represent the normalized MI gain amplitude in these figures. The results indicate that, there are two MI gain bands when the pump power Pp is lower than a certain value even if the gas pressure is different. One MI gain band is in the anti-Stokes region (see the left region of the vertical white dotted line in Fig. 3), while the other lies in the Stokes region (see the right region of the vertical white dotted line in Fig. 3). It is obvious that their total MI gain bandwidths from stokes and anti-Stokes region are ultraband, and they decrease gradually with the increase of Argon gas pressure in PCF fiber, which can be obtained by comparing their maximum MI gain bandwidth. One can see that their maximum MI gain bandwidths are 5065 nm, 4668 nm, 4297 nm, 3833 nm, respectively, when the Argon gas pressure in the PCF is 1 p0, 100 p0, 200 p0 and 300 p0. In fact, this result can also be obtained by observing the shift of the boundary wavelength of the MI gain spectrum. The boundary wavelength in the anti-stokes region shifts from 2500 nm to 2627 nm, while it shifts from 8050 nm to 6676 nm in the Stokes region. Then the 3
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Fig. 3. The variation of sideband wavelength with the pump power when pump wavelength is 3768 nm, and the gas pressure is 1 p0, 100 p0, 200 p0, 300 p0.
wavelength difference between two boundary points will decrease from 5571 nm to 4049 nm. It is worth noting that, these wavelength differences are different from the maximum MI gain bandwidth, which will be explained in the next section. Further analyzing the Fig. 3, one can also find an interesting phenomenon. When the pump power exceeds a certain value (see the power value at the intersection of the higher horizontal white line and the y-axis in Fig. 3), the broadband MI gain spectrum disappears completely. These special power values are 8450 W, 6700 W, 4900 W, 3060 W, respectively, for the gas pressure 1 p0, 100 p0, 200 p0, 300 p0. Meanwhile, a narrowband gain spectrum with an enough intensity generates for the different gas pressures, and they appear before the broadband gain spectra disappear. It indicates that those narrowband gain spectra start competing with the broadband gain spectra when the pump power reaches a bottom limitation. Those bottom limits are not the same for different gas pressures, and they are 5528 nm at 1 p0, 4722 nm at 100 p0, 2916 nm at 200 p0, and 1385 nm at 300 p0, respectively (see the power value at the intersection of the lower horizontal white line and the y-axis in Fig. 3). One can see the minimum power required to start competing with the other nonlinear effects reduces with the increase of the gas pressure. Therefore, the broadband and narrowband gain spectra can be obtained by controlling the gas pressure, and it is helpful to choose some suitable experimental parameters in the future.
5. Raman scattering competition and its effect on MI gain One knows from Fig. 3 that, when the pump power reaches the bottom limit required to start competing with the other nonlinear effects, the narrowband MI gain spectrum will appear and be enhanced while the broadband MI gain spectrum will become narrow quickly. There exists a competition phenomenon between a narrowband gain spectrum and a broadband MI gain spectrum. For explaining this novel phenomenon, it is necessary to analyze the variation characteristic of MI gain in detail. Fig. 4 shows the sideband wavelength variation with the pump power at gas pressure p = 400 p0, and three important regions are analyzed. It can be seen that, the MI gain bandwidth reaches its maximum at a certain power gradually, and then begin to decrease. This result is similar to those results observed at gas pressure p = 1p0, 100 p0, 200 p0, 300 p0. For Region 3 in Fig. 4(d), Fig. 4(c) indicates that four MI gain bands are generated when pump power Pp < 20 W. Among them, two gain bands appear in the Stokes region (λ > λp or ωs < ωp), while the other two lie in the anti-Stokes region (λ < λp or ωas > ωp). Moreover, two gain bands among them are away from pump wavelength 3768 nm: one gain band is at about 2750 nm in the anti-Stokes region, and the other is at about 5800 nm in the Stokes region. When pump power is increased to about 30 W, two gain bands merge in the Stokes and then a broadband gain profile is formed. Similar variation characteristics also occurs in the anti-Stokes region. However, the generated broadband MI gain in the two regions doesn’t 4
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Fig. 4. The power-dependent spectral properties of different regions from the MI gain in the As2Se3 PCF when gas pressure p = 400 p0: (a) the power-dependent single gain peak (region 1 in Fig. (d)); (b) the power-dependent competition process between MI and SRS (region 2 in Fig. (d)); (c) the power-dependent formation process of the ultra-broadband MI gain spectrum (region 3 in Fig. (d)); (d) the power-dependent MI gain versus wavelength at gas pressure p = 400 p0.
continue widening with the further increase of pump power. Inversely, when pump power is increased from 680 W to 1080 W, the two broadband gain spectra in the Stokes and anti-Stokes regions start narrowing gradually and disappearing. Then a narrow-band gain spectrum at about 3500 nm becomes more and more stronger in the anti-Stokes region (see Fig. 3(b)), while the small gain peak (about 4200 nm) close to pump light (3768 nm) in the Stokes region reduce gradually. Moreover, the anti-Stokes intensity of the narrow band peak is much higher than that in the Stokes region, which may result from the FWM in the fiber and the detailed reasons need further study. Finally, it is worth noting that, when pump power continues to increase from 1200 W to 1600 W (see Fig. 3(a)), the position of the narrow-band gain spectrum does not change. In fact, its central wavelength is still about 3450 nm although its intensity increases. In order to clarify the formation mechanism of the single narrowband gain spectrum, the variation of its frequency-dependent MI gain spectrum is shown in Fig. 5. When the pump power is increased from 930 W to 960 W, the amplitude of the MI gain spectrum decreases gradually and disappears at about Pp = 960 W finally without considering the Raman effect, while it changes slightly and doesn’t reduce to zero if considering the Raman effect (see the Fig. 5(a) and (b)). Moreover, with or without the Raman effect, there are two gain bands locating at about 19 THz (see the Fig. 5(a) and (b)), which indicates the other two gain peaks located at about 7 THz in Fig. 5(b) result from the Raman effect. Besides, the two gain peaks locating at about 19 THz disappear completely when the pump power is increased to over
1080 W (see the Fig. 4(d)), and then only one gain peak is remained (see the Fig. 4(d)), which is the single gain peak observed in this paper. Further analysis shows that the normalized single gain spectral characteristic in Fig. 5(c) is the same to the normalized Raman gain spectrum of the As2Se3 fiber material in Fig. 1. So we can be sure that the narrow-band single gain peak at about 7 THz only depends on Raman scattering effect when the pump power Pp > 1100 W and the argon gas pressure p = 400 p0. It also indicates the spectral competition phenomenon above mentioned only occurs between the Raman effect and the other nonlinear effect including FD, SS, FWM and SPM. Actually, this competitive behavior doesn’t only appear under the gas pressure of p = 400 p0, but also occurs under other gas pressure (see Fig. 3). However, there are still some differences for different gas pressures. For example, when gas pressure p = 1 p0 (see Fig. 3(a)), this competition behavior appears between 7500 W and 8450 W, while it occurs between 400 W and 1100 W for gas pressure p = 400 p0 (see Fig. 4(d)). Since the narrow-band gain spectrum only results from the Raman effect, the broadband MI gain mainly results from the combined operation of all nonlinear effects in the fiber. Moreover, The Raman scattering effect has a weak influence on the formation of broadband MI gain although it plays an important role on the MI gain for higher pump power (see Fig. 5(d)). Fig. 5(d) shows the broadband MI gain profile with or without considering SAS effect at pump power Pp = 100 W and gas pressure p = 400 p0. One can see that the MI gain is high without the SRS effect. Otherwise, it is low with the SAS effect, and two small peaks appear (see the area in the dot circle in Fig. 5(d)). Simultaneously, the 5
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Fig. 5. The single gain spectrum and the broadband gain spectra. (a) and (b) show the MI gain variation with or without Raman effect when the pump power is changed from 930 W to 960 W. (c) The single gain spectrum from MI at pump power Pp = 1400 W. (d) The broadband MI gain profile with or without considering Raman scattering effect.
gain amplitude and bandwidth are reduced slightly, and one of the reasonable explanations is that a part of pump power is transferred to Raman peak by SRS effect. Thus, for the As2Se3 fiber, the broadband MI gain mainly depends on the other nonlinear effects (including FD, SPM, FWM and SS) rather than SRS effect. The competition phenomenon between the SRS effect and the other nonlinear effects is related to the pump power, which is helpful for understanding the MI gain experiment in the future.
SRS effect starts competing with other nonlinear effects for the higher pump power, which results in the formation of a single gain peak with a narrow bandwidth. This novel phenomenon is helpful to gain a bandwidth-tunable gain spectrum by controlling the gas pressure in As2Se3 PCF. And it can be used into the fiber laser and fiber communication. Acknowledgments This work was supported by the Natural Science Foundation of Zhejiang Province (Grant No. LY15F050010), the Scientific Research Foundation of Zhejiang University of Technology (Grant No. 1401109012408), the National Natural Science Foundation of China (Grant No. 61605171, Grant No. 11604297).
6. Conclusion MI gain properties of the As2Se3 PCF filled with Argon gas are studied in this work. MI gain bandwidth expands gradually with the increase of pump power at a fix gas pressure, and then reduces gradually. But its gain amplitude continues to increase with the increasing power. When the gas pressure is changed from 1 p0 to 400 p0, the bandwidth of ultra-broadband MI gain decreases gradually, but their maximum gain bandwidths are different. In any case, a narrow MI gain band appears when pump power is increased to a special value, which is different for different gas pressure. Moreover, by analyzing the MI gain variation with the increase of pump power at gas pressure 400 p0, one finds that, there is an obvious competition between simulated Raman scattering and other nonlinear effects including FD, SPM, FWM and SS. When pump power is low, the MI gain mainly depends on the other nonlinear effects (including FD, SPM, FWM and SS) rather than SRS effect. However, after MI gain bandwidth reaches the maximum value,
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Raman competition from ultra-broadband modulation instability gain in the As2Se3 glass photonic crystal fiber filled with argon gas Helin Wanga, , Aijun Yangb, Bin Wua, Zhan Zhenga, Kan Lia ⁎
a b
Centre for Optics and Optoelectronics Research, Zhejiang University of Technology, Hangzhou, DC 310023, China College of Science, Zhejiang University of Technology, Hangzhou, DC 310023, China
HIGHLIGHTS
photonic crystal fiber filled with argon gas was designed. • APressure-dependent modulation instability gain was obtained. • Raman competition ultra-broadband from ultra-broadband modulation instability gain was found. • Provide a feasible way to get a bandwidth-tunable gain spectrum. • ABSTRACT
Based on the As2Se3 glass photonic crystal fiber (PCF) filled with Argon gas, pressure- and power-dependent modulation instability(MI) gain properties are studied. Two narrow MI gain bandwidths in the Stokes or anti-Stokes region expand gradually with the increasing light power, and merge into an ultra-broadband gain profile eventually by increasing the pump power from 1 W to 9000 W at a fixed gas pressure. Further increasing pump power will result in the reduction of the gain bandwidth and the formation of a constant narrow-band gain peak, which is a novel phenomenon and observed in our work for the first time. When the argon gas pressure is changed from 1 P0 to 400 P0, a similar gain characteristic is also found. However, their maximum gain bandwidths are reduced from 5065 nm to 3833 nm with the increase of the gas pressure, and their pump power required to generate the narrow-band gain spectrum are reduced from 8450 W to 3060 W. These results indicate that the formation of the ultra-broadband MI gain mainly depends on some nonlinear effects rather than stimulated Raman scattering (SRS) effect, while the single gain peak with a narrow bandwidth mainly results from the SRS effect. And they also show that, it is feasible to control the MI gain characteristic in the midinfrared region by changing the Argon gas pressure in the As2Se3 PCF, and this scheme can also be used in optical communication and spectral shaping in the future.
1. Introduction Modulation instability (MI) phenomenon, as a special four-wave mixing process, has attracted much attention, and it has been observed in some fields such as fluid dynamics, nonlinear optics and plasma physics [1–3]. As far as the physical mechanism is concerned, MI results from the interplay between the nonlinear and dispersive effect, and it can not only appear in the anomalous group velocity dispersion (GVD) regime [4], but also occur in the normal GVD regime in some cases, such as in the presence of higher even-order linear dispersions [5] and the co-propagation of two or more optical fields in an optical fiber [6]. Although MI may bring some negative effects in some systems such as optical communication, it is useful in wavelength conversion and optical amplification, and has been applied to produce optical pulse trains [7] and optical solitons [8], make MI laser [9], and generate supercontinuum (SC) [10] etc. For the fiber, our previous research results indicate that MI process can be used to generate a broadband gain, and ⁎
it depends on the fiber dispersion, the fiber nonlinearity and the stimulate Raman scattering (SRS) besides the pump light parameters [11–14]. Among them, the SRS effect is decided by the fiber substrate material (for example, As2Se3), which can’t be changed once the fiber is made. Thus, in order to manipulate MI gain effectively in experiment, it is convenient to adjust the fiber dispersion and nonlinearity by changing its effective mode refractive index and mode area. Considering that the photonic crystal fiber (PCF) can be designed flexibly [15], PCF is usually used to obtain a tunable MI gain by changing its geometry structure or choosing the different frequency light based on its zero dispersion wavelength (ZDW). These methods are feasible in the practical applications, but they may waste too much time and money on the experiment. Usually, it is hard to find the commercial light source satisfying the experiment requirement in practice. Moreover, the ZDW of PCF is invariable once it is made, which can’t be used to obtain a tunable MI gain. Thus, to avoid these difficulties, a PCF fiber filled with argon gas is considered in this work, which can be used to
Corresponding author. E-mail address: [email protected] (H. Wang).
https://doi.org/10.1016/j.optlastec.2019.105935 Received 10 November 2018; Received in revised form 2 July 2019; Accepted 28 October 2019 0030-3992/ © 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Helin Wang, et al., Optics and Laser Technology, https://doi.org/10.1016/j.optlastec.2019.105935
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get a tunable MI gain by changing the gas pressure rather than its structure or pump parameters. Besides, the substrate material of PCF plays an essential role on MI gain properties. Usually, the fiber material with a high nonlinearity and a high Raman response can make the MI gain observed easily, so the chalcogenide glass photonic crystal fibers such as As2Se3 PCF with the high Raman response and the high nonlinearity (almost 1000 times higher than the standard silica PCF) have attracted much attention in recent years [16–18]. Thus, in order to get a high, tunable and ultrabroadband MI gain, the effect of high Raman scattering and high nonlinearity on the MI gain in the As2Se3 glass PCF filled with Argon gas is discussed in detail, and some new MI properties including the bandwidth, amplitude and spectral peak position are found in this paper.
n0 =
+
For studying the MI gain characteristic of PCF fiber filled with Argon gas, the scalar nonlinear Schrödinger equation (NLSE) including all nonlinear phenomena such as fiber dispersion (FD), self-phase modulations (SPM), self-steepening (SS), stimulated Raman scattering (SRS) and four wave mixing (FWM) is adopted, and it can be expressed generally as follows [11]:
in 1 n n!
= i T U (z , t )
R ( )|U (z, t
nU
(
n02 1 pT0 n02 + 2 p0 T
1 2
1
,
n 02 1 pT0 n02 + 2 p0 T
+
1+
5.15 × 105 2 0
+
4.19 × 1011 4 0
,
4.32 × 10 23 8 0
) uu ((zz)) s
= iM 2 × 2
as
us (z ) , uas (z )
(2)
2×2
where M designates the stability matrix of the system. Suppose that us(z), u* as(z) ∝ exp(-iηz) is a solution of Eq. (2), then the MI phenomenon appears when η possesses a nonzero imaginary part. The perturbation parameter η depends on the pump wavelength, the pump power and the fiber parameters (including the fiber dispersion and its nonlinear coefficient), and it is determined by the eigenvalues of the matrix M2×2. Generally, the MI gain coefficient, defined as gMI(Ω) = 2|Im(η)|, reads:
(1)
where U(z, t) is the electric field amplitude, z is the longitudinal coordinate along the fiber. T = [1 + (i/ω)(∂/∂t)], α is the fiber loss. t is the time in a reference frame travelling. βn = ∂nβ(ω)/∂ωn|ω = ωp is the nth-order frequency-dependent dispersion coefficient of PCF filled with Argon gas at the pump frequency ωp. γ = n2ωp/cAeff is the nonlinear coefficient, n2 is the nonlinear refractive index of As2Se3 material, and Aeff is the effective mode area of the PCF fiber. It is worth noting that, the fiber nonlinearity γ and the dispersion βn depend on the refractive index of the fiber core and the gas hole in the fiber clad. Thus, if the gas hole of PCF fiber is filled with argon gas, the effective mode refractive index of PCF can be controlled by changing the pressure of argon gas filled into the clad. In the fiber, the refractive index of argon gas can be written as [19]:
nAr = 1 + 2
6 0
a a d us (z ) = i a11 a12 21 22 dz uas (z )
(z , t ) tn
)| d
4.09 × 1017
4
where nAr is the refractive index of Argon gas in the fiber clad at the gas pressure p and the temperature T, while n0 is the refractive index of Argon gas at the normal conditions (T0 = 273.15 K, p0 = 1 atm). λ0 is the pump wavelength. Usually, by restricting our analysis to the undepleted-dump approximation, MI in the NLSE is revealed by a line stability analysis of its continuous wave (CW) pumped with a power Pp, and the steady solution of Eq. (1) can be expressed by Ust(z) = (Pp)1/2exp(iγPpz). MI gain can be discussed by adding small perturbations into this steady solution. Now assume that the perturbation signals are us(z) and uas(z), then the real solution of Eq. (1) can be expressed by U(z) = Ust(z) + [us(z) exp(iΩt) + uas(z)exp(−iΩt)]exp(iγPpz), where Ω = ωp − ωs = ωas − ωp(Ω > 0) is the detuning frequency, ωs and ωas are the Stokes and Anti-Stokes frequency, respectively. After inserting U(z) into Eq. (1), two coupled linear ordinary differential equations on the perturbation signal us(z) and uas(z) are attained by picking out the Stokes term and the complex conjugate of the anti-Stokes term, and making them linearized. They can be written as:
2. MI gain of As2Se3 photonic crystal fiber
U (z , t ) + U (z , t ) + z 2
1 + 5.547 × 10
gMI ( ) = 2 Im{ PP R ( ) ± [D 2 ( ) + 2 PP R ( ) D ( ) +
2P 2 2R2 ( P
)]1 2 } (3)
, 2n
where Pp is the pump power, D(Ω)=∑β2nΩ /(2n)! (n ≥ 1) is the sum of the even order dispersion coefficients. For the As2Se3 chalcogenide glass, the nonlinear index can be estimated by the explicit KramersKronig transformation equation [20,21]. ξ = Ω/ωp. R(Ω) is the Fourier transform of Raman response function R(t) of As2Se3 PCFs [22,23]. The calculated Raman susceptibility and Raman gain spectrum are shown in Fig. 1. It indicates that, the detuning frequency of its Raman gain peak is 7 THz (the corresponding detuning angle frequency is 2π × 7 = 43.96 THz), and its Raman gain bandwidths is about 10 THz
1 2
,
Fig. 1. The Raman susceptibility and normalized Raman gain spectral of As2Se3 fiber material. 2
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Fig. 2. Optical parameters of the designed As2Se3 PCF filled with Argon gas. (a) The structure of PCF; (b) The nonlinear coefficient and effective mode area of PCF; (c) The dispersion of PCF; (d) The loss of PCF.
(the corresponding detuning 2π × 10 = 62.8 THz).
angle
frequency
bandwidth
is
fiber. Therefore, the pressure-dependent MI gain will be discussed in detail in the following sections when the pump wavelength is 3768 nm, which is close to the fiber ZDW and lies in the anomalous dispersion region of the fiber.
3. Characteristic for As2Se3 PCF filled with Argon gas To study the pressure-dependent tunable MI gain in a As2Se3 total internal reflection photonic crystal fiber (TIR-PCF) with a hexagonal structure, a TIR-PCF fiber filled with argon gas in the air hole is firstly designed with the finite-difference frequency-domain techniques. The fiber pitch Λ = 4 μm and its clad hole diameter d = 1.2 μm (see Fig. 2(a)). The fiber core diameter is 2Λ = 8 μm and its effective fiber core diameter is about 6.93 μm. During the simulation, the total dispersion only includes the material dispersion and waveguide dispersion due to the fundamental mode transmission and the non-birefringence characteristic of the designed fiber. When the Argon gas pressure filled into the PCF fiber increases from 1 P0 to 400 P0, the fiber dispersion decreases gradually (see Fig. 2(c)) at the same wavelength while its nonlinearity and loss increase gradually (see Fig. 2(b) and (d)). It is worth pointing out that, although the fiber loss increases with the gas pressure, it is very low when the pump wavelength λp < 4500 nm (see Fig. 2(d)), so it can be neglected in the calculation if the pump wavelength is set between 3700 nm and 3800 nm. Moreover, the zero dispersion wavelength (ZDW) of As2Se3 PCF fiber shifts from 3724 nm to 3766 nm (see Fig. 2(c)) with the increase of gas pressure, while MI gain depends on the fiber dispersion and its nonlinearity besides the Raman scattering (see Eq. (3)). Thus, the variation of the gas pressure can generate an important influence on the MI gain by changing the fiber dispersion and its nonlinearity. In other words, it is more convenient to obtain a tunable MI gain by controlling the gas pressure in the fiber than by changing the pump wavelength or the geometry structure of the
4. Pressure-dependent tunable MI gain properties In order to clarity the pressure-dependent tunable MI gain characteristic, the influence of gas pressure variation on MI gain in the case of the different pump power is analyzed in detail. Fig. 3(a)–(d) show that the MI gain varies with the sideband wavelength when the pump power Pp is changed from 1 W to 9000 W and the Argon gas pressure p = 1 p0, 100 p0, 200 p0, 300 p0, respectively. To better express these power-dependent MI gains, the color bar is used to represent the normalized MI gain amplitude in these figures. The results indicate that, there are two MI gain bands when the pump power Pp is lower than a certain value even if the gas pressure is different. One MI gain band is in the anti-Stokes region (see the left region of the vertical white dotted line in Fig. 3), while the other lies in the Stokes region (see the right region of the vertical white dotted line in Fig. 3). It is obvious that their total MI gain bandwidths from stokes and anti-Stokes region are ultraband, and they decrease gradually with the increase of Argon gas pressure in PCF fiber, which can be obtained by comparing their maximum MI gain bandwidth. One can see that their maximum MI gain bandwidths are 5065 nm, 4668 nm, 4297 nm, 3833 nm, respectively, when the Argon gas pressure in the PCF is 1 p0, 100 p0, 200 p0 and 300 p0. In fact, this result can also be obtained by observing the shift of the boundary wavelength of the MI gain spectrum. The boundary wavelength in the anti-stokes region shifts from 2500 nm to 2627 nm, while it shifts from 8050 nm to 6676 nm in the Stokes region. Then the 3
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Fig. 3. The variation of sideband wavelength with the pump power when pump wavelength is 3768 nm, and the gas pressure is 1 p0, 100 p0, 200 p0, 300 p0.
wavelength difference between two boundary points will decrease from 5571 nm to 4049 nm. It is worth noting that, these wavelength differences are different from the maximum MI gain bandwidth, which will be explained in the next section. Further analyzing the Fig. 3, one can also find an interesting phenomenon. When the pump power exceeds a certain value (see the power value at the intersection of the higher horizontal white line and the y-axis in Fig. 3), the broadband MI gain spectrum disappears completely. These special power values are 8450 W, 6700 W, 4900 W, 3060 W, respectively, for the gas pressure 1 p0, 100 p0, 200 p0, 300 p0. Meanwhile, a narrowband gain spectrum with an enough intensity generates for the different gas pressures, and they appear before the broadband gain spectra disappear. It indicates that those narrowband gain spectra start competing with the broadband gain spectra when the pump power reaches a bottom limitation. Those bottom limits are not the same for different gas pressures, and they are 5528 nm at 1 p0, 4722 nm at 100 p0, 2916 nm at 200 p0, and 1385 nm at 300 p0, respectively (see the power value at the intersection of the lower horizontal white line and the y-axis in Fig. 3). One can see the minimum power required to start competing with the other nonlinear effects reduces with the increase of the gas pressure. Therefore, the broadband and narrowband gain spectra can be obtained by controlling the gas pressure, and it is helpful to choose some suitable experimental parameters in the future.
5. Raman scattering competition and its effect on MI gain One knows from Fig. 3 that, when the pump power reaches the bottom limit required to start competing with the other nonlinear effects, the narrowband MI gain spectrum will appear and be enhanced while the broadband MI gain spectrum will become narrow quickly. There exists a competition phenomenon between a narrowband gain spectrum and a broadband MI gain spectrum. For explaining this novel phenomenon, it is necessary to analyze the variation characteristic of MI gain in detail. Fig. 4 shows the sideband wavelength variation with the pump power at gas pressure p = 400 p0, and three important regions are analyzed. It can be seen that, the MI gain bandwidth reaches its maximum at a certain power gradually, and then begin to decrease. This result is similar to those results observed at gas pressure p = 1p0, 100 p0, 200 p0, 300 p0. For Region 3 in Fig. 4(d), Fig. 4(c) indicates that four MI gain bands are generated when pump power Pp < 20 W. Among them, two gain bands appear in the Stokes region (λ > λp or ωs < ωp), while the other two lie in the anti-Stokes region (λ < λp or ωas > ωp). Moreover, two gain bands among them are away from pump wavelength 3768 nm: one gain band is at about 2750 nm in the anti-Stokes region, and the other is at about 5800 nm in the Stokes region. When pump power is increased to about 30 W, two gain bands merge in the Stokes and then a broadband gain profile is formed. Similar variation characteristics also occurs in the anti-Stokes region. However, the generated broadband MI gain in the two regions doesn’t 4
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Fig. 4. The power-dependent spectral properties of different regions from the MI gain in the As2Se3 PCF when gas pressure p = 400 p0: (a) the power-dependent single gain peak (region 1 in Fig. (d)); (b) the power-dependent competition process between MI and SRS (region 2 in Fig. (d)); (c) the power-dependent formation process of the ultra-broadband MI gain spectrum (region 3 in Fig. (d)); (d) the power-dependent MI gain versus wavelength at gas pressure p = 400 p0.
continue widening with the further increase of pump power. Inversely, when pump power is increased from 680 W to 1080 W, the two broadband gain spectra in the Stokes and anti-Stokes regions start narrowing gradually and disappearing. Then a narrow-band gain spectrum at about 3500 nm becomes more and more stronger in the anti-Stokes region (see Fig. 3(b)), while the small gain peak (about 4200 nm) close to pump light (3768 nm) in the Stokes region reduce gradually. Moreover, the anti-Stokes intensity of the narrow band peak is much higher than that in the Stokes region, which may result from the FWM in the fiber and the detailed reasons need further study. Finally, it is worth noting that, when pump power continues to increase from 1200 W to 1600 W (see Fig. 3(a)), the position of the narrow-band gain spectrum does not change. In fact, its central wavelength is still about 3450 nm although its intensity increases. In order to clarify the formation mechanism of the single narrowband gain spectrum, the variation of its frequency-dependent MI gain spectrum is shown in Fig. 5. When the pump power is increased from 930 W to 960 W, the amplitude of the MI gain spectrum decreases gradually and disappears at about Pp = 960 W finally without considering the Raman effect, while it changes slightly and doesn’t reduce to zero if considering the Raman effect (see the Fig. 5(a) and (b)). Moreover, with or without the Raman effect, there are two gain bands locating at about 19 THz (see the Fig. 5(a) and (b)), which indicates the other two gain peaks located at about 7 THz in Fig. 5(b) result from the Raman effect. Besides, the two gain peaks locating at about 19 THz disappear completely when the pump power is increased to over
1080 W (see the Fig. 4(d)), and then only one gain peak is remained (see the Fig. 4(d)), which is the single gain peak observed in this paper. Further analysis shows that the normalized single gain spectral characteristic in Fig. 5(c) is the same to the normalized Raman gain spectrum of the As2Se3 fiber material in Fig. 1. So we can be sure that the narrow-band single gain peak at about 7 THz only depends on Raman scattering effect when the pump power Pp > 1100 W and the argon gas pressure p = 400 p0. It also indicates the spectral competition phenomenon above mentioned only occurs between the Raman effect and the other nonlinear effect including FD, SS, FWM and SPM. Actually, this competitive behavior doesn’t only appear under the gas pressure of p = 400 p0, but also occurs under other gas pressure (see Fig. 3). However, there are still some differences for different gas pressures. For example, when gas pressure p = 1 p0 (see Fig. 3(a)), this competition behavior appears between 7500 W and 8450 W, while it occurs between 400 W and 1100 W for gas pressure p = 400 p0 (see Fig. 4(d)). Since the narrow-band gain spectrum only results from the Raman effect, the broadband MI gain mainly results from the combined operation of all nonlinear effects in the fiber. Moreover, The Raman scattering effect has a weak influence on the formation of broadband MI gain although it plays an important role on the MI gain for higher pump power (see Fig. 5(d)). Fig. 5(d) shows the broadband MI gain profile with or without considering SAS effect at pump power Pp = 100 W and gas pressure p = 400 p0. One can see that the MI gain is high without the SRS effect. Otherwise, it is low with the SAS effect, and two small peaks appear (see the area in the dot circle in Fig. 5(d)). Simultaneously, the 5
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Fig. 5. The single gain spectrum and the broadband gain spectra. (a) and (b) show the MI gain variation with or without Raman effect when the pump power is changed from 930 W to 960 W. (c) The single gain spectrum from MI at pump power Pp = 1400 W. (d) The broadband MI gain profile with or without considering Raman scattering effect.
gain amplitude and bandwidth are reduced slightly, and one of the reasonable explanations is that a part of pump power is transferred to Raman peak by SRS effect. Thus, for the As2Se3 fiber, the broadband MI gain mainly depends on the other nonlinear effects (including FD, SPM, FWM and SS) rather than SRS effect. The competition phenomenon between the SRS effect and the other nonlinear effects is related to the pump power, which is helpful for understanding the MI gain experiment in the future.
SRS effect starts competing with other nonlinear effects for the higher pump power, which results in the formation of a single gain peak with a narrow bandwidth. This novel phenomenon is helpful to gain a bandwidth-tunable gain spectrum by controlling the gas pressure in As2Se3 PCF. And it can be used into the fiber laser and fiber communication. Acknowledgments This work was supported by the Natural Science Foundation of Zhejiang Province (Grant No. LY15F050010), the Scientific Research Foundation of Zhejiang University of Technology (Grant No. 1401109012408), the National Natural Science Foundation of China (Grant No. 61605171, Grant No. 11604297).
6. Conclusion MI gain properties of the As2Se3 PCF filled with Argon gas are studied in this work. MI gain bandwidth expands gradually with the increase of pump power at a fix gas pressure, and then reduces gradually. But its gain amplitude continues to increase with the increasing power. When the gas pressure is changed from 1 p0 to 400 p0, the bandwidth of ultra-broadband MI gain decreases gradually, but their maximum gain bandwidths are different. In any case, a narrow MI gain band appears when pump power is increased to a special value, which is different for different gas pressure. Moreover, by analyzing the MI gain variation with the increase of pump power at gas pressure 400 p0, one finds that, there is an obvious competition between simulated Raman scattering and other nonlinear effects including FD, SPM, FWM and SS. When pump power is low, the MI gain mainly depends on the other nonlinear effects (including FD, SPM, FWM and SS) rather than SRS effect. However, after MI gain bandwidth reaches the maximum value,
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