Imaging with Diffractive Axicons Rapidly Milled on Sapphire by Femtosecond Laser Ablation

Fabrication of large area (sub 1 cm cross-section) micro-optical components in a short period of time (approximately 10 min) and with lesser number of processing steps is highly desirable and cost-effective. In the recent years, femtosecond laser fabrication technology has revolutionized the field of manufacturing by offering the above capabilities. In this study, a fundamental diffractive optical element, binary axicon i.e axicon with two phase or amplitude levels, has been designed in three configurations namely conventional axicon, photon sieve axicon (PSA) and sparse PSA and directly milled onto the Sapphire substrate. The fabrication results revealed that a single pulse burst fabrication can produce a flat and smooth profile than pulse overlapped fabrication which gives rise to surface damage and increased roughness. The fabricated elements were processed in IsoPropyl Alcohol and Potassium Hydroxide to remove debris and redeposited amorphous Sapphire. An incoherent illumination was used for optical testing of the components and a non-linear optical filter was used for cleaning the noisy images generated by the diffractive optical elements.


Introduction
Generation and precise control of optical fields is crucial in many optical instruments and imaging systems [1]. The optical fields can be engineered using optical components belonging to different categories such as refractive [2], diffractive [2], reflective [3], metaoptics [4][5][6] polarization optical components such as Q-plates [7] and liquid crystal optical components [8] based on the principle of operation. Depending upon the degree of control needed on the components of the optical field such as amplitude, phase and polarization, the complexity involved in design as well as fabrication varies. Next to the fully matured refractive optical technology, diffractive optics remains as the most widely used mode of beam control offering easier design, reasonable fabrication costs, and reaching diffraction efficiency closer to the values of refractive elements. Diffractive optics plays an important role in many areas of research such as optical trapping and tomography [9], computer generated holography [10], biomedical applications [11], integrated optics [12], display technology and serve also as components in augmented and mixed reality glasses [8].
The diffractive optical elements (DOEs) can be manufactured using different techniques such as photolithography [13], electron beam lithography [14], ion beam lithography [15], depending upon the feature sizes and area of the design. All the above methods are not only time-consuming but also have higher operating costs resulting in an increase in the cost of the DOEs. Except ion beam lithography, the other two methods cannot fabricate elements directly on to the substrate and therefore requires additional processing steps such as reactive ion etching to transfer the pattern from the resist layer to the substrate which increases the cost of manufacturing further. The electron beam and ion beam lithography methods are not suitable for fabrication of large area DOEs. Even though it may be argued that photolithography can transfer large area patterns, the time is still spent on fabrication of the mask. Therefore, except for mass production, even photolithography cannot be considered as a rapid fabrication system.
In the recent years, there has been a revolution in manufacturing of DOEs due to the advent of femtosecond ablation techniques [16][17][18][19]. They not only offer capabilities to manufacture large area DOEs but also can reach subwavelength features by the generation of harmonics.
Furthermore, the vacuum conditions and other environmental isolation needed for electron and ion beam optics can be avoided in the femtosecond fabrication system. All the above makes femtosecond fabrication systems superior to the existing methods on various technological and economical fronts.
In this study, the femtosecond fabrication by ablation has been implemented to rapidly fabricate two-level axicons directly on to Sapphire substrates. The axicons were realized in conventional ring as well as sieve configurations. This manuscript consists of four sections.
The calculation of diffracted intensity distribution and the simulation results are presented in the next section. The third section contains the fabrication procedure and characterization and optical experiments and beam cleaning procedure. The summary and conclusion are presented in the final section.

Methodology
In most of the earlier studies on beam characteristics of axicon only a coherent illumination is considered [20][21][22]. In this study, the beam characteristics are investigated for a spatially incoherent illumination as it is easily available at a low cost and will be highly relevant to large scale and astronomical applications where large area devices are often required which in turn can be manufactured using femtosecond fabrication systems. For a single point object, the behaviour of a coherent light is like that of an incoherent one [23]. However, when there is more than one point, then the optical system differs hugely between coherent and incoherent illuminations. The optical configuration is shown in Fig. 1 in which the light diffracted from a point object is incident on a the diffractive axicon and the intensity distribution is recorded. The distance between the object and the axicon is u and the distance between the axicon and the sensor is v. The binary axicon can be realized either as concentric rings or rings filled with discs similar to a photon sieve [24]. While the former with concentric rings has been widely used, only a few reports are available on axicons with sieve configuration. The rationale for comparing a conventional axicon with a photon sieve axicon (PSA) is that the fabrication using femtosecond ablation occurs point by point and it is much easier to realize an axicon as a PSA (disconnected discs) rather than in conventional type (overlapped discs). The radii of zones of a binary axicon can be given as where Λ is the period of the axicon and n is an integer. The phase of a binary axicon can be expressed as The phase of a PSA is mathematically complicated as it is made up of circular discs of the same diameter instead of rings of varying diameters as in an axicon. A two-step process is proposed to design the PSA with polar coordinates (R, φ) where = −1 ( / ). In the first step, a mask consisting of Delta functions located at the x and y coordinates of the centers of the discs of diameter Λ/2 is generated. The centers of the circular discs can be located only at specific radial values given as The circular discs cannot be tightly arranged in the ring of an axicon as the circumference given as 2 2− cannot be an integral multiple of Λ for all values of 2− . The difference between the above two values is given as and so, the increment in location needed for every disc is The values of the angles are given as where m varies from 0 to (4 2− /Λ). The mask consisting of Delta functions is given as At the end of the first step, the map of Delta functions is obtained for mounting the circular discs in the next step. To generate the mask of PSA (MPSA), it is just sufficient to carry out a 2D convolution of the Mask function with a Circ function which is a circular top hat with a radius given by where '' is a 2D convolutional operator. The phase of the PSA is given as Φ PSA = ( PSA ).
Let us consider a point object located at a distance of u from the diffractive axicon emitting light with an amplitude of √ . The complex amplitude reaching the axicon is given as where and C1 is a complex constant and there is no linear phase associated with a point object in this case as the optical axis is assumed to match with the centres of all the optical elements. The complex amplitude after the binary axicon and PSA is given as where Φ DOE = Φ Axicon when an axicon with concentric rings is used and Φ DOE = Φ PSA when an axicon with sieve configuration is used. The intensity pattern observed at a distance of v from the DOE is given as a convolution of the complex amplitude with the quadratic phase where, '' is a 2D convolution operator. From Eq (13), if the Φ DOE equals ( 2 / )(1/ + 1/ ), then in the observation plane a focused point is obtained as the other phase components are cancelled. But for an axicon, the behaviour is quite interesting as it has constant radial spatial frequency. Within the focal depth of the axicon, there is always a radial region of the axicon which has a phase distribution same as ( 2 / )(1/ + 1/ ) . This region is responsible for generating the central maximum of the Bessel function and the other radial regions which does not match the above phase distribution generate ring patterns around the central maximum. Higher the phase difference, the larger the diameter of the ring and the smaller is the energy density [25]. The intensity for a 2D object O can be expressed as Io=OIv.
Unlike a coherent source, where the complex amplitude is convolved, here only the intensity distribution is convolved as there is no spatial coherence present to generate the interference terms.
The images of the simulated axicon and PSA with same period are shown in Figs obtained for axicon is smaller than that of PSA. This behaviour is different from the observations made with a Photon sieve Fresnel lens [26], where an improvement in the resolution was observed. The difference may be accounted to the difference between the areas of the two-phase levels. In the previous simulation studies, the areas were made equal to obtain the maximum efficiency in addition to maintaining the phase difference between two levels at π for a two-level structure [2]. Due to the replacement of rings by circular discs, this condition cannot be met accurately. A previous study on PSA directly fabricated on the tip of optical fibre exhibited characteristics of an axicon [27]. To further evaluate the focal characteristics, the axial characteristics are studied for axicon and PSA under ideal conditions. The axial variation of the intensity distributions at (y=0) line for axicon and PSA are shown in Figs. 3(a) and 3(b) respectively. Comparing the above two figures, it is seen that the performances are similar.
From the simulation studies, it is seen that approximating a binary axicon using circular discs does not yield a better performance but reduces the contrast of the central maxima with respect to the surrounding ring patterns. However, the fabrication of PSA using femtosecond ablation is relatively easier and the fabrication of a conventional axicon.  The next step is to study the imaging performances of axicon and PSA using a test object.
All experiments for this study have been carried out at Nanolab, Swinburne University and so the test object has been selected as the letters of "Nanolab" in arial font as shown in Fig. 4(a).   5. The case for a diffractive lens is shown for reference. From this study, it seems that a PSA has a higher axial resolution than an axicon. The randomness associated with the distribution of disc suppresses the peaks around the central maximum. From an imaging point of view, the PSA has a better performance than an axicon. When imaging a thick object and focusing a plane of that object, the information from other planes will be of slightly lower intensity in the case of PSA when compared to an axicon.

Fabrication
The fabrication was carried out on sapphire substrate with a thickness of 500 μm and index of refraction ns = 1.76. In order to have π phase difference between the two levels the thickness needed to be milled is given as t = λ/2(ns-1) which is ~0.41 μm for λ = 617 nm which is the

Optical testing
The optical testing was carried out using a high-power LED from Thorlabs (M617L3, λc = 617 nm, full width at half maximum (FWHM) = 18 nm) and a spectral filter was used at 600 nm with a width of 10 nm to improve the temporal coherence. A pinhole with a size of 100 μm and a cross-shaped object were used for imaging. A  Therefore, it is feasible to use a sparse axicon instead of a regular axicon for imaging applications. By virtue of a wide spectral transparency of sapphire from UV to IR, optical elements with ~1 cm cross section can be made within ~10 min which allows for a wider use of such prototyping for more complex optical systems. It must be noted that the cleaning procedure does not require the recording of the intensity distribution for a pinhole exactly at the same location as the object owing to the high focal depth of the imager. For the same reason, while cleaning images of thick objects the planes that are out of focus will have a higher intensity as the plane in focus. Consequently, if the information of two planes overlap laterally then the information could not be perceived accurately. A recording at a plane can be used to clean the image for many distances. The image cleaning results for the intensity distributions recorded for the cross object at 5 cm ( Fig. 8(a)) and 6 cm ( Fig. 8(b)) using pinhole intensity distributions recorded at 5 mm are shown for axicon and PSA as shown in Fig. 8(c) and 8(d) respectively. Since the cross object is a simple object, the cleaning may not appear significant.
A synthetic object consisting of the letters 'APHB' was used next and the corresponding intensity distributions for PSA were synthesized by convolving the intensity distributions recorded for a pinhole with the object function. The intensity distribution of the synthetic object at two planes 5 cm and 6 cm are shown in Fig. 8(e) and 8(f) respectively. The cleaned images using the pinhole recording at 5 mm are shown in Figs. 8(g) and 8(h) respectively.

Conclusion
Rapid fabrication of large area DOEs using the femtosecond laser fabrication system has been investigated. A simpler design with the function of a binary axicon was used for the study. The fabrication time was only 10 minutes using the femtosecond fabrication method for a large area of 5 mm × 5 mm. The binary axicon was realized in three configurations namely conventional axicon, PSA and sparse PSA. It was noticed that when the beam overlaps during milling results in redeposition of material resulting in a lower depth than the case without beam overlap which goes against the common belief that overall higher exposure with beam overlap increases the depth. In addition, the case with beam overlap has a higher roughness value than the case without beam overlap. This increase in roughness is partly contributed by the redeposition and partly due to light-matter interaction at temperature changes caused by ablation by the previous pulse. Reduction of debris and mechanism of ripple-free deep ablation at high irradiance which exceeds ablation threshold more than an order of magnitude are directions for future investigations which were not practical at low repetition rates [35].
One interesting outcome of this study is that it seems it may be necessary to approximate the design of the DOE functions as in this case where a sieve configuration was used instead of rings to achieve a milling favorable design. As is seen in this study, the intensity distributions obtained for the sieve configuration is similar to that of the conventional axicon with a slight loss of lateral resolution and increase in axial resolution. Therefore, it should be possible to modify the design of DOEs without compromising the diffraction performances and at the same time achieve high fabrication accuracy without material redeposition and increase in surface roughness.
Some of the recent studies on controlling the distribution of debris using externally applied electric and magnetic fields offer opportunities to control the light-matter interactions and achieve better fabrication results [36][37]. We believe that the direction of research will enable the fabrication of large area structures suitable for incoherent illumination and astronomical imaging applications. With the introduction of spatial multiplexing [38], optimization [39] and redesigning of DOEs, the performance of the femtosecond fabrication system can be improved further. Astronomical imaging is one of the areas which require large area optical devices with capabilities to perform a high signal to noise ratio. We believe that the femtosecond ablation based rapid fabrication and computational optics in tandem will lead to the development of advanced astronomical imaging technologies. Some of the latest developments in astronomical spectral imaging technologies such as FOBOS [40] require numerous micro-optical devices for the successful implementation of free space to fiber bundle coupling for spectral imaging. We believe that the current work consisting of rapid fabrication and beam cleaning will support retrieving spatial information in addition to the recorded spectral information [41].
Funding. ARC LP190100505 is acknowledged for funding.