## Abstract

We have measured and characterized, over a wide range of doping levels, the UV-near-IR (190-2000-nm) absorption properties of Ti:sapphire crystals. We find that the strengths of absorption centered around 400-450 and 268 nm depend on the square of the Ti^{3+} doping level, suggesting an origin from pairs of Ti^{3+} ions. In addition, we have identified an absorption feature below 210 nm due to Ti^{3+}, rather than Ti^{4+} charge-transfer transitions. Finally, our data on 800-nm-peak, near-IR absorption shows a complex lineshape, with a lower limit set by Ti^{3+} pair absorption. Thus the maximum possible Figure-of-Merit for Ti:sapphire reduces as the doping level increases.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Lasers based on Ti:sapphire (Ti^{3+}:Al_{2}O_{3}) crystals remain among the most widely used devices for scientific and medical-research applications. Demonstration of the first laser operation from the material [1,2] was based on earlier publications showing broadband absorption [3] centered in the blue-green wavelength region (peaking at 490 nm) and associated broadband emission at near-IR wavelengths [4], the result of transitions between the crystal-field-split, 3d energy levels of the Ti^{3+} single outer-shell electron. Laser operation was possible by optical pumping into 490-nm peak band, referred to in the following as the pump band.

A subsequent publication [5] provided more details and included data on the spectroscopy of the material, including the full spectral extent of the near-IR emission, its lifetime, and estimates of the absorption and emission cross sections. The paper discussed other properties, including weak broadband absorption peaking around 800 nm and an intense UV absorption starting around 300 nm and increasing up to the 190-nm limit of the measurement. Based on other publications, [6,7] the paper presented two hypotheses for the 800-nm band: 1) Ti^{3+} ions at interstitial or defect sites or associated with other crystal defects and 2) pairs of Ti^{3+} and Ti^{4+} ions. Subsequent analysis [8] has shown support for the latter idea and it has become the most accepted explanation for the 800-nm absorption. Reference [5] noted prior work, particularly by Bessonova *et al.* [9] suggesting that the UV absorption was from Ti^{4+} ions. More recent work by Evans [10] and Wong *et al.* [11] also makes this claim. The latter paper concluded that a charge-transfer process, where an electron from a nearby O^{2-} ion moves to the Ti^{4+} ion to form a Ti^{3+} ion, gives rise to two strong absorption bands centered at 220 and 180 nm. They also examined a UV band peaking at 268 nm and attributed this to the creation of an exciton bound to a Ti^{3+} ion.

Observation of the spectral shape of absorption in typical lightly doped Ti:sapphire laser crystals shows that absorption beyond the blue extreme (around 400 nm) of the pump band does not return to zero, but rather reaches a minimum in the wavelength region between 300 and 400 nm (see Fig. 1 in [12] as an example). This effect could be interpreted as the result of the long-wavelength tails of the UV absorption overlapping the short-wavelength end of the pump band. Recent papers have described the operation of Ti:sapphire lasers pumped by 450-nm-region, InGaN diode lasers, where the pump light might then be absorbed by a combination of processes. Effects observed with the short-wavelength pumps include reduced efficiency compared to longer-wavelength sources (beyond that explained from the larger photon deficit), an increased (and reversible) crystal loss [13–15], and the need to operate with crystals at cryogenic temperature to obtain laser operation [16]. In some work, there was no evident increase in crystal loss [17].

An earlier attempt at understanding these phenomena employed Gaussian fits to the pump-band absorption spectrum, which showed qualitatively that the 450-nm-region absorption was not solely due to this 3d-3d band [18]. In the following, we extend that work to more fully examine the blue-near-UV wavelength-region absorption in Ti:sapphire for crystals with varying doping levels. We find significant changes in the spectral properties of absorption spectra in this region with increasing doping. We note that this was apparent in some early data on absorption spectra as a function of doping level [19], as well in a more recent publication [20]. More importantly, we acknowledge prior work that has perhaps not received proper recognition, showing a similar change in the spectral shape of the near-UV-blue absorption with increased doping, and making an assignment of this feature to a different species than Ti^{3+}. Specifically, the authors speculate the absorption is due to a complex of a pair of Ti^{3+} ions and a F center (oxygen vacancy with two trapped electrons) [21,22], or a single Ti^{3+} ion paired with a F_{2}^{+} center [23], the latter center being paired oxygen vacancies with three trapped electrons. One of our contributions in this paper is to characterize the spectral lineshape for this species, show an explicit dependence of the absorption intensity with Ti^{3+} concentration, and also show that the UV absorption band centered at 268 nm may well result from the same absorbing entity. Following some recent theoretical models, we present a hypothesis that the entity is pairs of Ti^{3+} ions. We measured absorption in the 200-nm region and identify a feature that may be due to a charge-transfer process originating with Ti^{3+} ions. We also investigated the infrared (IR) absorption and found complexity that has not been fully explored in the past. We provide some new hypotheses for its origins, specifically that in samples with low Ti^{4+} impuritiy concentrations the absorption is also from Ti^{3+}-ion pairs.

## 2. Crystals

We obtained Ti:sapphire crystals from a number of sources. An assortment of crystals was available at MIT Lincoln Laboratory. Suppliers included Union Carbide (no longer providing material), where crystals were grown by the Czochralski technique [6], followed by post-growth annealing techniques to reduce the level of Ti^{4+} ions [24]. Another supplier was Crystal Systems (now GT Advanced Technologies), providing material grown by the Heat Exchanger Method (HEM) [25], a variation of the Bridgman technique. In addition, we had samples grown at MIT Lincoln Laboratory by the gradient-freeze technique [19], another Bridgman variant. We tried as best as possible to sort the materials we had by source, but in some cases the origins were not clearly indicated.

We also characterized newly grown crystals, one set from Crytur Ltd. (Turnov, Czech Republic), which featured high doping levels. The material was grown by the Czochralski technique in tungsten crucibles. The pulling speed was 0.1-0.5 mm/hour and the growing interface was controlled by varying rotation speeds from 1.5 to 8 rpm. The post-growth annealing of material (to minimize the Ti^{4+} concentration) used a patented procedure [26] where crystals were heated to about 50°C below the melting point of sapphire and kept in a hydrogen atmosphere for 50-100 hrs.

Other new crystals were grown at Northrop Grumman Synoptics, also by the Czochralski technique, which generally involved melting Al_{2}O_{3} and Ti_{2}O_{3} in an iridium crucible with induction heating under a growth atmosphere of argon within a refractory furnace specifically designed for the work. The pulling system harnesses a rotating, a-axis oriented seed to initiate growth. Rotation speeds of 5 to 10 rpm were employed in order to achieve the desired interface, assuring quality growth. Growth rates of 0.25- to 0.50-mm/hour were ultimately established in order to avoid induced defects, such as bubbles and dislocations. The crystals were cooled in the same atmospheric conditions used for growth, over a period of 2 to 5 days. Post-growth anneals were performed with the appropriate atmosphere.

Table 1 lists the samples, showing the labels we employed, the source, sample thickness (along the measurement direction), and the pi (E||c)-polarized absorption strength (cm^{−1}) at 490 nm, for samples where it could be measured. This strength provides an indication of the Ti^{3+} concentration averaged along the sample length. We discuss below the complications in relating 490-nm absorption to doping level. For labels, the first two letters of the sample names indicate the source, then a number that increases with the measured concentrations. Sample names differing by letters at the end were from the same growth run, in some cases from different portions of the crystal boule. We employed a variety of thicknesses to better measure certain wavelength regions, with thin samples useful for UV-region measurements, where absorption coefficients range to beyond 100 cm^{−1}. Thick samples allowed us to better measure relatively weak, IR-absorption levels. Our crystal samples covered an order-of-magnitude range in doping level, and a wide range of IR-absorption-band strengths.

For absorption measurements we employed a Perkin-Elmer (PE), double-beam, Lambda 1050 UV/Vis spectrophotometer, which could provide data in the 190-2500-nm range. In order to make polarized measurements, we inserted a PE Model B050-5284 Glan-Thompson polarizer in the sample beam, with a practical short-wavelength limit of about 245 nm. In cases where we did not use a polarizer, the spectrophotometer also included a PE Model B050-1282 common-beam depolarizer to assure that the samples were probed by a highly depolarized beam. We corrected the transmission measurements to account for the Fresnel losses of the samples, employing a three-term Sellmeier equation for the refractive index of Al_{2}O_{3}, valid from 200 to 5000 nm [27].

## 3. Experimental data on absorption

#### 3.1 Overview

Figures 1 and 2 show data from three samples. Sample CT2a provided pi-polarized data for wavelengths longer than 300 nm, while CT2b, with nominally the same doping level, but about 5% of the thickness, was used for unpolarized data at shorter wavelengths. One sample, UC1a, with about 13% of the Ti^{3+} doping level of the CT2 samples, as determined by 490-nm absorption, provided data for all of the wavelengths. Data in Fig. 1 is plotted against wavelength, while Fig. 2 plots results in terms of frequency (wavenumber in cm^{−1}, with an energy scale in eV also included, for reference) and covers all of the measurement range into the infrared. Much of the data to be shown in the paper we plot against the more widely used wavelength scale, even though our spectral models are based on frequency calculations.

Figure 2 shows a division of our measurements into three regions. Region 1 covers the infrared to a 700-nm (≈14300 cm^{−1}) upper limit and includes the IR absorption that is known to vary greatly depending on sample preparation. The strength of this band is used in the so-called Figure of Merit (FOM) characterization of commercial materials, which is the ratio of pump-band absorption to peak IR absorption. For this paper we use a FOM definition [12] of the ratio of absorption at 514.5 nm to 820 nm, for pi-polarized light. Region 2, from 700 nm to 300 nm (≈33300 cm^{−1}) includes the pump band. The upper endpoint, as we will show below, marks the effective boundary between two absorption features we identify later in this paper. It is evident that the spectral shape of the absorption in Region 2 changes with increased Ti^{3+} concentration, which is also true for Region 3. The upper boundary for this Region, 190 nm (≈52600 cm^{−1}), is the upper limit for our measurement instrument. Sapphire remains transparent up to the band-edge limit of about 140 nm (≈71000 cm^{−1}) [28], so further absorption characterization of Ti:sapphire requires vacuum-UV instrumentation, and for high doping levels at least, very thin samples due to the strength of the absorption.

#### 3.2 Region 2 absorption data

Figures 3 and 4 present polarized Region 2 absorption data for samples with different doping levels, both actual as well as normalized to the value at 490 nm. The emergence of an additional absorption feature peaking around 375 nm is apparent with increasing doping level. The more profound effect on the shape of sigma (E⊥c) spectra indicates the feature does not have the same polarization characteristics as that of the pump band.

#### 3.3 Region 3 absorption data

Figure 5(a) presents unpolarized data, all, with the exception of UC1a, having about 0.5-mm thicknesses to allow accurate determination of high-value absorption coefficients. We limit our data to a maximum value around 130 cm^{−1}, as higher values lead to transmission levels below 0.05%, and increasing inaccuracy due to instrument noise. We show the same data in Fig. 5b plotted with a logarithmic scale to better present the spectral shape over a wide range of values.

Sample SY1 was unique in that the starting dopant material for the crystal was TiO_{2} (0.21 wt.%), and there was no post-growth annealing. In this situation, one might expect that a large fraction of the Ti ions would be in the 4+ state, and this is apparent in our measurements of the sample, which shows a very weak Ti^{3+}-related absorption at 490 nm (about 0.12 cm^{−1} in strength), and intense absorption in the UV from the Ti^{4+} charge-transfer transition. The spectral shape of SY1 absorption is a good match to that found earlier [9] and also observed in [11a] for one sample (A) with a high estimated Ti^{4+} concentration. It is evident for all the other samples that some of the absorption appears to be from the Ti^{4+} charge-transfer transition, but there are two features that differ from SY1a. First, there is an absorption band peaked at 268 nm that grows rapidly in intensity with the Ti^{3+} doping level. Following a prior study [11a] we refer to this as the “E Band.” Second, there is a rapid increase in absorption at wavelengths below 200 nm. In our subsequent analysis we identify this feature as the long-wavelength tail of a Ti^{3+} charge-transfer transition, which has been predicted [28] to peak around 7.82 eV (159 nm).

We were able to do polarized measurement for wavelengths longer than about 245 nm. The polarizer insertion losses became impractically high at shorter wavelengths, which limited the peak absorption coefficients to less than 80 cm^{−1} for our thin samples. Figures 6 and 7 show, respectively, pi- and sigma-polarized, 245-300-nm absorption data. We present both linear and log plots of absorption, the latter to provide better information on the absorption spectral shape for all samples. In general, the sigma-polarized absorption was much stronger for both the E band and the Ti^{4+} charge-transfer transition. In fact, the sigma-polarized E-band peak absorption exceeded our upper limit for the most heavily-doped samples (SY7d and CT2b), but we include the data for those samples up to that limit. We show data on the log scales for both sample SY1a and SY1b to provide a wide dynamic range of absorption coefficients.

Figure 8 shows pi- and sigma-polarized data for two samples to provide a direct comparison of the different absorption strengths. It is evident that the Ti^{4+} ions create a sloping background absorption that distorts the shape of the E band, and the background varies from one sample to another, even for those with comparable Ti^{3+} doping levels. In our analysis below, we provide results based on subtracting out this background to provide more accurate data on the band.

#### 3.4 Region 1 absorption data

Our data starts at 600 nm to include the long-wavelength extreme of the pump band, and terminates at 1800 nm, the practical limit caused by instrument baseline noise. We include measurements (Fig. 9(a)) on some samples with relatively high absorption in this region, *i.e.* a low FOM, but did not include them in data for Region 2 as the high IR-absorption levels impact the long-wavelength end of the Region 2 data, as we discuss in more detail below. These samples are from early growth efforts, where annealing (if done at all) to increase the FOM was not as effective as in more recently grown materials. For the data in Fig. 9(b), we employed relatively long samples, given the low absorption levels. It is evident that the spectral structure and polarization properties in the IR absorption region are complex and sample dependent.

## 4. Data analysis

As an introduction to the section, we discuss a number of absorption bands or regions beyond those we introduced above. Table 2 is a key to the approximate wavelength ranges involved.

#### 4.1 Region 2 absorption data

As a start to analyzing the data shown in Figs. 3 and 4 we seek to distinguish absorption due to the pump band from the absorption features that emerge with increasing Ti^{3+} doping level. Our approach is to fit the appropriate spectral model to the measured pump-band absorption, and then use that model to subtract the pump-band absorption from the sample data to show the remaining, or “residual” absorption.

The spectrally broad pump and emission bands of Ti:sapphire are the result of a strong coupling between the host lattice and the 3d energy levels, leading to a large change in the lattice as the Ti^{3+} ion is promoted from the ground (^{2}T_{2}) state to the excited (^{2}E) state. The standard, simplified calculation of the absorption and emission lineshapes employs a configuration coordinate model, which treats only one phonon frequency in the interaction. For Ti:sapphire this model becomes more complex due to Jahn-Teller splitting of the ^{2}E level, which leads to the double-peaked absorption lineshape.

In the simple configuration-coordinate model, the peak energy of the absorption and emission bands are equidistant from the zero-phonon-transition energy. In Ti:sapphire (ignoring details of the spin-orbit splitting of ground state) the low-temperature value for the latter is approximately 16200 cm^{−1} (617 nm) [4,30]. Based on data shown in [5], the emission band peak, *ν _{ep}*, around 13175 cm

^{−1}(759 nm), is 3025 cm

^{−1}lower than the zero-phonon line. The Huang-Rhys parameter [30] for emission,

*S*, can be approximately determined [31] through the formula:

_{e}*ν*is the frequency of the phonons used to characterize the ground-state vibrational levels. Based on multiple publications [4,29,32], this frequency falls in the 200-250 cm

_{g}^{−1}region, and thus S

_{e}is in the range 12-15. We expect a comparable value for the absorption Huang-Rhys parameter. Although the Pekarian [33] lineshape function for absorption in the simple model is appropriate based on phonon-wavefunction overlap theory, the Pekarian function approaches a Gaussian for high values of S, with 5% error for S as low as 4 [34], and we employed the latter to fit our pump-band absorption data. We used a two-Gaussian fit (in frequency) to accommodate the Jahn-Teller splitting of the excited state. Figure 10 plots our results for pi-polarized data for sample UC1a. We chose this sample as it had the lowest level of short-wavelength absorption, as well as unmeasurably low Region 1 absorption, thus providing the closest case to absorption due only to isolated Ti

^{3+}ions. The plot shows a least-squares-minimized, two-Gaussian fit (“high” and “low” bands), optimized for the wavelength range covering from 475 to 650 nm. The R-squared value for the fit differed from unity at the 1 × 10

^{−5}level, but this difference started to rapidly increase as the short-wavelength fitting range was reduced below 475 nm. The difference between data and fit as a function of wavelength shows a ± 0.2% deviation from 480 to 700 nm, with a rapidly increasing, positive deviation at shorter wavelengths.

Figure 11 plots similar data for sigma-polarized absorption. We used the same parameters derived for the two-Gaussian, pi-polarized data (peak position, relative peak strength and linewidth) and simply scaled the absorption strength to provide good fit. We found that the sigma/pi absorption intensity ratio was 0.424. The agreement is good, but with a notable deviation in the 500-550-nm region. We could improve the fit by developing a different set of parameters from the pi-polarized data, but that would imply some type of polarization–dependent change in the phonon-overlap functions, a violation of the Franck-Condon principle. As we show below, we believe the deviation is the result of the additional absorption that we are characterizing in this section.

Table 3 lists the parameters for the two Gaussian functions. Other two-Gaussian fittings [11,32] find high-band peaks in the 20400 to 20618 cm^{−1} range and low-band peaks in the 17987-18200 cm^{−1} range. Our lower value for the latter could be the result of the very low Region 1 absorption in our sample. Remarkably, if we take the average of the two peak energies (19211 cm^{−1}) we find the energy difference from the zero-phonon-line energy to be 3011 cm^{−1}, and thus the same, within experimental error, as the difference from the zero-phonon energy to the emission peak energy. In the configuration-coordinate model the larger linewidth for the high band is consistent with Franck-Condon transitions to a steeper-sloped region on the configuration-coordinate energy curve for the excited state.

We employed the fitting parameters of Table 3 to analyze the pi-polarized data from 16 different samples, and 10 samples for sigma-polarized data. We determined the best fit in terms of minimized difference from the data at pump-band wavelengths and subtracted the fit to generate residual absorption in the Region 2 range. Our pi-polarized results for 10 representative samples appear in Fig. 12, for both linear and logarithmic plots, the latter to provide a better view of the changing spectral shape from low to high doping levels. We show sigma-polarized results in Fig. 13 for a subset of 7 of the samples shown in Fig. 12. Data from samples not shown had similar spectral characteristics and dependence on doping level. To compare the polarization properties of the residual absorption we plot, for three samples, data for both polarizations together in Fig. 14.

The rising UV edge of the residual absorption is clearly the long-wavelength tail of the intense E band that is apparent in Figs. 6–8. We discuss in the next section how to subtract this tail to determine the short-wavelength properties of the residual absorption, for material where thin samples would allow us to determine the peak strength of the E band. Our data shows that, (Fig. 14) while the peak strength of residual absorption is comparable for both polarizations, the spectral shape is notably different, with sigma-polarized absorption extending to much longer wavelengths. Also, it is apparent that the spectral shape of the residual absorption varies with doping, with the peak value gradually shifting from 440 to 400 nm with increasing doping.

As an attempt to determine how the overall strength of the residual absorption varies with doping level, in Fig. 15 we plot, for all 17 samples, the ratio of the pi-polarized, residual-band absorption at 400 nm to the peak pi-polarized high-band absorption as a function of the peak pi-polarized absorption of the high band. The latter represents a good measure of the Ti^{3+}concentration, unencumbered by any residual absorption in the 490-nm-region. The high-band peak absorption is about 98% of the measured 490-nm, pi-polarized absorption for lightly doped samples, with a slight decrease to 97% at high doping. This is in contrast to sigma-polarized data at 490 nm, where, as is evident from Fig. 4b, the residual-band absorption has a larger effect on 490-nm-region absorption, even shifting the peak absorption to shorter wavelengths at high doping levels.

We distinguish data for three sample sets, annealed Synoptics samples (all but SY1) with a high FOM, other samples with a low FOM (< 30) and other samples with a higher FOM (>30). The high-FOM sets can be well-modeled with a linear dependence of the 400-nm peak values on Ti^{3+} doping levels, *i.e.* the residual absorption strength at 400 nm grows as the square of the Ti^{3+} concentration, with some apparent dependence on material growth and annealing details.

#### 4.2 Region 3 absorption data

We can use the spectral shape of the absorption data for Sample SY1a, with an appropriate scale factor, to subtract out the Region-3, Ti^{4+}-related absorption in other samples. It is generally accepted that the Ti^{4+} ions in Al sites are charge-compensated by Al vacancies [35] with one vacancy for every three Ti^{4+} ions. This leads to a distribution of possible environments for the ions, depending on the distance to the Al vacancy, with resultant inhomogeneity of the spectral properties of the ensemble of ions [11b]. Thus, unlike the case of the Ti^{3+} pump band, the subtraction process is an approximation, since the details of Ti^{4+} site distribution can vary from one crystal to another and change the spectral shape of the Ti^{4+} absorption.

Figure 16 shows results from this subtraction procedure with unpolarized data for two samples, based on the assumption that absorption in the 225-nm-region is entirely due to Ti^{4+} ions. The Ti^{4+}-corrected absorption shows two main features discussed above, the E band peaking at 268 nm, and absorption that shows rapidly increasing strength as the wavelength falls below 220 nm.

One of the results of this analysis is a measure of the relative Ti^{4+} concentration, with respect to that in sample SY1a. Figure 17 plots this Ti^{4+} scaling factor as a function of the high-band peak absorption coefficient for six different samples with fittings done to unpolarized data. We also show data for fitting to pi-polarized data for seven samples, where, for subtraction, we attempted to scale the SY1a pi-polarized absorption data matched to a 245-nm wavelength. In this case we include two samples not in the unpolarized data set. We note, for the same samples, good agreement between the scaling factor for unpolarized and pi-polarized data. The same was true for analysis of sigma-polarized data, not included in Fig. 17. While there is a broad trend to higher Ti^{4+} absorption with increasing Ti^{3+} concentration, there is considerable spread, reflecting the different growth and annealing conditions amongst the samples, even those grown by the same technology.

We did a least-squares fit (in frequency) to a Gaussian lineshape of the Ti^{4+}-corrected E band for a number of samples, using unpolarized, pi-, and sigma-polarized data in the 255- to 280-nm range. Results for two samples of widely differing doping levels appear in Fig. 18. It is evident that the data and fit deviate in the wings of the Gaussian lineshape. On the long-wavelength extreme this is, in part, due to the presence of the Region 2 residual absorption. We consider this in more detail below. Inspection of the difference between the data and the fit shows some structure around the peak of the absorption, which also appears directly in the polarized data in Figs. 6–8. We highlight this structure in Fig. 19, which shows the difference spectra between unpolarized data and Gaussian fit for three samples normalized to the peak absorption of the E band for each sample, hence providing the difference in terms of percentage. The data for sample CT2b has 1-nm spectral resolution, while for samples SY7d and UC1a the resolution was 0.2 nm, hence their higher noise level. The data indicates that a 1-nm resolution sufficed to resolve the structure. The similarity in spectra in both shape and amplitude is noteworthy, despite the large range of doping levels. We could not reproduce the structure by employing two overlapping Gaussian bands with similar widths but differing peak positions. Data for polarized absorption, due to losses in the polarizer, was considerably noisier and not shown here, but generally had larger amplitudes in the structure, ±2-3% and somewhat different spectra, perhaps leading to some cancellation of peak differences in the unpolarized data.

Figure 20 plots the ratio, for an assortment of samples, for polarized and unpolarized measurements, of the fitted peak absorption coefficient for the E band to the peak high-band absorption coefficient for the sample, as a function of the high-band peak absorption coefficient. The data set for sigma-polarized measurements was limited by the high value of the absorption for heavily doped samples, too high to measure even with thin samples. We observe that linear relations provide an excellent fit to the data. This implies, as for the residual absorption peaking at 400 nm, that the E-band intensity grows as the square of the Ti^{3+} concentration. The ratio of the sigma- to pi-polarized peak absorption is about 1.66.

Table 4 provides a listing of the average of the fitting parameters for the E band as a function of polarization, which includes data on the standard deviation of the peak positions for the data set. Note a slight shift in peak position for sigma data that appears to be outside of the experimental deviation.

Regarding analysis of the long-wavelength tail of the E band, we show in Fig. 21 a logarithmic plot of pi-polarized, 270-320-nm absorption. For the two highly doped samples the data is a composite of two thicknesses from the same growth run. For samples SY7b and SY7d we applied a scale factor (0.92) to the latter data to match absorption at 292 nm, while for samples SY4a and SY5a we matched the latter to the former at 280 nm with a scale factor of 1.04. The dotted lines shown are exponential functions (in frequency) of the form

where*α*is the absorption coefficient,

*α*is a scale factor,

_{0}*ν*is the frequency and frequencies

*E*and

_{0}*E*are fitting factors. We arbitrarily set

_{1}*E*to the frequency equivalent of 266 nm (37594 cm

_{1}^{−1}), and least-squares-fit

*α*and

_{0}*E*to the data, typically in the 282-290-nm range. Values for

_{0}*E*spanned 1058 (SY4a, SY5a) to 1232 (LL1) cm

_{0}^{−1}. The functional form in Eq. (2) is the Urbach tail [36]. This, over many orders-of-magnitude, provides a good model to the wavelength dependence of the long-wavelength absorption edge of band-edge transitions in disordered solids, single crystals including sapphire [37], as well as the long-wavelength extremes of absorption from localized excitons [38]. In the case of the E band, the Urbach tail appears at about 30% of the absorption peak, and partially explains the long-wavelength deviation from the fitted Gaussian lineshape plotted in Fig. 18. We show this lineshape fitted to sample SY7d (properly scaled) in Fig. 21, for reference. At wavelengths longer than 295 nm the measured absorption starts to increase above the Urbach tail, and we assert this is from the onset of the residual absorption peaking in Region 2. We note that Ti

^{4+}-related absorption at these long wavelengths is negligible, hence so are errors related to the use of our single model for the shape of the Ti

^{4+}absorption band.

For reference, we also include data in Fig. 21 for sample SY1b, showing the Urbach tail from the Ti^{4+} charge-transfer band, which remains linear on a log plot over a range of nearly 3 orders-of-magnitude, limited at the long-wavelength end by increasing baseline errors.

On the basis that the Urbach tail is the appropriate model for the long-wavelength end of the E band, we can subtract the tails fit to different samples and approximate the lineshape of the residual absorption in Region 2 for a subset of our samples, plotted in Figs. 22 and 23.

We attempted to use the sum of several Gaussian lineshapes to model the data in Figs. 22 and 23, but there was not a good fit for as many as three separate functions. In principle, a larger number of functions might work, but the result would likely be more mathematical than physical.

The remaining feature to discuss is the sharp rise in Ti^{4+}-corrected, unpolarized absorption evident at wavelengths below 220 nm, which we refer to as the deep-UV absorption. Figure 24 plots this data on a log scale, along with Urbach tails fitted in the 194 to 200-nm range. We arbitrarily set *E _{1}* to the frequency equivalent of 185 nm (53825 cm

^{−1}) and note good agreement over about an order-of-magnitude. Deviations at the short-wavelength end for some samples are likely the result of reaching the high-absorption measurement limit of our system, while deviation at the long-wavelength end could indicate the errors in our model for Ti

^{4+}absorption. For sample SY5a in particular, the data is the result a subtracting a very large Ti

^{4+}background absorption level, with a variation in the scaling factor of ± 2% leading to unphysical results (

*e.g*. negative absorption).

The deep-UV absorption strength scales roughly linearly with Ti^{3+} concentration, as an inspection of Fig. 16 shows. A more accurate determination is complicated since we are not able to observe the full lineshape of the absorption, rather we measure what appears to be an Urbach tail of a much stronger absorption peaking at wavelengths shorter than the measurement limit. To add to the challenge, the slope of the tail, as determined by the fitting shown in Fig. 24, decreases monotonically with increasing Ti^{3+} levels, with *E _{0}* ranging from 902 cm

^{−1}for sample UC1a to 1209 cm

^{−1}for CT1b. This could be an indication of an increasing level of disorder in the crystal, which causes a rise in the effective linewidth of the absorption in the tail region. When we corrected for the change in slope, under the assumption that the integrated absorption remained constant, over our sample set the deep-UV band intensity scales linearly with the Ti

^{3+}concentration, within a ± 20% deviation.

Based on the essentially linear variation of deep-UV absorption with doping, we estimated the amount of it for sample SY2a, our reference for Ti^{4+}-related absorption. We found the unpolarized SY2a absorption coefficient would be about 0.4% less at 196 nm, rising to 1.8% less at 190 nm, making any correction to our analysis less than the experimental uncertainty.

#### 4.3 Region 1 absorption data

IR absorption spectra in Region 1 shows a complex lineshape that varies with Ti^{3+} doping levels and sample. Before we attempt to analyze the lineshape, we consider a rough measure of the overall IR absorption strength by taking the pi-polarized value at 820 nm. Table 5 shows measured absorption strength and calculated FOM for a set of samples, ordered by increasing Ti^{3+} concentration. For some samples, the absorption was strong enough to determine the FOM for the specific sample, in other cases for the IR measurement we had to use thick samples from the same growth run, and thinner samples for the 514.5-nm data, as indicated in the table when we list two samples.

Figure 25(a) plots the 820-nm absorption values as a function of the high-band peak absorption coefficient and (b) shows the FOM values against the same parameter. Except for the specific samples marked, we found that the ratio (*R _{820/490}*) of IR to 490-nm absorption (

*α*) was well characterized as

_{490}^{2}value of 0.944. Hence, the IR absorption for a subset of samples (all of the Synoptics samples and one of the Crytur samples) follows a square-law dependence on Ti

^{3+}concentration. The other Crytur sample (CT1a) was from an earlier growth run, with perhaps less effective post-growth annealing. As we have noted, sample SY1b was different from the other Synoptics material, while the remaining samples marked were from the 1980’s, when annealing technology was less developed, if used at all. In Fig. 25 (b) we plot the FOM variation with high-band peak absorption coefficient, and include an inverse-law fit, consistent with the IR absorption square-law variation for most of the samples. Here the approximate relation was:

Based on this relation, it is reasonable that we could not determine the IR absorption in sample UC1b, as with effective annealing that the sample would have an absorption on the order of 1 × 10^{−3} cm^{−1}, below our measurement limit.

Prior work on IR absorption [8] has considered that the physical mechanism is due to pairs of Ti^{3+} and Ti^{4+} ions, thus we might expect absorption at, say, 820 nm, *α _{820}*, would follow the relation

*Ti*] and [

^{3+}*Ti*] are the concentrations of Ti

^{4+}^{3+}and Ti

^{4+}ions, respectively. On this basis, since the 514.5-nm absorption is proportional to [

*Ti*], we would expect We have data on the relative Ti

^{3+}^{4+}concentrations (scaling factor) for a number of samples, or sets of samples, where we also have FOM data. Figure 26 plots the FOM values against the inverse of the scaling factor for six samples, or sample sets. Also included is a zero-origin straight line fit to the data for samples CT1a, CT2a, SY2a,d, and SY7b. While there is support for the pair model with these samples, we note that there are three substantially deviant samples, SY1b, LL1 and SY5c,d. The latter is part of the annealed Synoptics subset, and otherwise fits well in trends on

*α*and the FOM, as plotted in Figs. 25(a) and 25(b). Thus, we find that the pair model does not universally explain the level of IR absorption in Ti:sapphire crystals.

_{820}In terms of the spectral shape of the IR absorption, shown in Figs. 9(a) and (b) for some of the samples, the structure evident, especially at low absorption levels, suggests the use of multiple absorption bands to fit the data, and we selected Gaussian lineshapes, a functional form in keeping with the pump and E bands in Ti:sapphire. Figure 27(a) shows a three-Gaussian fit (bands IR1-IR3) to pi-polarized data for sample SY2a, which had the weakest measurable absorption of all the annealed samples available, and also showed the most pronounced structure. A three-band fit to stronger, sigma-polarized absorption in sample SY7c appears in Fig. 27(b), based on essentially the same band peak positions as for SY2a. The fits are good but clearly deviate from the data at frequencies higher than 13,000 cm^{−1}, mostly due to the onset of the Ti^{3+} low-band absorption. Given the strength of the latter compared to the IR absorption, subtracting out the predicted Ti^{3+} absorption proved to be problematic, and that was true for nearly all of the samples with a high FOM. The exception was for sample UC3, where the strong IR absorption made it possible to do a plausible subtraction of Ti^{3+} absorption, as shown in Fig. 28(a), to obtain an estimate for the full spectral shape of the IR band. Figure 28(b) shows that a relatively good fit to the entire band can be accomplished through the addition of a fourth band (IR4) peaked around 17,000 cm^{−1}. For sigma polarization in UC3, there was a challenge in sorting out the long-wavelength extreme of the “residual” band from the short-wavelength edge of the IR absorption, and we did not attempt analysis.

It was possible for us to use low-noise data on IR absorption for the remainder of samples (with the exception of SY1b, discussed below) to obtain good, three-band fits for the spectral shape, away from the onset of Ti^{3+} absorption. Table 6 lists the samples we characterized, which covered a 40:1 range in IR3 absorption strength. The fits were similar in quality to that shown in Fig. 27.

We have chosen to plot the derived fits in terms of the peak absorption strength of the fitted pi-polarized IR3 band, which is a good approximation to the peak absorption of the entire IR band. The data derived for each band, for both polarizations, was peak energy, absorption at the peak energy, and linewidth. Figure 29(a) plots the peak energy by band for the samples in Table 6, while Fig. 29(b) plots the fraction of total fitting strength for each band. This was calculated by dividing the peak absorption for that band by the sum of peak absorption for all three bands. Figure 30(a) plots the fitted linewidth, while Fig. 30(b) plots the ratio of sigma to pi peak absorption strengths for the three bands.

Different style markers separate the data for Synoptics samples from all others, to highlight any trends for those samples, all grown and annealed by the same general process.

In Fig. 29(a) we observe a fairly constant, and near-polarization-independent peak position amongst the samples. We note a trend, in Fig. 29(b), for the IR3 band to assume a larger fraction of the total content of the absorption at higher absorption levels. Some caution is necessary in the interpretation of the fitting to the IR3-band data as the impact of the long-wavelength tail of the much stronger Ti^{3+} low-band absorption is hard to determine. In Fig. 30(a) the linewidths for the IR2 and IR3 bands evidence a significant increase with overall absorption, while the IR1-band linewidth remains nearly constant. Finally, Fig. 30(b) highlights a difference between Synoptics samples and other crystals. For the former, the sigma/pi ratios for the IR1 and IR2 bands were nearly constant, while the ratio for the IR3 band showed a monotonic decrease with IR3-band peak absorption. Data for the other crystals was more scattered in terms of trends with IR3-band absorption.

The IR absorption in sample SY2b showed a noticeably different spectral shape from the other samples, and we show this and fitting to pi- and sigma-polarized results in Figs. 31(a) and 31(b), respectively. For pi-polarized data we found a good fit with three Gaussian bands (IR1a, IR2a and IR3a), and for sigma data we found a good fit with the same (in terms of peak position and width) IR2a and IR3a bands.

Table 7 provides a comparison of band fitting data for sample SY1b and that for the samples (Others) listed in Table 6, where the latter data is an average for all of the samples, and we have rounded data to the nearest 10s. Clearly, the properties of the IR1a and IR2a bands are significantly different from the IR1 and IR2 bands, as they are higher in energy and spectrally narrower. The IR3a band matches the peak, within fitting error, of the IR3 band, and is about the same width, but we note, from Fig. 30(a) that the IR3 width, especially for pi-polarized data, undergoes a substantial increase with increasing IR3-band absorption. A comparison the pi- and sigma-polarized data in Fig. 31 shows a substantial difference, unlike other samples, as the IR2a band is more intense than the IR3 band for pi-polarized measurements, while the sigma-polarized data is almost entirely modeled by IR3a-band absorption, with no IR1a component and only a small amount of the IR2a band.

## 5. Discussion and implications

#### 5.1 Ti^{3+} and Ti^{4+} concentrations

Above, we have discussed Ti^{3+}- and Ti^{4+}- ion concentrations in terms of measured optical absorption levels. We can connect these to actual ion concentrations based on assumptions about the cross sections for the associated absorption bands.

For the Ti^{3+} pump band, we use a cross section, of 9.3 × 10^{−20} cm^{2} at 490 nm determined through magnetic susceptibility measurements by Aggarwal *et al*. [39], which implies a concentration (*n _{Ti}*) of 1.1 × 10

^{19}Ti

^{3+}/cm

^{3}for a peak high-band absorption coefficient of 1 cm

^{−1}. Consequently, our samples had concentrations ranging from about 1 × 10

^{18}(SY1) to 9.4 × 10

^{19}(CT1) cm

^{−3}. This corresponds, based on an Al

^{3+}ion density (

*n*) of 4.7 × 10

_{Al}^{22}cm

^{−3}, to molar Ti

^{3+}percentage doping levels from 0.0021% to 0.2%.

In terms of weight (wt.) % Ti_{2}O_{3} doping, this value is 1.41 times the molar percentage level. Commercial literature values [40] that connect peak 490-nm absorption and wt. % Ti_{2}O_{3} doping appear to employ the 490-nm cross section value (6.5 × 10^{−20} cm^{2}) from our original paper [5], where the calculated absorption for 0.1 wt. % Ti_{2}O_{3} doping would be 2.16 cm^{−1}. The Aggarwal cross section yields a value of 3.09 cm^{−1}, which we believe to more appropriate. In general, given the literature uncertainty in specifying percentage doping (molar %, wt. % Ti, Ti_{2}O_{3} and even TiO_{2}) we suggest that specifying material in terms of measured 490-nm, pi-polarized absorption is the least ambiguous method.

Evans [10] has estimated the cross section at 230 nm for the Ti^{4+} charge transfer band to be 2.8 × 10^{−17} cm^{2}. Thus, we calculate our reference sample SY1a has a Ti^{4+} concentration of 2.3 × 10^{18} cm^{3} and a Ti^{4+} to Ti^{3+} ratio of about 2.1. From Fig. 17, we note the high level of Ti^{4+} ions in sample SY5a, but there the same ratio is about 0.066.

#### 5.2 Square-law behavior and pair model

We have benefited from having a wide variety of samples with differing Ti^{3+} doping levels, and to a lesser extent, varying crystal-growth conditions. The most significant result of this is a clear indication that all (or a major portion) of the intensity of a number of absorption features in Ti:sapphire is related to the square of the Ti^{3+} concentration, with the latter measured by the strength of the pump-band absorption. Both the residual absorption, peaking around 400 nm, and the E band show this behavior, and for the subset of samples provided by Synoptics, and one from Crytur, the IR absorption does as well. The residual and IR absorption show complex and sample-dependent lineshapes, and we discuss this below as caused by different absorbing species in the same wavelength region. For the IR absorption, it is evident that sample growth and annealing history have an important effect on its intensity, and we see that behavior (Fig. 15) at low doping levels for residual absorption. The strength of deep-UV absorption feature appears to be linear with Ti^{3+} doping level.

There can be several physical explanations for the square-law behavior. We can eliminate one, the presence of Ti^{4+} ions, since we have a measure of the concentration of these ions (Fig. 17). We see square-law intensity dependence maintained among samples even when there is considerable scatter in the Ti^{4+} concentrations for samples with nominally the same Ti^{3+} concentrations. One could postulate that one or several square-law features could result from a near-neighbor pairing of Ti^{3+} ions with some native defect, such as a vacancy, or a more complex color center. The concentration of the defects would have to scale close-to-linearly with that of the Ti^{3+} ions and be insensitive to the growth technique as well as any subsequent post-growth annealing.

We choose to assume the simplest explanation, the square-law features result from pairs of Ti^{3+} ions, since the concentration of pairs should scale as the square of the concentration of Ti^{3+} in the crystal. Table 8 lists the calculated spacings of nearest- and near-neighbor Al^{3+} ions in Al_{2}O_{3}, as well as the number of equivalent ions at each spacing [41]. The closest spacing is for ions along the c axis, but there are ions at three sites with very similar spacing, at pair-axis angles away from c. We also include first-principles pseudopotential calculations [42] of the binding energies of Ti^{3+} ion pairs, which involves the two 3d-electrons for the pair. We also list the calculated spacing caused by the binding interaction, and the ratio of the calculated spacing to the assumed initial spacing. The calculations assume a simpler hexagonal crystal structure than the actual, corundum-structure Al_{2}O_{3} lattice, hence the difference between the actual spacings and the “Theory” spacing, and the missing theoretical values for NN2 and NN3 pairs. Other pseudopotential calculations [43] show a pair binding energy of 1.36 eV.

The theoretical calculations show that there is a favorable binding-energy condition for Ti^{3+} ions to pair up, and the energy rapidly decreases as the ion spacing increases. This could lead to a pairing process that produces a higher fraction of closely spaced pairs than predicted from purely random location of Ti^{3+} ions on Al^{3+} sites. We suspect that the details of the pairing process would depend on the exact nature of the crystal growth and subsequent annealing routines and will leave calculations of this to further studies. One estimate of pair concentrations [43] showed a very high fraction of Ti^{3+} ions formed pairs, but for the reducing growth/annealing conditions used for Ti:sapphire this applied at only for doping levels several orders-of-magnitude below the levels we examined. In the following we assume that pair concentrations follow from random Ti^{3+} ion substitutions.

#### 5.3 Oscillator strengths

One way to gain some understanding about the nature of the absorption features is to make an estimate of their oscillator strength. We can do that for the “square law” features through knowledge of their peak absorption coefficient, and the assumed concentration of the entities creating the feature, which allow us to estimate the feature absorption cross section. In addition, we need to know the spectral shapes of the features, and the degeneracies of their lower and upper levels. We start with the formula (in SI units) combining that derived in [44], equation A4.6, with the formula derived in [5], equation A10,

*g*and

_{i}*g*are the degeneracies of the initial (lower) and final (upper) levels, respectively,

_{f}*σ*and

_{σ}(ω)*σ*are the sigma- and pi-polarized absorption cross sections as a function of angular frequency,

_{π}(ω)*ω*,

*e*is the charge of an electron,

*ɛ*is the permittivity of free space,

_{0}*m*is the mass of an electron,

*c*is the speed of light,

*n*is the refractive index of the host crystal and

*f*is the oscillator strength of the transition. The term

*E*represents a correction factor to accommodate the difference in electric field from the light experienced by the absorbing species inside the host crystal compared to the average electric field in the crystal. The exact correction is a matter of some debate [45,46], as it depends on the spatial extent of the species, but for all of our calculations (given the amount of uncertainty about other parameters), we use the so-called Lorenz-Lorenz correction factor given by We note that Eq (7), calculated for the appropriate lineshape, and employing Eq. (8), is the basis for a specialized form of Smakula’s relationship [47], valid for uniaxial crystals, such as sapphire, and accounting for the degeneracies of the lower and upper levels of the absorbing transition. Where possible we have fit absorption lineshapes to Gaussian functions, and for those the integrated absorption is

_{loc}/E*σ*and

_{σmax}*σ*are cross sections at the peak of the absorption feature for sigma- and pi-polarized light and

_{πmax}*Δω*are the FWHM linewidths in angular frequency for sigma- and pi-polarized transitions. We note the square-root factor in Eq. 9 is about 1.064, hence the integrated cross section is, to a good approximation, the product of the peak absorption times the FWHM linewidth.

_{σ,π}For the concentration of Ti^{3+} pairs *n _{p}*, [8]

Table 9 lists the spectral data, estimated concentrations and resultant oscillator strengths for most of the absorption features we have measured and analyzed.

Our oscillator strengths for the high and low pump bands utilizes the high-band, pi-polarized cross section from [39], with sigma and low-band cross sections scaled according to our absorption measurements. A previous calculation [5] of *f* for the high band (1.94 × 10^{−4}) was based on smaller cross section (6.5 × 10^{−20} cm^{2}) and an incorrect assumption about the upper-level degeneracy *g _{f}* , which we now assert should be 1, rather than 2, because of the Jahn-Teller splitting of the upper

^{2}E state. With the updated cross section and degeneracy, the corrected value is 4.2 × 10

^{−4}, in good agreement with the value shown in Table 9. We derived data for the residual, E band and IR3 absorption from samples SY7b, SY5a and SY5d, respectively. Lacking any knowledge of the electronic structure of the absorbing species, we made the simplifying assumption of equal degeneracies for the initial and final states. Given the irregular spectral shape of the residual absorption, and the shape change with doping level, our calculations for this band are rough estimates.

The pump band oscillator strength is commensurate with calculated values (based on the Lorenz-Lorenz correction) for similar spin-allowed, 3d-3d transitions in Cr^{3+}:Al_{2}O_{3} [48], where the spatially averaged strengths for the ^{4}A_{2}→^{4}T_{2} (560-nm peak) and ^{4}A_{2}→^{4}T_{1} (410-nm peak) transitions are 2.0 × 10^{−4} and 4.7 × 10^{−4}, respectively. We note that our IR3-band estimate is consistent with 3d-3d transitions in sapphire as well.

#### 5.4 Absorption processes and energy levels

The implication of the high values for the oscillator strengths for the residual and E bands, approaching unity for the latter, is the transitions must involve not only 3d-orbital electrons from the Ti^{3+} ions but also electronic states in the top of the valence band (2p-like, associated with O atoms) or bottom of the conduction band (3s-like, associated with Al atoms) of Al_{2}O_{3} [49]. Several possible optical transitions for isolated Ti^{3+} and Ti^{4+} ions include:

^{4+}ions, with a reported oscillator strength of about 0.1 [11b]. In that process, the minimum energy needed to move a valence-band electron to what becomes a Ti

^{3+}impurity represents the position of the ground-state energy of that ion in the sapphire band-gap. Various calculations position the level at 4.66 [49], 4.22 [43] or 4.78 eV [50] above the top of the valence band, corresponding to wavelengths in the 293-259-nm region. This is consistent with our measurements showing the onset of strong absorption around 270 nm (4.6 eV). Additional calculations [49] predict that the lowest lying Ti

^{2+}ion level is 5.84 eV above the valence band, and thus we would expect to observe the charge-transfer transition starting from a Ti

^{3+}ion to appear at wavelengths shorter than 212 nm, consistent with data for the deep-UV feature plotted in Figs. 16 and 24. From this and the near-linear variation in feature strength with Ti

^{3+}concentration, we assert that the deep-UV feature is the long-wavelength tail of the charge-transfer process for Ti

^{3+}ions, which has its peak at shorter wavelengths than we could measure.

Earlier studies [11a] identified the E band as due to a related process, which creates an exciton (electron-hole pair) bound to a single Ti^{3+} ion, but our square-law data for this band invalidates this conclusion. The same reference has claimed that the Ti^{3+} donor/ionization process has a long-wavelength limit (threshold) wavelength around 263 nm, but we were not able to isolate this transition, as our samples all had background absorption from the Ti^{4+} charge-transfer band. The apparent weakness of ionization-process absorption at long-wavelengths may be the result of the low density of states at the lowest energies in the sapphire conduction band. Conversely, the intensity of the charge-transfer process results from the very high density of states at the highest energies in the sapphire valence band [51].

To consider the wavelengths/energies for equivalent processes for Ti^{3+} pairs, we show in Fig. 32 the estimated positions of the Ti^{3+} single and pair impurity levels in a simplified sapphire energy-level diagram. The bandgap for sapphire falls in the range 9.0-9.4 eV, determined by optical transmission and exciton reflectance peaks [51] or thermal excitation [52]. The energy separation of a single Ti^{3+} ion from the bottom of the conduction band, based on the ionization threshold of 263 nm, is 4.71 eV, and we noted above the estimated position above the valence band. We include the two crystal-field-split levels in Ti^{3+}-level diagram, with the separation corresponding to the zero-phonon line discussed above. The positions of the levels in the sapphire energy gap is one of the reasons for the success of Ti:sapphire as a laser material, since, for the laser wavelength region, there are no excited-state processes from the upper laser level to the conduction band, nor low-energy charge-transfer processes from the valence band.

For the Ti^{3+} pair levels, we show the results from two separate calculations [42,43], where we have estimated the positions of the calculated levels from Fig. 2 in [42] and Fig. 10(a) in [43]. We note the pseudopotential-based, energy-level calculations have large uncertainties, beyond their inability to predict the sapphire bandgap. For example, the calculated crystal-field splitting for the Ti^{3+} ion is 2.79 eV [43] compared to the measured 2.01 eV. The position of the lowest energy level of the pair system is subject to some uncertainty, as the two references provide different binding energies (1.23 and 1.36 eV, indicated in Fig. 32), and there are varied estimates of the position of the lowest energy level for isolated Ti^{3+} ions.

The pair system has two d-level electrons, but the site symmetry is so different from an octahedral environment that the standard Tanabe-Sugano diagram showing the effect of a crystal field on the energy levels is not appropriate, nor is it evident what basis functions would be suitable, likely not the “e” and “t_{2}” wavefunctions commonly used.

#### 5.5 3d-3d pair transitions

Regarding transitions amongst the levels, in Table 10 we show possible *3d-3d*-electron optical transitions from the (b) sets of levels in Fig. 32, which are more numerous but generally not that different in span or positions from the (a) levels. We tabulate transitions from the lowest energy level to higher-lying levels. The wavelengths shown would correspond to zero-phonon lines for the transitions.

Based on the levels in Table 10, we can attribute our derived IR1, IR2 and IR3 absorption bands centered at approximately 7400, 9570 and 12400 cm^{−1} to transitions from the lowest-lying level to states at 3.9, 4.2 and 4.4 eV, and the calculated separations between absorption peak and zero-phonon line would be, respectively, 1600, 1370 and 2400 cm^{−1}. In comparison, the separations for the pump-band zero-phonon line and the two absorption peaks are about 1760 and 4200 cm^{−1}. Given the calculation uncertainties in the levels in Fig. 32, we claim at least a plausible connection between IR absorption data and pair theory. Regarding the zero-phonon transition at 4000 cm^{−1} predicted by Table 10, that would be well outside of our measurement limits, while the absorption from higher-lying transitions is difficult to separate from the much stronger pump-band transitions, as well as the residual absorption and E band.

#### 5.6 High-oscillator strength pair transitions

In parallel to the high-oscillator-strength transitions for isolated Ti^{3+} ions, possible Ti^{3+}-pair transitions are

^{3+}ions, to be essentially impossible to observe given the intense absorption from charge-transfer transitions in that wavelength region. For the acceptor transition we would need to understand the position of the Ti

^{3+}-Ti

^{2+}pair state in the energy gap of sapphire to predict where absorption from the transition would begin appear. We also include a process that creates an exciton (electron-hole pair) bound to Ti

^{3+}pairs. This can be viewed as an excited state of the system produced by the acceptor transition. The electron transferred to the pair de-localizes, forming an exciton with the hole, but “trapped” around the Ti

^{3+}pair. As we noted, the E band was, we believe, incorrectly attributed [11a] to an exciton bound to an isolated Ti

^{3+}ion. The absorption from such an entity must appear at shorter wavelengths than we have been able to measure.

We speculate that the residual absorption is the result of an acceptor transition to Ti^{3+} pairs, while the E band results from the creation of an exciton bound to pairs, but further theoretical and experimental study is required to support this or establish another set of explanations.

In general, we note that there is not one, but a variety of pair sites, following the listings in Table 8. Thus, we expect that any absorption resulting from pairs would exhibit a more complex line shape than we see for, say, the pump band. For example, while the E band can be fit well to a Gaussian over a wide region, there are noticeable deviations from the fit on the “wings” of the absorption (Fig. 18) as well as a small but measurable amount of structure around the peak (Fig. 19).

While there is strong evidence that Ti^{3+} pairs create a large fraction of the intensity in residual-absorption wavelength region, at least at high doping levels, and, for some samples, are a major contributor to the IR absorption, we need to consider what other species may create absorption at these wavelengths. We know that, besides Ti^{3+} and Ti^{4+} ions in our samples, for charge-compensation of the latter the assumption is there will be one Al^{3+} (cation) vacancy (VAl^{3-}) per three Ti^{4+} ions [35]. Table 11 lists some possible defect complexes in Ti:sapphire beyond the Ti^{3+} pairs and also lists the pseudopotential-theory-calculated binding energies for each, for nearest-neighbor configurations. Ti^{4+} ions have a strong binding to vacancies, but it is interesting to note that calculations indicate the strongest binding in the Table is with Ti^{3+} pairs and a vacancy, while the second strongest is in the Ti^{3+}-VAl^{3−}-Ti^{4+} complex.

#### 5.7 Other absorbing species - IR absorption

For IR absorption, prior work [8] has presented evidence that Ti^{3+}-Ti^{4+} pairs are responsible for the absorption. In the work here, we have been able, for some samples, to estimate both the Ti^{3+} and Ti^{4+} concentrations and test this theory. For example, for sample SY1b the estimated Ti^{3+} and Ti^{4+} concentrations are about 1.3 × 10^{18} and 2.3 × 10^{18} ions/cm^{3}, respectively. Based on the assumption of a random distribution of dopant ions, and the use of Eq. 10, where we substitute the product of Ti^{3+} and Ti^{4+} concentrations for the square of the Ti^{3+} concentrations, we calculate concentrations of 1.3 and 2.3 × 10^{14} for Ti^{3+}-Ti^{3+} and Ti^{3+}-Ti^{4+} pairs, respectively. Based on the measured sample SY1b IR peak absorption of about 0.015 cm^{−1} (Fig. 31), we can estimate a peak absorption cross section of 6.7 × 10^{17} cm^{2} for the Ti^{3+}-Ti^{4+} pairs, which (from Table 9), is about 400x larger than the estimated cross section for the IR3 band, based on the Ti^{3+}-pair model.

We would expect that a Ti^{3+}-Ti^{4+} pair would share the single *3d* electron from the Ti^{3+} ion in a low-symmetry environment, but it would be surprising if that could induce a more-than-2-orders-of-magnitude increase in the oscillator strength, compared to the *3d*-electron transitions in Ti^{3+} pairs. The latter are also in a low-symmetry environment. An alternative explanation, that the pairing is highly preferential (especially compared to Ti^{3+} pairs), rather than random, looks to be implausible given that, from Table 11, the binding energy is less than that for Ti^{3+} pairs.

It is reasonable that defect complexes involving VAl^{3-} would involve some mixing in of valence and conduction band *2p*, *3s* and *3p* electronic states, creating high-oscillator-strength electronic transitions. In addition, we expect a major level of lattice distortion around the vacancies, leading to a reduction in nearby site symmetries, and suggest that the apparent large cross sections that characterize IR absorption in sample SY1b are connected to the presence of VAl^{3-} sites. It is not evident, given the closed-shell [Ar] electronic configuration of a Ti^{4+} ion, how IR absorption transitions could arise from a Ti^{4+}-VAl^{3-} complex, nor how a pairing of Ti^{3+}-VAl^{3-} sites would produce IR bands, since we might expect larger crystal field surroundings for the single *3d* electron of the Ti^{3+} ion near a vacancy. A more plausible situation is that complexes involving Ti^{3+}-Ti^{3+} and Ti^{3+}-Ti^{4+} pairs and VAl^{3-} would produce high-oscillator-strength transitions in the IR region. Since the IR absorption in sample SY1b shows different peak positions and linewidths for the IR1 and IR2 bands compared to absorption in more heavily doped materials, we may conclude absorption is not simply from enhanced-cross section Ti^{3+}-Ti^{3+} pairs.

Given that the vacancy creates a large electrostatic field, the range over which an interaction with other species is relatively large compared to, say, the exchange interactions that lead to Ti^{3+}-Ti^{3+} binding, and thus the density of vacancy-perturbed absorbing species may be higher than we calculate for NN1-NN3 pairs. Also, given the large calculated binding energies for 3 of the 4 triple complexes in Table 11, we may have to consider preferential, rather than random distributions of defect centers.

Table 12 provides a summary of data for all of the samples where we had data on both Ti^{3+} and Ti^{4+} concentrations. Included is the ratio of the 800-nm absorption cross section calculated from the measured absorption and the assumed Ti^{3+}-Ti^{3+} pair concentration, to the cross section we estimate for the IR3 band, from Table 9. The closer this ratio is to unity, the better is our model of IR3 absorption being entirely predicted from the concentration of Ti^{3+} pairs, without enhanced oscillator strengths. We order the table from the highest ratio to the lowest. With one exception, the agreement with the pair model improves as the ratio of Ti^{4+} to Ti^{3+} concentrations (and hence relative concentrations of VAl^{3-}) becomes smaller. We have no clear explanation for the anomalous behavior of samples SY5a,b, which have a larger fractional concentration of Ti^{4+} ions, but fit moderately well to a square-law model. The material did undergo post-growth annealing, and perhaps that has some influence on clustering of complexes with VAl^{3-} defects.

In general, with the exception of sample SY1b, given the Ti^{4+} to Ti^{3+} ratios we would expect, on a random distribution basis, that there would be 15-100x higher Ti^{3+}-Ti^{3+} pair concentrations than for Ti^{3+}-Ti^{4+} pairs. Except for sample SY1b we were able to fit IR absorption spectral data to our IR1-IR3 band model, which tends to enforce the idea that the major IR absorption is from Ti^{3+} pairs, in some cases with cross sections enhanced by the presence of Al vacancies.

To summarize, our claim, based on our ability to estimate independently both Ti^{3+} and Ti^{4+} concentrations, is that the model that Ti^{3+} and Ti^{4+} pairs create IR absorption needs revision. Both ions do play a role, but we assert that the important effect of Ti^{4+} ions is from their associated Al vacancies, which lead to enhanced IR absorption, likely from several possible defect complexes. Annealing works to reduce the Ti^{4+} and vacancy concentrations and hence increase the FOM. Even in well-annealed samples, there will still be Ti^{3+} pairs that create IR absorption, and this may be an upper limit to the FOM set by the Ti^{3+} doping level. Whether fundamental crystal-growth techniques and/or post-growth processing can also affect the probability of Ti^{3+} pairing remains to be determined.

#### 5.8 Other absorbing species - residual absorption

In parallel with absorption in the IR region, residual absorption for samples with a low FOM show a variation from square-law behavior at low Ti^{3+} concentrations (Fig. 15).

First consider the most extreme case, sample SY1b. It has residual absorption peaked around 435 nm, with a FHWM spectral width of about 3750 cm^{−1}, a ratio of peak pi-polarized absorption to sigma of about 1.35, and a FOM of 12. Based on the measured absorption, an assumption of equal degeneracies and those numbers, we calculate the oscillator strengths for the 435-nm-peak band to be about 0.13 and 0.22 if the absorbing species are, respectively, randomly formed Ti^{3+}-Ti^{4+} and Ti^{3+}-Ti^{3+} pairs. The strengths are about 7.5x higher than shown in Table 9 for the Ti^{3+}-pair model, and 4.3x higher for Ti^{3+}-Ti^{4+} pairs.

We hypothesize that the intensity of the residual band is enhanced in low-FOM samples, where the ratio of Ti^{4+} to Ti^{3+} concentrations is high, and we expect that the relatively high VAl^{3-} concentrations perturb the absorbing species. In comparing spectra for low-FOM samples with other samples, we do not observe distinctly different spectral features, and for the same Ti^{3+} doping level we do not see a notable difference in the spectral shape. For example, from Figs. 12 and 13 we note the similarity in the peak residual absorption for samples SY1b and UC1a, but the latter has a Ti^{4+} to Ti^{3+} ratio of about 0.04 and such low IR absorption that we could not determine the FOM. From Fig. 15, both samples have about the same ratio of 400-nm absorption to high-band absorption, but sample UC1a has about 7x the high-band absorption and the ratio fits well to the square-law behavior. Thus, it appears the effect of vacancies is primarily absorption-strength enhancement, but not a spectral shift. Also, given the low relative amount of Ti^{4+} in sample UC1a, the model of absorbing species connected to Ti^{3+} pairs is further validated.

We do not have an overall theory to explain all of the spectral features of the residual band. In common with the IR absorption region, the spectral shape changes with increasing peak intensity but in contrast we are not able to model the spectrum as the sum of a limited number of Gaussian-lineshape bands. We cannot rule out that some of the weak components in the spectra are from 3d-3d transitions from Ti^{3+} pairs, as predicted by Table 10, but based on the oscillator strength for these transitions from Table 9, we would expect these to be nearly two-orders-of magnitude weaker than the peak residual absorption. If the absorption is from a charge-transfer type of transition, the spectral behavior would be related to the energy of the absorbing species above the energy gap. Given the distribution of pair sites with different ion spacings, along with the energy spread of electrons in the valence band, which is small, but non-zero, we would expect the more complex absorption spectral shape we measure, rather than a simple Gaussian band.

## 6. Summary and conclusions

Based on near-IR-to UV absorption measurements on Ti:sapphire crystals with a wide range of Ti doping levels, we have been able to better characterize absorption features beyond those from the blue-green, pump-band, d-d transitions of isolated Ti^{3+} ions and UV charge-transfer transitions of Ti^{4+} ions. Through subtraction of those known features from our data, we identify both the spectral shapes and dependence on Ti^{3+} concentration of a polarization-dependent, complex-lineshape, “residual absorption” spanning from about 550 to 300 nm, an intense “Band E,” near-Gaussian-lineshape absorption peaking at 268 nm, and an Urbach-tail, deep-UV absorption at wavelengths shorter than about 210 nm. The strength of the first two show a square-law dependence on the Ti^{3+} doping, with, for the residual band, some variation on the relation depending on sample growth and annealing history. We have also more fully characterized the near-IR absorption in Ti:sapphire, and, as expected, find a wide variation in intensity in our samples, with a spectral shape that can be well characterized as the sum of 3-4 distinct, Gaussian-lineshape bands. Among a subset of samples with relatively weak near-IR absorption, *i.e*. a high FOM, we also find a square-law dependence on the Ti^{3+} doping level.

Our subsequent analysis of the absorption data hypothesizes that the square-law-dependent features are due to pairs of Ti^{3+} ions. This theory has support from prior published models showing a positive binding energy for pairs, as well as energy levels that could explain the near-IR absorption spectra. Based on an assumed concentration of pairs produced by random lattice locations for Ti^{3+} ions, we have calculated oscillator strengths for the different pair-created features. For high-FOM samples the oscillator strengths for the near-IR bands are commensurate with d-d-transitions in sapphire, but 2-3-orders-of-magnitude higher for the residual absorption and the E band. We theorize that these transitions must involve the valence and/or conduction bands of sapphire, possibly some type of charge-transfer mechanism.

We also note that the presence of Ti^{4+} ions in samples can enhance both the near-IR and residual absorption strengths and theorize that the mechanism is not directly from these ions, but from the associated Al vacancies, which can form complexes that have high oscillator strengths.

For the deep-UV feature that we have isolated, through subtraction of Ti^{4+}-charge-transfer absorption, we theorize it is the long-wavelength tail of the Ti^{3+}-ion charge-transfer band.

We conclude that a wavelength- and doping-dependent fraction of the light absorbed in the optical pumping wavelength region of Ti:sapphire is due to residual absorption. In a companion article [53] we consider the effect that residual absorption has on optical pumping with InGaN diode lasers, and attempt to explain some unexpected effects observed when 450-nm-wavelength diodes are used as pumps. In addition, we provide guidance on laser designs that account for the presence of residual absorption.

Another conclusion from our work is that there is a fundamental upper bound to the FOM for Ti:sapphire crystals. Even in material with low Ti^{4+} impurity levels, obtained typically by post-growth annealing, the presence of Ti^{3+} pairs leads to weak 3d-3d transitions from the pairs in the laser wavelength region. Practically, even at the highest doping levels we examined, the FOM is still on the order of 100, and the effect on laser performance would be minimal for typical output-coupler transmissions.

Our work was limited to absorption measurements only and does not include the vacuum-UV region. Measurements there could better characterize the deep-UV feature we assign to the Ti^{3+} charge-transfer transition. Further observations of the doping- and temperature-dependent fluorescence spectra and decay times emitted through absorption by features we have discussed would help in better understanding the nature of the absorbing species. Temperature-dependent excitation spectroscopy would additionally help in understanding how excitation transfers in the material, and better characterize the possible creation of transient color centers. From a theoretical standpoint, further modeling of the energy levels of different complexes in Ti:sapphire would help to prove or disprove our Ti^{3+} pair models, and better explain the effects of other impurities/defect such as Ti^{4+} ions and Al vacancies.

## Funding

Office of the Assistant Secretary for Research and Technology (OST-R) (FA8702-15-D-0001).

## Acknowledgements

Any opinions, findings, conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the Assistant Secretary of Defense for Research and Engineering. For the MIT Lincoln Laboratory work we acknowledge the skilled absorption measurements by Jonathan Wilson with support from Peter O’Brien, and sample polishing by Patrick Hassett. We benefited from editing and comments by T.Y. Fan, R. Aggarwal and A. Sanchez. At Crytur, we cite the efforts of Karel Bartos, Jan Polak and Martin Klejch. Finally, we acknowledge work by Adam Lindsey, Chris Oles, Patricia Cajas and Mario Lopez at Synoptics.

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