Abstract

It was experimentally found that electronic structures of Russell-Saunders manifolds in Nd:YAG depended on the Nd3+-doping concentration (C Nd) and its fabrication process. Both of the bandwidth and the branching ratio in fluorescent transitions in Nd:YAG varied almost linearly depending on C Nd, and a fabrication process has its own diluted limit of the bandwidth and the branching ratio. Also dependences of Stark splitting in Nd:YAG were also observed. Nd3+-doping causes 1.9% and 4.5% reduction in the stimulated emission cross section of Nd:YAG per 1at.% of C Nd at 1.064 μm and 1.319 μm, respectively.

© 2011 OSA

1. Introduction

The ceramic laser technology has become an indispensable technological element in solid-state lasers due to its advantages such as higher fracture toughness [1] and larger crystal size [2]. Recent advances of this technology realized a demonstration of more than 100-kW cw laser oscillation with TEM00 transverse mode by using of YAG ceramics [3]. From the viewpoint of developing high power solid-state lasers, the ceramic laser technology can supply laser media with high concentration of luminous rare-earth ions that have larger ionic radii than replaced ions: such as Nd3+ in YAG. While this advantage enables the higher pump absorption efficiency [4], it brings the quenching of radiative quantum efficiency [5]. Therefore, it is important for optimizing laser oscillators to determine the best doping concentration of luminous ions in each laser host material.

Since Nd:YAG is the most useful solid-state laser medium, there have been many researches for the definition of the figure of merit for Nd3+-doping concentration (C Nd) in Nd:YAG. In these previous works it was assumed that the cross section of optical transitions was invariant. Only quenching of fluorescence lifetimes has been considered as the origin of C Nd-dependence in spectroscopic properties of Nd:YAG by using of Dexter-Förster theory [6,7]. On the contrary, a few works that paid attentions to the change of cross section induced by Nd3+-doping exist [5], where the accordance of the radiative quantum efficiency between a theory and experiments suggest that total line strength of decay process is invariant. Also authors reported that there was a certain dependence of fluorescent profiles in concentrated Nd:YAG [8]. However, all these researches were still processed on the assumption that Einstein's A and B coefficients of optical transitions in Nd:YAG were invariant.

In addition, even though the influences of the fabrication process to spectroscopic properties are often ignored, there are some reports that referred to the manifold splitting variation due to the crystal field variation caused by the elaboration procedure [9]. This fact strongly suggests that there should be differences in spectroscopic properties between single-crystalline laser media and ceramic laser media.

In this work, detailed fluorescent characteristics of Nd:YAG fabricated by different processes were studied by spectroscopic evaluations with high resolution, and we discussed the variation of the stimulated emission cross section in Nd:YAG due to the structural changes of Russell-Saunders manifolds introduced by not only C Nd but also different fabrication processes.

2. Theory and method

The stimulated emission cross section σ(ν) at frequency ν is expressed with Einstein's B coefficient and the line-shape function g(ν) by

σ(ν)=hνcBg(ν),
where h and c are Plank's constant and the speed of light, respectively. Einstein's B coefficient for the optical transition accompanied from an electronic initial state i to a final state f is defined by Fermi's golden rule as
Bif=8π33h2λ|ψf|μλ|ψi|2,
where |ψj and μ are the wave-vector of the electronic state j and the electric dipole moment operator, respectively. λ indicates polarization component. Though C Nd is not included in Eqs. (1) and (2) definitely, perturbation to |ψj can be induced by the modified crystal field V C influenced by C Nd or other factors as crystal point defects. If the expansion V C in terms of the strain ε is defined by
VC=V0+V1ε+V2ε2+,
Stark splitting Vif between |ψi and |ψf is given by
Vif=|ψi|V0|ψiψf|V0|ψf|,
where |ψi and |ψf must belongs to the same Russell-Saunders manifold 2 S +1 LJ. Higher order terms of V C give the spectral line broadening caused by nonlinear interactions between electronic states and lattice vibrations. If Raman effects are dominant for determining the bandwidth Δνif of the transition from |ψi to |ψf, Δνif is given as FWHM in the fluorescence by
Δνif=92π3cν2h2ρ2v10|ψf|V1|ψi|2(2πkΘh)70ΘTx6ex(ex1)2dx,
where k, ρ, v, and Θ are Boltzmann's constant, the density, the velocity of the phonon, and Debye temperature of the crystal, respectively [10]. Since Θ of Nd:YAG is considered to be independent on C Nd [11], |ψf|V1|ψi|2/Δνif becomes a constant from Eq. (5).

By using of the assumption that σ(ν) has a lorentzian spectral profile, the line-shape function gif (ν) of the transition from the state i to f is expressed by

gif(ν)=Δνif2π1(Δνif/2)2+(ννif)2,
where νif is the center frequency of the transition from i to f. By considering Eq. (1) and the relation between fluorescent intensities and σ(ν) [12], the fluorescent intensity Iif(ν) of the transition from level i to f is
Iif(ν)=ηn2(ν)ν4ΔνifBif(Δνif/2)2+(ννif)2,
where η is the efficiency of a measurement defined by the experimental setup. Similarly the fluorescent intensity Ijk(ν) of the transition from manifold j to k is the summation of Iif(ν) as
Ijk(ν)=ijfkηfin2(ν)ν4ΔνifBif(Δνif/2)2+(ννif)2,
where fi is the fractional population of level i in the manifold j. Equation (8) directly gives the branching ratio bif of transition from level i in manifold j to level f in manifold k, and bif is

bif=fiBif/ij,fkfiBif.

The maximum value of the cross section σifm of the transition from level i to level f is the most important value for cavity designs of high power lasers. From Eqs. (1) and (6) σifm is

σifm=2πΔνifhνifcBif.

Δνif, νif, and Bif can be obtained by the least square fitting of experimentally measured fluorescent spectra to Eq. (7), and they give the relative values of the matrix elements of μ, V 0, and V 1 between optical transitions in each sample.

3. Experimental setup

There are two types of transparent YAG ceramics used in this work. Six samples in the first group were fabricated by sintering of carbonated precursor from wet synthesized powder (WS) with C Nd from 0.4 to 4.8at.%, and other four samples were fabricated by sintering with solid-state reaction (SS) between raw sesquioxide powders with C Nd from 1.1 to 5.4at.%. Also twelve YAG single crystals grown by Czochralski method (CZ) were evaluated in this work with C Nd from 0.06 to 1.4at.%. Manufactures of the samples in this work were Konoshima Chemicals Co., Ltd (WS), World-Lab Co., Ltd. (SS), and Scientific materials Co (CZ). These samples had parallel mirror polished surfaces with the thickness of 1 mm. Typical grain sizes of WS and SS ceramics were about several-μm and several tens-μm, respectively.

C Nd of these samples were calibrated by the linearity of the absorption coefficient at 808 nm with the resolution of 0.2 nm measured by a spectrophotometer (U-3500, Hitachi Co.). These samples were considered to have no clustering of Nd3+ from the dependence of the fluorescent lifetime on C Nd [5].

In order to obtain fluorescence spectra of Nd:YAG, samples were pumped by 10-mW radiation with 100Hz repetition rate and 10% duty ratio from a fiber coupled 808-nm laser diode (LIMO40-F400-DL808, LIMO GmbH). Fluorescences were analyzed by a monochromater (TRIAX-550, JOBIN YVON) with less than 0.05-nm resolution, and were detected by linear-InGaAs-array detector (IGA512-1x1, JOBIN YVON). The calibration of the wavelength was done within an error of 0.2 cm−1 by Ne lamp (Pencil style calibration lamp 6032, Oriel), Xe lamp (Pencil style calibration lamp 6033, Oriel) and Hg lamp (Pen-Ray Mercury lamp, UVP). Samples were sandwiched by copper plates of which temperature was maintained at 23°C by a thermo-electric controller (LDT-5948, ILX Lightwave Co.).

In order to evaluate the center wavelengths, the bandwidths, and line intensities of fluorescence, the least square fitting was carried out with typical errors of 0.02 cm−1, 1%, and 2%, respectively. The accuracy of evaluations in this work is summarized in Table 1 , where δA means a reproducible error of A.

Tables Icon

Table 1. The Accuracy of Evaluations in This Work

4. Results

When we compare the spectral shapes of various samples, it can be assumed the invariant line strength as stated in the introduction. Therefore, comparing of spectral profiles normalized by their integral with wavelength should be appropriate. Figure 1(a) shows normalized fluorescent intensities of Nd3+-diluted ceramics (WS) and Nd3+-concentrated ceramics (SS). 0.8at.% WS sample has the spectrum width of 0.7 nm at 1.064 μm, and it is 63% of 1.1 nm that is a well-known value for Nd:YAG [13]. In the case of 1.319 μm shown in Fig. 1(b), observed bandwidth 0.7 nm of 0.4at.% WS samples is 58% of 1.2 nm reported in [13]. Since measured spectral bandwidths depends on C Nd, peak intensities of 5.4at.% SS sample at 1.064-μm and 1.319-μm were decreased 17.2% and 30.9% from above sample, respectively.Even though in the same C Nd conditions, there is a difference in the bandwidth between single crystal and ceramics as shown in Fig. 2(a) . As a result of this difference in line broadenings, 1.8% and 3.2% smaller peak intensities of 1.3at.% Nd:YAG single crystal than ceramics (WS) at 1.064-μm and 1.319-μm, respectively, were detected experimentally. The label of each Stark level that is used in this work is summerized in Fig. 2(b).

 

Fig. 1 Normalized fluorescent intensities of 0.8at.% Nd:YAG ceramics (WS) and 5.4at.% ceramics (SS) at 1.06-μm (a), and 0.4at.% Nd:YAG ceramics (WS) and 5.4at.% ceramics (SS) at 1.32-μm (b).

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Fig. 2 The dependence of the bandwidth of several fluorescence lines of Nd:YAG on C Nd (a), and the label of each Stark level in manifolds 4 F 3/2, 4 I 13/2, and 4 I 11/2 of Nd3+(b). Circle, triangle, and square indicate the evaluated value of the fluorescent bandwidth of CZ, SS, and WS, respectively. Labels R i, X i and Y i indicates the i-th lower Stark level of 4 F 3/2, 4 I 13/2, and 4 I 11/2 manifolds, respectively.

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5. Discussions

5.1 Matrix element of 0th order term of crystal field

Information of Stark-splitting in Russell-Saunders manifolds of 4 F and 4 I in Nd3+-doped laser materials is very important, because laser excitation and oscillation wavelengths coincide to the intervals between these levels. Stark splitting defined by the matrix elements of V 0 with Eq. (4) can be evaluated by the difference of νif.

Stark splitting of 4 F 3/2 in Nd:YAG is shown in Fig. 3(a) . This splitting was defined by the difference between the frequencies of the transition from R i to Y 1, where generally R i, Y i, and X i are used as labels of the i-th lower Stark level of 4 F 3/2, 4 I 11/2, and 4 I 13/2 manifolds, respectively. Though changes of Stark splitting by introducing Nd3+ can be observed with the order of a percent, the existence of the differences in splitting between single crystals and ceramics is quite important because R 1 and R 2 are the laser upper levels of Nd:YAG.

 

Fig. 3 Stark splitting in Nd:YAG. The splitting of 4 F 3/2 is estimated from the transition from R i to Y 1 (a), and similarly the splitting of X 1 and X 2 are calculated from the transition from R i to X 1 and X 2 (b). The right axes indicate the ratio of the splitting variation, where ν min is the minimum value of the measured splitting in each graph. Circle, triangle, and square indicate the evaluated value of the Stark splitting in CZ, SS, and WS, respectively.

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Also Fig. 3(b) shows Stark splitting of 4 I 13/2 in Nd:YAG calculated from the transition from R i to X 1 and X 2, which are the lower laser levels of 1.319-μm oscillation. While the splitting of X 1 and X 2 in single crystals is similar to that of WS ceramics, the difference between WS and SS ceramics was observed. The obtained variation of this splitting among our samples reached to 14%.

The variation of matrix elements of V 0 should imply the change of branching ratio of otical transitions according to Judd-Ofelt optical transition scheme [14,15]. However, changes of νif in fluorescence from Nd:YAG depending on C Nd or fabrication process are negligible because crystal field effects discussed in this section are much smaller than the splitting between 2 S +1 LJ manifolds.

5.2 Matrix element of 1st order term of crystal field

Matrix element of V 1 brings the line broadening of the fluorescence due to Raman interactions as described in Eq. (5), where the ratio of |ψf|V1|ψi|2 among different samples coincides to the ratio of Δνif. Figures 4(a) and 4(b) show the ratio of |ψf|V1|ψi|2 in the transition from 4 F 3/2 to 4 I 11/2, where the differences in |ψf|V1|ψi|2 between fabrication process of CZ, WS, and SS are found. The linear dependence of |ψf|V1|ψi|2 on C Nd is also discovered experimentally.

 

Fig. 4 Ratios of fluorescent bandwidths under the final states Y i to Y 1 in Nd:YAG, which coincide with ratios of |ψf|V1|ψi|2. Initial states are R 1 (a) and R 2 (b). Circle, triangle, and square indicate the evaluated ratios of |ψf|V1|ψi|2 in CZ, SS, and WS, respectively.

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These variations of |ψf|V1|ψi|2 directly relate σifm as in Eq. (10). Therefore, what are discussed in this section give the crucial influence to the laser oscillation characteristics of Nd:YAG. The variation of σifm is summarized in Sect. 5.5.

5.3 Matrix element of dipole moment

As discussed in sect. 5.1 matrix elements of a dipole transition moment in each sample should also be influenced by Nd3+-doping and fabrication processes, and Figs. 5(a) and 5(b) show the ratio of |ψf|μ|ψi|2 of each sample in the transition from 4 F 3/2 to 4 I 11/2. The dependence on fabrication processes of the branching ratio in Nd:YAG was indicated clearly in Fig. 5(a), where branching ratios of single crystals and ceramics are invariant and variant, respectively. On the contrary, dependence of |ψf|μ|ψi|2 on C Nd in 4 F 3/2 to Y 3 without differences between fabrication process is found in Fig. 5(b).

 

Fig. 5 Ratios of |ψf|μ|ψi|2 (Bi - f) from the initial states R 2 to R 1 in Nd:YAG. Final states are Y 1 (a) and Y 3 (b). Circle, triangle, and square indicate the evaluated ratios of |ψf|μ|ψi|2 in CZ, SS, and WS, respectively.

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These differences of |ψf|μ|ψi|2 among samples directly indicate that Einstein's A and B coefficients of one sample can be different from those of another sample. The reason why the variation of the ratios is so different between Figs. 5(a) and 5(b) is not revealed definitely, while it can be due to the different representation of the crystal field in each Stark level.

5.4 Branching ratio

Since Einstein's A and B coefficients are not invariant as discussed above, branching ratios should also be influenced by the amount of doped Nd3+. Figure 6(a) shows the C Nd-dependence of branching ratio in transitions from 4 F 3/2 to 4 I 11/2 of Nd:YAG. Although in other peaks C Nd-dependence was hardly observed, branching ratios of the transition from R 2 to Y 3 (1064.1 nm) and from R 1 to Y 2 (1064.5 nm) transitions have strong C Nd-dependence. This variation in the branching ratio should enhance the reduction of σ em at 1.064-μm due to the line-broadening introduced by Nd3+-doping. On the contrary, the change of branching ratio suppresses the reduction of σ em at 1.319-μm transition as shown in Fig. 6(b).Past spectroscopic evaluations of rare-earth doped laser media by use of the concentration quenching analysis [16] have assumed the invariance of branching ratios. Hence, there is a possibility of errors under designing of advanced laser oscillators by use of the physical properties value that has been believed until nowadays. This possibility is expressed numerically by newly introduced parameter A(λ) in the next section.

 

Fig. 6 C Nd-dependence of branching ratio of Nd:YAG. The main peak of 1.1-μm emission (R 2 to Y 3) and of 1.3-μm emission (R 2 to X 1) have alternate dependences on C Nd. Circle, triangle, and square indicate the evaluated branching ratios of CZ, SS, and WS, respectively.

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5.5 Stimulated emission cross section

The maximum value of the cross section σifm contains Δνif, νif, and Bif that depend on C Nd as expressed in Eq. (10). We can evaluate the relative value of σifm by using of the Δνif, νif, and the relative value of Bif that were experimentally obtained in this work. Figures 7(a) and 7(b) show the estimated variation of σifm of Nd:YAG at 1.064-μm and 1.319-μm, respectively. The stimulated emission cross section σ em(λ) can be approximated by

σem(λ,CNd)=[1A(λ)CNd]σem0(λ),
where λ is wavelength, σ em 0(λ) is a diluted limit of σ em(λ), and A(λ) is a fitting parameter. While σ em 0(λ) depends on synthesizing processes of samples, A(λ) is hardly influenced by their fabrication processes. At 1.064-μm and 1.319-μm, A(λ) is 0.019/at.% and 0.045/at.%, respectively.

 

Fig. 7 The estimated variation of σifm of Nd:YAG at 1.064-μm (a) and 1.319-μm (b). Circle, triangle, and square indicate σem/σem0 of CZ, SS, and WS, respectively.

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When we estimate laser performances with highly concentrated Nd:YAG, it is important to consider not σ em but the product of σ em and fluorescent lifetime τ. In the case of 5at.% Nd:YAG, the radiative quantum efficiency is 21% [5]. Therefore, σ em τ of 5at.% Nd:YAG is decreased to 19% and 16% at 1.064-μm and 1.319-μm, respectively.

6. Conclusion

We experimentally confirmed that wave-vectors of electronic states in Russell-Saunders manifolds 4 F and 4 I in Nd:YAG depended on both of C Nd and its fabrication process. An error of C Nd in this work was below 0.4%. This dependence should be due to the perturbation introduced by the crystal field V C that is influenced by C Nd or other factors such as crystal defects. These variations of electronic states cause following changes in spectroscopic characteristics of Nd:YAG:

  • 1. Bandwidth of each fluorescence peak in Nd:YAG varies linearly depending on C Nd.
  • 2. Bandwidth of ceramics (SS) is narrower than other samples with same C Nd.
  • 3. Branching ratio of some transitions in Nd:YAG depends on C Nd severely.

In this work at the utmost 16% and 29% reduction of σ em at 1064 μm and 1319 μm were experimentally found respectively, while the deviation in evaluations of σ em was below 3%. Therefore, it is necessary for constructing laser oscillators with highly concentrated Nd:YAG to consider not only degradation of radiative quantum efficiency for the sake of concentration quenching but also the reduction of laser gain due to decreased σ em by Nd3+-doping.

Since the temperature was fixed at 23°C in this work, the evaluation of temperature dependence should be a future work in order to validate the presented results. Also the detailed studies on the branching ratio between Russell-Saunders manifolds will give interesting characteristics for physics of the solid-state laser materials. Above studies are also very useful for future detailed analyses of absorption bands in order to improve the diode-pumped solid-state lasers.

It is important for improving the material performance to clarify the relationship between results of this work and Nd3+-local environments such as defect, disorder, impurities, homogeneity, and etc. In order to solve this problem, the development of the detection method for every distinct environment of Nd3+ in laser media must be required. Deep theoretical discussions based on the evaluated Nd3+-local environments will bring the complete design rule of Nd3+-doped laser media.

Acknowledgments

This work was partially supported by Genesis Research Institute, and by the Special Coordination Funds for Promoting Science and Technology of the Ministry of Education, Culture, Sports, Science, and Technology of Japan.

References and links

1. A. A. Kaminskii, M. S. Akchurin, R. V. Gainutdinov, K. Takaichi, A. Shirakava, H. Yagi, T. Yanagitani, and K. Ueda, “Microhardness and fracture toughness of Y2O3- and Y3Al5O12-based nanocrystalline laser ceramics,” Crystallogr. Rep. 50(5), 869–873 (2005). [CrossRef]  

2. R. M. Yamamoto, B. S. Bhachu, K. P. Cutter, S. N. Fochs, S. A. Letts, C. W. Parks, M. D. Rotter, and T. F. Soules, “The use of large transparent ceramics in a high powered, diode pumped solid-state laser,” Proceedings of OSA Topical meeting on Advanced Solid-State Photonics 2008, WC5, Nara, Japan (Jan. 2008).

3. S. J. McNaught, H. Komine, S. B. Weiss, R. Simpson, A. M. F. Johnson, J. Machan, C. P. Asman, M. Weber, G. C. Jones, M. M. Valley, A. Jankevics, D. Burchman, M. McClellan, J. Sollee, J. Marmo, and H. Injeyan, “100 kW coherently combined slab MOPAs,” in Proceedings of Conference on Quantum Electronics and Laser Science Conference on Lasers and Electro-Optics, CLEO/QELS, CThA1, Baltimore, MA, USA (2009).

4. I. Shoji, S. Kurimura, Y. Sato, T. Taira, A. Ikesue, and K. Yoshida, “Optical properties and laser characteristics of highly Nd3+-doped Y3Al5O12 ceramics,” Appl. Phys. Lett. 77(7), 939–941 (2000). [CrossRef]  

5. V. Lupei, A. Lupei, S. Georgescu, T. Taira, Y. Sato, and A. Ikesue, “The effect of Nd concentration on the spectroscopic and emission decay properties of highly doped Nd:YAG ceramics,” Phys. Rev. B 64(9), 092102 (2001). [CrossRef]  

6. D. L. Dexter, “A theory of sensitized luminescence in solids,” J. Chem. Phys. 21(5), 836–850 (1953). [CrossRef]  

7. T. Förster, “Intermolecular energy migration and fluorescence,” Ann. Phys. 2, 55–75 (1948).

8. Y. Sato, T. Taira, and A. Ikesue, “A study on influences of Nd3+-doping concentration upon spectroscopic properties of Nd:Y3Al5O12 ceramics,” in Proceedings of OSA Topical meeting on Advanced Solid-State Photonics 2009, WB18, Denver, CO, USA (Feb. 2009).

9. F. S. Ermeneux, C. Goutaudier, R. Moncorge, M. T. Cohen-Adad, M. Bettinelli, and E. Cavalli, “Comparative optical characterization of various Nd3+:YVO4 single crystals,” Opt. Mater. 13(2), 193–204 (1999). [CrossRef]  

10. T. Kushida, “Linewidths and thermal shifts of spectral lines in neodymium-doped yttrium aluminum garnet and calcium fluorophosphate,” Phys. Rev. 185(2), 500–508 (1969). [CrossRef]  

11. Y. Sato, J. Akiyama, and T. Taira, “Effects of rare-earth doping on thermal conductivity in Y3Al5O12 crystals,” Opt. Mater. 31(5), 720–724 (2009). [CrossRef]  

12. T. Kushida, H. M. Marcos, and J. E. Geusic, “Laser transition cross section and fluorescence branching ratio for Nd3+ in yttrium aluminum garnet,” Phys. Rev. 167(2), 289–291 (1968). [CrossRef]  

13. T. Taira, “RE3+-ion-doped YAG ceramic lasers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 798–809 (2007). [CrossRef]  

14. B. R. Judd, “Optical absorption intensities of rare-earth ions,” Phys. Rev. 127(3), 750–761 (1962). [CrossRef]  

15. G. S. Ofelt, “Intensities of crystal spectra of rare-earth ions,” J. Chem. Phys. 37(3), 511–520 (1962). [CrossRef]  

16. A. Blumen and J. Manz, “On the concentration and time dependence of the energy transfer to randomly distributed acceptors,” J. Chem. Phys. 71(11), 4694–4702 (1979). [CrossRef]  

References

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  1. A. A. Kaminskii, M. S. Akchurin, R. V. Gainutdinov, K. Takaichi, A. Shirakava, H. Yagi, T. Yanagitani, and K. Ueda, “Microhardness and fracture toughness of Y2O3- and Y3Al5O12-based nanocrystalline laser ceramics,” Crystallogr. Rep. 50(5), 869–873 (2005).
    [CrossRef]
  2. R. M. Yamamoto, B. S. Bhachu, K. P. Cutter, S. N. Fochs, S. A. Letts, C. W. Parks, M. D. Rotter, and T. F. Soules, “The use of large transparent ceramics in a high powered, diode pumped solid-state laser,” Proceedings of OSA Topical meeting on Advanced Solid-State Photonics 2008, WC5, Nara, Japan (Jan. 2008).
  3. S. J. McNaught, H. Komine, S. B. Weiss, R. Simpson, A. M. F. Johnson, J. Machan, C. P. Asman, M. Weber, G. C. Jones, M. M. Valley, A. Jankevics, D. Burchman, M. McClellan, J. Sollee, J. Marmo, and H. Injeyan, “100 kW coherently combined slab MOPAs,” in Proceedings of Conference on Quantum Electronics and Laser Science Conference on Lasers and Electro-Optics, CLEO/QELS, CThA1, Baltimore, MA, USA (2009).
  4. I. Shoji, S. Kurimura, Y. Sato, T. Taira, A. Ikesue, and K. Yoshida, “Optical properties and laser characteristics of highly Nd3+-doped Y3Al5O12 ceramics,” Appl. Phys. Lett. 77(7), 939–941 (2000).
    [CrossRef]
  5. V. Lupei, A. Lupei, S. Georgescu, T. Taira, Y. Sato, and A. Ikesue, “The effect of Nd concentration on the spectroscopic and emission decay properties of highly doped Nd:YAG ceramics,” Phys. Rev. B 64(9), 092102 (2001).
    [CrossRef]
  6. D. L. Dexter, “A theory of sensitized luminescence in solids,” J. Chem. Phys. 21(5), 836–850 (1953).
    [CrossRef]
  7. T. Förster, “Intermolecular energy migration and fluorescence,” Ann. Phys. 2, 55–75 (1948).
  8. Y. Sato, T. Taira, and A. Ikesue, “A study on influences of Nd3+-doping concentration upon spectroscopic properties of Nd:Y3Al5O12 ceramics,” in Proceedings of OSA Topical meeting on Advanced Solid-State Photonics 2009, WB18, Denver, CO, USA (Feb. 2009).
  9. F. S. Ermeneux, C. Goutaudier, R. Moncorge, M. T. Cohen-Adad, M. Bettinelli, and E. Cavalli, “Comparative optical characterization of various Nd3+:YVO4 single crystals,” Opt. Mater. 13(2), 193–204 (1999).
    [CrossRef]
  10. T. Kushida, “Linewidths and thermal shifts of spectral lines in neodymium-doped yttrium aluminum garnet and calcium fluorophosphate,” Phys. Rev. 185(2), 500–508 (1969).
    [CrossRef]
  11. Y. Sato, J. Akiyama, and T. Taira, “Effects of rare-earth doping on thermal conductivity in Y3Al5O12 crystals,” Opt. Mater. 31(5), 720–724 (2009).
    [CrossRef]
  12. T. Kushida, H. M. Marcos, and J. E. Geusic, “Laser transition cross section and fluorescence branching ratio for Nd3+ in yttrium aluminum garnet,” Phys. Rev. 167(2), 289–291 (1968).
    [CrossRef]
  13. T. Taira, “RE3+-ion-doped YAG ceramic lasers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 798–809 (2007).
    [CrossRef]
  14. B. R. Judd, “Optical absorption intensities of rare-earth ions,” Phys. Rev. 127(3), 750–761 (1962).
    [CrossRef]
  15. G. S. Ofelt, “Intensities of crystal spectra of rare-earth ions,” J. Chem. Phys. 37(3), 511–520 (1962).
    [CrossRef]
  16. A. Blumen and J. Manz, “On the concentration and time dependence of the energy transfer to randomly distributed acceptors,” J. Chem. Phys. 71(11), 4694–4702 (1979).
    [CrossRef]

2009

Y. Sato, J. Akiyama, and T. Taira, “Effects of rare-earth doping on thermal conductivity in Y3Al5O12 crystals,” Opt. Mater. 31(5), 720–724 (2009).
[CrossRef]

2007

T. Taira, “RE3+-ion-doped YAG ceramic lasers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 798–809 (2007).
[CrossRef]

2005

A. A. Kaminskii, M. S. Akchurin, R. V. Gainutdinov, K. Takaichi, A. Shirakava, H. Yagi, T. Yanagitani, and K. Ueda, “Microhardness and fracture toughness of Y2O3- and Y3Al5O12-based nanocrystalline laser ceramics,” Crystallogr. Rep. 50(5), 869–873 (2005).
[CrossRef]

2001

V. Lupei, A. Lupei, S. Georgescu, T. Taira, Y. Sato, and A. Ikesue, “The effect of Nd concentration on the spectroscopic and emission decay properties of highly doped Nd:YAG ceramics,” Phys. Rev. B 64(9), 092102 (2001).
[CrossRef]

2000

I. Shoji, S. Kurimura, Y. Sato, T. Taira, A. Ikesue, and K. Yoshida, “Optical properties and laser characteristics of highly Nd3+-doped Y3Al5O12 ceramics,” Appl. Phys. Lett. 77(7), 939–941 (2000).
[CrossRef]

1999

F. S. Ermeneux, C. Goutaudier, R. Moncorge, M. T. Cohen-Adad, M. Bettinelli, and E. Cavalli, “Comparative optical characterization of various Nd3+:YVO4 single crystals,” Opt. Mater. 13(2), 193–204 (1999).
[CrossRef]

1979

A. Blumen and J. Manz, “On the concentration and time dependence of the energy transfer to randomly distributed acceptors,” J. Chem. Phys. 71(11), 4694–4702 (1979).
[CrossRef]

1969

T. Kushida, “Linewidths and thermal shifts of spectral lines in neodymium-doped yttrium aluminum garnet and calcium fluorophosphate,” Phys. Rev. 185(2), 500–508 (1969).
[CrossRef]

1968

T. Kushida, H. M. Marcos, and J. E. Geusic, “Laser transition cross section and fluorescence branching ratio for Nd3+ in yttrium aluminum garnet,” Phys. Rev. 167(2), 289–291 (1968).
[CrossRef]

1962

B. R. Judd, “Optical absorption intensities of rare-earth ions,” Phys. Rev. 127(3), 750–761 (1962).
[CrossRef]

G. S. Ofelt, “Intensities of crystal spectra of rare-earth ions,” J. Chem. Phys. 37(3), 511–520 (1962).
[CrossRef]

1953

D. L. Dexter, “A theory of sensitized luminescence in solids,” J. Chem. Phys. 21(5), 836–850 (1953).
[CrossRef]

1948

T. Förster, “Intermolecular energy migration and fluorescence,” Ann. Phys. 2, 55–75 (1948).

Akchurin, M. S.

A. A. Kaminskii, M. S. Akchurin, R. V. Gainutdinov, K. Takaichi, A. Shirakava, H. Yagi, T. Yanagitani, and K. Ueda, “Microhardness and fracture toughness of Y2O3- and Y3Al5O12-based nanocrystalline laser ceramics,” Crystallogr. Rep. 50(5), 869–873 (2005).
[CrossRef]

Akiyama, J.

Y. Sato, J. Akiyama, and T. Taira, “Effects of rare-earth doping on thermal conductivity in Y3Al5O12 crystals,” Opt. Mater. 31(5), 720–724 (2009).
[CrossRef]

Bettinelli, M.

F. S. Ermeneux, C. Goutaudier, R. Moncorge, M. T. Cohen-Adad, M. Bettinelli, and E. Cavalli, “Comparative optical characterization of various Nd3+:YVO4 single crystals,” Opt. Mater. 13(2), 193–204 (1999).
[CrossRef]

Blumen, A.

A. Blumen and J. Manz, “On the concentration and time dependence of the energy transfer to randomly distributed acceptors,” J. Chem. Phys. 71(11), 4694–4702 (1979).
[CrossRef]

Cavalli, E.

F. S. Ermeneux, C. Goutaudier, R. Moncorge, M. T. Cohen-Adad, M. Bettinelli, and E. Cavalli, “Comparative optical characterization of various Nd3+:YVO4 single crystals,” Opt. Mater. 13(2), 193–204 (1999).
[CrossRef]

Cohen-Adad, M. T.

F. S. Ermeneux, C. Goutaudier, R. Moncorge, M. T. Cohen-Adad, M. Bettinelli, and E. Cavalli, “Comparative optical characterization of various Nd3+:YVO4 single crystals,” Opt. Mater. 13(2), 193–204 (1999).
[CrossRef]

Dexter, D. L.

D. L. Dexter, “A theory of sensitized luminescence in solids,” J. Chem. Phys. 21(5), 836–850 (1953).
[CrossRef]

Ermeneux, F. S.

F. S. Ermeneux, C. Goutaudier, R. Moncorge, M. T. Cohen-Adad, M. Bettinelli, and E. Cavalli, “Comparative optical characterization of various Nd3+:YVO4 single crystals,” Opt. Mater. 13(2), 193–204 (1999).
[CrossRef]

Förster, T.

T. Förster, “Intermolecular energy migration and fluorescence,” Ann. Phys. 2, 55–75 (1948).

Gainutdinov, R. V.

A. A. Kaminskii, M. S. Akchurin, R. V. Gainutdinov, K. Takaichi, A. Shirakava, H. Yagi, T. Yanagitani, and K. Ueda, “Microhardness and fracture toughness of Y2O3- and Y3Al5O12-based nanocrystalline laser ceramics,” Crystallogr. Rep. 50(5), 869–873 (2005).
[CrossRef]

Georgescu, S.

V. Lupei, A. Lupei, S. Georgescu, T. Taira, Y. Sato, and A. Ikesue, “The effect of Nd concentration on the spectroscopic and emission decay properties of highly doped Nd:YAG ceramics,” Phys. Rev. B 64(9), 092102 (2001).
[CrossRef]

Geusic, J. E.

T. Kushida, H. M. Marcos, and J. E. Geusic, “Laser transition cross section and fluorescence branching ratio for Nd3+ in yttrium aluminum garnet,” Phys. Rev. 167(2), 289–291 (1968).
[CrossRef]

Goutaudier, C.

F. S. Ermeneux, C. Goutaudier, R. Moncorge, M. T. Cohen-Adad, M. Bettinelli, and E. Cavalli, “Comparative optical characterization of various Nd3+:YVO4 single crystals,” Opt. Mater. 13(2), 193–204 (1999).
[CrossRef]

Ikesue, A.

V. Lupei, A. Lupei, S. Georgescu, T. Taira, Y. Sato, and A. Ikesue, “The effect of Nd concentration on the spectroscopic and emission decay properties of highly doped Nd:YAG ceramics,” Phys. Rev. B 64(9), 092102 (2001).
[CrossRef]

I. Shoji, S. Kurimura, Y. Sato, T. Taira, A. Ikesue, and K. Yoshida, “Optical properties and laser characteristics of highly Nd3+-doped Y3Al5O12 ceramics,” Appl. Phys. Lett. 77(7), 939–941 (2000).
[CrossRef]

Judd, B. R.

B. R. Judd, “Optical absorption intensities of rare-earth ions,” Phys. Rev. 127(3), 750–761 (1962).
[CrossRef]

Kaminskii, A. A.

A. A. Kaminskii, M. S. Akchurin, R. V. Gainutdinov, K. Takaichi, A. Shirakava, H. Yagi, T. Yanagitani, and K. Ueda, “Microhardness and fracture toughness of Y2O3- and Y3Al5O12-based nanocrystalline laser ceramics,” Crystallogr. Rep. 50(5), 869–873 (2005).
[CrossRef]

Kurimura, S.

I. Shoji, S. Kurimura, Y. Sato, T. Taira, A. Ikesue, and K. Yoshida, “Optical properties and laser characteristics of highly Nd3+-doped Y3Al5O12 ceramics,” Appl. Phys. Lett. 77(7), 939–941 (2000).
[CrossRef]

Kushida, T.

T. Kushida, “Linewidths and thermal shifts of spectral lines in neodymium-doped yttrium aluminum garnet and calcium fluorophosphate,” Phys. Rev. 185(2), 500–508 (1969).
[CrossRef]

T. Kushida, H. M. Marcos, and J. E. Geusic, “Laser transition cross section and fluorescence branching ratio for Nd3+ in yttrium aluminum garnet,” Phys. Rev. 167(2), 289–291 (1968).
[CrossRef]

Lupei, A.

V. Lupei, A. Lupei, S. Georgescu, T. Taira, Y. Sato, and A. Ikesue, “The effect of Nd concentration on the spectroscopic and emission decay properties of highly doped Nd:YAG ceramics,” Phys. Rev. B 64(9), 092102 (2001).
[CrossRef]

Lupei, V.

V. Lupei, A. Lupei, S. Georgescu, T. Taira, Y. Sato, and A. Ikesue, “The effect of Nd concentration on the spectroscopic and emission decay properties of highly doped Nd:YAG ceramics,” Phys. Rev. B 64(9), 092102 (2001).
[CrossRef]

Manz, J.

A. Blumen and J. Manz, “On the concentration and time dependence of the energy transfer to randomly distributed acceptors,” J. Chem. Phys. 71(11), 4694–4702 (1979).
[CrossRef]

Marcos, H. M.

T. Kushida, H. M. Marcos, and J. E. Geusic, “Laser transition cross section and fluorescence branching ratio for Nd3+ in yttrium aluminum garnet,” Phys. Rev. 167(2), 289–291 (1968).
[CrossRef]

Moncorge, R.

F. S. Ermeneux, C. Goutaudier, R. Moncorge, M. T. Cohen-Adad, M. Bettinelli, and E. Cavalli, “Comparative optical characterization of various Nd3+:YVO4 single crystals,” Opt. Mater. 13(2), 193–204 (1999).
[CrossRef]

Ofelt, G. S.

G. S. Ofelt, “Intensities of crystal spectra of rare-earth ions,” J. Chem. Phys. 37(3), 511–520 (1962).
[CrossRef]

Sato, Y.

Y. Sato, J. Akiyama, and T. Taira, “Effects of rare-earth doping on thermal conductivity in Y3Al5O12 crystals,” Opt. Mater. 31(5), 720–724 (2009).
[CrossRef]

V. Lupei, A. Lupei, S. Georgescu, T. Taira, Y. Sato, and A. Ikesue, “The effect of Nd concentration on the spectroscopic and emission decay properties of highly doped Nd:YAG ceramics,” Phys. Rev. B 64(9), 092102 (2001).
[CrossRef]

I. Shoji, S. Kurimura, Y. Sato, T. Taira, A. Ikesue, and K. Yoshida, “Optical properties and laser characteristics of highly Nd3+-doped Y3Al5O12 ceramics,” Appl. Phys. Lett. 77(7), 939–941 (2000).
[CrossRef]

Shirakava, A.

A. A. Kaminskii, M. S. Akchurin, R. V. Gainutdinov, K. Takaichi, A. Shirakava, H. Yagi, T. Yanagitani, and K. Ueda, “Microhardness and fracture toughness of Y2O3- and Y3Al5O12-based nanocrystalline laser ceramics,” Crystallogr. Rep. 50(5), 869–873 (2005).
[CrossRef]

Shoji, I.

I. Shoji, S. Kurimura, Y. Sato, T. Taira, A. Ikesue, and K. Yoshida, “Optical properties and laser characteristics of highly Nd3+-doped Y3Al5O12 ceramics,” Appl. Phys. Lett. 77(7), 939–941 (2000).
[CrossRef]

Taira, T.

Y. Sato, J. Akiyama, and T. Taira, “Effects of rare-earth doping on thermal conductivity in Y3Al5O12 crystals,” Opt. Mater. 31(5), 720–724 (2009).
[CrossRef]

T. Taira, “RE3+-ion-doped YAG ceramic lasers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 798–809 (2007).
[CrossRef]

V. Lupei, A. Lupei, S. Georgescu, T. Taira, Y. Sato, and A. Ikesue, “The effect of Nd concentration on the spectroscopic and emission decay properties of highly doped Nd:YAG ceramics,” Phys. Rev. B 64(9), 092102 (2001).
[CrossRef]

I. Shoji, S. Kurimura, Y. Sato, T. Taira, A. Ikesue, and K. Yoshida, “Optical properties and laser characteristics of highly Nd3+-doped Y3Al5O12 ceramics,” Appl. Phys. Lett. 77(7), 939–941 (2000).
[CrossRef]

Takaichi, K.

A. A. Kaminskii, M. S. Akchurin, R. V. Gainutdinov, K. Takaichi, A. Shirakava, H. Yagi, T. Yanagitani, and K. Ueda, “Microhardness and fracture toughness of Y2O3- and Y3Al5O12-based nanocrystalline laser ceramics,” Crystallogr. Rep. 50(5), 869–873 (2005).
[CrossRef]

Ueda, K.

A. A. Kaminskii, M. S. Akchurin, R. V. Gainutdinov, K. Takaichi, A. Shirakava, H. Yagi, T. Yanagitani, and K. Ueda, “Microhardness and fracture toughness of Y2O3- and Y3Al5O12-based nanocrystalline laser ceramics,” Crystallogr. Rep. 50(5), 869–873 (2005).
[CrossRef]

Yagi, H.

A. A. Kaminskii, M. S. Akchurin, R. V. Gainutdinov, K. Takaichi, A. Shirakava, H. Yagi, T. Yanagitani, and K. Ueda, “Microhardness and fracture toughness of Y2O3- and Y3Al5O12-based nanocrystalline laser ceramics,” Crystallogr. Rep. 50(5), 869–873 (2005).
[CrossRef]

Yanagitani, T.

A. A. Kaminskii, M. S. Akchurin, R. V. Gainutdinov, K. Takaichi, A. Shirakava, H. Yagi, T. Yanagitani, and K. Ueda, “Microhardness and fracture toughness of Y2O3- and Y3Al5O12-based nanocrystalline laser ceramics,” Crystallogr. Rep. 50(5), 869–873 (2005).
[CrossRef]

Yoshida, K.

I. Shoji, S. Kurimura, Y. Sato, T. Taira, A. Ikesue, and K. Yoshida, “Optical properties and laser characteristics of highly Nd3+-doped Y3Al5O12 ceramics,” Appl. Phys. Lett. 77(7), 939–941 (2000).
[CrossRef]

Ann. Phys.

T. Förster, “Intermolecular energy migration and fluorescence,” Ann. Phys. 2, 55–75 (1948).

Appl. Phys. Lett.

I. Shoji, S. Kurimura, Y. Sato, T. Taira, A. Ikesue, and K. Yoshida, “Optical properties and laser characteristics of highly Nd3+-doped Y3Al5O12 ceramics,” Appl. Phys. Lett. 77(7), 939–941 (2000).
[CrossRef]

Crystallogr. Rep.

A. A. Kaminskii, M. S. Akchurin, R. V. Gainutdinov, K. Takaichi, A. Shirakava, H. Yagi, T. Yanagitani, and K. Ueda, “Microhardness and fracture toughness of Y2O3- and Y3Al5O12-based nanocrystalline laser ceramics,” Crystallogr. Rep. 50(5), 869–873 (2005).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

T. Taira, “RE3+-ion-doped YAG ceramic lasers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 798–809 (2007).
[CrossRef]

J. Chem. Phys.

G. S. Ofelt, “Intensities of crystal spectra of rare-earth ions,” J. Chem. Phys. 37(3), 511–520 (1962).
[CrossRef]

A. Blumen and J. Manz, “On the concentration and time dependence of the energy transfer to randomly distributed acceptors,” J. Chem. Phys. 71(11), 4694–4702 (1979).
[CrossRef]

D. L. Dexter, “A theory of sensitized luminescence in solids,” J. Chem. Phys. 21(5), 836–850 (1953).
[CrossRef]

Opt. Mater.

Y. Sato, J. Akiyama, and T. Taira, “Effects of rare-earth doping on thermal conductivity in Y3Al5O12 crystals,” Opt. Mater. 31(5), 720–724 (2009).
[CrossRef]

F. S. Ermeneux, C. Goutaudier, R. Moncorge, M. T. Cohen-Adad, M. Bettinelli, and E. Cavalli, “Comparative optical characterization of various Nd3+:YVO4 single crystals,” Opt. Mater. 13(2), 193–204 (1999).
[CrossRef]

Phys. Rev.

T. Kushida, “Linewidths and thermal shifts of spectral lines in neodymium-doped yttrium aluminum garnet and calcium fluorophosphate,” Phys. Rev. 185(2), 500–508 (1969).
[CrossRef]

T. Kushida, H. M. Marcos, and J. E. Geusic, “Laser transition cross section and fluorescence branching ratio for Nd3+ in yttrium aluminum garnet,” Phys. Rev. 167(2), 289–291 (1968).
[CrossRef]

B. R. Judd, “Optical absorption intensities of rare-earth ions,” Phys. Rev. 127(3), 750–761 (1962).
[CrossRef]

Phys. Rev. B

V. Lupei, A. Lupei, S. Georgescu, T. Taira, Y. Sato, and A. Ikesue, “The effect of Nd concentration on the spectroscopic and emission decay properties of highly doped Nd:YAG ceramics,” Phys. Rev. B 64(9), 092102 (2001).
[CrossRef]

Other

R. M. Yamamoto, B. S. Bhachu, K. P. Cutter, S. N. Fochs, S. A. Letts, C. W. Parks, M. D. Rotter, and T. F. Soules, “The use of large transparent ceramics in a high powered, diode pumped solid-state laser,” Proceedings of OSA Topical meeting on Advanced Solid-State Photonics 2008, WC5, Nara, Japan (Jan. 2008).

S. J. McNaught, H. Komine, S. B. Weiss, R. Simpson, A. M. F. Johnson, J. Machan, C. P. Asman, M. Weber, G. C. Jones, M. M. Valley, A. Jankevics, D. Burchman, M. McClellan, J. Sollee, J. Marmo, and H. Injeyan, “100 kW coherently combined slab MOPAs,” in Proceedings of Conference on Quantum Electronics and Laser Science Conference on Lasers and Electro-Optics, CLEO/QELS, CThA1, Baltimore, MA, USA (2009).

Y. Sato, T. Taira, and A. Ikesue, “A study on influences of Nd3+-doping concentration upon spectroscopic properties of Nd:Y3Al5O12 ceramics,” in Proceedings of OSA Topical meeting on Advanced Solid-State Photonics 2009, WB18, Denver, CO, USA (Feb. 2009).

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Figures (7)

Fig. 1
Fig. 1

Normalized fluorescent intensities of 0.8at.% Nd:YAG ceramics (WS) and 5.4at.% ceramics (SS) at 1.06-μm (a), and 0.4at.% Nd:YAG ceramics (WS) and 5.4at.% ceramics (SS) at 1.32-μm (b).

Fig. 2
Fig. 2

The dependence of the bandwidth of several fluorescence lines of Nd:YAG on C Nd (a), and the label of each Stark level in manifolds 4 F 3/2, 4 I 13/2, and 4 I 11/2 of Nd3+(b). Circle, triangle, and square indicate the evaluated value of the fluorescent bandwidth of CZ, SS, and WS, respectively. Labels R i, X i and Y i indicates the i-th lower Stark level of 4 F 3/2, 4 I 13/2, and 4 I 11/2 manifolds, respectively.

Fig. 3
Fig. 3

Stark splitting in Nd:YAG. The splitting of 4 F 3/2 is estimated from the transition from R i to Y 1 (a), and similarly the splitting of X 1 and X 2 are calculated from the transition from R i to X 1 and X 2 (b). The right axes indicate the ratio of the splitting variation, where ν min is the minimum value of the measured splitting in each graph. Circle, triangle, and square indicate the evaluated value of the Stark splitting in CZ, SS, and WS, respectively.

Fig. 4
Fig. 4

Ratios of fluorescent bandwidths under the final states Y i to Y 1 in Nd:YAG, which coincide with ratios of | ψ f | V 1 | ψ i | 2 . Initial states are R 1 (a) and R 2 (b). Circle, triangle, and square indicate the evaluated ratios of | ψ f | V 1 | ψ i | 2 in CZ, SS, and WS, respectively.

Fig. 5
Fig. 5

Ratios of | ψ f | μ | ψ i | 2 (Bi - f ) from the initial states R 2 to R 1 in Nd:YAG. Final states are Y 1 (a) and Y 3 (b). Circle, triangle, and square indicate the evaluated ratios of | ψ f | μ | ψ i | 2 in CZ, SS, and WS, respectively.

Fig. 6
Fig. 6

C Nd-dependence of branching ratio of Nd:YAG. The main peak of 1.1-μm emission (R 2 to Y 3) and of 1.3-μm emission (R 2 to X 1) have alternate dependences on C Nd. Circle, triangle, and square indicate the evaluated branching ratios of CZ, SS, and WS, respectively.

Fig. 7
Fig. 7

The estimated variation of σif m of Nd:YAG at 1.064-μm (a) and 1.319-μm (b). Circle, triangle, and square indicate σ em / σ em 0 of CZ, SS, and WS, respectively.

Tables (1)

Tables Icon

Table 1 The Accuracy of Evaluations in This Work

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

σ ( ν ) = h ν c B g ( ν ) ,
B i f = 8 π 3 3 h 2 λ | ψ f | μ λ | ψ i | 2 ,
V C = V 0 + V 1 ε + V 2 ε 2 + ,
V i f = | ψ i | V 0 | ψ i ψ f | V 0 | ψ f | ,
Δ ν i f = 9 2 π 3 c ν 2 h 2 ρ 2 v 10 | ψ f | V 1 | ψ i | 2 ( 2 π k Θ h ) 7 0 Θ T x 6 e x ( e x 1 ) 2 d x ,
g i f ( ν ) = Δ ν i f 2 π 1 ( Δ ν i f / 2 ) 2 + ( ν ν i f ) 2 ,
I i f ( ν ) = η n 2 ( ν ) ν 4 Δ ν i f B i f ( Δ ν i f / 2 ) 2 + ( ν ν i f ) 2 ,
I j k ( ν ) = i j f k η f i n 2 ( ν ) ν 4 Δ ν i f B i f ( Δ ν i f / 2 ) 2 + ( ν ν i f ) 2 ,
b i f = f i B i f / i j , f k f i B i f .
σ i f m = 2 π Δ ν i f h ν i f c B i f .
σ em ( λ , C Nd ) = [ 1 A ( λ ) C Nd ] σ em 0 ( λ ) ,

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