First, note that X = Omega4/Omega6 as a measure of 'Spectroscopic Quality' is only valid if |U(2)|^2 is zero or very small compared to |U(4)|^2 and |U(6)|^2 so that it can be neglected. This generally tends to be the case, but not always. Some times these quality factors can be dependent on Omega2/Omega6 and Omega4/Omega6. These factors are, in fact, related to the branching ratios.

Be aware that magnetic dipole contribution must be retained also if not negligible and considered in the calculations. Many people forget this and just use the formula above. The original concept of this 'Spectroscopic Quality' parameter was introduced by Kaminskii in 1974 for the 4F3/2 manifold of Nd. For Er on the 4I13/2 manifold it does has a magnetic dipole contribution, unlike Nd on 4F3/2.

Peak cross sections are dependent on the Omega parameters and spectral bandwidth, which are affected by structural changes in the host. The matrix elements affect the cross section too, but don't change with host, however, sometimes |U(4)|^2 > |U(6)|^2 and sometimes |U(4)|^2 < |U(6)|^2, and depends on the transitions. For high intensity one generally likes both Omega4 and Omega6 to be as large as possible, assuming the bandwidth remains relatively constant. With X = Omega4/Omega6, the other conditions would be:

Highest intensity will occur when X is minimum if |U(4)|^2|U(6)|^2

This is generally how it works. Anyway, it's not cookbook science and each transition must be examined carefully for size of |U(2)|^2, |U(4)|^2, and|U(6)|^2 and the size of the magnetic dipole matrix element too. The spectroscopic quality (X) is often used in this way to assess laser potential of a given transition in a rare earth ion.

Thank you very much for your informative answer. In the case of the 4I13/2 to 4I15/2 transition in an Erbium-doped glass what is the most favorable condition? Here the Omega2 is usually large than two others, where |U(4)|^2 ~ 0.1173 and |U(6)|^2 ~ 1.4316 ... Does one need smaller X to obtain higher intensity for this transition? Unfortunately, some authors have written something in contrary to this scenario and made the confusion. I also need a reference to refer in future. I would be thankful by receiving some comments.

I sympathize with your confusion and you are not alone in that. For the Er 4I13/2 to 4I15/2 the |U(2)|^2 = 0.0195 so you can neglect it compared to |U(4)|^2 and |U(6)|^2 and since |U(4)|^2

Thank you very much. Now it is clear.

Thanks for the references as well.

Dear Dr  M.Reza Dousti and Dr Brian M Walsh

Actually both of you  were too expertly in the Laser materials and laser output arena. And I'm too far from your research game and naive. I have did in marine geochemistry REE on  tetrad effect  nature and nephelauxetic phenomena. Seen your research very interesting and sound likely effect to this nature. Could Pm3+ used for the doped to making the laser output become more capability.?

Thank you

Regard's

Zolhizir bin Daud

Promethium (Pm) was investigated as a laser material in the early 1970's by W.F. Krupke and found to be similar to neodymium (Nd) lasers. A Pm solid state laser was eventually demonstrated in the late 1980's. This work was all done at Lawrence Livermore National Laboratory. Special handling was necessary because Pm has no stable isotopes, resulting in radioactivity. Needless to say, there has not been a great deal of interest in Pm as a laser ion for this reason.

Dear Dr. Brian M Walsh and Dr M. Reza

Thank you for the answer. Now it become clearer. Thank you again

Best regard's

Zolhizir bin Daud