1. Introduction

Ultrafast lasers have generated substantial interest as an alternative platform for fabricating miniaturized photonic components due to the ability to induce permanent, highly localized refractive index changes inside bulk glass [1]. While the underlying mechanism of this phenomenon is reasonably well understood, there is still debate over the effect of ultrafast-laser polarization. A number of groups have reported the formation of nano-gratings in ultrafast-laser-modified regions in fused silica, where the alignment of the nano-gratings have been shown to be dependent on the polarization of the ultrafast laser beam [2–4]. Dynamics of plasma formation and the interference of plasmonic waves and the incoming ultrafast-laser pulse have been cited as explanations for this observed behavior.

Under conditions typical of writing waveguides (astigmatic beam-shaping, ~0.6 NA focusing objective), it has been observed that waveguides written with circularly polarized ultrafast beams can exhibit substantially different refractive changes than waveguides written with linearly polarized ultrafast beams under otherwise identical writing conditions [5,6]. To date, a satisfactory explanation for this behavior has not yet been proposed. Manipulating the polarization of the ultrafast writing beam has become an increasingly common technique to alter or enhance the way in which ultrafast lasers modify different materials [7–9]. Explanations into the underlying cause of this behavior will allow researchers to systematically optimize the polarization for any given application over a range of materials and writing systems.

In this paper, the influence of the polarization of the ultrafast pulses on the induced refractive index is investigated in Schott BK7 borosilicate glass and Kigre Yb:QX phosphate glass. Drawing on these observations, and those made in previous studies in fused silica, it is proposed that glasses exhibit refractive index changes that depend on the polarization of the ultrafast pulses because photo-ionization cross-sections are different for linearly and circularly polarized beams. Depending on the irradiance of the ultrafast writing beam and the energy band gap of the glass, ultrafast modification will fall in one of three regimes. In the first regime, linearly polarized pulses yield a higher refractive index change, in the second, circularly polarized pulses yield a higher refractive index change while in the third regime, both polarizations yield similar refractive index changes.

2. Theory

Ultrafast-laser induced modification of glass is the end result of a cascade of physical processes initiated by nonlinear optical absorption and the generation of free carriers (photoionization) [10]. Higher pulse energies almost always result in a larger refractive index change (whether positive or negative). Larger refractive index changes are therefore indicative of more energy being absorbed by the glass undergoing modification.

It has been observed previously that ultrafast-laser written waveguides exhibit a different refractive index change depending on whether the polarization of the ultrafast laser pulses was linear or circular under otherwise identical writing conditions [5,6]. This behavior implies that the energy absorbed is polarization-dependent. Here, we hypothesize that the polarization dependence of photoionization cross-sections is the origin of this behavior. To test this hypothesis, the polarization-dependency on the refractive index change is measured for several glasses and checked for consistency with photoionization theory.

It has been established theoretically and experimentally that the relative photoionization cross-sections for linearly and circularly polarized beams (denoted σ_l and σ_c respectively) is most strongly dependent on the multiphoton order, N, which is defined as the number of photons an electron is required to absorb and cross the energy band gap of the glass to become a free carrier [11,12]. Reiss identified three behavioral regimes; First, if N = 1, 2 then σ_c / σ_l is approximately 1. Second, if N = 3, 4 then σ_c / σ_l > 1 and third, if N ≥ 5, then σ_c / σ_l < 1. In the context of ultrafast-laser modification, these behavioral regimes can be identified by comparing the refractive index change induced by linearly and circularly polarized pulses. In the first regime (N = 1, 2) the refractive index change induced by each polarization should be equal, while for the second and third regimes (N = 3, 4 and N ≥ 5) the polarization that yields the largest refractive index change should be circular and linear respectively.

2.1 Correcting for the Strong Laser Field

According to this scheme, the only factor influencing whether circularly or linearly polarized pulses yield a higher index change is N, however this is contradicted by previous work showing that the highest refractive index change is obtained using linearly polarized pulses for some powers and circularly polarized pulses for others [6]. To reconcile this, the nature of photoionization at higher irradiances must be considered.

A photoionization event can be described in terms of two contributing ionization mechanisms, multiphoton ionization and tunneling ionization. Multiphoton ionization is the direct absorption of N photons by an electron, resulting in a free carrier being generated and is the dominant mechanism at low irradiances. Tunneling ionization is the tunneling of an electron through the potential barrier that is formed between the valence and conduction bands of the glass when distorted by a strong electric field and is the dominant mechanism at high irradiances. The relative dominance of multiphoton and tunneling ionization is quantified using the Keldysh parameter

To correct for the contribution due to tunneling ionization that arises from the strong laser field, a modified form for the multiphoton order is required. If we take the point at which multiphoton effects become sufficiently negligible as γ = 0.5, then we express the multiphoton order after correcting for the strong laser field to be

3. Experiment

Waveguides were fabricated using a regeneratively-amplified Ti:Sapphire laser operating at a wavelength of 800 nm, a FWHM pulse duration of 120 fs and a pulse repetition rate of 1 kHz. Waveguides were written in two different glasses, Schott BK7 borosilicate glass and Ytterbium-doped Kigre QX phosphate glass. Data for Schott Lithosil fused silica was taken from a previous study [6]. A Berek compensator was used to switch between linearly and circularly polarized beams. A slit was used to astigmatically shape the beam to obtain circularly symmetric waveguides [13]. A 40 × objective (NA = 0.6) was used to focus the ultrafast-laser beam inside the glass samples, which were translated at 25 μm. This objective was chosen to minimize nonlinear beam distortion. Waveguides were written using pulse energies of 300, 400, 500, 600 and 700 nJ, corresponding to peak irradiances of 42, 56, 70, 84 and 98 TW/cm², with one waveguide written at each polarization for each pulse energy used. Peak irradiances were calculated assuming a sech² pulse shape and a Gaussian beam with measured 1/e² beam diameters of 10.1 μm and 1.14 μm. After processing, glass samples were ground back and polished at both end-facets.

Refractive index profiles of ultrafast-laser written waveguides were measured using a refractive near-field profilometer (Rinck Electronik) operating at 637 nm. The purpose of measuring the refractive index profile was threefold; to measure the peak refractive index contrast of the waveguide cross-section, to verify that no significant differences in waveguide morphology existed between waveguides written with different polarizations under otherwise identical conditions and to verify that nonlinear distortion of the ultrafast beam was not present. Note that peak refractive index changes quoted in this paper refer to positive refractive index changes unless stated otherwise.

Figure 1 below shows the measured refractive index profiles of waveguides written in Kigre Yb:QX phosphate glass and Schott BK7 borosilicate glass with circularly and linearly polarized ultrafast pulses possessing a pulse energy of 700 nJ. The circular symmetry of the modified regions verified that spatial distortion of the ultrafast writing beam was negligible. While there is a slight asymmetry in the waveguide shape along the x-direction, this is caused by depletion of the pulses as they pass through focus due to photo-ionization, resulting in a greater amount of energy being absorbed before focus than after. For the waveguides depicted in Fig. 1, the measured 1/e² diameters were 7.7 ± 0.2 μm and 12.6 ± 0.2 μm in X and Y respectively for waveguide a) (Kigre Yb:QX; circularly polarized pulses), 7.5 ± 0.2 μm and 13.0 ± 0.2 μm in X and Y for waveguide b) (Kigre Yb:QX; linearly polarized pulses), 5.0 ± 0.2 μm and 7.5 ± 0.2 μm for waveguide c) (Schott BK7; circularly polarized pulses) and 5.0 ± 0.2 μm and 7.9 ± 0.2 μm for waveguide d) (Schott BK7; linearly polarized pulses). Similar agreement was found for waveguides written at other pulse energies. This verified that no significant differences in morphology existed between waveguides written with different polarizations under otherwise identical writing conditions.

 

Fig. 1 (Color online) Refractive index profiles of waveguides written in a) Kigre Yb:QX with circularly polarized pulses, b) Kigre Yb:QX with linearly polarized pulses, c) Schott BK7 with circularly polarized pulses, d) Schott BK7 with linearly polarized pulses. All waveguides in this image were written with a pulse energy of 700 nJ. The ultrafast writing beam propagated from left to right.

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4. Results

The peak refractive index change of waveguides written in Schott BK7 borosilicate glass as a function of the peak irradiance of the writing beam is shown in Fig. 2 . The refractive index change induced by linearly polarized pulses was very slightly greater than the refractive index induced by circularly polarized pulses, in most cases the difference was less than the experimental uncertainty.

 

Fig. 2 Peak refractive index change induced in BK7 by circularly and linearly polarized ultrafast pulses as a function of the peak irradiance.

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The peak refractive index change of waveguides written in Kigre Yb:QX phosphate glass is shown in Fig. 3 . In contrast to the results in BK7 glass, circularly polarized pulses clearly induce a higher refractive index change for all the tested pulse energies except at 300 nJ, where the difference was negligible.

 

Fig. 3 Peak refractive index change induced in Yb:QX by circularly and linearly polarized ultrafast pulses as a function of the peak irradiance. The data points at 42 TW/cm² overlap with one another.

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5. Discussion

Experimental observations can be explained through calculation of the modified multiphoton order and linking it back to one of the three regimes of polarization-dependent behavior identified by Reiss. By calculating the band gap for each glass and using Eq. (3). N’ was calculated as a function of irradiance. A summary of these calculations is shown below in Fig. 4 .

 

Fig. 4 N’ (solid lines) and (2E_g – U_p)/2E_photon (dashed lines) as a function of irradiance for Schott Lithosil fused silica, Kigre Yb:QX phosphate glass and Schott BK7 borosilicate glass.

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For Schott BK7 borosilicate glass, the energy band gap was calculated from the transmission spectrum of the glass to be 4.28 eV using the method outlined by Mott and Davis [14]. Assuming a reduced mass close to that of fused silica (m* = 0.64m_e [15]) for irradiances of 42-98 TW/cm², N’ is calculated to be 2. This corresponds to the regime where σ_c / σ_l ≈1 and the refractive index change induced by linearly and circularly polarized ultrafast pulses is roughly equal, which agrees with what is observed when writing waveguides in BK7.

For Kigre Yb:QX phosphate glass, the energy band gap was calculated to be 5.31 eV from the cut-off wavelength measured by Yoneda et al. [16]. Assuming the same value of m* as before, for irradiances from 42 to 69 TW/cm², N’ is calculated to be 3, whereas for irradiances of 69-98 TW/cm², N’ is calculated to be 2. As σ_c / σ_l > 1 when N’ = 3, 4, it is expected that circularly polarized pulses will induce the highest refractive index change up to a peak irradiance of around 69 TW/cm². For irradiances above this, it is expected that the refractive index change induced by linearly and circularly polarized pulses is roughly equal. It is observed however that circularly polarized pulses induce a higher refractive index change all the way up to 98 TW/cm². This difference between prediction and observation can be resolved by considering the presence of optical losses due to photo-ionization and inverse Bremsstrahlung absorption which would act to reduce the peak irradiance of the focused ultrafast pulse, potentially reducing it below 69 TW/cm², even for the highest pulse energy used.

A second disagreement between prediction and observation at 42 TW/cm², where the refractive index induced by circularly and linearly polarized pulses is equal, can be attributed to the presence of defect states in the band structure of Yb:QX; energy states that lie between the valence and conduction bands, which potentially reduces the size of the electronic band gap, reducing N’ to 2 or lower [16]. As defect states are sparsely populated, they are quickly depleted and so this effect diminishes at higher irradiances.

In light of this reasonably strong agreement between prediction and observation, it is worth re-examining past experimental work done in Schott Lithosil fused silica by the authors [6]. It was observed in fused silica that for peak irradiances less than 42 TW/cm² a higher refractive index change was obtained using linearly polarized pulses, whereas for peak irradiances greater than 42 TW/cm² a higher refractive index change was observed using circularly polarized pulses. The behavior indicates that there is a switch in regime from σ_c / σ_l < 1 (N’ ≥ 5) to σ_c / σ_l > 1 (N’ = 3, 4). For fused silica, with a band gap of 7.76 eV calculated from the transmission spectrum, the irradiance at which N’ switches from 5 to 4 is about 48 TW/cm², which is in close agreement to the observed value of 42 TW/cm².

It should be noted that there is substantial amount of uncertainty associated with these calculations, mainly regarding to the value of the reduced mass of the electron-hole pair in the various glasses. Other factors such as optical loss and the non-parabolic band structure of the glasses will also influence these figures. Despite this, strong qualitative agreement is observed between experiment and the predicted behavior.

There are some important practical implications that arise from this explanation for the observed dependence of ultrafast-laser induced refractive index change on the polarization of the ultrafast pulses. Firstly, for large band gap media such as fused silica and crystalline materials the polarization of the ultrafast pulses is expected to strongly influence the resultant modification (at a wavelength of 800 nm) since N is large and thus falls into a regime where the modifications induced by linearly and circularly polarized pulses is very different. In contrast, for small band gap media such as chalcogenide, the polarization of the ultrafast pulses is not expected to influence the resultant modification since N is small.

For ultrafast lasers with shorter wavelengths, the corresponding N for these materials will be lower, which will tend to shift the process into regimes where circularly polarized pulses yield greater modification (N = 3, 4) or where both polarizations yield the same polarization (N = 1, 2). Note too that the ponderomotive energy depends on λ ² so the polarization-dependence is much slower to switch to different regimes with increasing irradiance. For ultrafast lasers with longer wavelengths, the corresponding N will be larger which will tend to shift the process into regimes where linearly polarized pulses yield greater modification. This however, is offset by the fact that the ponderomotive energy increases more rapidly with irradiance, which will tend to push the process into the N = 3, 4 or N = 1, 2 regimes. A recent paper by Eaton et al. found when writing waveguides in fused silica with a 532 nm wavelength laser, circularly polarized pulses yielded waveguides with a smaller mode-field diameter than those written with linearly polarized pulses [17]. This is indicative of a higher refractive index change being induced by circularly polarized pulses, which is consistent with prediction, as N’ = 3, 4 up to an irradiance of 218 TW/cm² for 532 nm in fused silica. It is anticipated that future studies comparing waveguides written with circularly and linearly polarized ultrafast pulses at wavelengths other than 800 nm will continue to test the predictions outlined herein.

6. Conclusion

The dependence of ultrafast-laser induced refractive index on the polarization of the ultrafast pulses was explained using the regimes of polarization-dependency of photoionization identified by Reiss. The contrasting behavior in three different glasses; Schott BK7 borosilicate glass, where the refractive index change induced by circularly and linear pulses was equal; Kigre Yb:QX phosphate glass, where the highest refractive index change was induced by circularly polarized pulses and Schott Lithosil fused silica, where the highest refractive index change induced was induced by linearly polarized pulses below 42 TW/cm² and circularly polarized pulses above 42 TW/cm², was successfully reconciled using this explanation, with good agreement between observation and prediction in all three glasses. This insight allows researchers to take a more systematic approach when optimizing the polarization of the ultrafast laser for their chosen applications.