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Doping Concentration

Author: the photonics expert

Definition: the concentration of some dopant, e.g. of laser-active ions in a laser gain medium

Categories: article belongs to category optical materials optical materials, article belongs to category laser devices and laser physics laser devices and laser physics

Units: %, m−3, ppm

Formula symbol: <$N_\textrm{dop}$>

DOI: 10.61835/0fd   Cite the article: BibTex plain textHTML

An important parameter of a laser gain medium such as a rare-earth-doped laser crystal or glass (possibly in the form of a fiber), is the doping concentration. The doping concentration can be quantitatively specified in different ways:

  • the molar (atomic) percentage of the dopant (“at. %” or “% at.”), also often given in molar ppm (parts per million)
  • the percentage by weight (more precisely: by mass) of the dopant, also often given in ppm wt. (parts per million with respect to weight)
  • the number density <$N$> of the laser-active ions, i.e., the number of ions per cubic meter or cubic centimeter

These specifications and the relations between them are discussed more in detail in the following.

Number Density

The number density <$N$> (in units of m−3) is useful for many calculations. In particular, combined with cross-section data it makes it easy to calculate absorption and gain coefficients (see below). It is also a type of specification which leaves no room for ambiguities.

Molar Percentage

As an example, we consider laser crystal made of Nd:YAG. While undoped YAG (yttrium aluminum garnet, Y3Al5O12) would simply be transparent to the laser radiation, one can obtain laser gain by replacing some of the yttrium (Y3+) ions with laser-active Nd3+ ions. As those have about the same size and the same charge state, they nicely fit into the YAG lattice.

The molar (or atomic) percentage of doping is the fraction of yttrium (Y3+) ions which have been replaced with Nd3+ ions. It is important to take into account that each garnet unit Y3Al5O12 contains three yttrium ions, so that in principle up to three neodymium ions could be incorporated per unit cell. For the conversion between molar percentage and number density, the volume of one Y3Al5O12 unit needs to be known, or alternatively the mass density (here: 4.55 g/cm3), the chemical formula and the atomic masses. The number density of neodymium ions for a 1 at. % doping level can be calculated as

$${N_{{\rm{Nd:YAG, }}1\% }} = \frac{{3 \cdot 4.55\,{\rm{g/c}}{{\rm{m}}^{\rm{3}}}}}{{\left( {{\rm{3}} \cdot {\rm{88}}.9 + 5 \cdot {\rm{27}}{\rm{.0}} + {\rm{12}} \cdot {\rm{16}}{\rm{.0}}} \right) \cdot 1.66 \cdot {{10}^{ - 24}}{\rm{g}}}} \cdot 0.01 = 1.38 \cdot {10^{20}}{\rm{/c}}{{\rm{m}}^{\rm{3}}}$$

The denominator is the mass of the Y3Al5O12 unit, calculated from the relative atomic masses and the atomic mass unit (which is close to the proton mass). The inherent assumption is that the average size of the unit cell is not modified by the doping; this is a reasonable approximation as long as the doping concentration is low.

For Nd:YVO4, the calculation is slightly different because only a single neodymium ion can occupy a YVO4 unit:

$${N_{{\rm{Nd:YV}}{{\rm{O}}_{\rm{4}}}{\rm{, }}1\% }} = \frac{{1 \cdot 4.22\,{\rm{g/c}}{{\rm{m}}^{\rm{3}}}}}{{\left( {{\rm{88}}{\rm{.9}} + {\rm{50}}{\rm{.9}} + {\rm{4}} \cdot {\rm{16}}{\rm{.0}}} \right) \cdot 1.66 \cdot {{10}^{ - 24}}{\rm{g}}}} \cdot 0.01 = 1.25 \cdot {10^{20}}{\rm{/c}}{{\rm{m}}^{\rm{3}}}$$

Nevertheless, the resulting number densities are similar because the larger number of Nd3+ ions per unit in YAG is offset by the larger volume of that unit.

Fraction of Weight

The fraction of weight (specified in “% wt.” = weight percent or in “ppm wt.”) is easily understood in principle, but it should be made clear exactly which weight is counted. For example, if a laser glass is doped with neodymium oxide (Nd2O3), either the weight of the neodymium only or the weight of the oxide may be specified. If the weight of the neodymium only is taken, mass percentages of 0.73% for Nd:YAG and 0.71% for Nd:YVO4, respectively, are obtained, again assuming an atomic doping density of 1%. Weight fractions are most popular in the context of glass materials, including fibers, and then often count the weight of whole compounds (e.g. oxides) rather than of the laser-active ions only.

Obviously, if one specifies a percentage (or similarly a ppm value) without saying whether it means molar (atomic) percent or percentage by weight, it is not clear what the number means. Unfortunately, this is very frequently done by crystal or fiber suppliers and even in the scientific literature. In most but not all cases, percentage numbers then refer to atomic percent, and ppm are probably more often ppm of weight. Another frequently encountered problem is the specification by weight fractions without saying what exactly is counted (see above).

Importance of Dopant Concentration in Laser Gain Media

The doping concentration of a laser gain medium is a very important parameter, as it has an impact on different phenomena:

  • It determines the (unbleached) absorption coefficient <$\alpha$> for the pump light (assuming optical pumping) according to <$\alpha = N \: \sigma_\rm{abs}$>.
  • Similarly, the gain coefficient (in 1/m) is <$g = n_2 \: N \: \sigma_\rm{em}$>, where <$n_2$> is the achievable fraction of excitation and a four-level gain medium is assumed. (In the case of optical fibers and other waveguides, there is an additional overlap factor, describing the partial overlap of the light field with the doped region.)
  • High doping densities often lead to quenching of the upper-state lifetime. This can result from energy transfer between ions, which can be strongly enhanced by clustering of ions (the extent of which can depend on the fabrication conditions). Apart from the average doping density and the homogeneity of the dopant distribution, an important parameter for energy transfers is the number density for 100% doping, which is related to the size of the unit cell: there are crystalline media with a large distance between laser-active dopants (even if these are located in adjacent unit cells), so that energy transfer processes are weak.
  • High doping densities also increase the dissipated power per unit volume and may thus increase the temperature (possibly reducing the laser efficiency) and the temperature gradients, and also their effects such as mechanical stress and thermal lensing. However, this is not always true; for thin-disk lasers, for example, higher doping allows the use of a thinner disk, which can be more efficiently cooled.

Typical doping concentrations of rare-earth-doped bulk crystals are between 0.1 at % and 3 at. %. There are some special cases where much higher doping concentrations, in principle up to 100 at. %, are possible without strong quenching effects. An example of this is the ytterbium-doped tungstate Yb:KYW = Yb:KY(WO4)2, which becomes KYbW = Yb:KYb(WO4)2 for 100% ytterbium doping. Such high doping concentrations can in some cases be utilized with advantages for laser design, particularly in the context of thin-disk lasers.

Rare-earth-doped Fibers

Rare-earth-doped fibers typically have only the fiber core doped with a laser-active substance. Typically, they have doping densities between a few hundred and several thousand ppm wt., i.e., in most cases lower than for typical laser crystals. Limits are usually set by the tendency for quenching at higher doping concentrations, and sometimes for high-power fiber lasers and amplifiers by the acceptable heat generation per unit length. The combination of a limited doping concentration with the typically low transition cross-sections of glass materials limits the gain per unit length. Nevertheless, the overall gain can be large due to the typically long length. The limitation by quenching is more severe for erbium-doped fibers (particularly with silicate glasses) than for, e.g., ytterbium-doped fibers.

The doping concentration in the fiber core is often not spatially constant, particularly in fibers. For example, fibers made in an MCVD process often exhibit a dip of the doping concentration along the fiber axis. Calculations e.g. of the amplifier gain and saturation power are often based on the simplifying assumption of a constant doping concentration throughout the fiber core, even when this is actually not realistic.

“Confined doping” in a fiber usually means that only part of the fiber core (typically the inner part) is doped. That can be helpful e.g. for suppressing the excitation of higher-order modes, but also often increases the required doping density. Another variant is ring doping [1], i.e., doping only a ring around the fiber core.

Measuring or Calculating Doping Concentrations

There are various methods for determining doping concentrations. Some of them are based on what is known during the fabrication, for example how much of the dopant material added. However, that may not always work, since one may not be sure that all added material is dissolved in a sample.

Alternatively, it is common to use absorption spectra, when the absorption transition cross-sections are known. In the case of optical fibers, where one often has the dopant only in the fiber core, one may also need to take into account an overlap factor of the light the fiber core.

More to Learn

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[1]J. Nilsson et al., “Ring-doped cladding-pumped single-mode three-level fiber laser”, Opt. Lett. 23 (5), 355 (1998); https://doi.org/10.1364/OL.23.000355
[2]H. Kühn et al., “Method for the determination of dopant concentrations of luminescent ions”, Opt. Lett. 36 (23), 4500 (2011)

(Suggest additional literature!)

Questions and Comments from Users


I have a Nd:glass rod made of P2O5. I only know the Nd number density, which is 2 · 1020 cm−3. How to calculate the molar percentage?

The author's answer:

The glass does not only contain P2O5, and you need to know the full composition to do that calculation. Anyway, the neodymium number density is all you need to calculate the relevant optical properties.

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