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6 International & 27 All India Manufacturing Technology, Design and Research Conference (AIMTDR-2016), December 16-18, 2016 at College of Engineering., Pune, Maharashtra, INDIA.

Focused Ion Beam Fabrication: Process Development and Optimization Strategy for Optical Applications Vivek Garg1, Rakesh G. Mote2, Jing Fu3 1

IITB Monash Research Academy, Mumbai-400076, India Department of Mechanical Engineering, IIT Bombay, Mumbai-400076, India

2

2

[email protected]

3

Department of Mechanical and Aerospace Engineering, Monash University, Clayton-3800, Australia

Abstract— This work presents optimization of focused ion beam (FIB) milling process for fabrication of optical elements. Focused ion beam is a nanofabrication process involving ion-beam material interaction at atomic/molecular levels. We have investigated the ion-solid interactions for the fabrication of desired 3D geometries. Focused ion beam dwell time is an important parameter and determines the final geometry and accuracy of fabrication. An algorithm has been developed to optimize the ion-beam dwell time for the desired geometry fabrication maintaining the high accuracy. The algorithm has been simulated using MATLAB. The proposed algorithm can be used for fabrication of 3D optical elements and micro/nano structures. Binary Fresnel Zone Plates (FZPs) have been fabricated using the optimized simulation results for demonstration. Keywords— Focused Ion Beam, Optical Applications, Fresnel Zone Plate, Optimization and Simulation, Micro/Nano Fabrication, Nanotechnology

1. INTRODUCTION Focused ion beam is an important and advanced technology in the field of micro/nano fabrication. It has tremendous capabilities ranging from milling and deposition to imaging at micro/nano scale. The distinct advantage of FIB lies in the fact that it enables mask-less direct fabrication, making it suitable for applications in the field of nanotechnology. Focused ion beam instruments, when combined with imaging technologies like scanning electron microscope (SEM), offer a wide range of applications for fabrication and characterization of micro/nano structures. A great detail of research exists in the literature for FIB principle, its applications and limitations [1]–[4]. Focused ion beam is a specialized process and involves several complexities in terms of beam control and fabrication accuracy. Accurate FIB fabrication of optical elements and 3D micro/nano structures for specific applications require deep insight into the process. The accuracy of FIB fabricated profiles depends on number of

parameters like scanning strategy, beam dwell time, beam current, acceleration voltage etc. Out of these parameters, ion-beam dwell time is the most important parameter that determines the final geometry and accuracy of fabricated 3D structures. A better understanding of the milling mechanisms and material removal at micro/nano level is required for accurate fabrication and improved performance of these elements. This will allow accurate fabrication of optical elements or any other 3D structure for specific applications in X-ray optics, micro/nano photonics etc. Fresnel Zone Plates (FZPs) are very popular optical components for applications in X-ray microscopy [5] with the capability of focusing X-rays. X-rays have the potential for high penetration depths and resolution for material characterization [6]. As a result, FZPs become very important optical elements for applications in microscopic imaging and nanotechnology. FZPs are diffractive optical elements and these are made of alternate opaque and transparent zones. The FZPs are designed according to, 𝑟𝑛 = √𝑛𝜆𝑓 + 𝑛2 𝜆2 ⁄4 (1) where, rn is the radius of nth zone, λ is the wavelength and f is the focal distance [6]. The resolution of FZP depends on the outermost zone and is given by, 𝛥𝑟𝑅𝑎𝑦𝑙𝑒𝑖𝑔ℎ = 1.22𝛥𝑟 (2) where, ΔrRayleigh is the Rayleigh resolution limit and Δr is the width of outermost FZP zone [6]. Researchers all over the world have presented a variety of FIB fabrication methods for FZPs [7]–[12]. Experimental demonstrations of FZPs [13] as well as subwavelength focusing technique using metallic coatings also exist [14]. However, there is still a need to explore the method for control and improvement of fabrication accuracy. In the present work, optimization of focused ion beam nano-fabrication is presented and simulations have been carried out for optimization of ion-beam

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6 International & 27 All India Manufacturing Technology, Design and Research Conference (AIMTDR-2016), December 16-18, 2016 at College of Engineering., Pune, Maharashtra, INDIA.

dwell time. An optimization problem is formulated for FIB fabrication and has been solved in MATLAB by an iterative approach for an optimal solution. The algorithm minimizes the deviation of a simulated geometry obtained by FIB fabrication from the desired geometry. The proposed optimization strategy can be further used to fabricate optical elements or any 3D micro/nano structures. The ion-beam dwell time has been optimized for the desired geometry fabrication, maintaining the required accuracy. This optimization algorithm has further been used to demonstrate the simulation of FZP for specific design. Experiments have been carried out to validate the optimization model. Further, FZPs have been fabricated based on the optimization results. 2. THE OPTIMIZATION MODEL Focused ion beam process planning is important for fabrication of desired micro/nano structures from accuracy point of view. Focused ion beam is a complex process and involves material interaction of ion-beam during the process. The accuracy of fabricated profile depends on number of parameters like scanning strategy, beam dwell time beam current, acceleration voltage etc. An optimization model from the literature [15] is adapted here to optimize the dwell time for accuracy and fabrication of desired profile using FIB milling. A. Scanning for FIB Milling A charged beam of ions is scanned over the substrate for machining in FIB milling. The substrate is divided into a grid of pixels for beam control and machining accuracy. The machining depth depends upon the dwell time, i.e. the time for which a particular pixel is scanned by FIB. However, when the beam is scanned over a certain pixel (say (xj, yj)), it results in milling at nearby pixel also (say (xi, yi)) due to the Gaussian distribution of beam. The effect of nearby pixels being milled due to FIB scanning at certain pixels is shown in Fig. 1. It is also clear from Fig. 1, if a particular pixel is scanned for a longer duration (more dwell time), it will result in more depth, and thus fabrication profile can be varied by controlling the dwell time. The standard normal distribution is given by, 𝐹(𝑥) =

1 √2𝜋𝜎

[𝑒 2

(𝑥−µ) 2𝜎2

−

]

(3)

where, σ is the standard deviation of the beam. The total depth being machined due to FIB milling can be written as, 𝑧𝑖 =

∑𝑀 𝑗=1 𝑌𝑠 𝑡𝑗

[𝑒

2 2 [(𝑥𝑖 −𝑥𝑗 ) +(𝑦𝑖 −𝑦𝑗 ) ] 2𝜎2

−

]

(4)

where, M is the number of pixels to be machined, zi is the machined depth at ith pixel, Ys is the erosion

Fig. 1 Effect of beam shape and dwell time on fabrication depth at different pixels (color intensity represents the fabrication depth)

rate and tj is the dwell time (which is decision variable) at jth pixel. The above equation can be written as, 𝑧𝑖 = ∑𝑀 (5) 𝑗=1 𝑎𝑖𝑗 𝑡𝑗 = 𝑨𝒊 𝒕, 𝑓𝑜𝑟 𝑖 = 1,2, … 𝑁 where, N is the total number of pixels on substrate, aij is the milling coefficient and A is an N×M matrix to measure the milling coefficients. B. The Objective Function The decision variable dwell time t should be optimized such that it results in material removal At (i.e. fabricated profile) close to the desired geometry profile zo. A quadratic optimization problem is set up to minimize the error between At and zo. minimize 𝐹(𝒕) = ‖𝑨𝒕 − 𝒛𝒐 ‖22 subject to, 𝒕 ≥ 𝟎 (6) The objective function in F(t) is rewritten in the following form to represent a standard quadratic programming problem, minimize 𝐹(𝒕) = 𝒕𝑇 𝑸𝒕 + 2𝒑𝑇 𝒕 + 𝑟 subject to, 𝒕 ≥ 𝟎 (7) where, 𝑸 = 𝑨𝑇 𝑨, 𝒑 = −𝑨𝑇 𝒛𝒐 , 𝑟 = (𝒛𝒐 )𝑇 𝒛𝒐 (8) C. Solution A multiplicative updates algorithm from the literature [16] is used for solution of the above non negative quadratic programming problem. If t0 is an initial value of dwell time t, and tk (k>0) is the value from the kth update, algorithm can be written as, 𝑡𝑗𝑘+1 ←

max[−𝑝𝑗 ,0] 𝑘 𝑡𝑗 (𝑸𝒕𝑘 )𝑗

(9)

3. SIMULATION RESULTS The optimization algorithm from the previous section is used to simulate a spherical cavity. MATLAB is used for the simulation. The beam current is taken as 20pA, X=Y=600nm, and the pixel size is 3nm. Results are shown in Fig. 2 (a-c) on a 200 by 200 grid (since pixel size is 3nm). The beam diameter at this current is 17nm (Zeiss Auriga dual beam FIB-

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6 International & 27 All India Manufacturing Technology, Design and Research Conference (AIMTDR-2016), December 16-18, 2016 at College of Engineering., Pune, Maharashtra, INDIA.

(a)

(a)

(b)

(b)

(c)

(c)

Fig. 2 (a-c) Simulation results for a spherical profile obtained from optimization algorithm at a beam current of 20pA and pixel size of 3nm (a) desired spherical profile, (b) simulated spherical profile, (c) error between the desired and simulated profile

Fig. 3 (a-c) Simulation results for fabrication of Fresnel Zone Plate obtained from optimization algorithm at a beam current of 500pA and pixel size of 200nm (a) desired profile, (b) simulated profile, (c) error between the desired and simulated profile

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6 International & 27 All India Manufacturing Technology, Design and Research Conference (AIMTDR-2016), December 16-18, 2016 at College of Engineering., Pune, Maharashtra, INDIA.

SEM system) and step size is taken as 3nm. The plot in Fig. 2 (a) is the desired spherical shape with zo=60nm. Fig. 2 (b) shows the simulation results. The difference between At (simulated geometry) and zo (desired geometry) i.e. error is shown in Fig. 2 (c). It can be seen from the figure that the error is significant only at the boundaries. It is because no actual machining takes place at boundary pixels, and the depth obtained is only because of the milling at nearby pixels. Thus, the optimization model predicts quite accurate results. Several other simulations have also been run using the optimization model for different profiles to check the validity of optimization algorithm, and the results obtained for all the profiles are fairly accurate. Further, optimization of dwell time for desired and accurate 3D optical elements can be obtained using the developed algorithm. Fig. 3 (a-c) show the simulation results for FZP as an example. The simulation results shown here are for a binary FZP of diameter 100µm, focal length of 150µm and for a wavelength of 660nm. The beam current for the simulation is taken as 500 pA, X=Y=100µm, and a large value of pixel size 200nm is chosen. Results are shown in Fig. 3 (a-c) on a 500 by 500 grid (since pixel size is 200nm). The beam diameter at this current is 50nm for Zeiss Auriga dual beam FIB-SEM system. The reason for selecting large pixel size for FZP simulation is to reduce the simulation time maintaining accuracy, as the size of FZP is large (100µm). The results from Fig. 3 (c) indicate a very small value of error; it is due to the binary nature of simulated FZP, as it did not involve any curved features. Next, an attempt is made to validate the optimization results with experiments. 4. FOCUSED ION BEAM FABRICATION RESULTS Experiments have been carried out on Zeiss Auriga dual beam FIB-SEM system using Gallium ion beam. This system allows smallest spot size of 8nm at a beam current of 1pA. It allows milling, deposition and imaging using ion and electron beam. Focused ion beam milling is a very powerful tool and it enables micro/nano fabrication with great accuracy. Simulation profile of FZP in the previous section is fabricated using FIB milling for demonstration. The system allows feature based milling and bitmap images consisting of designed pattern can be milled by choosing desired beam current, dwell time and acceleration voltage. A bitmap image consisting of ring patterns based on the FZP design is created in MATLAB. This bitmap image has been used on FIBSEM system for fabrication. Fig. 4 (a) shows the bitmap image used for a binary FZP of diameter 100µm, focal length of 150µm for a wavelength of 660nm. Focused ion beam is scanned only over the black areas in the

(a)

(b) Fig. 4 (a-b) Binary Fresnel Zone Plate fabrication by Focused Ion Beam milling (a) bitmap image used in the Zeiss Auriga dual beam FIB-SEM system, (b) Scanning electron microscopic image of fabricated FZP

bitmap image and milling takes place at these areas only. Scanning electron microscopic (SEM) image of FZP fabricated on silicon using FIB milling is shown in Fig. 4 (b). The fabrication has been done using Gallium ion beam at a beam current of 500pA and acceleration voltage of 30kV. Next, atomic force microscopic (AFM) characterization of fabricated FZP is done. Only a portion of FZP is scanned due to the scanning limits of AFM. Fig. 5 (a-c) show the AFM characterization results of fabricated FZP. 2D and 3D profiles of FIB fabricated FZP in Fig. 5 (a) and Fig. 5 (b) respectively demonstrates the fine

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6 International & 27 All India Manufacturing Technology, Design and Research Conference (AIMTDR-2016), December 16-18, 2016 at College of Engineering., Pune, Maharashtra, INDIA.

accuracy and capability of the process. Fig. 5 (c) represents the surface profile of fabricated FZP.

(a)

(b)

5. CONCLUSIONS This work presented focused ion beam milling process development and optimization for fabrication of optical applications. An optimization model has been developed and used to simulate the FIB milling process for fabrication of desired profiles. Ion-beam dwell time is an important parameter for accurate fabrication, which is optimized for required accuracy and an example has been presented for a spherical profile. Simulations results have been able to predict the fabricated geometry fairly well. The amount of error between the desired geometry and simulated geometry is quite low, indicating good results and validity of optimization algorithm. Also, simulation results for optimization of FZP fabrication are presented, which have been able to predict the required fabrication geometry very well at the given beam parameters. The error between the desired geometry and simulated FZP geometry is very less due to its binary nature. Further, experiments have been carried out and fabrication of FZP on silicon is demonstrated as an example. This approach can be further extended to fabrication of 3D optical elements and complex micro/nano structures by FIB milling, which are otherwise difficult to achieve. These will find potential applications in optics, photonics, microfluidics etc. in the field of nanotechnology. REFERENCES [1]

[2]

[3]

[4]

[5]

[6] [7]

[8]

(c) Fig. 5 (a-c) Atomic Force Microscopic (AFM) results of focused ion beam milled Fresnel Zone Plate (a) 2D AFM image, (b) 3D AFM image, (c) Variation of surface profile

[9]

J. Melngailis, “Focused ion beam technology and applications,” J. Vac. Sci. Technol. B, vol. 5, no. 2, pp. 469–495, 1987. S. Reyntjens and R. Puers, “A review of focused ion beam applications in microsystem technology,” J. Micromechanics Microengineering, vol. 11, no. 4, pp. 287–300, 2001. C. Lehrer, L. Frey, S. Petersen, and H. Ryssel, “Limitations of focused ion beam nanomachining,” J. Vac. Sci. Technol. B Microelectron. Nanom. Struct., vol. 19, no. 6, p. 2533, 2001. C. S. Kim, S. H. Ahn, and D. Y. Jang, “Review: Developments in micro/nanoscale fabrication by focused ion beams,” Vacuum, vol. 86, no. 8, pp. 1014–1035, 2012. H. M. Quiney, a. G. Peele, Z. Cai, D. Paterson, and K. a. Nugent, “Diffractive imaging of highly focused X-ray fields,” Nat. Phys., vol. 2, no. 2, pp. 101–104, 2006. D. Attwood, Soft X-rays and Extreme Ultraviolet Radiation. Cambridge University Press, 1999. K. Keskinbora, C. Grévent, and U. Eigenthaler, “Rapid Prototyping of Fresnel Zone Plates via Direct Ga+ Ion Beam Lithography for High-Resolution X-ray Imaging,” ACS Nano, vol. 7, no. 11, pp. 9788–9797, 2013. K. Keskinbora, C. Grévent, M. Bechtel, M. Weigand, E. Goering, A. Nadzeyka, L. Peto, S. Rehbein, G. Schneider, R. Follath, J. Vila-Comamala, H. Yan, and G. Schütz, “Ion beam lithography for Fresnel zone plates in X-ray microscopy.,” Opt. Express, vol. 21, no. 10, pp. 11747–56, 2013. A. Vijayakumar, U. Eigenthaler, K. Keskinbora, G. M.

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[10]

[11]

[12]

Sridharan, V. Pramitha, M. Hirscher, J. P. Spatz, and S. Bhattacharya, “Optimizing the Fabrication of Diffractive Optical Elements Using a Focused Ion Beam System,” Micro-Optics 2014, vol. 9130, pp. 1–8, 2014. U. T. Sanli, K. Keskinbora, K. Gregorczyk, J. Leister, N. Teeny, C. Grévent, M. Knez, and G. Schütz, “Highresolution high-efficiency multilayer Fresnel zone plates for soft and hard x-rays,” Proc. SPIE, vol. 9592. p. 95920F–95920F–8, 2015. A. Vijayakumar and S. Bhattacharya, “Design of multifunctional diffractive optical elements,” Opt. Eng., vol. 54, no. 2, p. 24104, 2015. R. S. Rodrigues Ribeiro, P. Dahal, A. Guerreiro, P. A. S. Jorge, and J. Viegas, “Compact solutions for optical fiber tweezers using Fresnel zone and phase lenses fabricated using FIB milling,” Proc. SPIE, vol. 9764. p. 97640C–97640C–7, 2016.

[13]

[14]

[15]

[16]

R. G. Mote, S. F. Yu, A. Kumar, W. Zhou, and X. F. Li, “Experimental demonstration of near-field focusing of a phase micro-Fresnel zone plate (FZP) under linearly polarized illumination,” Appl. Phys. B, vol. 102, no. 1, pp. 95–100, 2010. R. G. Mote, S. F. Yu, B. K. Ng, W. Zhou, and S. P. Lau, “Near-field focusing properties of zone plates in visible regime--new insights.,” Opt. Express, vol. 16, no. 13, pp. 9554–9564, 2008. R. Qin, J. Fu, Z. Yin, and C. Zheng, “Large-scale process optimization for focused ion beam 3-D nanofabrication,” Int. J. Adv. Manuf. Technol., vol. 64, no. 1–4, pp. 587– 600, 2013. Z. Yang and J. Laaksonen, “Multiplicative updates for non-negative projections,” Neurocomputing, vol. 71, no. 1–3, pp. 363–373, 2007.

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6 International & 27 All India Manufacturing Technology, Design and Research Conference (AIMTDR-2016), December 16-18, 2016 at College of Engineering., Pune, Maharashtra, INDIA.

Focused Ion Beam Fabrication: Process Development and Optimization Strategy for Optical Applications Vivek Garg1, Rakesh G. Mote2, Jing Fu3 1

IITB Monash Research Academy, Mumbai-400076, India Department of Mechanical Engineering, IIT Bombay, Mumbai-400076, India

2

2

[email protected]

3

Department of Mechanical and Aerospace Engineering, Monash University, Clayton-3800, Australia

Abstract— This work presents optimization of focused ion beam (FIB) milling process for fabrication of optical elements. Focused ion beam is a nanofabrication process involving ion-beam material interaction at atomic/molecular levels. We have investigated the ion-solid interactions for the fabrication of desired 3D geometries. Focused ion beam dwell time is an important parameter and determines the final geometry and accuracy of fabrication. An algorithm has been developed to optimize the ion-beam dwell time for the desired geometry fabrication maintaining the high accuracy. The algorithm has been simulated using MATLAB. The proposed algorithm can be used for fabrication of 3D optical elements and micro/nano structures. Binary Fresnel Zone Plates (FZPs) have been fabricated using the optimized simulation results for demonstration. Keywords— Focused Ion Beam, Optical Applications, Fresnel Zone Plate, Optimization and Simulation, Micro/Nano Fabrication, Nanotechnology

1. INTRODUCTION Focused ion beam is an important and advanced technology in the field of micro/nano fabrication. It has tremendous capabilities ranging from milling and deposition to imaging at micro/nano scale. The distinct advantage of FIB lies in the fact that it enables mask-less direct fabrication, making it suitable for applications in the field of nanotechnology. Focused ion beam instruments, when combined with imaging technologies like scanning electron microscope (SEM), offer a wide range of applications for fabrication and characterization of micro/nano structures. A great detail of research exists in the literature for FIB principle, its applications and limitations [1]–[4]. Focused ion beam is a specialized process and involves several complexities in terms of beam control and fabrication accuracy. Accurate FIB fabrication of optical elements and 3D micro/nano structures for specific applications require deep insight into the process. The accuracy of FIB fabricated profiles depends on number of

parameters like scanning strategy, beam dwell time, beam current, acceleration voltage etc. Out of these parameters, ion-beam dwell time is the most important parameter that determines the final geometry and accuracy of fabricated 3D structures. A better understanding of the milling mechanisms and material removal at micro/nano level is required for accurate fabrication and improved performance of these elements. This will allow accurate fabrication of optical elements or any other 3D structure for specific applications in X-ray optics, micro/nano photonics etc. Fresnel Zone Plates (FZPs) are very popular optical components for applications in X-ray microscopy [5] with the capability of focusing X-rays. X-rays have the potential for high penetration depths and resolution for material characterization [6]. As a result, FZPs become very important optical elements for applications in microscopic imaging and nanotechnology. FZPs are diffractive optical elements and these are made of alternate opaque and transparent zones. The FZPs are designed according to, 𝑟𝑛 = √𝑛𝜆𝑓 + 𝑛2 𝜆2 ⁄4 (1) where, rn is the radius of nth zone, λ is the wavelength and f is the focal distance [6]. The resolution of FZP depends on the outermost zone and is given by, 𝛥𝑟𝑅𝑎𝑦𝑙𝑒𝑖𝑔ℎ = 1.22𝛥𝑟 (2) where, ΔrRayleigh is the Rayleigh resolution limit and Δr is the width of outermost FZP zone [6]. Researchers all over the world have presented a variety of FIB fabrication methods for FZPs [7]–[12]. Experimental demonstrations of FZPs [13] as well as subwavelength focusing technique using metallic coatings also exist [14]. However, there is still a need to explore the method for control and improvement of fabrication accuracy. In the present work, optimization of focused ion beam nano-fabrication is presented and simulations have been carried out for optimization of ion-beam

th

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6 International & 27 All India Manufacturing Technology, Design and Research Conference (AIMTDR-2016), December 16-18, 2016 at College of Engineering., Pune, Maharashtra, INDIA.

dwell time. An optimization problem is formulated for FIB fabrication and has been solved in MATLAB by an iterative approach for an optimal solution. The algorithm minimizes the deviation of a simulated geometry obtained by FIB fabrication from the desired geometry. The proposed optimization strategy can be further used to fabricate optical elements or any 3D micro/nano structures. The ion-beam dwell time has been optimized for the desired geometry fabrication, maintaining the required accuracy. This optimization algorithm has further been used to demonstrate the simulation of FZP for specific design. Experiments have been carried out to validate the optimization model. Further, FZPs have been fabricated based on the optimization results. 2. THE OPTIMIZATION MODEL Focused ion beam process planning is important for fabrication of desired micro/nano structures from accuracy point of view. Focused ion beam is a complex process and involves material interaction of ion-beam during the process. The accuracy of fabricated profile depends on number of parameters like scanning strategy, beam dwell time beam current, acceleration voltage etc. An optimization model from the literature [15] is adapted here to optimize the dwell time for accuracy and fabrication of desired profile using FIB milling. A. Scanning for FIB Milling A charged beam of ions is scanned over the substrate for machining in FIB milling. The substrate is divided into a grid of pixels for beam control and machining accuracy. The machining depth depends upon the dwell time, i.e. the time for which a particular pixel is scanned by FIB. However, when the beam is scanned over a certain pixel (say (xj, yj)), it results in milling at nearby pixel also (say (xi, yi)) due to the Gaussian distribution of beam. The effect of nearby pixels being milled due to FIB scanning at certain pixels is shown in Fig. 1. It is also clear from Fig. 1, if a particular pixel is scanned for a longer duration (more dwell time), it will result in more depth, and thus fabrication profile can be varied by controlling the dwell time. The standard normal distribution is given by, 𝐹(𝑥) =

1 √2𝜋𝜎

[𝑒 2

(𝑥−µ) 2𝜎2

−

]

(3)

where, σ is the standard deviation of the beam. The total depth being machined due to FIB milling can be written as, 𝑧𝑖 =

∑𝑀 𝑗=1 𝑌𝑠 𝑡𝑗

[𝑒

2 2 [(𝑥𝑖 −𝑥𝑗 ) +(𝑦𝑖 −𝑦𝑗 ) ] 2𝜎2

−

]

(4)

where, M is the number of pixels to be machined, zi is the machined depth at ith pixel, Ys is the erosion

Fig. 1 Effect of beam shape and dwell time on fabrication depth at different pixels (color intensity represents the fabrication depth)

rate and tj is the dwell time (which is decision variable) at jth pixel. The above equation can be written as, 𝑧𝑖 = ∑𝑀 (5) 𝑗=1 𝑎𝑖𝑗 𝑡𝑗 = 𝑨𝒊 𝒕, 𝑓𝑜𝑟 𝑖 = 1,2, … 𝑁 where, N is the total number of pixels on substrate, aij is the milling coefficient and A is an N×M matrix to measure the milling coefficients. B. The Objective Function The decision variable dwell time t should be optimized such that it results in material removal At (i.e. fabricated profile) close to the desired geometry profile zo. A quadratic optimization problem is set up to minimize the error between At and zo. minimize 𝐹(𝒕) = ‖𝑨𝒕 − 𝒛𝒐 ‖22 subject to, 𝒕 ≥ 𝟎 (6) The objective function in F(t) is rewritten in the following form to represent a standard quadratic programming problem, minimize 𝐹(𝒕) = 𝒕𝑇 𝑸𝒕 + 2𝒑𝑇 𝒕 + 𝑟 subject to, 𝒕 ≥ 𝟎 (7) where, 𝑸 = 𝑨𝑇 𝑨, 𝒑 = −𝑨𝑇 𝒛𝒐 , 𝑟 = (𝒛𝒐 )𝑇 𝒛𝒐 (8) C. Solution A multiplicative updates algorithm from the literature [16] is used for solution of the above non negative quadratic programming problem. If t0 is an initial value of dwell time t, and tk (k>0) is the value from the kth update, algorithm can be written as, 𝑡𝑗𝑘+1 ←

max[−𝑝𝑗 ,0] 𝑘 𝑡𝑗 (𝑸𝒕𝑘 )𝑗

(9)

3. SIMULATION RESULTS The optimization algorithm from the previous section is used to simulate a spherical cavity. MATLAB is used for the simulation. The beam current is taken as 20pA, X=Y=600nm, and the pixel size is 3nm. Results are shown in Fig. 2 (a-c) on a 200 by 200 grid (since pixel size is 3nm). The beam diameter at this current is 17nm (Zeiss Auriga dual beam FIB-

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6 International & 27 All India Manufacturing Technology, Design and Research Conference (AIMTDR-2016), December 16-18, 2016 at College of Engineering., Pune, Maharashtra, INDIA.

(a)

(a)

(b)

(b)

(c)

(c)

Fig. 2 (a-c) Simulation results for a spherical profile obtained from optimization algorithm at a beam current of 20pA and pixel size of 3nm (a) desired spherical profile, (b) simulated spherical profile, (c) error between the desired and simulated profile

Fig. 3 (a-c) Simulation results for fabrication of Fresnel Zone Plate obtained from optimization algorithm at a beam current of 500pA and pixel size of 200nm (a) desired profile, (b) simulated profile, (c) error between the desired and simulated profile

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6 International & 27 All India Manufacturing Technology, Design and Research Conference (AIMTDR-2016), December 16-18, 2016 at College of Engineering., Pune, Maharashtra, INDIA.

SEM system) and step size is taken as 3nm. The plot in Fig. 2 (a) is the desired spherical shape with zo=60nm. Fig. 2 (b) shows the simulation results. The difference between At (simulated geometry) and zo (desired geometry) i.e. error is shown in Fig. 2 (c). It can be seen from the figure that the error is significant only at the boundaries. It is because no actual machining takes place at boundary pixels, and the depth obtained is only because of the milling at nearby pixels. Thus, the optimization model predicts quite accurate results. Several other simulations have also been run using the optimization model for different profiles to check the validity of optimization algorithm, and the results obtained for all the profiles are fairly accurate. Further, optimization of dwell time for desired and accurate 3D optical elements can be obtained using the developed algorithm. Fig. 3 (a-c) show the simulation results for FZP as an example. The simulation results shown here are for a binary FZP of diameter 100µm, focal length of 150µm and for a wavelength of 660nm. The beam current for the simulation is taken as 500 pA, X=Y=100µm, and a large value of pixel size 200nm is chosen. Results are shown in Fig. 3 (a-c) on a 500 by 500 grid (since pixel size is 200nm). The beam diameter at this current is 50nm for Zeiss Auriga dual beam FIB-SEM system. The reason for selecting large pixel size for FZP simulation is to reduce the simulation time maintaining accuracy, as the size of FZP is large (100µm). The results from Fig. 3 (c) indicate a very small value of error; it is due to the binary nature of simulated FZP, as it did not involve any curved features. Next, an attempt is made to validate the optimization results with experiments. 4. FOCUSED ION BEAM FABRICATION RESULTS Experiments have been carried out on Zeiss Auriga dual beam FIB-SEM system using Gallium ion beam. This system allows smallest spot size of 8nm at a beam current of 1pA. It allows milling, deposition and imaging using ion and electron beam. Focused ion beam milling is a very powerful tool and it enables micro/nano fabrication with great accuracy. Simulation profile of FZP in the previous section is fabricated using FIB milling for demonstration. The system allows feature based milling and bitmap images consisting of designed pattern can be milled by choosing desired beam current, dwell time and acceleration voltage. A bitmap image consisting of ring patterns based on the FZP design is created in MATLAB. This bitmap image has been used on FIBSEM system for fabrication. Fig. 4 (a) shows the bitmap image used for a binary FZP of diameter 100µm, focal length of 150µm for a wavelength of 660nm. Focused ion beam is scanned only over the black areas in the

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(b) Fig. 4 (a-b) Binary Fresnel Zone Plate fabrication by Focused Ion Beam milling (a) bitmap image used in the Zeiss Auriga dual beam FIB-SEM system, (b) Scanning electron microscopic image of fabricated FZP

bitmap image and milling takes place at these areas only. Scanning electron microscopic (SEM) image of FZP fabricated on silicon using FIB milling is shown in Fig. 4 (b). The fabrication has been done using Gallium ion beam at a beam current of 500pA and acceleration voltage of 30kV. Next, atomic force microscopic (AFM) characterization of fabricated FZP is done. Only a portion of FZP is scanned due to the scanning limits of AFM. Fig. 5 (a-c) show the AFM characterization results of fabricated FZP. 2D and 3D profiles of FIB fabricated FZP in Fig. 5 (a) and Fig. 5 (b) respectively demonstrates the fine

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6 International & 27 All India Manufacturing Technology, Design and Research Conference (AIMTDR-2016), December 16-18, 2016 at College of Engineering., Pune, Maharashtra, INDIA.

accuracy and capability of the process. Fig. 5 (c) represents the surface profile of fabricated FZP.

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5. CONCLUSIONS This work presented focused ion beam milling process development and optimization for fabrication of optical applications. An optimization model has been developed and used to simulate the FIB milling process for fabrication of desired profiles. Ion-beam dwell time is an important parameter for accurate fabrication, which is optimized for required accuracy and an example has been presented for a spherical profile. Simulations results have been able to predict the fabricated geometry fairly well. The amount of error between the desired geometry and simulated geometry is quite low, indicating good results and validity of optimization algorithm. Also, simulation results for optimization of FZP fabrication are presented, which have been able to predict the required fabrication geometry very well at the given beam parameters. The error between the desired geometry and simulated FZP geometry is very less due to its binary nature. Further, experiments have been carried out and fabrication of FZP on silicon is demonstrated as an example. This approach can be further extended to fabrication of 3D optical elements and complex micro/nano structures by FIB milling, which are otherwise difficult to achieve. These will find potential applications in optics, photonics, microfluidics etc. in the field of nanotechnology. REFERENCES [1]

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(c) Fig. 5 (a-c) Atomic Force Microscopic (AFM) results of focused ion beam milled Fresnel Zone Plate (a) 2D AFM image, (b) 3D AFM image, (c) Variation of surface profile

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R. G. Mote, S. F. Yu, A. Kumar, W. Zhou, and X. F. Li, “Experimental demonstration of near-field focusing of a phase micro-Fresnel zone plate (FZP) under linearly polarized illumination,” Appl. Phys. B, vol. 102, no. 1, pp. 95–100, 2010. R. G. Mote, S. F. Yu, B. K. Ng, W. Zhou, and S. P. Lau, “Near-field focusing properties of zone plates in visible regime--new insights.,” Opt. Express, vol. 16, no. 13, pp. 9554–9564, 2008. R. Qin, J. Fu, Z. Yin, and C. Zheng, “Large-scale process optimization for focused ion beam 3-D nanofabrication,” Int. J. Adv. Manuf. Technol., vol. 64, no. 1–4, pp. 587– 600, 2013. Z. Yang and J. Laaksonen, “Multiplicative updates for non-negative projections,” Neurocomputing, vol. 71, no. 1–3, pp. 363–373, 2007.