Rectangular apertures have been used as a simple means to approximate elliptical Gaussian beams in femtosecond direct writing systems to correct for the elongated focus inherent in low numerical aperture (NA) systems. In this work it is recognized that the rectangular aperture, more accurately functions as a diffractive element and hence redistributes the intensity gradient around the focus in accordance to the physical effects of diffraction. A diffractive model for the technique was proposed and confirmed experimentally to investigate the effects of diffraction and the extent of its influence on the focus shape over different conditions. It was found that because of diffraction, the radius of curvature for the leading edge of the focal spot is dissimilar from its trailing edge. However this effect is mitigated when lower processing energy is used and circular waveguides can be obtained.
© 2005 Optical Society of America
In recent years, femtosecond lasers have been found to be able to perform micro-fabrication within the bulk of transparent glass materials through a nonlinear multi-photon absorption process. [1, 2] These ultra-short laser pulses last for only a few femtoseconds (a quadrillionth of a second) and their tremendously short pulse-width makes it easy to attain high peak intensities with low averaged pulse energies. The ultra-short duration of the pulse fundamentally changes some aspects of the laser-matter interaction and thus has opened up many new possibilities in laser direct-writing of optical devices and micro-fluidics [3–6].
When a femtosecond pulse is focused into glass, there exists a spatial correlation between the region of index change and the shape of the intensity distribution around the focus. The field at the focus is not infinitesimally small as predicted by geometrical optics but has a 3D intensity distribution whose contours trace out an ellipsoidal volume. This elongated depth of focus is present in all optical systems and interested readers are directed to  for a detailed treatise on the focusing of scalar waves. Here it suffice to say that when the glass material is exposed to femtosecond laser pulses, the index of refraction is locally modified along an ellipsoidal shape whose form is defined by the N.A. of the optical system and by the laser parameters (beam waist, energy, etc.). This can result in structures with a strong asymmetric transverse profile (see Fig. 1) when the focus is translated perpendicularly to the direction of beam propagation; a direct writing method known in the field as the side-writing technique.
The elongation can be compressed using high numerical aperture (N.A.) oil-immersion objectives to shorten the longitudinal direction of the light distribution at the focus.  However the working distance would then be in the order of several hundred micrometers and the laser induced index change will only be formed close to the substrate surface, thus reducing the writing freedom for 3D structures. Furthermore ultra-short pulses are highly susceptible to spatiotemporal chromatic dispersion that occurs in high N.A. objectives. 
It was recognized that when focused, the different Rayleigh lengths for the major and minor axes of an elliptical Gaussian beam would allow for an improvement in the waveguide aspect ratio (AR). In consequence, a technique using cylindrical lenses to shape a symmetrical Gaussian beam into an elliptical one also allowed the spot size to be changed in the tangential and sagittal planes as well as the relative position of the beam waist, thereby giving adjustable waveguide cross sections and size.  A simpler approach was to place a rectangular slit before the focusing objective, thus approximating the elliptical beam shape.  This had the advantage of having only a single element to align and the theoretical model based on Gaussian beam focusing to model the shape of the region around the focus was further studied.  The element can be easily incorporated into existing systems with minimal modifications; making it an attractive means of preserving the working depth of femtosecond lasers configured for side-writing while eliminating the asymmetric waveguide profiles typically obtained.
In this work it is recognized that a small rectangular aperture positioned just before the objective, more accurately functions as a diffractive element and hence any theoretical modeling of such a setup should take into account the physical effects of diffraction. This diffractive spread introduced by the aperture will compete with the extent of focusing by the lens to influence the final 3-dimensional (3D) shape of the focus. The following sections of this paper give an account of the theoretical and experimental work done to investigate the effects of diffraction and the extent of its influence on the focus shape.
2. Comparison of theoretical models
2.1 Diffraction model
The three-dimensional light distribution near the focus of lens has been and still is the subject of classical studies with many practical applications in a modern engineering context. This shape of the optical field at the focus cannot be computed using geometrical optics or ray tracing, instead diffraction theory is used.
For regions around the focal point of the setup in Fig. 2, the optical field may be analyzed using a linear systems (known also as the angular spectrum) approach, where the complex amplitude distribution at the focus Uf(x,y) is the convolution between the freespace response f(x,y) and the field just after the lens lA(x,y). This is described mathematically as
Where the freespace transfer function is
In this work the incident optical disturbance is modulated by a rectangular transmission mask described as
The field lA(x,y) just after the lens is product of the incident optical disturbance A(x,y) modulated by the rectangular mask and the lens function . Therefore the Fourier transform of the transmitted field just after the lens is
A thorough discussion of the angular spectrum approach can be found in , here we directly present the results obtained from the simulation. Figure 3 shows the calculated intensity distribution around the focus for various rectangular widths wx in a low numerical aperture (N.A of 0.1366) system having a focal length of 2900 units and lens diameter of 800 units. The length wy was selected to be much larger than the beam diameter. The arbitrary units represent multiples of wavelengths and the parameters were selected to meet the paraxial approximation requirement for the angular spectrum approach.
The consequence of adding an aperture in front of the lens is to introduce a diffractive spread that competes with the focusing effect of the lens. The beam intensity is thus redistributed in the transverse plane defined by the U axis of Fig. 2 and the longitudinal propagation direction. When this transverse beam spread is comparable to the longitudinal depth of focus, then the AR of the focus spot can be nearly unity (Fig. 3(c)). The extent of the spread depends on the diffraction introduced; a smaller aperture gives a larger diffraction (Fig. 3(a) and 3(b)) while a large aperture gives smaller diffraction (Fig. 3(d)–3(f)). Here the AR is defined as the width of the beam in the vertical axis of Fig. 3 divided by the length of the focus along the direction of propagation.
The distribution of intensity around the focus is revealed with greater detail using a series of contour plots shown in Fig. 4, where the distribution is observed to be asymmetric about the vertical. Observing the regions of highest intensity, corresponding to the inner most contour line in Fig. 4, we find that the radius of curvature for the leading edge of the focal spot is dissimilar from its trailing edge. Even at the near unity AR condition of Fig. 3(c), the curvatures do not match perfectly and the intensity distribution can resemble an asymmetric ellipsoid. Here a simple parameter r is used to describe the beam irregularity about the vertical, where r is the ratio of the horizontal contour edge distance (l1 and l2) to the point of focus. The model shows that parameter r increases with an increase in the width of the aperture (Fig. 4(a)–4(f)), however the focus will elongate in the direction of propagation similar to the focus of an unmodified system. We then examined the intensity distribution using the Gaussian beam focusing model.
2.2 Elliptical Gaussian beam focusing model
If the physical effects of diffraction from the rectangular aperture are ignored, then the intensity distribution around the focal point of the lens can also be obtained from the equation describing elliptical Gaussian beam focusing. Here we adopt similar notations to review the model proposed in [11, 12], where the intensity can be expressed analytically as:
Here wy0 and zy0 is the beam waist at the focus and Rayleigh length of the major axis of the beam respectively. They are related to the beam waist wx0 and Rayleigh length zx0 of the minor beam axis through wx0 = (Ry/Rx)wy0. Once these parameters are determined, the intensity is calculated from a distance -z to +z around the focus which has been selected as the origin. The term Ry/Rx was used as an indication of the transverse AR of the elliptical beam before focusing. Note: this differs from the definition used in the diffractive model which refers to the AR of the region around the focus. The physical dimensions of the rectangular slit were then tailored to match the required beam ellipticity as defined by Ry/Rx. Hence as the ratio decreases, it implies that the width of the rectangular slit increases correspondingly in comparison to its length. Figure 5 illustrates the longitudinal intensity distributions and corresponding contour plots obtained using this model for various input beam ellipticity in a 0.1366 N.A. system.
As seen in Fig. 5, the elliptical Gaussian beam focusing model gives intensity distributions that have perfect symmetry about the vertical axis and the parameter r = 1for the plots. This behavior is obvious from the coordinate symmetry imposed by Eq. (5); the Gaussian beam waist and curvature are identical on both sides of the origin. In contrast, the radius of curvature for the leading edge of the focal spot in the diffraction model was found to be dissimilar from its trailing edge. The question then arises if this diffractive effect is significant during the direct laser writing process and what is its influence on the structural morphology in the material.
3. Experiment procedure
This work used a 800 mW regenerative amplified Ti:Sapphire laser (Clark MXR) emitting at 775 nm with pulse duration of 150 fs, repetition rate of 1 kHz. The substrates used were commercially available photostructurable glass (Foturan glass) from Schott Glass Corporation. Foturan was selected as the base substrate because of its potential for fabricating both microfluidic and integrated optical devices. A set of 8 mm long diffractive apertures were made from laser ablation of metal sheets, their individual widths covered a range of widths from 120 μm to 300 μm. The diffractive apertures where placed just before the entrance of 50x 0.55 NA and 20x 0.45 NA objectives to modulate the femtosecond beam which originally has a TEM00 transverse profile.
The substrate was placed on a 3-axis motorized stage and the beam focus was adjusted to a depth of 500 μm from the top surface. The sample was then translated along the axis V of Fig. 2 (parallel to the major axis of the aperture) at a speed of 0.1 mm/s over a distance of 10 mm. This configuration is that of an amplified low repetition rate system in a side writing geometry, which inherently has produced asymmetric waveguides. For each writing session at a specific aperture width, multiple lines where written in the substrate bulk with each line drawn at decreasing beam power. The average power of the laser was controlled by a variable attenuator over a range from 140 to 460 mW and we estimate the injected pulse energies range from ~3 to 30 μJ by taking only into account the transmission efficiencies of the various apertures. Although high pulse energies result in damaged waveguides with strong scattering losses in FOTURAN, the higher beam powers were intentionally used for this experiment to allow for a better visual contrast between the regions of material change and rest of the glass. The end surfaces of the written structures where subsequently polished and measurements of the major and minor axis of the profiles observed under a calibrated optical microscope where then used to calculate the structure’s AR.
4. Experiment results
Each panel in Fig. 6 shows the typical cross sectional shape obtained in Foturan glass for each writing session, the profile mirrors the trend seen in the simulated beam profiles when different aperture widths are used for the diffraction model. The laser induced structures also possess different radius of curvature for their leading and trailing edges; they show the same asymmetry about the vertical axis similar to the effect predicted by the rectangular aperture diffraction model seen in Fig. 4. Preliminary experiments conducted with fused silica also indicate similar trends.
Plots of AR against diffractive aperture width used for both the 50x 0.55 NA and 20x 0.45 NA objectives are presented in Fig. 7. The experimentally obtained data points for the two sets of focusing conditions are curve-fitted by a nonlinear regression solution a+b/x1.5+c/x2. It is seen that the AR of the induced structures are higher then unity value for small aperture widths, this is similar to both theoretical models, where an elliptical focal shape orientated in the vertical direction is obtained at small aperture widths. However when the aperture width increases pass an optimum dimension, the focal shape returns to its elongated profile orientated in the propagation direction. The trend seen from the graph suggests that a near-unity AR for a circular focus shape can be obtained for a 150 μm aperture width when using a 50x NA 0.55 objective. At the looser focusing condition of the 20x NA 0.45 objective, the same near-unity AR is achieved using an aperture width of approximately 240 μm.
It was observed from the experiment that the induced structures in glass are also influenced by the pulse energy. The various cross sections obtained using the 50x Na 0.55 objective is shown in Fig. 8 where it is evident that the overall size of the structures decreases at lower pulse energies. The estimated injected pulse energies were calculated based on the transmission efficiencies of the various apertures only. Typical diameters of the laser written structures (for cross sections showing AR≈1) ranged from approximately 8 μm to 35 μm. At the higher pulse energies the substrate showed evidence of micro-cracking and material damage, these are absent at lower (~ 6 μJ) energy and distinct regions are obtained that can act as waveguides. However no visible change in foturan glass was observed at pulse energies less than 4 μJ. Another trend observed was that the structures show an improvement in the AR and profile symmetry for some aperture dimensions (Fig. 8(b), 8(c) and 8(d)) when processing with lower pulse energies.
5. Analysis and discussion
The similarities between the simulated intensity profiles seen in Fig. 3 with the cross sectional shape obtained during the experiment (Fig. 6 and 8) confirms that there is a spatial correlation between the index change region and the shape of the intensity distribution around the focus. Of note is that in Fig. 3 the radius of curvature for the leading edge of the focal spot in the diffraction model is dissimilar from its trailing edge, hence the focal profile has a slight asymmetry about the vertical axis. This asymmetry is also reflected in the structural profiles seen in Fig. 6 and 8. The elliptical Gaussian beam model however does not show this trend.
This spatial correlation is defined by the threshold required for nonlinear material processing which is itself dependant upon the material properties of the glass, thus only portions of the beam focus that have intensity gradients above the threshold will induce the localized refractive index changes. Consequently by varying the critical aperture dimension (wx) in both the simulation and experiment allows for a redistribution of intensity around the focus; in essence ‘molding’ the 3-dimensional intensity gradient around the region of focus. This intensity redistribution is influenced by the effects of diffraction which can cause the focal spot to possess different radius of curvature for the leading and trailing edges. The optical micrographs of Fig. 6 and 8 also show this asymmetry, hence providing evidence that the rectangular aperture more accurately functions as a diffractive element rather then just a means of approximating an elliptical Gaussian beam.
This diffractively obtained 3-dimensional focal volume then forms the virtual ‘tool-tip’ for femtosecond laser direct writing and can give a near unity AR for the focus at an optimum dimension where the diffractive effect of the aperture balances the focusing power of the lens. The aperture dimensions are themselves determined by the focusing conditions used as seen in Fig. 9 where the AR was modeled for a range of aperture widths and NA using the angular spectrum approach. The same nonlinear regression solution was also used to curve fit the data points. This provides further confirmation that the model accurately predicts the physical experiment.
The simulated plots in Fig. 9 show the AR decreasing according to a general trend described by the curve a+b/x1.5+c/x2 as the width of the diffracting aperture increases. This is similar to the experimentally obtained data where the transverse spread of the intensity profile around the focal region can be larger then the longitudinal depth of focus at small aperture widths. This gives rise to a greater-than-unity AR and an elliptical intensity distribution with major axis orientated along the axis V of Fig. 2. At larger aperture widths, the AR is less than unity and the elliptical intensity distribution lies along the beam propagation axis instead. The graphs indicate a cross over point between the trends where it is possible to obtain an AR of nearly 1 for the different focus used. The optimum aperture dimension required is smaller for higher NA objectives and larger at looser focusing. This implies that the diffraction competes with the focusing power of the lens, hence tighter focusing requires a greater diffractive spread (a smaller aperture width) to balance the finite dimensions of the focal spot.
The influence of pulse energy was modeled by selectively mapping the intensity contours of the region around the focus and the results are presented in Fig. 10. As femtosecond laser processing is a threshold dependant process, we believe that this is a suitable method to examine the evolution of the focal shape over a range of beam powers and the corresponding pulse energy.
Similar to the experimentally obtained data, the model shows that the overall size of the focal shape decreases when the beam is modeled at lower power and the AR may improve for some aperture dimensions (Fig. 10(c) and 10(d)). In addition, the lower powers appear to mitigate the effects of diffraction on the focal shape; the leading and trailing edges now have approximately the same radius of curvature. This was also observed experimentally in Fig. 8(b) and 8(c); suggesting that the core of the focus has a circular distribution even though the outer regions have intensity profiles modulated by the rectangular aperture. One implication is that at lower pulse energies, there may not be a stringent requirement for a specific aperture size to obtain perfectly circular waveguides. This margin of error serves to underscore the robustness and ease of employing a rectangular aperture to diffractively beam shape a femtosecond laser for material processing.
Preliminary results for a near field profile at the exit surface of such a waveguide written in FOTURAN glass is shown in Fig. 11 where the output of a 632.8 nm HeNe laser was butt-coupled by a corresponding single mode fiber into the polished surface of the waveguide. The writing parameters of 160mw laser power (estimated ~5 μJ), 0. 1mm/s writing speed and a 150 μm aperture width, were used to fabricate a circular waveguide of approximately 8 μm diameter. The beam at the exit surface of the waveguide was observed to have a near Gaussian profile in both the horizontal and vertical directions. For this preliminary result, the index change was approximated by observing the divergence angle from the waveguide and was found to be in the order of 10-4 for the stated process parameters (assuming a step-index profile). However femtosecond written waveguides have index profiles that are influenced by the focal shape , therefore even though waveguide AR may be improved using diffractive apertures, such modulated intensity distributions (Fig. 4) may result in corresponding graded index profiles that are non-axially symmetric.
As noted earlier, the diameter of femtosecond laser induced structures ranged from approximately 8 to 35 μm in the substrate with the processing parameters of ~5 to 20 μJ using the 50x NA0.55 objective. This range of specifiable structure sizes can be used for laser processing of Foturan glass which is suitable for creating micro-fluidic channels after proper annealing procedures and subsequent etching with HF acid. [5, 15] However the effects of diffraction on channel cross sections will be increasingly apparent as the structure scales with increasing pulse energies. Such irregularities in the cross sections can be significant for micro-fluidic applications where reagents, solvents or biological fluids are delivered to a reaction site in nanoliter volumes. Hence knowledge of channel shapes will be useful in accurately predicting and controlling fluid flow in the required quantities.
In summary, the aspect ratio of waveguides or micro-fluidic channels made using a femtosecond direct write process can be improved by redistributing the intensity distribution at focus with the aid of a small rectangular aperture. This was previously modeled as a means to approximate elliptical Gaussian beams which when focused will give circular waveguide profiles. In this work it is recognized that the rectangular aperture, more accurately functions as a diffractive element and hence redistributes the intensity gradient around the focus in accordance to the physical effects of diffraction. A diffractive model for the technique was proposed and confirmed experimentally to investigate the effects of diffraction, the extent of its influence on the focus shape and results over different focusing conditions and pulse energy. It was found that this more accurately models the intensity profile at the focus and the morphology of the femtosecond laser written structures.
References and links
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