王兴军--2015 暑期学校PPT-光发射1 硅基集成中的光发射技术

1、http:/ Emission in Silicon Photonics王兴军王兴军 Beijing 2015.7.13 http:/ Emission and detector http:/ Photonic Integration and Microsystems:Combining electrical and optical components on the same silicon-based substrates used in the fabrication of a semiconductor chip, which employs numerous integrated o

2、ptics devices, including lasers, photodetectors, splitters, isolators, filters, modulators, switches, et. al.Silicon photonicsIntroduction (Intel and IBM)http:/ 硅基光源是硅基光电子学元器件中的重中之重。l 由于硅是间接带隙的半导体，发光效率不高，内量子效率约为10-5-10-6。l 因此硅一直以来被认为不适合制作光源材料。 http:/ 上世纪90年代，多孔硅的室温发光 2000年，纳米硅的增益 2004年，铒掺杂微环激光器 2005

3、年，硅拉曼激光器 2006年，III-V族-硅混合激光器 2010年，锗硅激光器http:/ l=hc/Eg=1.24eV/EgLight emissionhttp:/ 一个电子或空穴吸收另一电子与空穴复合时放出的能量，跃迁到更高的能量状态的过程称为俄歇复合过程。 这种过程的几率与复合的载流子浓度和接受能量的载流子浓度乘积成正比，所以载流子浓度高的材料俄歇复合过程更容易。l自由载流子吸收l多声子复合 晶体中电子和空穴复合时可以发射多个声子来释放能量称之为多声子复合。http:/ 自发辐射是指处于激发态的原子中的电子在激发态能级上只能停留一段很短的时间，就自发地跃迁到较低能级中去，同时辐射出一个

4、光子。l受激辐射 当原子处于激发态E2时，如果恰好有能量等于（E2-E1）的光子入射进来，在入射光子的影响下，原子会发出一个同样的光子而跃迁到低能级E1上去，这种辐射叫做受激辐射。这种方式就是产生激光的基本原理。http:/ 1917年 爱因斯坦在“关于辐射的量子理论”论文中提出，解释黑体辐射现象光子与原子的三种相互作用：自发辐射、受激辐射、受激吸收自发辐射、受激辐射、受激吸收 自发辐射：自发辐射：电子-空穴对复合时产生的光在波长、相位等特性彼此互不关联，自发性的行为。光谱较宽，相位不一致，没有偏振性，光输出功率较弱，例：发光二极管 受激辐射：受激辐射：已有的传播光子诱发产生一个光子能量、相位

5、、偏振等特性与前一个光子完全相同。这个辐射复合过程为受激辐射。光谱窄、相位一致、有偏振方向、光输出功率大。例：激光器 EcEvSpontaneous emissionStimulated emissionhttp:/ 在正常状态下，电子处于基态E1，在入射光作用下，它会吸收光子的能量跃迁到激发态E2上，这种跃迁称为受激吸收。电子跃迁后，在基态留下相同数目的空穴。受激吸收是受激辐射的逆过程。 设在单位物质中，处于E1和E2的原子数分别为N1和N2。当系统处于热平衡状态时，存在下面的分布 http:/ Emission Rate and Einstein Coefficients The upwa

6、rd transition depends on the number of atoms N1 and the energy density in the radiation:Need to determine the controlling factors for the rates of stimulated emission, spontaneous emission, and absorption in two energy system.E1E2h(a) Absorptionh(b) Spontaneous emissionh(c) Stimulated emissionInhOut

7、hE2E2E1E1Absorption, spontaneous (random photon) emission and stimulatedemission. 1999 S.O. Kasap, Op to electro n ics (Prentice Hall)B12, A21, and B21 are the Einstein coefficients for absorption, spontaneous emission and stimulated emission.The downward transition depends on the number of atoms N2

8、 and the energy density in the radiation:http:/ Emission Rate and Einstein Coefficients At the thermal equilibrium, we can assume up and down transitions are equal and the atom numbers at energy levels are determined by Boltzmann statistics:Also at the thermal equilibrium, the radiated photon energy

9、 density from atoms must follow the Plancks black body radiation distribution law:http:/ Emission Rate and Einstein Coefficients Based on above assumptions and equations, the Einstein coefficients can be determined as:The ratio of stimulated to spontaneous emissions can be determined as:orhttp:/ Emi

10、ssion Rate and Einstein Coefficients The ratio of stimulated emission to absorption can be determined as:From above equations, two requirements need to be met to have stronger stimulated emission over spontaneous emission and absorption (Lasing): (1) large photon concentration (optical cavity) and (

11、2) N2 N1 (population inversion). http:/ (wnr+wr)之比。因此，只有当nrr，才能获得高效率的光子发射。l 对间接复合为主的半导体材料，一般既存在发光中心，又存在其他复合中心，通过前者产生辐射复合，后者产生非辐射复合。因此，要使辐射复合占优势，必须使发光中心浓度远大于其他杂质浓度。发光效率http:/ 光的放大主要由材料的增益谱决定，对于半导体材料，它是由态密度（(h)）、费米函数 (fg(h)和辐射寿命r决定的。L.Pavesi, Review Article:Silicon-Based Light Sources for Silicon Inte

12、grated Circuits Advances in Optical TechnologiesVolume 2008 其中，drstim or drabs是一定光子能量h下的受激发射和受激吸收率，g(h)是增益系数，d是光子流量的变化。 fe和和fh是电子是电子-空穴对的热分布函数，空穴对的热分布函数，是光子流密度，是光子流密度，EFe和和 EFh是电子和空穴的准费米能是电子和空穴的准费米能级，当没有外泵浦的情况下，费米级，当没有外泵浦的情况下，费米函数函数减少到简单的费米态，也就是对于一个空的导带和减少到简单的费米态，也就是对于一个空的导带和填满的价带，增益系数小于吸收系数，填满的价带，增

13、益系数小于吸收系数，fgh，满足粒子数反转条件，满足粒子数反转条件，fg0。这意味上面的公式为正值，。这意味上面的公式为正值，因此系统也显示正的增益。从上面公式可以看出，辐射寿命因此系统也显示正的增益。从上面公式可以看出，辐射寿命r也是一个关键的参数也是一个关键的参数,寿命越寿命越短，增益越大。短，增益越大。 http:/ http:/ I=I0egx, g=Jm, g为增益系数，J为电流密度，为增益因子，对于同质m=1，异质m=2.8。光传输公式： I=I0e(g-)x R1R2e2(g- )L=1 g= +ln(1/R1R2)/2L左边光吸收引起的损耗包括体内所有损耗：本征光吸收,自由载流

14、子吸收等右边为端面透射损耗.http:/ 阈值的物理意义：阈值的物理意义：在激光物质中，要实现受激辐射的光放大，必须其内部增益足够大。 g= +ln(1/R1R2)/2Ll尽量提高电注入效率，使增益尽可能大，受激辐射尽可能高。l通过高质量的外延生长获得高质量的晶体，使内部吸收非常低。l通过镀反射膜来提高反射率，减少透射损耗。http:/ 能产生激光的物质 直接带隙的半导体发光效率比间接带隙高3个数量级，一般只有直接带隙才能制备激光器，对比：GaAs：0.5；Si：10-5(2) 粒子数反转 光照、电流注入、化学反应等泵浦方式（p-n结电注入载流子是最简单方式，效率高）(3) 谐振腔 对频率一定

15、、方向一致的光产生正反馈，使其获得足够大的增益，克服内部和端面的损耗，从而发生谐振，产生激光。 直接带隙材料、电注入实现粒子数反转和谐振腔直接带隙材料、电注入实现粒子数反转和谐振腔三大要素构三大要素构成了成了半导体激光器半导体激光器的基本支柱。的基本支柱。http:/ 1.1 eV 两个强非辐射跃迁过程： 俄歇复合 自由载流子吸收。http:/ limits: 1l 在硅中，电子-空穴对的辐射寿命长（毫秒量级），一个电子-空穴对需要毫秒才能复合，典型的非辐射寿命是纳秒量级，因此内量子效率约为10-5-10-6。 在此期间，电子和空穴移动的体积达到10m3。这样他们很容易遇到缺陷或俘获中心，载流

16、子就会发生非辐射复合。int = wr/(wnr+wr)=nr /(nr + r )=1-r /(nr + r )http:/ limits: 2硅的俄歇复合 一旦多的载流子被激发，这种机理就非常严重。一个俄歇复合的概率是和激发的载流子数n的平方成正比，和禁带宽度成反比。因为半导体中有很多的载流子，所以俄歇复合是很强的。l其中C为常数,和材料的掺杂浓度有关。对于硅，为10-30cm6s-1,当n为1019cm-3，非辐射复合寿命为10ns。因此，对于高载流子注入硅，俄歇复合是非辐射复合的主要机制。http:/ limits: 3自由载流子吸收l自由载流子吸收系数是和硅的自由载流子浓度nfc以及

17、光波长有关an 10-18 nfc 2 当nfc=1019cm-3, =1.55m，n为24cm-16。对于重掺杂硅，这也是产生激光的主要限制，然而对于本征硅，除非nfc非常高，否则这种贡献比较小。 http:/ process:The anodization of Si wafers at low current densities in HF-based solution can be used to generate an array of extremely small holes that run orthogonal to the surface.Porous silicon (P

18、L) They observed strong visible PL at room temperature. The have been interpreted as arising from free standing Si quantum wires wherein two-dimensional Quantum confinement of carriers has appreciably widened the Si band gap.1990, Canham et al. APL 57, 1046http:/ The demonstration of bright and effi

19、cient EL from porous silicon layers at low bias voltages is very encouraging. It shows that electron-hole pairs can be created electrically with subsequent radiative recombination. 1992, Canham et al. APL 61, 2583Porous silicon (EL)http:/ nanocrystal fabricationPavesi et al. Advances in Optical Tech

20、nologies 2008 Beat the indirect band gap and avoid non-radiative recombinationshttp:/ 目前对低维纳米硅基材料发光机理的研究出现了各种各样的物理模型，各有优缺点，总体上看，目前主要存在以下3种发光机理解释的模型。l量子限制发光模型l与氧有关的缺陷发光模型l量子限制效应-发光中心发光模型http:/ Xe 4f n n=1-14 .4f n 5s2 5p66s2Optical doping with lanthanide ionshttp:/ levels of lanthanide ions1.5 mEgap(

21、Si)http:/ dopingEnhanced co-doping Er silicate(a) Si:Er( ) (N1018cm-3)(b) Si:Er( )/O( ) (N1019cm-3)(e) Er2SiO5 (N1022cm-3) Er-O polyhedra SiOx crystalline matrix(c) SiO2:Er( ) (N1019cm-3)(d)SiO2:Er( )ncSi( )(N1020cm-3)Schematics of related light source materials embedded with Er ionsSchematics of re

22、lated light source materials embedded with Er ions Er concentration 10dB gainEDFA 1018cm-3 10mEDWA 1020cm-3 3cmEr2SiO5 1022cm-3 1 mm ?!G(dB)=4.43 (s semN2-s sabsN1)L s semNEr Ls sem: emission cross sections sabs: absorption cross section: optical confinement factorL: waveguide lengthDoping Constitut

23、ional elementEDFAhttp:/ Erbium Doping SiliconPLELhttp:/ processEr ion implantation bulk SiIon implantation process(能量为100keV-10MeV量级) Advantage: ion implantation is a materials engineering process by which ions of a material can be implanted into another solid, thereby changing the physical properti

24、es of the solid. 1. pure doping 2. enlarge solubility of Er doping 3 controlling Er ion concentration and deep 4. large areahttp:/ and EL PropertiesPL Ion implantation350KeV, 1*1017-5*1018 Er ions/cm3EL Ion implantation20 KeV, 5.6*1018 Er ions/cm3 http:/ dependent on microstructure 1400 1450 1500 15

25、50 1600 1650 1700 654nm30mWRT PL intensity arb. unitsWavelength nmEr2SiO5 Er2O3 Er:Al2O3http:/ (Edge X-ray Absorption Fine Structure) Local structure around Er in glassesM.A. Marcus et al., J. of Non-Cryst. Solids 136, 260 (1991) http:/ Erbium Doping and Er CompoundsIntroduction to about rare earth

26、ionsSi based Er light source historyEr + Silicon (si nanocrystal) riched silicon oxide Er compoundEr doping Al2O3 materialsSome new structure conclusionshttp:/ + Silicon (si nanocrystal) riched silicon oxide http:/ of Er2SiO5 structure 0.86nmHR-TEM of Er2SiO5 crystal 0.86 nmHRTEM05010015020025030002

27、0406080100 O, Si, Er atomic%Depth nm O Si Erhttp:/ of Er2SiO5 structure at 300K PL at RT1400 1450 1500 1550 1600 1650 1700 654nm30mWRT PL intensity arb. unitsWavelength nm1528 nmhttp:/ mW4I13/24I15/2Er3+87PL at different measurement temperaturehttp:/ J F et al. Opt. Exp. 15: 11272 (2007)Energy band

28、engineering of Ge136 meV115 meVGeSi 0.25%的张应变会使的张应变会使与与L谷的带隙差减小到谷的带隙差减小到115meV，这一差别可，这一差别可以通过以通过n型掺杂来进一步弥补。型掺杂来进一步弥补。1019的的n掺杂可以填满掺杂可以填满L谷并使电子谷并使电子开始填充开始填充谷，从而获得直接带隙跃迁发光和显著的增益，谷，从而获得直接带隙跃迁发光和显著的增益，锗的带隙结构：(a)体材料；(b)0.25%张应变；(c)0.25%张应变加n掺杂http:/ resultshttp:/ 实现其高效率和高稳定度的发光。二是从器件实用化角度考虑,如何实现硅-LED 在室温

29、下的电致发光。l人们已尝试了三种硅基纳米材料用于高效率硅-LED 的制作, 即高纯体单晶硅，硅纳米量子点和掺Er硅。http:/ Si light-emitting diodes M. A. Green, et al, Nature, 412: 805, 2001.以区熔法生长的具有晶格完整以区熔法生长的具有晶格完整性好的硅单晶作为基底性好的硅单晶作为基底, 利用利用适宜的蚀刻技术使表面加以构适宜的蚀刻技术使表面加以构型型, 把硅表面设计成锯齿状光把硅表面设计成锯齿状光学图形，使入射角小于全反射学图形，使入射角小于全反射角，光的输出效率可以达到角，光的输出效率可以达到99%以上以上并对其进行高

30、质量的热氧化和并对其进行高质量的热氧化和表面钝化表面钝化, 以有效地减少载流以有效地减少载流子的非辐射复合速率子的非辐射复合速率,可使量子可使量子效率得以明显提高效率得以明显提高, 其室温下其室温下的电注入有效量子效率可达的电注入有效量子效率可达1 %以上，开启电压小于以上，开启电压小于1V。http:/ silicate-V+p-SiErYb/Y silicaten-Sip-SiErYb/Y silicateITO台阶结构台阶结构p-i-n结构结构MIS结构结构铒镱铒镱/钇硅酸盐薄膜容易击穿，钇硅酸盐薄膜容易击穿，电流无法与铒离子作用电流无法与铒离子作用http:/ C C退火退火11001

31、100 C C退火退火直接带隙吸收拟合直接带隙吸收拟合12gEaww 5gEeV绝缘体绝缘体电流注入困难电流注入困难http:/ silicateSiNx/SiONITO加入加入SiNx/SiON限流层限流层防止硅酸盐薄膜的破坏性击穿防止硅酸盐薄膜的破坏性击穿 FN隧穿产生热载流子，碰撞激发铒离子隧穿产生热载流子，碰撞激发铒离子Alp-SiErYb/Y silicateITOhttp:/ -Fowler-Nordheim隧穿隧穿2expJBAEEAlp-SiErYb silicate 60nmSiNx 60nmITO2expJBAEEAlp-SiErYb silicate 60nmITOAlp

32、-SiSiNx 60nmITO限流层辅助实现了限流层辅助实现了FN隧穿，隧穿，有可能产生热载流子有可能产生热载流子http:/ EL-V-IAlp-SiErYb silicate 60nmSiNx/SiON 60nmITO1.53um EL谱谱http:/ SiON做限流层做限流层Alp-SiErYb silicate 60nmSiON 60nmITO1.53m EL寿命寿命1.5ms1.53m EL碰撞激发截面碰撞激发截面ErYb silicate: 310-14cm2Er-doped SiO2: 110-14cm211risedecaysErYb silicate 的碰撞激发截面较大！的碰

33、撞激发截面较大！http:/ 硅基激光器的研制是硅基光电子学领域中的一个最具有魅力、最富挑战性的前沿课题。制备出具有光增益、光放大和受激辐射的有源区材料或结构,能够实现粒子数的反转,具有适宜结构形式的光学谐振腔,能够实现电注入条件下的受激辐射。http:/ et al. Nature materials 4: 888 (2005) 刻蚀了一种周期性的纳米孔阵列结刻蚀了一种周期性的纳米孔阵列结构，孔直径为构，孔直径为110nm。在连续。在连续1.5W的的514.5nm Ar离子激光泵浦下获得离子激光泵浦下获得了了1.278m的连续光。这是由于周的连续光。这是由于周期性的纳米孔阵列产生了高密度的期

34、性的纳米孔阵列产生了高密度的A型陷阱中心，这些缺陷中心作为光型陷阱中心，这些缺陷中心作为光激活中心发生反转产生受激发射和激活中心发生反转产生受激发射和光增益。但主要缺点是只有在低于光增益。但主要缺点是只有在低于80K的温度才能产生受激发射的温度才能产生受激发射Nanopatterned siliconhttp:/ Although these experimental observations strongly suggest that significant optical gain and stimulated emission can be achieved in periodic na

35、nopatterned crystalline silicon, a complete understanding or analysis of the observed phenomena is not readily available at this early stage. For one, the sub-bandgap emission at 1278 nm can be attributed to the so-called A-centre mediated radiative recombination. It has been established that an A-c

36、entre defect state, located 0.17 eV below the conduction band edge of silicon, allows direct (phononless) recombination between trapped electrons and free holes. The exact nature and origin of the A-type trapping centres have remained a subject of inquiry, but are more often attributed to silicon va

37、cancies.http:/ Raman scattering is an inelastic light scattering process, whereby the energy of an incident photon is modified by an inelastic interaction with a molecule.http:/ Raman spectrumhttp:/ Raman laser级联拉曼光纤激光器是利用光纤的非线性效应,产生红外激光的一种新型激光器。在光纤通信中，可作为拉曼光纤放大器和远程掺铒光纤放大器的泵浦光源。在其他领域也有广泛应用。原理上只要泵浦功率

38、足够强，就可以在红外范围实现高功率、高质量激光光束输出，应用前景广泛。http:/ Another nonlinear optical effect that is particularly strong in semiconductor. The effect results in pump depletion and generation of the free carriers. TPA-induced free-carrier absorption depends on the free carriers concentration through the relation a=1.4

39、5*10-17(l/1.55)2N N=Ip2eff/(2hv) hv is pump photon energy, eff is effective recombination lifetime for free carriers; Ip: pump intensityTwo photon absorption (TPA)http:/ approacheslOne method for diminishing these losses is to reduce the free carrier lifetime through lateral scaling of waveguide mod

40、al area. lAnother approach for reducing free carrier losses is to use pulsed pumping.l Using a reverse biased p-i-n diode embedded in a silicon waveguide to remove the carriershttp:/ main challenge in Raman laser is TPA that competes with Gainhttp:/ Bulk Si: Several tens ns. SOI: blow several ns. lT

41、he lifetime is determined by the combination of diffusion and interface/surface recombination currents between top Si and buried oxide layer in a bare SOI waveguide, the geometry of the waveguide plays a significant role in determining the carrier lifetimes.http:/ Raman silicon laser based on a ring

42、-resonator-cavity configuration. (2007-Intel) A cascaded silicon Raman laser (2008- Intel)Si Raman laser developmenthttp:/ silicon laser pulses with lifelGain: Raman amplificationlLoss: free carrier absorption due to TPAlSolution 1: pulsed operationlPulse width carrier pulse periodO. Boyraz and B. J

43、alali, “Demonstration of a silicon Raman laser,” Opt. Express 12, 5269 (2004).Pulsed Si Raman laserhttp:/ 25ps Pulsed pumping. To the extent that the pulse width is much less than the carrier lifetime and the pulse period is much larger than the lifetime.http:/ Threshold: 9 W pump pulse powerhttp:/

44、Pulse operation is necessary in order to avoid accumulation of free carriers that are generated due to TPA. The results show that free carrier induced limitations can be solved by using pulsed pumping. http:/ TPA-induced FCA in silicon can be significantly reduced by introducing a reverse biased p-i

45、-n diode embedded in a silicon waveguide. The laser cavity is formed by coating the facets of the silicon waveguide with multilayer dielectric films.rib width: 1.5 m; height (H):1.55 m; etched depth (h):0.7 m. The waveguide was formed in an S-shaped curve with a total length of 4.8 cm and a bend rad

46、ius of 400 mRong et al, Nature 433, 725 (2005).http:/ a reverse bias voltage is applied to the p-i-n diode, the TPA-generated electronhole pairs can be swept out of the silicon waveguide by the electric field between the p- and n-doped regions. Thus the effective carrier lifetime, representing the l

47、ifetime of the free carriers interaction with the optical mode in the waveguide region, reduces with increased bias voltage. At a reverse-bias voltage of 25 V, the effective carrier lifetime is reduced to 1ns.http:/ the parameterslThe performance of this silicon Raman laser could be further improved

48、 by optimizing cavity mirror and cavity length design. lThe threshold power could be reduced by using a waveguide with smaller cross-sectional dimensions and/or by introducing a larger cavity enhancement for the pump beam.lThe fibre to waveguide coupling efficiency could be improved by adding a mode

49、 converter in the waveguide. lIn addition, with optimization of the p-i-n diode design, it may be possible to further reduce the effective carrier lifetime to below 1 ns. http:/ silicon laser based on a ring-resonator-cavity configuration A racetrack-shaped ring laser cavity. A bus waveguide is conn

50、ected to the ring cavity through a directional coupler, which couples both pump and signal laser light into and out of the cavity. The coupling ratio depends on the wavelength and polarization and can be varied by changing the gap or length of the coupler or both to achieve the desired coupling rati

51、os for pump and lasing wavelengths. The gap (d) between the two waveguides in the coupler was 0.7 mm, Rong et al, Nature Photonics 1, 235 (2007).http:/ reducing the linear loss to 0.2 dB cm-1 and the carrier lifetime to 0.4 ns.http:/ The realization of low-threshold and zero-power-consumption silico

52、n Raman lasers represents a major leap towards producing practical silicon-based lasers. Threshold can be reduced to 16 mW.http:/ cascaded silicon Raman laser (2008- Intel)Rong et al, Nature Photonics 2, 170 (2008).http:/ The pump power reaches a threshold of 80 mW, first-order lasing takes place an

53、d the laser output power continues to increase with increasing pump power. When the coupled pump power is increased to 120 mW, the intracavity power of the first-order laser becomes high enough to generate sufficient optical gain at the second-order Stokes wavelength, and second-order Raman lasing b

54、egins.http:/ applications Methane, one of the major greenhouse gases, and water vapour, which needs to be controlled tightly in high-yield semiconductor manufacturing processes. These molecules have characteristic absorption patterns in the regions covered by the cascaded silicon Raman laser.http:/

55、summary, cascaded Raman lasing in silicon has been demonstrated. Using a pump beam of 1,550 nm. A stable, single-mode, first- and second-order continuous-wave lasing at 1,686 nm and 1,848 nm, respectively was observed. lThey have been able to resolve the rotationalvibrational IR absorption spectra o

56、f methane and water vapour molecules in two separate spectral regions over 160 nm apart.l The realization of the second-order silicon Raman laser paves the way towards higher-order cascaded Raman lasing, and opens a new path to producing low-cost, compact, room temperature, high-performance mid-IR l

57、asers http:/ Hybrid laserA novel laser that utilizes a silicon waveguide bonded to AlGaInAs quantum wells is demonstrated. r using low temperature oxygen plasma-assisted wafer bonding. The optically pumped1538 nm laser has a pulsed threshold of 30 mW and an output power of 1.4 mW.John E. Bowers, et

58、al. Opt. Exp. 13, 9460 (2005)http:/ laser The fabrication is done in four major parts. First, the silicon waveguides are formed on the SOI wafer. Next, the III-V epitaxial layer structure is transferred to the SOI wafer through oxide plasma assisted wafer bonding. The III-V layers are then processed

59、 to control the flow of current to the optical mode. Finally the devices are diced and polished to create high quality mirror facets and define the cavity length. Browers et al. 4th International Conference on Group IV Photonics, Tokyo Japan, Sep. 2007http:/ Reported the first observation of optical

60、 gain and laser in epitaxial Ge-on-Si at room temperature by using tensile strain and n-type doping for band engineering. Absorption spectra of the n+ Ge mesa sample under 0 and 100 mW optical pumping. Negative absorption coefficients corresponding to optical gain are observed in the wavelength rang