## Abstract

We present studies of the optical properties of the new nonlinear material BiB_{3}O_{6} for second harmonic generation from the visible to infrared. We have determined the phase-matching conditions and effective nonlinear coefficients in the three principal optical planes, acceptance bandwidths, spatial and temporal walkoff, group velocity dispersion and double phase-matching behaviour. We also report on experimental studies in this material, where efficient, high-average-power second harmonic generation of femtosecond pulses into the blue is demonstrated. Using 130-fs fundamental pulses at 76 MHz, single-pass second harmonic average powers as much as 830 mW at greater than 50% conversion efficiency have been generated over a tunable range of 375–435 nm. Using cross-correlation measurements in a 100-µm β-BaB_{2}O_{4} crystal second harmonic pulse durations of 220 fs are obtained. Our theoretical findings are verified by experimental data, where excellent agreement between the calculations and measurements is obtained. Direct comparison of BiB_{3}O_{6} with β-BaB_{2}O_{4} also confirms improved performance of this new material for second harmonic generation of femtosecond pulses.

©2004 Optical Society of America

## 1. Introduction

In the field of nonlinear frequency conversion there remains a continued need for new materials with improved linear and nonlinear optical characteristics, including higher nonlinear coefficients, wider transmission range, more flexible phase-matching properties, enhanced optical damage tolerance, chemical, thermal, and mechanical stability.

For the visible and ultraviolet (UV) spectral range, the nonlinear crystals of β-BaB_{2}O_{4} (BBO) and LiB_{3}O_{5} (LBO) have been established as materials of choice for a range of frequency conversion applications. They offer wide transmission range well into the UV and high optical damage threshold. However, the relatively low nonlinear optical coefficients confine the use of BBO and LBO predominantly to applications involving high pulse energies or limited phase-matching geometries.

Bismuth borate, BiB_{3}O_{6} (BIBO), is a newly developed nonlinear material [1,2] with unique optical properties for frequency conversion in the visible and UV. It combines the advantages of UV transparency and high optical damage threshold, as in BBO and LBO, with enhanced optical nonlinearity, as in KTP. The optical transmission of BIBO extends from 2500 nm in the infrared down to approximately 280 nm in the UV and, as a biaxial crystal, it offers versatile phase-matching characteristics. The effective nonlinear coefficient of BIBO has been reported as 3.2 pm/V at 1079 nm [2], which is larger than those of BBO and LBO, and comparable to that in KTP. Such a combination of properties makes BIBO an attractive nonlinear material for a wide range of frequency conversion applications in the visible and UV spectral regions.

Since the first introduction of BIBO by Hellwig *et al.* [1, 2], a number of frequency conversion experiments have been performed in this material. These include internal second harmonic generation (SHG) of continuous-wave (cw) radiation at 1.06 µm [3], single-pass SHG of pulsed laser at 1.06 µm [4], Q-switched internal SHG at 1.06 µm [5], internal frequency-doubling of cw Nd:YAG laser [6], and photo-induced SHG in partially crystallized BIBO glass [7]. We also recently reported efficient SHG of high-repetition-rate femtosecond pulses into the blue in this material [8].

However, many of the important nonlinear optical properties of BIBO have not yet been fully investigated, limiting the scope for optimum exploitation of this material for a wide range of nonlinear frequency conversion applications. In this paper, we report studies of the important properties of BIBO pertinent to SHG throughout its transparency range from the visible to the near-infrared. We present numerical calculations for the important properties of this material which, to our knowledge, have not been previously reported. We also present full characterisation of our experiments on single-pass SHG of tunable femtosecond pulses into the blue in BIBO, where average powers of as much as 830 mW at conversion efficiencies exceeding 50% have been generated.

## 2. Phase-matching directions and effective nonlinear coefficients

By using the Sellmeier equations and nonlinear optical coefficients reported in [2], we calculated phase-matching (PM) directions for SHG and the corresponding effective nonlinear coefficients, *d*
_{eff}, of BIBO in the principal optical planes, which we designate as (*xyz*). The *d*
_{ijk} coefficients in [2] were measured in crystallographic coordinates system (*XYZ*) and because of monoclinic symmetry of BIBO the axes in this coordinates system are not parallel to the axes in optical coordinates system (*xyz*). Therefore, in order to find the nonlinear optical coefficients in *xyz* coordinates, we transformed the *d*
_{ijk} tensor elements given in the crystallographic coordinates in [2] by the prescribed rotation φ(λ) around *Y* axis and then re-naming the axes according to *X*→*z*,*Y*→*x* and *Z*→*y*. Following this rotation, the nonlinear tensor elements, *d*
_{ijk}, in the *XYZ* coordinates are transformed to ${d}_{\text{ijk}}^{\prime}$ in the optical coordinates, *xyz*, with values of the non-zero elements given in Table 1. The calculations were performed at a wavelength of 810 nm, but given the small wavelength independence of the ${d}_{\text{ijk}}^{\prime}$ tensor there should be little variation in the values of nonlinear coefficients.

The calculations of phase-matching were performed in the optical *xyz* coordinates, with θ representing the polar angle relative to the optical *z*-axis and φ representing the azimuthal angle measured from the *x*-axis in the optical *xy* plane. For simplicity, and in analogy with uniaxial crystals, for calculations of phase-matching in each plane we designate the polarization direction normal to the plane by *o* and the other allowed polarization direction parallel to the plane by *e*. In the *xy* plane, type I(*o*+*o*→*e*) phase-matching is available over limited fundamental wavelength region of 540–610 nm in the visible, as shown in Fig. 1(a). The effective nonlinear coefficient for this phase-matching scheme can be shown to be:

The variation of the effective nonlinearity is also shown in Fig. 1(a), ranging from *d*
_{eff}~0.3 to 2 pm/V across the tuning range. Type II (*o*+*e*→*e*) phase-matching is also available in the *xy* plane, again over a limited fundamental wavelength range of 690–790 nm, as shown in Fig. 1(b). The effective nonlinear coefficient in this case is given by:

The effective nonlinearity varies between *d*
_{eff}~0.3 and 1.65 pm/V across the tuning range, as also shown in Fig. 1(b).

In the *yz* plane, type I (*e*+*e*→*o*) phase-matching is available for angles between θ~90° and θ~180°, over a fundamental wavelength range from 0.542 to 1.18 µm, as shown in Fig. 2(a). The same phase-matching scheme is available for angles between θ~160° and ~180° in this plane over a fundamental wavelength range from 2.33 to 3 µm, as shown in Fig. 2(b). The effective nonlinear coefficient for this phase-matching scheme is given by:

The variation of the effective nonlinearity across the tuning range for this phase-matching configuration is also shown in Figs 2(a)–(d). The effective nonlinearity has a relatively large value, *d*
_{eff}>3 pm/V, across most of the tuning range in both schemes, making *yz* the most important plane for nonlinear optical applications. It is also interesting to note that in the *yz* plane, for a given fundamental wavelength, there can be two possible phase-matching solutions for SHG at angles θ and 180-θ, with different effective nonlinear coefficients. This can be seen from comparison of Fig. 2(a) with Fig. 2(c) and Fig. 2(b) with Fig. 2(d). However, due to the monoclinic symmetry of BIBO, type II phase-matching in *yz* plane is not possible, because of vanishing *d*
_{eff}.

In the *xz* plane, there are three phase-matching possibilities. For small angles of approximately 1°<θ<11°, type I (*o*+*o*→*e*) phase-matching over a fundamental range of ~1.18 to 2.33 µm is possible, as shown in Fig. 3(a). The corresponding effective nonlinear coefficient is given by:

The variation of the effective nonlinearity is also shown in Fig. 3(a). It can be seen that relatively large effective nonlinearity, *d*
_{eff}~3 pm/V, is available across the tuning range. For larger angles of around 35°<θ<90°, alternative type I (*e*+*e*→*o*) interaction is also available for fundamental wavelengths from ~610 nm to 3 µm, but at reduced effective nonlinearity with a maximum value, *d*
_{eff}~1.6 pm/V, as in Fig. 3(b). The effective nonlinearity in this case is given by:

Finally, type II (*o*+*e*→*o*) phase-matching in the *xz* plane is also available for fundamental wavelengths between 795 nm and 3 µm, with *d*
_{eff} >2 pm/V across most of the tuning range, as shown in Fig. 3(c). The effective nonlinear coefficient for this interaction is given by:

From the above, it may be concluded that the most effective phase-matching schemes for SHG are type I (*e*+*e*→*o*) in the *yz* plane, Figs 2(a) and 2(b), and type I (*o*+*o*→*e*) in the *xz* plane, Fig. 3(a). The combination of these two schemes provides tuning coverage across the full transparency of the BIBO, while at the same time maximising the effective nonlinearity at 2.8<*d*
_{eff}<3.5 pm/V. Therefore, in the discussion to follow, we will focus only on these two phase-matching configurations and will present studies of the parameters relating to these two schemes. We also report experimental results on type I (*e*+*e*→*o*) SHG in the *yz* plane.

It is also interesting to note that type I (*o*+*o*→*e*) phase-matching in *xz* plane is in fact the continuation of type I (*e*+*e*→*o*) interaction in *yz* plane, so that the phase-match directions for wavelengths shorter than ~1.18 µm lie within the *yz* plane and for longer wavelengths enter the *xz* plane, until they return to the *yz* plane for wavelengths longer than ~2.33 µm. This behaviour, and the corresponding effective nonlinearity, is shown in Fig. 4, where the phase-match directions in the two planes are presented in a single plot.

## 3. Experiments

The SHG experiments were performed using a Kerr-lens mode-locked Ti Sapphire laser (Coherent, Mira 900) as the fundamental pump source. The laser delivered pulses with durations of ~130 fs at 76 MHz repetition rate, with an average power of up to 1.8 W over a tunable range 750–950 nm.

## 3.1. Tuning characteristics

Several BIBO crystals of different lengths between 0.4 mm and 1.4 mm were used in our experiments. The crystals were grown using the top-seeded technique and crystallization was carried out under conditions of low temperature gradients [7]. As-grown BIBO single crystals were confined with flat crystallographic faces and had dimensions of 20×20×15mm. They were cut for type I phase-matching (*e*+*e*→*o*) phase-matching in *yz* plane (ϕ=90°) at internal angles close to θ=156° at normal incidence, and the facets were uncoated. The crystals were mounted on a precision rotation stage with one arc minute accuracy and for each sample the focusing condition, fundamental wavelength and crystal orientation were optimised to yield maximum SHG power. All experiments were performed at room temperature. Figure 5 shows the experimental data obtained for a 1.4-mm crystal and the corresponding tuning curve calculated from the Sellmeier equations [2]. We were able to achieve a wavelength tuning range of from 375 to 435 nm in the blue, limited by the crystal aperture at larger angles away from normal incidence. It is also evident from Fig. 5 that excellent agreement between the experimental data and the calculated tuning range is confirmed.

## 3.2. Output power and conversion efficiency

In addition to the effective nonlinearity and fundamental intensity, the attainment of maximum SHG output power and efficiency is dependent on a number of other factors. A particularly important parameter is spatial walkoff, which causes angular separation of orthogonally polarised *o* and *e* waves within the nonlinear crystal, effectively reducing the gain length for SHG. By using the formalism for biaxial crystals [9], we calculated walkoff angles corresponding to the different types of phase-matching in the three optical planes of BIBO. The results for type I phase-matching in the *yz* and *xz* planes are shown in Figs. 6 and 7, respectively. In the *yz* plane, the walkoff angle for type I (*e*+*e*→*o*) SHG varies between ~0 mrad and ~75 mrad for fundamental wavelengths between 0.57 and 1.17 µm, as shown in Fig. 6. As can also be seen from the plot, over the fundamental range of ~2.3 to 3 µm the walkoff angle varies from 0 mrad to ~40 mrad. For type I (*o*+*o*→*e*) phase-matching in the *xz* plane, the walkoff angle varies between 0 and ~30 mrad for fundamental wavelengths between ~1.18 and 2.33 µm, as shown in Fig. 7.

Of particular interest in these experiments is type I (*e*+*e*→*o*) SHG in *yz* plane, where the walkoff angle varies between ~40 mrad and ~65 mrad over the tuning range of our fundamental laser (750–950nm). This implies that for a focused beam waist radius *w*
_{0}~50µm, say, the generated SHG pulses produced at the focus will be completely separated from the fundamental after approximately 2–4 mm of propagation length. However, this so-called aperture length will also depend on the exact position of the focus within the crystal, so that maximisation of efficiency for a given crystal length will require the optimisation of focusing strength and position within the crystal. Therefore, in order to obtain the optimum conditions for maximum SHG conversion efficiency and output power, we used several BIBO samples of varying lengths from 0.4 mm to 1.4 mm, in combination with different focusing conditions using various lenses with focal length between 30 mm and 200 mm. The crystal sample and the lens were each placed on a translation stage, allowing the beam waist location within the sample to be optimised.

Using this optimisation procedure, we found that the SHG conversion efficiency and output power improved with increasing crystal length. The highest SHG output power and efficiency was obtained with the longest crystal, 1.4 mm in length. The results at a second harmonic wavelength of 406 nm are shown in Fig. 8.

The maximum average second harmonic power generated in this crystal was 830 mW for 1.65 W of fundamental input power, corresponding to a conversion efficiency of 50.3%. At the maximum fundamental power, there is evidence of saturation in SHG efficiency, with the value is remaining close to 50%. However, there could be further scope for increase output powers at this efficiency by using increased fundamental power and larger the crystal lengths. We also compared the performance of BIBO with that of BBO crystal. Using the two BIBO crystals of lengths 0.4 mm and 0.7 mm, we compared the second harmonic output power with a BBO crystal of length 0.5 mm, under the same experimental conditions. Optimised focusing using a *f*=80 mm focal length lens was used to ensure maximum output power. The results are shown in Fig. 9, where the superior performance of BIBO is clearly evident. The data correspond to a second harmonic wavelength of 406 nm. The maximum average SHG power generated with the 0.4-mm BIBO crystal was 450 mW, compared to 336 mW obtained with BBO, implying ~34% power enhancement despite a shorter crystal length.

## 3.3. Double phase-matching

According to monoclinic symmetry of BIBO, for a given fundamental wavelength there can be two possible phase-matching directions for type I (*e*+*e*→*o*) SHG. This is shown in Fig. 10 for a range of fundamental wavelengths from 700 nm to 1000 nm. The behaviour is similar for other fundamental wavelengths. Depending on the azimuthal angle, ϕ, the magnitude of *d*
_{eff} for the two directions can be different, as shown in Fig. 11 for the fundamental wavelength of 812 nm. In the *xz* plane (ϕ=0), the magnitude of *d*
_{eff} for both directions is the same, but as we depart this plane and move towards *yz* plane along the phase-match direction (ϕ>0), the values of *d*
_{eff} for the two directions depart. As can be seen from Fig. 11, the magnitude of *d*
_{eff} for the 90°<θ<180° directions are substantially larger than those for the 0°<θ<90° directions over nearly all values of ϕ>0. In the *yz* plane (ϕ=90°), the magnitude of the *d*
_{eff} for the 90°<θ<180° direction is ~3.2 pm/V compared with only ~0.5 pm/V for the 0°<θ<90° direction.

For experimental observation of this property in the yz plane, a number of BIBO crystals were cut at (θ,ϕ)=(180°,90°), so that we could measure SHG power for two phase-match angles, θ and 180°-θ, in the same sample and with the same amount of reflections. For a 1-mm thick crystal, we measured SHG power as a function of the fundamental power at a wavelength of 812 nm in two phase-matching directions, θ=152.3° and θ=27.7°, under identical focusing conditions. The results are shown in Fig. 12. For a low fundamental power of ~90 mW, we measured a SHG power of 3 mW for the θ=152.3° phase-match direction, compared to only ~0.2 mW in the θ=27.7° direction, indicating a ratio of 15 times in conversion efficiency. At high fundamental power of 1.18 W, the SHG power for the θ=152.3° phase-match direction was 430 mW, compared to 85 mW for the θ=27.7° direction, implying a difference in conversion efficiency of 5 times. From numerical calculations, *d*
_{eff}~3.2 pm/V for θ=152.3°, whereas and *d*
_{eff}~0.54 pm/V for θ=27.7°. Therefore, our measurement results are consistent with the predicted phase-matching behaviour. The smaller difference in efficiency values between the two directions at higher fundamental powers is a consequence of saturation effects setting in at the higher powers.

## 3.4. Phase-matching acceptance bandwidths

Another important parameter in the attainment of high conversion efficiency and output power is spatial and spectral phase-matching acceptance bandwidths for the SHG process. These parameters define the tolerance of phase-matching to the spatial and spectral spread of the input beam. The results of our SHG internal angular acceptance bandwidth calculations for type I (*e*+*e*→*o*) phase-matching in *yz* plane and type I (*o*+*o*→*e*) phase-matching in *xz* plane are shown in Figs. 13 and 14, respectively. The values were calculated from the sensitivity of phase-mismatch, Δ*k*, to the spectral spread and angular divergence of the fundamental using the appropriate Sellmeier equations [2].

We were able to experimentally verify our angular acceptance calculations through direct measurements of SHG power in the 1.4-mm crystal. In order to eliminate any errors due to the broad bandwidth of femtosecond pulses, we used the fundamental laser in cw operation.

In addition, to minimise any phase-mismatch effects due to strong focusing, diffraction, and beam divergence we employed long distance focusing using a *f*=200 cm lens. We further apertured the input fundamental before the crystal to eliminate any residual phase-mismatch due to angular spread of the beam.

Using this arrangement, we were able to measure SHG powers in the nW-µW range and accurately determine the angular acceptance for the process. The result at a fundamental wavelength of 812 nm is shown in Fig. 15, together with our numerical calculation. The measured FWHM internal angular acceptance bandwidth is 0.34 mrad.cm [10]. As evident from the plot, there is excellent agreement between the experimental data and calculation.

Considerations of spectral acceptance bandwidths are also highly important in the presence of femtosecond pulses with large spectral bandwidths. This parameter can have a dramatic effect on the second harmonic power and efficiency, as well as the output pulse duration. The results of our spectral acceptance calculations for type I (*e*+*e*→*o*) phase-matching in *yz* plane and type I (*o*+*o*→*e*) phase-matching in *xz* plane are shown in Figs. 16 and 17, respectively.

We similarly determined the spectral acceptance bandwidth for type I (*e*+*e*→*o*) SHG in *yz* plane through direct experimental measurements. By using the fundamental laser in cw operation and similar focusing arrangement as in angular acceptance measurements, we were able to measure SHG powers in the nW-µW range and accurately determine the spectral acceptance for phase-matching. The result for the 1.4-mm crystal at the fundamental wavelength of ~812 nm is shown in Fig. 18, together with the numerical culation. As can be seen, there is excellent agreement between the data and numerical curve.

The measured FWHM spectral acceptance is 1.8 nm, corresponding to a normalised value of 0.25 nm.cm. This value is also in excellent agreement with the calculated value of 0.24 nm.cm.

We also repeated our spectral acceptance measurements with crystals of different lengths and consistently obtained identical results corresponding to normalised values of ~0.25 nm.cm, hence confirming the accuracy of our calculations and measurements. It is interesting to note that for type I (*o*+*o*→*e*) SHG in *xz* plane (Fig. 17), the spectral acceptance bandwidth in the 1.7-2 µm region can be very large, approaching ~80 nm.cm. This opens up the possibility of generating ultrashort second harmonic pulses at high efficiencies by exploiting this phase-matching scheme with suitable fundamental laser sources in this region.

## 3.5. Temporal characteristics

In considering the temporal characteristics of second harmonic pulses, it is important to account for the effects of group velocity walkoff (GVW) and dispersion (GVD) in the BIBO crystal. Group velocity walkoff can affect output pulse duration and can be a limiting factor in the attainment of maximum efficiency in nonlinear frequency conversion processes. In the SHG process, however, the key role played by GVW is in temporal evolution of the second harmonic pulses. Similarly, GVD can lead to the broadening of fundamental and second harmonic pulses, with consequent effect on the temporal characteristics of the output.

We calculated the magnitude of GVW and GVD for phase-matched SHG in the *yz* and *xz* planes of BIBO. The variation of GVW between the fundamental and second harmonic pulses for type I (*e*+*e*→*o*) SHG in *yz* plane and type I (*o*+*o*→*e*) SHG in *xz* plane across the transparency range of BIBO is shown in Fig. 19.

For type I (*e*+*e*→*o*) SHG in yz plane used in these experiments, the magnitude of GVW varies from 328 fs/mm to 528 fs/mm across the second harmonic tuning range of 375 to 435 nm. The calculated GVD corresponding to the fundamental and second harmonic pulses for the two types of phase-matching in the yz and xz plane are also shown in Fig. 20. For type I (*e*+*e*→*o*) SHG in *yz* plane, the GVD values vary between 141 fs^{2}/mm and 185 fs^{2}/mm for the fundamental and between 389 fs^{2}/mm and 558 fs^{2}/mm for the second harmonic, across the tuning range.

Temporal characterisation of the output was also preformed using cross-correlation measurements between the fundamental and second harmonic pulses in a thin crystal of BBO. The measurements in this case were carried out for two Brewster-cut BIBO samples of lengths 0.4 mm and 2 mm at a second harmonic wavelength of 406 nm. From consideration of group velocity dispersion (GVD~73, 210, 420 fs^{2}/mm for 812, 406, 271 nm pulses) and mismatch (GVM~315 fs/mm between 812 nm and 406 nm pulses) in BBO, we chose a 100-µm thick cross-correlation crystal to ensure minimal pulse broadening for 130-fs fundamental pulses in the sample. Background-free cross-correlation intensity profiles were obtained using an AlGaN detector, with the results shown in Fig. 21. For the 0.4-mm crystal, we obtained a blue pulse duration of ~220 fs, indicating significant broadening from the fundamental pulse. The result for the 2-mm crystal is also shown in Fig. 21, where a blue pulse duration of ~500 fs is obtained. The cross-correlation intensity profile in this case has an asymmetric shape due to the increased effects of GVM in the longer crystal. As the blue and fundamental pulse propagate together through the crystal, the trailing edge of the blue pulse is increasingly stretched, since it is generated by progressively more depleted, less intense fundamental pulse [11]. This leads to the characteristic asymmetry and longer second harmonic pulse length.

The corresponding spectra for the two crystals are shown in Fig. 22, which together with the measured pulse durations result in time bandwidth products of 0.84 and 0.73 nm for the 0.4-mm and 2-mm samples, respectively, hence confirming significant chirp content in both cases. We also performed cross-correlation measurements at other wavelengths, but observed no significant variation across the second harmonic tuning range.

## 4. Conclusions

In conclusion, we have presented studies of nonlinear optical properties of the new nonlinear material BIBO for efficient second harmonic generation throughout the transparency range of this material. We have also demonstrated efficient generation of high-repetition-rate femtosecond pulses tunable over 375–430 nm in the blue at average powers of up to 830 mW and efficiencies in excess of 50%. We attribute the high output power and efficiency in these experiments to the relatively large spectral acceptance bandwidth and low temporal walkoff of BIBO for second harmonic generation of femtosecond pulses. With the use of longer crystals with antireflection-coated end facets and higher fundamental powers, there is further scope for increased second harmonic output powers at high efficiencies close to ~50%. Our results confirm that BIBO is a highly attractive nonlinear material for frequency conversion of femtosecond pulses into the visible and UV.

## Acknowledgments

We thank A. Majchrowski, E. Michalski and I. V. Kityk for the providing the BIBO crystals. M. Ghotbi is highly grateful to Dr S. Nourazar and Amir-Kabir University of Technology for their support. M. Ebrahim-Zadeh gratefully acknowledges personal support of the Institucio Catalana de Recerca I Estudis Avancats (ICREA), Spain.

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