Tunable lasers handbook phần 6 ppsx

234 Norman P. Barnes Z a b C e d Z FIGURE 6 Orbits in octahedral symmetry. (a, u orbit. (b) 1' orbit. (cj x orbit. (d) y orbit. (e) :orbit. 6 Transition Metal Solid-state Lasers 35 S=-K(T)(Y2,(6.$) + &&$))/2': , ;5 j Electron o'rbits described by these linear combinations of functions are graphed in Fig. 6. As can be seen, the 3dT orbits are maximized along the .I, y, and I axes. that is, the orbits are directed ton ard the positions of the nearest neighbors. On the other hand, the 3d~ orbits are maximized at angles directed between the nearest neighbors. Because the nearest neighbors usually have a net negative charge, it is logical that the orbits directed toward the nearest neighbors uould have a higher energy. In essence, the electrons are being forced to go where they are being repulsed. A calculation of the energies of the molecular bonding orbits must include the effects of the mutual repulsion. Mutual repulsion energy contributions can be expressed in terms of the Racah parameters, A. B, and C Racah parameters, in turn. are expressed in term5 of Slater integrals: however, it is beyond the scope of xhis chapter to delve into the details. Suffice it to say that the 4 term is an additive term on all of the diagonal elements. When only energy differences are to be calculated. this term drops out. The B and C energy terms occur on many off-diagonal elements. However. Tanabe and Sugano observed that the ratio of C/B is nearly constant and in the range of 4 to 5. A slight increase of this ratio is noted as the nuclear charge increases while the number of electrons remains constant. A. ratio of C/B of 3.97 was expected based on Slater integral formalism. Thus. the mutual repulsion contribution to the energy levels can be approxi- mated if only a single parameter is known. Usually this parameter is the Racah parameter B. Hence, many of the Tanabe-Sugano calculations are normalized by this parameter. Crystal field contributions to the energy of the molecular orbits can be described by the parameter Dq. Remember that lODq is the energy difference between the 3dT and the 3~1e levels for a single 3d electron. Consider the case where there are N electrons. These electrons can be split between the 3dT and 3d~ orbits. Suppose II of these electrons are in the 3de orbits. leaving N-n of them in the 3dT orbits. Crystal field effect contributions to the energy can be approximated as (6N - 1On)Dq. Crystal field energy contributions. in this simpli- fied approach, occur only for diagonal energy matrix elements. Energy differences between the various levels have been calculated for all combinations of electrons in octahedral symmetry and are presented in Tanabe- Sugano diagrams. Such diagrams often plot the energy difference between vari- ous energy levels, normalized by the Racah B parameter. as a function of the crystal field parameter, again normalized by the Racah B parameter. A Tanabe-Sugano diagram for three electrons in the 3d subshell is presented in 236 Norman P. Barnes Fig. 7. For this diagram, the ratio of C/B was assumed to be 4.5. Triply ionized Cr is an example of an active atom with three electrons in the 3d subshell. Ener- gies are calculated by diagonalizing the energy matrix. However, as the Dq term becomes large, the energy differences asymptotically approach a constant or a term that is linearly increasing with the parameter Dq. Such behavior would be expected since, for large values of Dq, the diagonal terms dominate and the crys- tal field energy contributions only appear on diagonal terms. Note that a Tanabe- Sugano diagram is valid only for one particular active atom since other active atoms may not have the same ratio of C/B. Absorption and emission occur when an electron makes a transition between levels. The energy difference between the initial and final levels of the electron is related to the energy of the absorbed or emitted photon. In purely electronic transitions, all of the energy between the two levels is taken up with the emitted or absorbed photon. However, as will be explained in more detail, some of the energy can appear as vibrations associated with the crystal lattice, that is, phonons, in the vicinity of the active atom. Selection rules indicate the strength of the transition between two levels of different energy. Obviously, a transition that is allowed will have stronger absorption and emission spectra than a transition that is not allowed. Two selec- tion rules are particularly germane to the transition metals, the spin selection 1234 Dq/B FIGURE 7 Tanabe-Sugano diagram for d3 electrons. 6 Transition Metal Solid-state lasers 37 rule and the Laporte selection rule. According to the spin selection rule, a transi- tion can only occur between levels in which the number of unpaired electrons in the initial and final levels is the same. In cases where a single electron undergoes a transition, the spin must be the same for the initial and final levels. According to one formulation of the Laporte selection rule, a transition is forbidden if it involves only a redistribution of electrons having similar orbitals v, ithin a single quantum shell. This formulation is particularly relevant to transition metals because transitions tend to be between different 3d levels but within the same quantum shell. For example, transitions involving only a rotary charge displace- ment in one plane would be forbidden by this selection rule. Selection rules were also considered by Tanabe and Sugano. Usually the strong interaction that allows a transition between levels with the emission of a photon is the electric dipole interaction. However, for the 3d electrons, all transi- tions between the various levels are forbidden since all levels have the same par- ity. Consequently, three other transition interactions were considered: the electric dipole interaction coupled with a vibration, the electric quadrupole interaction, and the magnetic dipole interaction. The strengths of these various interactions LA ere estimated. From these estimations. it was concluded that the electric dipole transition coupled with vibration, that is, a vibronic transition, u as the strongest interaction. Vibronic transitions involve emission or absorption of a photon and a quantized 3mount of lattice vibrations referred to as a phonon. Vibronic interac- tions were estimated to be about 2 orders of magnitude stronger than the nexl strongest interaction, the magnetic dipole interaction. McCumber [ 101 investigated the absorption and emission that results from vibromc interactions. Terminology used in the original paper refers to phonon- terminated absorption and emission rather than vibronic transitions. McCumber analyzed the absorption, emission. and gain of the transition metal Ni in the ini- tial paper. Emission spectra from Ni:MgF, were characterized by sharp emission lines and a broad emission spectra on &e long-wavelength side of the sharp emission lines. Sharp lines were associated nith electronic transitions, whereas the long-wavelength emission was associated with vibronic emission. Since then. this general analysis has been extended to many of the transition metals. Through the use of an analysis similar to the McCumber analysis, the gain characteristics of an active atom can be related to the absorption and emission spectra. Relating the gain to the absorption and emission spectra is of consider- able practical importance since the gain as a function of wavelength is a more difficult measurement than the absorption and emission. Emission and absorp- tion spectra often display relatively sharp electronic, or no phonon. transitions accompanied by adjacent broad vibronic transitions associated with the emission and absorption of phonons. General absorption and emission processes appear in Fig. 8. At reduced temperatures only phonon emission is observed since the average phonon population is low. In this case. the vibronic emission spectra 238 Norman P. Barnes extends to the long-wavelength side of the electronic transitions. On the other hand, the vibronic absorption spectra extends to the short-wavelength side of the electronic transition. In some cases, the absorption spectra and emission spectra are mirror images of each other. Although in general this is not true. at any wavelength the absorption, emission, and gain are related by the principle of detailed balance. Several assumptions must be met in order for the McCumber analysis to be valid. Consider a system consisting of an upper manifold and a lower manifold. As before, the term manifold will be used to describe a set of closely spaced levels. To first order approximation, levels within the manifold can be associ- ated with a simple harmonic motion of the active atom and its surrounding atoms. While the simple harmonic oscillator energy level spacings of the upper and lower manifolds may be the same. in general they do not have to be. Fur- thermore, the position of the minimum of the simple harmonic potential wells may be spatially offset from each other due to the difference in size of the active atom in the ground level and the excited level. Population densities of these manifolds are denoted by N, and N,. One of the assumptions used by the theory is that a single lattice temperature-can describe the population densities of these manifolds. For example, suppose the upper manifold consists of a series of levels commencing with the lowest energy le\7el which is an energy hvZp above the ground level. Levels within the manifold are separated by an energy hvv where this energy represents a quantum of vibrational energy asso- ciated with the simple harmonic motion of the upper level. According to this assumption, the active atoms in the upper manifold will be distributed among the various vibrational levels associated with the upper manifold according to a simple Boltzmann distribution. In turn. the Boltzmann distribution can be char- acterized by a single temperature T. Thus, with all of the vibrational levels equally degenerate, the population of any particular vibrational level will be given by N,exp (-JhvJkT) (1 - exp (-kv, /W)) where J is the integer denoting the energy ievel, k is Boltzmann's constant, and T is the lattice temperature. The last factor simply normalizes the distribution since it represents the summation over all levels within the manifold. Furthermore, the same temperature can describe the relative population of the levels comprising the lower manifold. Another assumption is that the time interval required for thermal equilibrium for the various population densities is very short compared with the lifetime of the upper level. For example. suppose all of the population of the upper mani- fold may be put initially in a single level by utilizing laser pumping. The sec- ond assumption says, in essence, that the closely spaced levels achieve thermal equilibrium in a time interval short with respect to the lifetime of the upper manifold. A third assumption is that nonradiative transitions are negligible compared with the transitions that produce the absorption or emission of a pho- ton. Although this is not always true. the lifetime of the upper level may be 6 Transition Metal Solid-state Lasers decomposed into components representing a radiative lifetime and a nonradia- tive lifetime. Given the population densities of the upper and lower manifolds, the absorption and emission cross sections can be related to the absorption and emission coefficients, up(k,v) and ep(k.v). respectively. In these expressions. k is the wave vector indicating the direction of propagation and v is the frequency. A subscript y is utilized since the absorption and emission may depend on the polarization p. Given the absorption and emission coefficients, absorption and emission cross sections can be defined by the relations Using the principle of detailed balance. the absorption and emission cross sec- tions are related by In this expression, hp is the energy required to excite one active atom from the lower level to the upper level while maintaining the lattice temperature T, In the lowtemperature limit for any system and for any temperature in a mirror image type of system, the parameter p is the frequency of the no phonon transition. Using these relations, the gain coefficient g,,(k,v) as a function of awe- length is given by While either of these expressions could be utilized to determine the gain coeffi- cient, the relation using the emission cross section is usually of the greater prac- tical importance. In general, the absorption cross section is too small to be mea- sured in a practical situation. On the other hand, the stimulated emission cross 240 Norman P. Barnes section can be readily deduced from a single fluorescence spectrum if the laser material is isotropic or fluorescence spectra if the material is not isotropic. McCumber's theory yields a practical method of deducing the emission cross section from the emission spectrum or spectra. To establish this relation, a function fp(k,v) is introduced. When multiplied by an incremental solid angle dokp and a unit frequency interval dv, this function represents the average intensity of emitted photonslsecond in the direction k, with frequency v, and with polarization p. One of the prime values of this function is that it can be easily measured and normalized. Normalization can be obtained through another easily measured quantity, the radiative lifetime of the upper manifold, T, by the relation Using this function. the stimulated emission cross section can be expressed as In this expression, c is the speed of light and II is the refractive index. In general, the refractive index will depend on the direction of propagation k, as well as the polarization. Combining these equations leads to the primary result of the McCumber analysis, That is, the gain can be related to the measurable quantities, the fluorescence spectrum or spectra, and the radiative lifetime. Although McCumber's theory laid the foundation for the determination of the gain, most experimental measurements are made in terms of watts per unit wavelength interval rather than photons per second per unit frequency interval. However, the change can be made in a straightforward manner. To change from fp(k,v) in units of photons per second per unit frequency interval to g,(k,v) in units of watts per unit wavelength interval, 6 Transition Metal Solid-state Lasers 241 where h is the wavelength associated with the frequency v. In a practical labora- tory system. only a fraction of the emitted radiation is collected by the fluores- cence measurement device. If this fraction collected, R, is independent of the wavelength, then where G,(k.v) is the measured quantity. Using the preceding relations, the quan- tity R can be determined using the relation between the radiative lifetime and the fluorescence spectrum. With the measured spectrum. the emission cross sec- tion becomes where I,, is defined by the relation Z,,,= -== rhGp(k,h) d31 . (17) In Eq. (271, it has been tacitly assumed that the material is isotropic. If the mate- rial is not isotropic, the extension to take into account the effects of anisotropy is straightforward. While McCumber related the gain of a transition metal to the absorption or emission spectra, Struck and Fonger [l 11 presented a unified theory of both the radiative emission and nonradiative decay processes. Previously, two disparate theories had described nonradiative decay processes. One of these theories, referred to as the activation energy relation, described the nonradiative decay process by the relation 1 L = A,, exp (-%) . (18) In this expression, rn, is the nonradiative lifetime. An2 is a rate constant. El is an activation energy, k is Boltzmann's constant, and T is the temperature. It can be loosely interpreted as the number of times per second that the excited active atom tries to escape from a potential well times the probability that it will have energy to effect its escape. Another theory is referred to as the niultiphonon emission fornzula. In this formulation, the nonradiative decay rate is given by 242 Norman P. Barnes where A, is a rate constant. E is a coupling constant, p is the number of phonons required to span the gap between the manifolds. and is the thermal occupation factor for the phonons, vp being the phonon frequency. In this formulation, the first two factors are nominally temperature independent so that the temperature dependence is carried by the thermal occupation factor for the phonons. To reconcile these two theories. Struck and Fonger relied on a single config- urational coordinate model. In the simplest application of the single configura- tional coordinate model. the interaction of the active atom and its nearest neigh- bors is considered to be described by a single configurational parameter. A configurational parameter can describe one aspect of the geometrical configura- tion of the active atom with its nearest neighbors. As an example, a configura- tional parameter for an active atom in a position of octahedral symmetry could be the average distance between the active atom and its six nearest neighbors. As the single configurational coordinate changes, the average distance between the active atom and its six nearest neighbors expands or contracts. In this case, the expansion and contraction is reminiscent of the breathing motion; consequently, it is often referred to as the breathing mode. Energies associated with different manifolds are dependent on this single configurational coordinate. Typically. energy as a function of the configuration coordinate appears as a parabola as shown in Fig. 8. Equilibrium positions are found near the lowest point in the parabola. That is, it would require energy to either expand or contract the configurational coordinate. For example, as the length between the active atom and its nearest neighbors contracts, the mutual repulsion of like charges would tend to dominate and push the nearest neighbors away. The strength of the interaction can be gauged from the shape of the para- bolic curves. If the energy depends strongly on the configurational coordinate, the parabola will be more strongly curved. Conversely, if the parabola is weakly curved, the energy depends only weakly on the configurational coordinate. Although the curvature of the parabolas for different manifolds can be different, a case can be made for them being roughly equal. The curvature of the parabolas describing the energy versus configurational coordinate determines the energy spacing between adjacent energy levels within the manifold. If a particle is trapped in a potential well described by a parabolic form. the particle will undergo simple harmonic motion. For the atoms involved in the configurational coordinate model, the harmonic motion must be described using quantum mechanics. For this reason, Struck and Fonger refer to a quantum mechanical single configurational coordinate. Quantizing the simple harmonic motion introduces two effects not found in classical simple harmonic oscillators, 6 Transition Metal Solid-state Lasers 243 V I I U FIGURE 8 Configuration coordinate energy-level diagram. discrete energy levels and a zero point energy. Differences between discrete energy levels associated with a quantum mechanical parabola are hv, where 17 is Planck’s constant and v,. is a frequency. Parabolas associated with different mani- folds can have different curvatnres with different frequencies. To describe the different curvatures, ari angle 8 is introduced and defined by where the subscripts 1’ and II denote the upper and lower parabolas, respectively. In terms of the discrete energy difference. the zero point energy associated wifn the v parabola is hv1,/2. Parabolas for manifolds with different energies may be offset from each other. Manifolds having different energies have different electronic charge con- figurations. For these different electronic charge configurations, the equilibrium position of the nearest neighbors can be different. For example, an electronic charge distribution that has the electrons appear between the active atom and its nearest neighbors may result in a stronger repulsion and consequently a longer distance between them. A difference in the equilibrium position can affect the energy of the manifold. In general, the active atom and its surrounding neighbors will prefer to reside in a configurational coordinate position. which minimizes [...]... c axis Thermal expansion a axis c axis Refractive index a axis c axis Refractive index variation a axis c axis Optical transparency Melting point Value Units 399 .6 963 .6 2989 938 1 .6 5.1 31 3 .6 1.3902 1.3889 10 -6/ K -4.2 -4 .6 Pm 825 'C 266 Norman P Barnes LiCaAlF, and LiSrAlF, are birefringent materials with relatively low refractive indices Refractive indices have been measured for LiCaAlF, at nine... crystal field on the transition probability 6 Transition Metal Solid-state lasers 0 267 v) 2 -1 .6 s - v) v) '5 -0.8 Lu 0 6 N E w 4 % 7 x 2 C 0 c 0 :: L o 4 e 0 c 0 - 2 0 0.30 0.50 0.70 3 E W 0.90 Wavelength (micrometers) FIGURE 22 Absoqtim and emission spectra of Cr:LiSrAlF, (Courtesy of S A Payne L ~wrence : Livermore National Laboratory.) Although Cr:LiCaA1F6 and Cr:LiSrAlF, have quite similar absorption... near the peak emission wavelengths, excited state absorption is a small effect 6 Transition Metal Solid-state Lasers 269 Normal mode thresholds for flashlamp-pumped Cr:LiSrA1F6 are considerably lower than they are for Cr:LiCaA1F6,reflecting the higher gain of the former laser material Experimental results are available for 6, 35-mm-diameter laser rods The length of the Cr:LiCaAlF, was 80 mm: the length... expansion a axis b anis c axis Refracti\Fe index a axis b axis c axis Refractive index variation a axis b anis c axis Optical transparency Melting point Units Pm 547 .6 112.7 3700 830 23 hg/m' Jkg-K 1VIm-K 10 -6 1.4 6. 8 6. 9 1.7421 1.7478 1.7401 10-6K 9.1 8.3 15.7 0.23-* 1870 Pm "C *Long wavelength cut off is unavailable on the order of 200 m-1 Typical of the Cr absorption spectra, these broad absorption bands... radiation longitudinal pumping is often employed Because the beam quality of the pump can be relatively good, the pump 6 Transition Metal Solid-state Lasers 4.0 261 -1 L I I I I 100 200 300 400 Temperature (" K) FIGURE 1 8 Upper laser level lifetime of Ti:Xl10, versus temperature a b 0 .6 0.7 0.8 09 1.0 Wavelength (micrometers) FIGURE 1 9 Wavelength (micrometers) Absorption and emission spectra of Ti:,41,0,... [12,14] One of these absorption bands lies in the blue region 6 Transition Metal Solid-state Lasers 249 TABLE 1 Physical Properties of A1,0, Parameter \%he Units Lattice constants a axis c axis Density Heat capacity Thermal conductivity a axis c axis 1 76. 3 1300.3 3990 775 33 35 Thermal expansion a axis c axis 1.8 5.3 Refractive index a axis c axis 1. 765 1 1.7573 Refractive index variation a axis 13.1 c axis... leads to efficient flashlamp pumping Absorption spectra for Cr:LiSrA1F6 are quite similar to absorption spectra for Cr:LiCaA1F6.Peaks occur at nearly the same wavelengths and the relative strengths of the peaks are also similar However, the absolute strengths for Cr:LiSrA1F6 are roughly twice as strong as the strengths of Cr:LiSrA1F6 Absorption spectra are shown in Fig 21 and 22 for these two laser materials... operating in the normal mode, as opposed to the Q-switched mode, because less energy is stored in the upper laser manifold with normal mode operation 6 Transition Metal Solid-state Lasers 300 Cr Concentration = 2.2 x 1019 ions:cm3 A (0.00 063 atomic) 0.40 0.50 0 .60 0.70 Wavelength (micrometers) FIGURE 1 4 Absorption spectra of Cr:BeAl,O, (a) a axis absorption (b) b axis absorption (ci c axis absorption (Courtesy... from Cr:LiCaAIF6 peaks about at 0. 76 pm and has a linewidth of about 0.132 pm for both polarizations [41] On the other hand emission from Cr:LiSrAlF, peaks about at 0.84 p m and has a linewidth of about 0.197 pm for both polarizations [42] For Cr:LiCaA1F6, the 7c polarized emission is approximately 1.5 times as intense as the CJ polarization The n polarized emission spectrum for Cr:LiSrA1F6 is approximately... around the peak gain wavelength of this laser material [ 26] As in the case of Cr:A1,0,, mercury-arc lamps were employed Threshold was high, somewhat over 20 06 W, but the slope efficiency was also reasonably high about 0.01 In this case, the laser could be tuned from less than 0.74 pm to beyond 0.78 pm 6 Ti:AI2O3 Ti:A1,0, is a laser material tunable over much of the near infrared, which has both a . FIGURE 6 Orbits in octahedral symmetry. (a, u orbit. (b) 1' orbit. (cj x orbit. (d) y orbit. (e) :orbit. 6 Transition Metal Solid-state Lasers 35 S=-K(T)(Y2, (6. $) +. spin selection 1234 Dq/B FIGURE 7 Tanabe-Sugano diagram for d3 electrons. 6 Transition Metal Solid-state lasers 37 rule and the Laporte selection rule. According to the spin selection. ton. Although this is not always true. the lifetime of the upper level may be 6 Transition Metal Solid-state Lasers decomposed into components representing a radiative lifetime and a nonradia-