1、,Summary,Blackbody radiation,Einstein Coefficients,Homogeneous Broadening:,Limited lifetime in the upper level,Lorentzian lineshape,Indistinguishable interaction to atoms,Linewidth usually in MHz range,Inhomogeneous Broadening:,Doppler broadening,Gaussian lineshape,Distinguishable interaction to ato
2、ms,Linewidth usually in GHz range,Population Inversion,4.4 The Significance of c(n),In such way, we can ignore the physical origin of c(w) and treat the wave propagation in the medium with e.,A plane, single frequency EM wave :,Lecture 7,4.4 The Significance of c(n),Additional phase delay due to ato
3、mic polarization :,Amplitude variation due to atomic polarization :,An alternative way to derive above equations:,and,while,and,where,4.5 Derivation of c(n),Kramers-Kronig Relation (KK relation),P: Cauchy Principal Value,In practice, we can obtain c(n) by measuring absorption coefficient a(n), which
4、 is proportional to c”(n). Then we can derive c(n) from c”(n) through KK relation.,4.5 Derivation of c(n),Cauchys Residue Theorem:,4.5 Derivation of c(n),Taking R - infinity and e - 0 :,The integral over C2 is proportional to R-2 - 0,The first and third terms on the right side is principal value of
5、the integral:,The integral over C1 is :,4.5 Derivation of c(n),Integral over C1,Residue,and,4.5 Derivation of c(n),A typical susceptibility plot for a resonant 2 level system. Note at the resonant point, imaginary part gets maximum, real part is zero.,4.6 Gain Saturation in Homogeneous Laser Media,I
6、n practice, the population inversion is caused by a “Pumping” process, such as electric current in injection lasers, flashlamp light in pulsed ruby lasers, or the energetic electrons in plasma-discharge gas lasers ,Gain Constant,Now, we are going to derive gain constant in presence of an optical fie
7、ld,4.6 Gain Saturation in Homogeneous Laser Media,I. Rate Equations,At steady state:,If the optical field is absent, i.e.:,4.6 Gain Saturation in Homogeneous Laser Media,In an efficient laser system:,and,Saturation intensity :,Gain constant :,Small signal gain constant :,II. Population Inversion Sat
8、uration,4.6 Gain Saturation in Homogeneous Laser Media,:,Population inversion has no relation with intensity,:,At saturation intensity point, induced radiation rate is comparable with other relaxation processes, and causes the inversion to drop to one half of its nonsaturated value.,4.6 Gain Saturat
9、ion in Homogeneous Laser Media,and,and,That is, the more closer to resonant, the smaller the saturation intensity,While at the same incident intensity, the smaller saturation intensity means the population inversion drops more, the saturation is more serious. So when incident laser frequency is clos
10、e to the resonant, the saturation effect gets stronger; while laser frequency is far away from the resonant, the saturation effect gets weaker.,4.6 Gain Saturation in Homogeneous Laser Media,Lorentzian lineshape population inversion,Only in this region, saturation effect is obvious,III. Gain Saturat
11、ion,4.6 Gain Saturation in Homogeneous Laser Media,Small signal gain constant,:,Gain constant in Lorentzian lineshape homogeneous laser media,Gain constant uniformly drops in homogeneous broadening case,4.6 Gain Saturation in Homogeneous Laser Media,Now we consider a higher intensity light with freq
12、uency n1, how does another weaker light with frequency n2 gain constant change?,The higher intensity light not only changes itself gain constant, but also changes other frequency weaker light gain constant in same degree.,4.7 Gain Saturation in Inhomogeneous Laser Media,Assume the inhomogeneous medi
13、um is made of classes of atoms each designated by a continuous variable x. Moreover, define a function f(x) so that a priori probability that an atom has its parameter between x and x+dx is p(x)dx, and satisfy:,The atoms within a given class x are such considered as homogeneously broadening, having
14、a lineshape function that is normalized as:,So,4.7 Gain Saturation in Inhomogeneous Laser Media,I. Rate Equations,Steady state solution:,4.7 Gain Saturation in Inhomogeneous Laser Media,Summing over all the classes:,Total power emitted by induced transitions per unit volume by atoms in class x:,Or,I
15、I. Gain Constant,4.7 Gain Saturation in Inhomogeneous Laser Media,Small signal gain constant:,In the absence of saturation the gain constant of a homogenous and an inhomogeneous atomic system are identical,Other cases:,In each class, all the atoms are identical and are homogeneous broadening,Substit
16、ute above equation to the gain expression for inhomogeneous media,4.7 Gain Saturation in Inhomogeneous Laser Media,In the extreme inhomogeneous case, the width of p (n) is much larger than that of g x (n).,Gain constant in inhomogeneous laser media,4.7 Gain Saturation in Inhomogeneous Laser Media,II
17、I. Gain Saturation,Small signal gain constant,:,If the inhomogeneous broadening is Doppler Broadening, then,Gain constant in a Doppler broadening laser media,Saturation intensity,4.7 Gain Saturation in Inhomogeneous Laser Media,IV. Spectral Hole Burning Effect,对于非均匀加宽，表现中心频率为n的粒子发射一条中心频率为v，线宽为Dv的均匀加
18、宽谱线。因而这部分粒子的饱和行为可以用均匀加宽情况下的增益饱和来描述。,When,and,When,saturation effect gets weaker,When,and,saturation effect can be ignored,4.7 Gain Saturation in Inhomogeneous Laser Media,因此在频率为v1，光强为Iv1的强光作用下，非均匀加宽介质中的反转布居数密度在v1 处产生局部的饱和，即在以下频率范围有显著的饱和效应，并形成一个以v1为中心的孔，而在此范围以外的粒子没有影响。以上现象称为反转布居数得“烧孔效应”。,孔的宽度：,孔的深度：,Spectral hole burning effect in the laser cavity,
