1 Introduction

Direct laser writing (DLW) in thick evaporated films of arsenic trisulfide (As\(_2\)S\(_3\)) allows for the creation of arbitrary three-dimensional (3D) microstructures that have large linear and nonlinear refractive indices [1–3] as well as good transmission in the near- to mid-infrared [4]. Two-dimensional (2D) structures in As\(_2\)S\(_3\) such as waveguides have been generally produced by the conventional process involving lithography followed by wet or dry etching [5, 6]. Recently, thermal nano-imprint lithography has also been shown to be effective for waveguide fabrication in As\(_2\)S\(_3\) [7]. These techniques are, however, limited to 2D structures and must be repeated multiple times if 3D structures need to be formed [8]. In comparison, the highly localized photo-modification produced by the focused femtosecond laser used in DLW allows a higher degree of freedom in structure fabrication and design [9–11], as the focus can be easily translated in any arbitrary direction [12–14].

The refractive index of the As\(_2\)S\(_3\) is, however, much larger than that of the immersion oils needed when using a high numerical aperture (NA) microscope objective to focus the laser beam inside the sample. This leads to severe spherical aberration of the laser wavefront [15, 16] which elongates the focal intensity distribution and hence reduces the resolution of the fabricated structures [12–14, 17]. To counteract the elongation, multiple lines spaced at equidistant lateral positions can be written, so that when combined they create a more symmetric rod [11, 12]. While this has the advantage of being simple to implement into a DLW system, it has a number of shortcomings. Primarily, it does not eliminate the reduced resolution due to the aberration and additionally the total fabrication time is increased by a factor equal to the number of repeated lines.

In order to reduce the elongation as well as to obtain the optimum resolution and fabrication speed, it is necessary to compensate for the aberration using adaptive optics. This method has been demonstrated in low refractive index materials [18, 19] and has recently been extended to a high refractive index lithium niobate crystal [20, 21]. By incorporating adaptive optics into the DLW systems, the elongation can be reduced to that produced by a diffraction-limited focal spot. The peak intensity of the focal spot should consequently increase, and any shift in the focal region due to refraction can also be removed.

Here, we describe the use of a liquid crystal spatial light modulator (SLM) to compensate for the aberration caused by the refractive index mismatch in As\(_2\)S\(_3\) thick films. We detail the effect of the aberration on the focal quality, i.e., the point spread function, at different fabrication depths, and then investigate the axial response as a measure for aberration compensation. Finally, we experimentally demonstrate the compensation of the aberrations due to the refractive index mismatch in As\(_2\)S\(_3\) with an SLM.

2 Effect of spherical aberration in As\(_2\)S\(_3\)

Fig. 1
figure 1

a Calculated focal intensity distributions when a plane wave is focused into depths of 0, 10 and 20 \(\upmu \)m in As\(_2\)S\(_3\). b Comparison of the intensity along the axial direction in a. c– e Plot of the evolution of peak intensity, FWHM, and focal shift, respectively, as the depth is increased from 0 to 100 \(\upmu \)m

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Due to the high NA of objectives used in DLW, the effect of depolarization on the focal intensity distribution must be taken into account. A full vectorial solution of the electric field in the focal region, called the point spread function, can be obtained through a Debye integral across the interface where the refractive index mismatch occurs [15], such that the electric field for the three orthogonal polarization directions are calculated independently and superimposed to form the total intensity distribution at the focus. The resulting phase imparted onto the beam by the refractive index mismatch can then be easily determined in terms of a spherical aberration function. In the case of a planar interface between isotropic immersion (n 1) and fabrication (n 2) refractive indices, the spherical aberration function is given by [15, 16]:

$$\psi = k d(n_2 \cos \theta _2 - n_1 \cos \theta _1) $$
(1)

where \(\theta _1\) and \(\theta _2\) are the angles of convergence in the immersion and fabrication media, respectively, and \(d\) is the nominal focusing depth when refraction is not considered. The factor \(k\) is the wavenumber given by \(2 \pi /\lambda \) where \(\lambda \) is the wavelength in vacuum.

For an objective with NA = 1.4 and an immersion oil (\(n_1\)) of refractive index 1.52, focused into depths of 0, 10 and 20 \(\upmu \)m of an as-deposited As\(_2\)S\(_3\) film with refractive index (\(n_2\)) 2.35 at a wavelength of 800 nm, the effect of the aberration on the focal intensity distributions in the ZX plane is shown in Fig. 1a. The distributions have been normalized to the same peak intensity and the beam is polarized along the x axis. It is clearly seen that significant aberration is present when the laser beam is focused through the interface. Because the volume of the focal intensity distribution is increased, the peak intensity within the focus is reduced as shown in the plots along the axial direction in Fig. 1b. Figure 1c–e shows the evolution of the peak intensity; the full width at half maximum (FWHM) in the axial and radial directions; and the focal shift, respectively, as the nominal focal depth is increased from 0 to 100 \(\upmu \)m. A dramatic drop in the peak intensity and an increase in the axial FWHM occur across the first 20 \(\upmu \)m, while the radial FWHM increases only slightly. A focal shift is also present which increases approximately linearly with the focal depth.

This degradation in the focal intensity distribution poses a problem to the use of DLW in As\(_2\)S\(_3\) as the quality of fabrication is now sensitive to the focusing depth. Regions of a structure written close to the refractive index interface receive high intensity with a symmetrical focal point, while regions further away have reduced intensity, higher aspect ratio and may experience the effect of side lobes that cause splitting of the written point. To ensure that the highest quality structures can be fabricated in this material, it is necessary to remove this aberration from the fabrication system. This can be achieved through the introduction of an additional phase generated by a liquid crystal-based SLM [21], to the converging laser wavefront that cancels out the phase produced by the refractive index mismatch.

3 Axial response as a measure of aberration correction

For thick films comprising two or more reflective surfaces, the axial response method can be used to detect the focal distortion as the spherical aberration predominantly affects the focus along the optical axis. This is achieved by aligning a detector confocally with the objective and by scanning the focus through the film [22]. At each interface, part of the focus is reflected and detected, allowing for the level of aberration and effectiveness of aberration compensation to be observed. The axial response can be modeled theoretically for comparison with experimental results [22]. For a single homogeneous dielectric film of thickness d, the axial response is given by:

$$ I= \left | \int\limits_0^\alpha P^2(\theta _1)R(\theta _1) \exp (-2 i k z n_1 \cos {\theta _1})\exp (-2ik\phi ) \sin {\theta _1} \cos {\theta _1} \;d\theta _1 \right |^2$$
(2)

where \(\alpha \) is the maximum convergence angle of the objective given by \(\sin {\alpha }=NA/n_1\) and \(z\) is the axial coordinate with origin at the first interface. \(n_2\) is the refractive index of the film, while \(n_1\) and \(n_3\) are the refractive indices of the materials before and after it, respectively. The function \(P(\theta _1)\) is the apodization function of the objective, which for a sine condition objective is given by \(P(\theta _1)=\sqrt{\cos {\theta _1}}\). Reflection from the film's two interfaces as well as optical propagation between them is encapsulated in the complex reflection coefficient \(R(\theta _1)\) given by:

$$R(\theta _1) = (r_{||} + r_\perp )/2 $$
(3)

The parallel and perpendicular components \(r_{||}\) and \(r_{\perp }\) can be found via [23]:

$$r = \frac{r_{12} + r_{23} \exp {(2 i k n_2 d \cos {\theta _2})}}{1 + r_{12} r_{23} \exp {(2 i k n_2 d \cos {\theta _2})}}$$
(4)

where \(r_{12}\) and \(r_{23}\) are the Fresnel reflection coefficients [23] for the two refractive index interfaces. The refractive index mismatch aberration is incorporated in Eq. 3, while the presence of the SLM is incorporated by the addition of the phase parameter \(\phi \) in Eq. 2. When \(\phi \) is set equal to the aberration function \(\psi \) in Eq. 1, the effect of the aberration can be completely canceled.

Fig. 2
figure 2

a Calculated axial response of an As\(_2\)S\(_3\) film of thickness 15 μm when aberration is uncorrected (dash), corrected with defocus (solid) and corrected without defocus (dash–dot). bEvolution of the As\(_2\)S\(_3\)–oil signal intensity change as the level of aberration compensation is increased. c Evolution of the axial FWHM change as the level of aberration compensation is increased from 0 to 200 %

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To match the experimental samples, we considered a 15-μm thick film of As2S3 (\(n_2\) = 2.35) deposited on a glass coverslip (\(n_1\) = 1.52) and surrounded by oil (\(n_3\) = 1.52) as depicted in Fig. 3b. The refractive index of the objective immersion oil was assumed to match that of the glass coverslip and hence the related mismatching can be ignored. The NA and the wavelength are as described above. Figure 2a shows the calculated axial response under three conditions. The dashed curve shows the response when no aberration compensation is applied, the solid curve when full aberration compensation is applied and the dash–dot curve when only the spherical aberration component of the aberration function is applied. The peaks on the left correspond to the first interface (glass–As2S3) and are not of interest because they are unaffected by the As2S3 film. The peaks on the right correspond to the second interface (As\(_2\)S\(_3\)–oil) and are affected by the aberration caused by the refractive index mismatch. As such, when no aberration compensation is applied (dashed curve) the reflection has reduced peak intensity as the energy is dispersed into a number of axial side lobes. The location of the detected signal also differs from the location of the As\(_2\)S\(_3\)–oil interface as refraction shifts the focus deeper into the As\(_2\)S\(_3\), causing it to reflect from the interface earlier. When full aberration compensation is applied (solid curve), the reflection from the As\(_2\)S\(_3\)–oil interface is now unaberrated as the aberration from the refractive index mismatch cancels out the phase produced by the SLM. A small drop in intensity is seen due to the unavoidable Fresnel reflection from the glass–As\(_2\)S\(_3\) interface. When only the spherical aberration component of the aberration function is compensated (dash–dot curve), the reflection is diffraction limited with the same focal size as the full compensation case but maintains the early reflection due to the presence of the defocus term. Figure 2b, c show the evolution of the peak intensity and the FWHM of the As\(_2\)S\(_3\)–oil reflection, respectively, as the amount of the aberration compensation is increased from 0 to 200 %. The intensity climbs to a maximum only when compensation is at 100 %, and then drops down as the aberration is overcompensated up to 200 %. The FWHM also reaches a minimum at the 100 % compensation, where it matches the diffraction-limited value from the glass–As\(_2\)S\(_3\) interface.

4 Experimental measurement of aberration compensation

Figure 3a shows how an SLM can be incorporated into a DLW system. A femtosecond laser (Vitesse, Coherent Scientific) operating at a wavelength of 800 nm, a pulse width of 70 fs and a repetition rate of 80 MHz was expanded to fill the active area of a reflection type of liquid crystal SLM. The SLM (Boulder Nonlinear Systems HSPDM5120785-PCIe) altered the wavefront of the beam according to the computer input, and the beam was then compressed down to match the back aperture of the microscope objective (Olympus UPLANSAPO 100XO) via the f = 300/200 mm 4f lens combination. A second 4f system comprising two 150-mm lens was used to relay the beam to the objective. The use of the 4f systems minimizes any diffraction of the SLM phase pattern or pixel structures, and allows for spatial separation of the SLMs diffractive orders by applying a phase ramp onto the SLM. Movement of the objectives focal point was achieved via a 3D piezo-electric stage. The confocal detection system comprised a photomultiplier tube (PMT), a 100-mm lens and a 10-μm pinhole. As the laser beam passes through the film both in the forward and backward directions, the refractive index mismatch aberration is imparted twice and doubles the aberration detected by the PMT. To account for this, the PMT was placed before the SLM so that the reflected wavefront from the As\(_2\)S\(_3\) film passed the SLM a second time and corrected for the additional aberration. A 0–20\(\pi \) phase ramp was applied to the SLM to separate the diffractive orders, with the first order being aligned along the optical axis.

Fig. 3
figure 3

a Experimental setup used for confocal axial response with an SLM. b Schematic of the sample structure and objective position

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Fig. 4
figure 4

Axial response of a 15-μm thick as-deposited As2S3 film without (dash) and with (solid) compensation of the refractive index mismatch aberration

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Fig. 5
figure 5

Axial response of a 15-μm thick film of as-deposited As2S3 when aberration compensation is applied at a level equivalent to a film thicknesses of a 0 μm, b 7 μm, c 14 μm and d 22 μm. The signal is reflected from the As2S3–oil interface

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To observe the axial response of an As2S3 film experimentally, a 15-μm thick film of As2S3 that had been deposited using thermal evaporation onto a 170-μm thick glass coverslip was used to test aberration compensation using the liquid crystal-based SLM. Details of the deposition procedure have been described elsewhere [13]. The film was arranged so that the glass coverslip was located between the As\(_2\)S\(_3\) film and the objectives immersion media (see Fig. 3b). The laser power was set to 14 \(\upmu \)W so that no change occurred in the axial response over a number of scans at the same point on the film. To prevent the total internal reflection from the second As\(_2\)S\(_3\) interface, a thick film of oil was applied to the bottom of the As\(_2\)S\(_3\) to displace any air. Figure 4 shows the axial response recorded when no aberration compensation was applied (dashed) and when aberration compensation was set to compensate for a thickness of 14 \(\upmu \)m (solid). A dramatic improvement in the detected signal from the As\(_2\)S\(_3\)–oil interface is seen, along with a shift of the location of the reflection to 15 \(\upmu \)m position, which matches the film thickness. This indicates both the spherical aberration and the focal shift have been effectively compensated. The small discrepancy between the applied correction which corresponded to 14-μm thickness and the actual film thickness of 15 μm is due to the presence of additional spherical aberration in the optical components used in our system that slightly cancels the spherical aberration caused by the refractive index mismatch. This aberration can be seen as small side lobes in the uncorrected (dashed) glass–As\(_2\)S\(_3\) reflection in Fig. 4.

Taking a closer look at the As\(_2\)S\(_3\)–oil reflections, Fig. 5a shows the detected axial response as the laser focus was scanned through the As\(_2\)S\(_3\) film with no compensation of aberration. The response is quite broad and exhibits a distinct side lobe that indicates strong spherical aberration is present. Aberration compensation was then applied by slowly increasing the strength of phase modulation on the SLM until compensation of an equivalent thickness roughly double that of the As\(_2\)S\(_3\) film thickness was achieved. Figure 5b–d shows the axial response when the strength of phase compensation was equivalent to film thicknesses of 7, 14 and 22 μm, respectively. When correcting for aberration equivalent to a thickness of 7 μm, only partial aberration compensation was achieved, resulting in a slight improvement in peak intensity and some improvement in the axial FWHM. When compensating for aberration equivalent to a film thickness of 14 μm, the peak intensity reached a maximum value more than double that of the fully aberrated axial response. The FWHM at this correction level was also improved by almost a factor of 2.2. When the correction corresponded to an equivalent film thickness of 22 \(\upmu \)m, the aberration was overcompensated and the axial response exhibited the effects of spherical aberration of the opposite sign.

In order to compare to the improvement of the theoretically determined levels, the peak axial response and FWHM values were recorded as the level of aberration compensation was increased. To provide an accurate comparison of the axial response intensity, the measured and calculated values were normalized to the measured and calculated peak intensity of the fully uncorrected cases, respectively. The experimental correction levels were scaled so that the optimal correction, corresponding to a film thickness of 14 μm, was positioned at 100 %. Figure 6a shows the experimental improvement in peak intensity compared with the predictions. As the correction approaches the optimal value, the intensity increases by a factor of up to 2.4: close to the theoretical limit of 2.75. If the correction is increased beyond the optimal level, overcorrection occurs and the intensity enhancement decreases as expected. The evolution of the FWHM is shown in Fig. 6b. As the aberration correction approaches the optimal level, the FWHM drops close to the diffraction limited minimum value, 2.2 times smaller than the original aberrated FWHM. Again as the correction is increased further, the FWHM increases as expected due to overcompensation. The differences between the theoretical and experimental curves are due to limitations in the experiment due to the SLM and small alignment errors in the system that introduce residual aberrations. The limited spatial resolution of liquid crystal SLMs also means that correction accuracy is more difficult with larger phase amplitudes such as those needed when overcorrecting the spherical aberration.

Fig. 6
figure 6

a Comparison of measured (cross) and calculated (solid) improvement of the peak axial response as the aberration correction level is scaled from 0 to 200 % for a 15-μm thick film of as-deposited As\(_2\)S\(_3\). b Comparison of the measured (cross) and calculated (solid) axial FWHM as the aberration correction level is scaled from 0 to 200 %

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5 Conclusion

We have demonstrated compensation for the refractive index mismatch aberration in an As\(_2\)S\(_3\) thick film. We have shown that for a film of As\(_2\)S\(_3\) 15 μm thick, the peak intensity of the focal intensity distribution can be increased by a factor of 2.4 and the axial FWHM reduced by a factor of 2.2. The improved focal shape will allow for the SLM-assisted DLW of more symmetric 3D structures. Furthermore, we have demonstrated that the axial response technique provides an accurate measure for the compensation for spherical aberration in a thick film.