#### Diffusion Theory

The Yb concentration profiles obtained in this work can be well approximated utilizing the one-dimensional diffusion theory described by Fick’s second law of diffusion.

where

D is the diffusion coefficient and

y quantifies the direction normal to the surface. Assuming an inexhaustible source (i.e., a thick film) of Yb on the surface, the solution of Fick’s law (1) can be approximated by

with

C_{max} being the concentration of Yb at the crystal surface and the diffusion depth

${d}_{erf}=\sqrt{4Dt}$. In the thick-film diffusion regime.

C_{max} describes the solubility of Yb in LiNbO

_{3}. This value is given by

Using the law of mass conservation, the time needed for the Yb source to be depleted can be calculated with the metallic Yb density

ρ, the atomic mass

m_{Yb}, and the thickness

τ of the initial Yb layer

For sufficiently thin films and/or sufficiently long in-diffusion times (

t >

t_{d}), the Yb source is depleted and the diffusion profile can be approximated by a Gaussian function

In this case,

${d}_{\mathrm{exp}}=\sqrt{4Dt}$ describes the 1/e-depth of the profile. The surface concentration

C_{0} is below the solubility and given by

The temperature dependence of the diffusion coefficient

D is given by the Arrhenius relation

Here, D_{0} is the diffusion constant, E_{A} the activation energy, k_{B} the Boltzmann constant, and T is the temperature used for in-diffusion.

As the solubility describes the equilibrium between Yb in LiNbO

_{3} and the pure metallic Yb on the surface, the concentration at the surface can also be written in terms of an Arrhenius-type equation as

where ∆

H is the mixing enthalpy and

$\widehat{C}$ is a pre-factor.

Figure 2 shows the high-resolution images of the crystal surfaces obtained using a confocal microscope and applying the differential interference contrast method for the two crystal orientations. When the temperature used for annealing is less than 1130 °C, an increased surface roughness caused by insufficient diffusion is visible. For the x-cut surfaces, the arithmetical mean height (

Sa) slightly decreases from 12 to 10 nm when the temperature increases from (a) 1000 to (b) 1060 °C.

For the z-cut surface shown in (d) and (e), the Sa is reduced from 11 to 7 nm. Annealing for 30 h at 1130 °C results in a smooth surface with Sa = 2 nm for both (c) x- and (f) z-cut surfaces, indicating that the Yb ion source at the surface is completely exhausted, and the Yb profile can be described by the thin-film diffusion regime of Equation (5).

For calibration of the SNMS data, the samples annealed at 1130 °C were used assuming a linear relation between ion yield and concentration as well as mass conservation in the diffusion process. For lower temperatures, the experimentally obtained profiles can be best fitted by complementary error functions, as can be seen in

Figure 3.

The profiles obtained using the highest temperature were fitted by Gaussian functions; see

Figure 4. The results obtained for the surface concentration and for the depths

d_{erf} or

d_{exp} are summarized in

Table 1.

With the values for the diffusion depth, the diffusion coefficients

D are calculated. The temperature dependence of the diffusion coefficient is given by Equation (7).

Figure 5 shows the corresponding Arrhenius plots for the two crystal-cut directions used. By correlating the fitted lines with Equation (7), the values for the diffusion coefficients and the activation energy for the x- and z-direction were obtained. The results are summarized in

Table 2.

The temperature dependence of the surface concentration is presented in

Figure 6. As can be seen, the measured values for the surface concentration of Yb for the three lowest temperatures indicate that, in contrast to the diffusivity, the solubility of Yb in LiNbO

_{3} shows no anisotropic behavior. The solid line is a fit to the data with Equation (8). This way, the values of ∆

H = (1.04 ± 0.23) eV and

$\widehat{C}$ = (1.6 ± 0.35) × 10

^{24} cm

^{−3} were found.

With these characteristic values for the diffusion of Yb in LiNbO

_{3}, a first test was performed for diffusion doping of a ridge waveguide prepared in an x-cut LiNbO

_{3} substrate oriented parallel to the y-axis by diamond blade dicing.

Figure 7a shows the simulated Yb concentration profile in the ridge for the case of two 22 nm thick Yb layers being coated under ±60° with respect to the x-axis obtained by numerically solving Fick’s second law. For in-diffusion, the sample is annealed for 216 h at 1125 °C. The concentration profile also shows that at the upper right and left corners of the ridge, the expected concentration is below the solubility limit. Thus, no additional surface defects are to be expected. To achieve waveguiding, the upper part of the ridge is additionally doped by in-diffusion of Ti.

Figure 7c shows the expected intensity distribution of the quasi-TE modes at 800 nm and the intensity distribution measured at 796 nm. It is clearly visible that a good overlap between the Yb profile and the intensity distribution of the mode is achieved. The simulated concentration profiles shown in

Figure 7a,b prove that ridge waveguides can be almost homogeneously doped by three-side in-diffusion while avoiding any surface damage that might be caused by exceeding the maximum solubility of Yb in LiNbO

_{3}.

The transmission spectrum of such a 10 µm wide and 2.2 cm long ridge waveguide was measured using unpolarized white light coupled into the waveguide by means of a 40× microscope objective. At the output facet, the transmitted light was collected and coupled into a spectrometer. To measure absorption as a function of polarization, a polarizer was placed in front of the collimator of the spectrometer. The measured absorption spectra (π and σ polarization) show three main peaks at 918, 956, and 980 nm (see

Figure 8a), which is in good agreement with the absorption spectra reported for bulk-doped Yb:LiNbO

_{3} [

22] and Yb:Ti:LiNbO

_{3} channel waveguides [

14]. The losses of the waveguide of 0.8 dB/cm were determined at 1064 nm using the Fabry–Perot method. Fluorescence spectra (π and σ polarization) were measured for pumping with a titanium-sapphire laser at 918 nm.

Figure 8b shows the polarized fluorescence spectrum. In agreement with previously reported results for Yb:LiNbO

_{3} waveguides [

14,

20], we observed the expected four main peaks around 960, 980, 1008, and 1062 nm.