This Master thesis was accomplished within the Erasmus Mundus Joint Master Degree “Photonic Integrated
Circuits, Sensors and NETworks (PIXNET)”.
Coordinating Institution: Scuola Superiore di Studi Universitari e di Perfezionamento Sant'anna
Partners
Osaka University
Aston University
Technische Universiteit Eindhoven
Project Data
Start: 01-09-2017 - End: 31-08-2022
Project Reference: 586665-EPP-1-2017-1-IT-EPPKA1-JMD-MOB
EU Grant: 3.334.000 EUR
Website: http://pixnet.santannapisa.it
Programme: Erasmus+
Key Action: Learning Mobility of Individuals
Beamforming in nanophotonic membrane integrated
circuit for free-space sensing and metrology
M. Colantonio
Abstract
Optical wafer metrology for semiconductor devices manufacturing relies on laser-based phase-grating sensors to measure the alignment of the wafer to the lithography system. This sensor can guarantee sub-nanometer accuracy but lacks the ability to measure multiple wafer locations at the same time. Here, a first effort was made to provide a solution that can map the wafer grid in a single step, thus increasing the throughput, by employing photonic integrated circuits. The main focus of this work was on developing optical integrated antennas as a key enabling device of such circuits. Analytic model and simulation results of antennas based on sub-wavelength gratings showed the possibility to produce beam widths from 6◦ to 0.1◦ and arbitrary sidelobe ratios along one direction in a single lithography step. The beam shape in the orthogonal direction was controlled through a phased array of these antennas. These results indicate that there is a possibility of building alignment sensors based on the simulated antennas, such as the circuits proposed at the end of this article.
Index Terms
sub-wavelength gratings, metamaterial, optical antennas, alignment, optical metrology
I. INTRODUCTION
P
HOTONIC technologies play a fundamental role in the fields of telecommunications, imaging, sensing and metrology. Nowadays, industries are showing increasing interest in the possibility to reduce size, weight and power consumption through integration of the already established solutions [1]–[3] into Photonic Integrated Circuits (PICs) that allow light manipulation within the chip and out-of-plane free-space radiation through grating couplers, potentially removing the need for micro-lenses and off-chip optics.Specifically, in optical wafer metrology for semiconductor devices manufacturing, a widely employed technique to achieve nanometer-scale overlay accuracy makes use of the laser-based phase-grating alignment [4]. In this type of sensors, a laser beam is diffracted by a grating-based marker on the wafer surface; the diffracted light is then imaged by an optical system on top of a reference mask before being detected by a photodiode. This sensor performs a convolution between the reference mask and the image of the grating as the laser beam scans the surface of the wafer so that the detected photocurrent is maximum when the grating on the wafer is perfectly aligned to the reference mask. The optical system is made of bulk optical components susceptible to mechanical vibrations, that can limit the measurement accuracy. Moreover, due to in-plane wafer distortions caused by fabrication processes, the alignment sensor has to measure many locations (≈ 40) to create a map of the deformed wafer grid, while guaranteeing high throughput. An alignment sensor based on a grid of PICs can satisfy these requirements due to their extremely small size, improved mechanical stability and, in the case of InP-based platforms, monolithic integration of light sources. Each PIC measures the alignment of a single marker, generating one point of the wafer grid map. It is, thus, possible to continuously monitor the alignment and deformation across the entire wafer while it is moved and rotated.
For such application, free-space coupling antennas offering precise control of the shape and the direction of the optical beams are necessary for the sensor to work at different heights from the wafer and to correct for fabrication errors. Previous research extensively focused on gratings for fiber coupling and LIDAR. The former aims at maximizing the overlap of the generated beam with the fundamental mode of a single-mode fiber (≈ 10 µm beam-waist), which is placed in contact with the PIC surface. In the latter, on the other hand, the aim is to minimize the full-width half-maximum (FWHM) of the free-space beam, which is inversely proportional to the area of the antenna, to increase the cross-range resolution of the system while maximizing the steerable solid angle, i.e. the field of view (FoV) of the system.
These parameters are also used to describe the requirements of alignment sensors. For a sensor placed at a distance of 1 cm from a marker of 30 µm×30 µm, the required FWHM of the beam is ≈ 0.2◦. At 1.55 µm free-space wavelength, this translates to a uniform antenna aperture of 0.16 mm2 (since the field distribution of a grating antenna is not uniform, the actual size is expected to be larger). As the FWHM is small, it scales linearly with the distance from the marker (doubling the distance halves the FWHM and quadruples the aperture of the antenna) and, thus, differently from LIDAR systems, a beam shaping control is needed for optimizing the overlap between the beam and the marker. Furthermore, the required FoV depends on the diffraction angles of the marker. For a 6 µm pitch grating at 1.55 µm free-space wavelength, the first diffraction orders angles are ±15◦. Capturing higher diffraction orders increases the resolution of the sensor but requires antennas with wider FoV.
In LIDAR circuits, two-dimensional beam steering is achieved either through phase-shifters or by wavelength tuning [11].
TABLE I
PERFORMANCE COMPARISON OF ELECTRONICALLY TUNABLE OPTICAL PHASED ARRAYS
Platform Aperture (mm2) FoV Power con-sumption (W) Integrated light source Beam steering Beam shap-ing Chung (2018) [5] SOI 0.007 45 ◦ 55 No thermo-optic phase-shifters No Hutchinson (2016) [6] SOI 0.5 80 ◦× 17◦ 10 No thermo-optic phase-shifters and wavelength tuning No Miller (2020) [7] SOI 0.4 70 ◦× 6◦ 1.9 No multi-pass thermo-optic phase-shifters and wavelength tuning No Poulton (2019) [8] SOI 10 56 ◦× 15◦ 0.001 No electro-optic phase-shifters and wavelength tuning No Dostart (2020) [9] SiN 0.4 36 ◦× 5.5◦ 0 No wavelength tuning No Fatemi (2019) [10]
SOI 0.023 16◦× 16◦ 1.35 No thermo-optic
phase-shifters No
Alignment
sensor 1 IMOS 0.35* 30
◦× 1◦* 2.7* Yes
thermo-optic phase-shifters and wavelength tuning
Yes Alignment
sensor 2 IMOS 0.35* 1
◦* 0 Yes wavelength tuning Yes
(fixed) *theoretical values
In [5], 1024 thermo-optic phase shifters were used to steer the beam generated by a linear phased-array of 1024 antennas in an FoV of 45◦, consuming 55W. In [6], a 0.5 mm2 128-element linear phased array achieved an FoV of 80◦× 17◦ with uneven
inter-element spacing, wavelength steering and total power consumption of 10W. To reduce power consumption, Miller et al. demonstrated the use of multi-pass thermo-optic phase-shifters in [7], achieving 1.9W dissipation in a 0.4 mm2 512-elements
linear phased-array with an FoV of 70◦× 6◦. Poulton et al. further reduced the energy requirements of integrated phased-array
by using electro-optic phase-shifters, demonstrating less than 1mW consumption by a 10 mm2 512-elements phased-array with an FoV of 56◦× 15◦[8]. In [9], a 0.4 mm2serpentine phased array that achieves two-dimensional beam steering only through
wavelength tuning with a 36◦× 5.5◦ FoV removed the need for phase shifters. Finally, a system based on the stacking of a
metalens on top of a PIC with a switchable vortex-beam emitter matrix was proposed in [12]. Since only one emitter is active at a given time, the number of active phase shifters scales logarithmically with the number of emitters, rather than linearly. These systems were fabricated in Silicon-based platforms and required either external laser sources or III-V optical amplifiers on-chip through hybrid or heterogeneous integration.
Indium-Phosphide (InP) Membrane on Silicon (IMOS) added value lies in the possibility of monolithic integration of active and passives devices, providing a complete set of optical functionalities [13]–[15], as well as enhanced efficiency of the optical antennas thanks to the high-index contrast and to double-sided processing [16], [17], thus merging the advantages of Silicon-on-Insulator and conventional InP.
These results, summarized in Table I, show that PICs can offer wide-range finely-tuning beam steering in addition to on-chip optical processing. Additionally, the table also reports the theoretical parameters of the antennas for two alignment sensors that will be discussed in this work, in which beam steering is used to point the antennas to the marker with angles given by the diffraction orders. The received diffraction orders are then manipulated in the PIC such that they interfere with each other before being detected by an integrated photodiode, thus achieving the same functionality of the bulk phase-grating alignment. Furthermore, beam shaping is a highly desirable functionality that allows to optimize the overlap between the beam spot on the wafer and the marker and to suppress side lobes, thus reducing the noise generated by the wafer topology.
While there are other optical beam steering and shaping technologies [18] [19], (e.g. galvanometer mirrors, liquid crystals (LC) on silicon and LC evanescent waveguide overlap, MEMS mirrors, lenslets, holographic glasses, birefringent prisms or gratings, electro-wetting, diffractive gratings, AWG routers), they all require an out-of-plane incident beam or fibers, resulting in a bulk systems that would hardly achieve the parallelism enabled by PICs.
This work focuses on developing optical antennas with arbitrary shape of the coupled free-space beam as key elements of an integrated alignment sensor for semiconductor device manufacturing. The rest of the paper is organized as follows. In Section II, 1D beam steering antennas are analyzed. Because of the dispersive nature of the gratings, the far-field pattern can be steered in one direction by wavelength tuning. A review of the theoretical behavior of leaky-wave grating antennas is given in the Appendix and the results are summarized in this section. Then, Sub-Wavelength Grating (SWG)-based antennas are proposed to achieve the beam shaping needed for optimal overlap between the grating-based marker and the spot size of the beam on the wafer. In Section III, a 2D beam-steering antenna is obtained through a linear optical phased array (OPA) of 1D beam steering antennas in which the phase of the elements of the array can be controlled through thermo-optic phase-shifters. The novelty
relies on employing Mach-Zender Modulators (MZMs) in the branching tree of the OPA to achieve beam shaping. Finally, Section IV offers a proposal for the implementation of a marker alignment sensor for semiconductor device manufacturing based on these antennas.
II. GRATINGANTENNAS
In this section, the theory of grating-based optical antennas will be addressed in order to derive a model that relates the design requirements to the geometrical parameters of the antenna, such as its width, length and the coupling strength of the grating.
Then, sub-wavelength gratings (in literature also referred to as metamaterials) are investigated as an alternative to already employed grating structures to massively extend the available design space thanks to the possibility of varying the coupling strength in a single lithography step.
A. Radiation Pattern of Grating Antennas
A Grating Antenna (GA) is a particular case of periodic leaky-wave antenna [20]. The geometry of the GA and the coordinate system is shown in Fig. 1. The GA is a periodically perturbed waveguide of period p, that goes from −L/2 to L/2 along x, from −W/2 to W/2 along y, from −H to 0 along z. The waveguide can be of any type, the one illustrated in the figure is buried in a low-refractive-index material and the structure can also include a reflective plane below it. In the spherical coordinate system used in the following discussion, θ denotes the steering along the x-axis and ψ along y. More precisely, ψ is the complementary of the angle between the positive y-axis and a vector, whereas θ denotes the angle between the z-axis and the projection of the vector on the xz-plane. A detailed derivation of the GA behaviour is presented in the Appendix A. In the following, the main results are summarized.
When the period p is chosen such that
β0+ k0ncl 2 < 2π p < β0+ k0ncl 2π p 6= β0 (1)
with β0 the propagation constant of the mode traveling inside the waveguide, k0 the free-space wavenumber and ncl the
refractive index surrounding the antenna, and the antenna is excited by the fundamental quasi-TE mode from −L/2, then the GA produces a far field pattern having a maximum when [A(18)]
k0cos ψ sin θ = β0−
2π
p (2)
which returns the same results of the well-known grating equation for ψ = 0 [A(20)]. However, when ψ 6= 0, the maximum angle θ increases. This equation, in fact, describes a circle, as y2+ z2= 1 − cos2ψ sin2
θ. This has to be taken into account when designing and characterizing optical phased array.
In any plane θ = const, the radiation maximum occurs at ψ = 0 and the FWHM is given by [A(25)]
FWHM ≈ 2 arcsin 1.87 λ0 πWe (3)
where λ0 is the free-space wavelength and Weis the effective width of the fundamental TE mode, which can be approximated
by W for high-index contrast waveguides.
The FWHM of the far field pattern in the plane ψ = 0 is given by [A(21)]
FWHM ≈ λ0
L cos θmax
(4)
for an antenna of finite length, under the assumption that FWHM θmax, and by
FWHM = 2α
k0cos θmax
(5)
for an antenna of infinite length, where α is the attenuation constant of the field inside the waveguide that characterizes all the loss mechanisms, including the loss due to radiation. In the latter case, the radiation pattern is a Lorentzian function [A(22)], meaning that, even if there are no sidelobes, the field decays slowly and a considerable part of the energy is not confined inside the FWHM. Comparing (4) and (5), one finds that the condition of infinite length becomes true when the length of the GA exceeds the value L ≈ π/α. Indeed, at this length, the intensity of the light inside the GA is exp(−2π), three orders of magnitude less than at the input.
In Fig. 2, the blue line reports the value of the parameter u3dB = k0L/2(sin θ3dB − sin θmax) as function of the antenna
parameter αL/2. This parameter can be seen as a normalized angle and the exact value of the FWHM is obtained as 2(θ3dB−
θmax). Thus, this graph can be used as a design tool that relates the beam divergence to the length and the attenuation constant
of any grating antenna. For a fixed value of α, when the antenna is short (L is small), u3dB increases less than linearly with
respect to L, meaning that the FWHM is decreasing at increasing length since the emitting area of the antenna is increasing. Instead, when L is large, u3dB has a linear dependence on L, and, thus, the FWHM does not change because the power left
inside the antenna is too weak to influence the far field radiation. The point at which the dependence of u3dB on αL/2 changes
is slightly lower than 2, confirming that the optimal GA length is around π/α. As an example of common values, in the same figure, it is also reported the FWHM obtained from the normalized angle u3dB for three values of the coupling strength α,
assuming maximum emission at θmax= 10◦ and 1.5 µm free-space wavelength.
Once the attenuation, or coupling strength, is known as a function of the geometry and materials of the GA, it is possible to modulate it along the length of the antenna by changing the geometry of each period of the grating. The required profile of the coupling strength that produces the desired far field pattern is [21]:
2α(x) = |A(x)| 2 P (−L 2) P (−L 2)−P (L2) RL2 −L 2 |A(ξ)|2dξ −Rx −L 2 |A(ξ)|2dξ (6)
where L is the length of the antenna, P (−L/2) and P (L/2) are the input power and the power left at the end of the antenna, respectively, and |A(x)|2is the near field intensity, whose Fourier transform is the desired far field.
Thus, it is possible to obtain arbitrary beam shaping along θ whenever a wide range of coupling strengths can be achieved by changing the geometry or the materials of each grating period. For example, in Fig. 3, (a) shows the comparison between three far field patterns and (b) the respective coupling strength profiles that generates them. All the antenna parameters are normalized and the only assumption concerns the power left inside the waveguide at the end of the antenna, which, according to the previous discussion about the optimal length of the GA, is set to 0.18%. In Fig 3, the solid green curves correspond to a constant coupling strength, the dashed orange to a constant near field intensity (i.e. A(x) = const in (6)), and the dotted blue to a far field with a side lobe ratio (SLR), i.e. the ratio between the peak intensity to the highest side lobe, of −30 dB. From
0 1 2 3 4 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1
Fig. 2. Normalized -3dB angle u3dB= k0L/2(sin θ3dB− sin θmax) as a function of the normalized GA length αL/2 and FWHM in the case of coupling
u
x/L
L/2
Fig. 3. (a) Far field and (b) coupling strength profile along the GA length for three different cases: solid green represents a constant coupling strength, dashed orange a constant near field intensity, dotted blue far field with −30dB SLR. The far field is symmetric to u = 0. The product αL is set to its optimal value π and u = k0L/2(sin θ − sin θmax).
Fig. 3a, we see that for a constant near field along the entire GA, the far field exhibits an SLR=13.26 dB and the smallest FWHM possible for a given antenna’s length, FWHM = 4c/k0L cos θmax, where c ≈ 1.4 is obtained solving |sinc(c)|2= 0.5.
Comparing this value with that in (5) shows that chirping the coupling strength as shown by the orange line in Fig.3b gives up to 2c/αL ≈ 1.4 times narrower beams. However, if the GA length decreases from the ideal value and, consequently, the power left at the end of the GA increases, the far field of constant coupling strength (solid green) tends to the far field of constant near field intensity (dashed orange). Indeed, if more power is left at the end of the antenna, it means that there is less difference between the radiated power at the beginning and the end of the GA, so that it resembles a constant near field intensity.
To summarize, for sensing applications, it is extremely important to confine the radiation in a well-defined beam with low sidelobes to suppress noise from the environment. As shown by the dotted blue line in Fig. 3a, this can be achieved at the expenses of increased beamwidth for a GA with a fixed length when an appropriate modulation of the coupling strength is applied. For this reason, the following section focuses on sub-wavelength gratings that allow controlling the coupling strength in a single lithography step.
Closed-form expressions [22] of α and computations based on Coupled Mode Theory [23], [24] are available only for gratings made of small perturbations. For more complex structures, one needs to rely on simulations of the complete vector
field. Finite-Difference Time-Domain (FDTD), Rigorous Coupled Wave Analysis (RCWA), and Eigenmode Expansion (EME) are the most used Maxwell’s equations solver engines for grating couplers. An FDTD solver has been used in this work because of the reliability of its results, even if computationally demanding.
B. Sub-Wavelength Grating GA
The attenuation constant α of the GA depends on the shape, dimension and material used to form the grating by periodic perturbation of a waveguide. There are several approaches to achieve antennas of hundreds of micrometers to a few millimeters length and FWHM well below one degree in high-index contrast PICs, such as IMOS. One possibility is to etch the surface of the waveguide by only a few tens of nanometers [6]. However, this approach requires a dedicated semiconductor etching step, is sensitive to etch rate variations and the range of achievable coupling strength is limited. Defining the gratings on the sidewalls of the waveguides [7] was also demonstrated as a viable option, with the restriction that the waveguide cross-section has to be close to the diffraction limit and the sidewall roughness negligible to suppress phase errors. Thus, this is a valid approach only for optical phased arrays, in which a single antenna has a wide divergence angle in ψ. An alternative, robust to etching variations, was demonstrated in the IMOS platform [17], in which the grating was defined in the silica layer cladding the antenna. Such GAs can efficiently emit very narrow beams (FWHM below 0.1◦) as the highest coupling strength is below 2 mm−1, which gives a minimum optimal length above 1.5 mm.
Here, the feasibility of sub-wavelength gratings (SWG) to engineer the coupling strength across a much wider range is investigated. These types of gratings were demonstrated for fiber grating couplers, with an effective length of ≈ 10 µm, to reduce the reflection in the waveguide [25], [26], but can potentially provide a way to fabricate much longer antennas in the same fabrication steps already used for other devices.
1) Fabrication: Fig. 4 shows the geometry of an SWG GA, where the antenna has been cut normally to the propagation direction of the guided mode. This antenna is characterized by two periods, Λx and Λy, as shown in the figure, and by the
etch depth. Thus, the coupling strength can be modulated by changing the fill-factors of the periods and producing shallower or deeper holes. The thickness of the InP membrane is 300 nm, the layers below it are 100 nm of SiO2, 1.8 µm of BCB
polymer, 50 nm of SiO2 and the silicon carrier.
The schematic of the fabrication is depicted in Fig. 5. An InGaAsP etch-stop layer and a 300 nm-thick InP layer are epitaxially grown on an InP wafer. Then, thin SiO2layers are deposited on the InP wafer and on a silicon wafer, as shown in
Fig. 5a, to improve the adhesion of the BCB that is used to bond the two wafers. Once the InP wafer is flip bonded to the Si wafer as shown in Fig. 5b, the InP substrate and the InGaAsP etch-stop layer are chemically removed with selective wet etching, ensuring a smooth top surface of the InP membrane (Fig. 5c). The ridge waveguides, the fiber couplers used to feed the antenna and the SWG antenna itself are defined through two electron beam lithography steps and semiconductor dry etching with etching depth of 300 nm (deep) and 120 nm (shallow). Finally, the membrane is covered by a 1 µm-thick polyimide layer as shown in Fig. 5d. The minimum reproducible etched gap is 100 nm and this is taken as the critical dimension of the
SWG. The feeding waveguide has a width of 400 nm and an adiabatic taper is used to transition the mode to the width of the GA.
2) Coupling strength and fabrication tolerance: The pitch of the SWG in the y-direction, Λy, has to be small enough not
to couple the fundamental mode to higher orders modes, guided or unguided. This condition translates to 2π Λy > k0ns+ ky,0 Λy< ns λ0 + 1 2We −1 (7)
where ns is the 1D effective index of the fundamental mode of the InP slab (≈ 2.74 at 1.55 µm wavelength), λ0 is the
free-space wavelength, k0 the respective free-space wavenumber, We is the effective width of the GA, ky,0 = π/We the
wavenumber of the fundamental TE mode in the y-direction.
The effective index of the SWG has been calculated through Effective Medium Theory (EMT) and it has been compared to the effective index found through Finite Difference Eigenmode (FDE) solver. According to the EMT [27], an heterogeneous layered medium can be seen as an homogeneous medium with an effective refractive index nemt that is found by solving the
transcendental equation γcl n2 cl tan γclb 2 = −γco n2 co tanγcoa 2 (8) where γco/cl= k q n2 co/cl− n 2
emt, a = F FyΛy, b = (1 − F Fy)Λy. The solution of such equation has been verified with that
of a 1D unit cell with periodic boundaries, returning the same result. Once the homogeneous effective index has been found, it is possible to calculate the effective index of the fundamental TE mode through the effective index method, substituting the SWG with an equivalent homogeneous slab.
In Fig. 6, the SWG effective refractive index calculated using (8) and the effective index method is compared with that
Fig. 5. Process flow of the standard fabrication of IMOS passive components
obtained from a 2D unit cell of width Λy with periodic boundaries using an FDE solver for four different deeply etched SWG
periods, showing an exact overlap only for the SWG with a pitch below 300 nm. This can be explained as the first approach fails to satisfy the boundary conditions at the corners of the SWG, resulting in a different field distribution near the top and the bottom of the SWG. The dotted vertical lines indicate the maximum allowed fill factor when the smallest reproducible etch width is 100 nm for the four SWG pitches reported in the legend. Being the coupling strength related to the index contrast between the InP slab and the SWG section, a pitch of 500 nm has been chosen for the design of SWG GA as it allows the lowest index contrast with the InP slab. Moreover, SWGs with larger pitches are more wavelength dispersive, thus increasing the steerable range in the θ direction. This SWG pitch also dictates a minimum GA width of 3 µm to satisfy (7) above 1.5 µm free-space wavelength.
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131.8143.8148.1147.1132.8112.778.5 52.9 29.4 15.3 5.9 104.3116.3121.2112.997.5 84.7 56.6 41.3 22.6 12.2 4.2 87.1 94.4 94.9 95.0 81.5 70.1 49.0 35.1 20.1 9.6 3.9 68.0 72.5 72.8 69.1 62.3 48.1 37.9 25.8 14.1 7.3 3.1 54.1 56.7 59.6 54.3 51.4 40.0 30.5 21.3 11.1 5.7 3.1 39.2 40.9 41.1 39.5 34.0 27.0 21.4 12.4 7.6 3.9 1.9 28.7 30.5 30.4 30.4 25.3 21.5 16.9 10.1 6.4 3.9 1.8 18.2 18.8 19.2 17.4 15.3 12.7 10.3 5.4 3.6 2.2 0.9 10.2 10.4 10.4 9.5 8.4 7.1 4.3 2.9 2.2 1.2 0.50.5
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130.5 132.0 126.6 113.4 100.4 74.5 50.3 30.0 14.3 126.2 126.8 121.6 108.7 96.2 71.6 48.3 28.9 13.6 119.8 120.2 114.5 101.8 90.0 67.2 45.4 27.1 12.5 112.0 111.7 105.7 93.6 82.5 61.7 41.9 24.9 11.3 102.6 101.6 95.5 84.3 74.2 55.8 37.8 22.5 10.393.8 92.5 86.6 75.6 67.4 50.7 34.7 20.5 9.4 83.6 82.0 76.4 72.7 59.5 44.8 30.7 18.1 8.2 73.0 71.3 66.4 62.8 51.6 39.0 26.7 15.7 7.1 62.4 60.6 55.7 53.4 43.8 33.1 22.7 13.4 6.0 53.0 51.4 47.4 45.3 37.4 28.5 19.5 11.5 5.1 44.6 43.0 39.7 38.1 31.4 24.0 16.6 9.6 4.3 36.4 35.2 35.3 31.1 25.8 19.7 13.6 8.0 3.6 29.5 28.4 28.4 25.1 20.8 16.0 11.1 6.4 2.8 23.0 22.1 22.1 19.5 16.2 12.4 8.4 4.9 2.1 18.3 18.9 17.7 15.7 13.0 10.0 6.8 4.0 1.8 13.9 14.4 13.4 11.8 9.9 7.5 5.2 3.0 1.3 10.8 10.4 9.7 8.6 7.1 5.4 3.7 2.1 0.97.1 6.8 6.3 5.6 4.6 3.4 2.3 1.3 0.5 4.3 4.1 3.8 3.4 2.8 2.1 1.4 0.8 0.320
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y(1
FF
y) [
nm
]
Coupling strength (mm
1)
Fig. 7. Attenuation constant of the (a) deep SWG and (b) shallow SWG as a function of the fill factors. The SWG period Λyis 500 nm and the etch width
corresponding to the fill factor F Fyis reported on the right axes. The grating period Λx is calculated such that the emission angle is kept to 10◦and the
0.05
0.23
0.41
0.59
0.77
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FF
y
1.6
1.8
2.0
2.2
2.4
2.6
n
ef
f
500nm
450nm
400nm
300nm
Fig. 6. Effective refractive index of the fundamental TE mode of the SWG section for SWG periods of 500 nm(red), 450 nm(green), 400 nm(blue), 300 nm(yellow). The vertical dashed lines indicates the maximum fill factor F Fyafter which the width of the etch is less than 100 nm. The solid lines are
obtained through EMT and effective index method, while the crosses through FDE simulation.
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0.7 0.9 1.1 1.4 1.6 2.0 1.9 2.6 3.2
0.9 1.0 1.1 1.2 1.6 1.7 2.0 2.7 3.5
1.0 1.0 1.3 1.3 1.9 1.7 2.3 3.2 3.3
1.0 1.1 1.3 1.4 1.7 1.9 2.5 3.2 3.3
1.2 1.5 1.4 2.0 2.0 2.2 3.4 2.9 3.2
1.4 1.4 1.7 1.9 1.9 2.3 2.9 2.7 2.7
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0.6 0.7 0.7 1.2 1.5 1.9 2.5
0.7 0.8 1.0 1.3 1.6 2.0 2.6
0.8 0.9 1.0 1.4 1.6 2.0 2.6
0.8 0.9 1.0 1.4 1.6 2.0 2.7
0.9 0.8 1.1 1.4 1.7 2.1 2.8
0.9 0.8 1.3 1.5 1.8 2.2 2.8
1.0 1.1 1.4 1.6 1.9 2.3 3.0
0.9 1.2 1.4 1.7 2.0 2.4 3.1
0.9 1.3 1.4 1.7 2.0 2.4 3.0
1.2 1.5 1.7 1.9 2.2 2.6 3.3
1.6 1.7 1.9 2.2 2.5 3.0 3.7
2.0 2.2 2.4 2.6 3.0 3.4 4.2
0
2
4
6
8
401
329
x(1 FF
258
x) for FF
190
y= 0.8 [nm]
125
61
250
200
150
100
50
309
x(1 FF
246
x) for FF
184
y= 0.8 [nm]
122
61
500
450
400
350
300
250
200
150
100
50
y(1
FF
y) [
nm
]
/ (%/nm)
Fig. 8. Sensitivity of the coupling strength of the (a) deep SWG and (b) shallow SWG as a function of the fill factors.
To obtain the attenuation constant as a function of the fill factors, a 20 µm-long SWG GA has been simulated through 3D FDTD for several combinations of these parameters while maintaining the maximum emission angle at 10◦by adapting the grat-ing pitch Λx. It was proven [28] that a 3D FDTD simulation of such short section provides results in agreement with the full-scale
simulation. The coupling strength can, thus, be obtained from the power left in the waveguide, as P (20 µm) = P (0)e−2α·20 µm. The simulation results are reported in Fig. 7a for deep etched SWG and in Fig. 7b for shallow etched SWG, in which the axes also report the values Λx(1 − F Fx), Λy(1 − F Fy) corresponding to the dimensions of the areas to be etched. The number
inside each cell indicates the value of the coupling strength. From these results, the sensitivity has been obtained and it is reported in Fig. 8, where, again, the number in the cells indicates the value ∆α/α in percentage per nanometer of variation. As expected, small features are more sensitive to dimensions variations and a deviation of 20 nm can result in up to 70% variation
of the coupling strength from its expected value. For a minimum reproducible etch width of 100 nm × 100 nm, the smallest achievable coupling strength is 4 mm−1 for the deep etched SWG and 2 mm−1 for the shallow etched SWG, corresponding respectively to a minimum FWHM of 0.12◦and 0.06◦ at 1.55 µm wavelength and 10◦ maximum emission, using the infinite length approximation (5). An error of +20 nm would change the FWHM to 0.17◦ and 0.086◦, respectively. The radiation efficiency of these structures has been obtained from the 3D simulation by comparing the power directed upwards to the power in the ideal case (all the power lost due to desired radiation), i.e. ηr= Pup/ (P0(1 − exp (−2αL))). The values were between
50% and 70%. Nevertheless, this efficiency can be consistently improved in the IMOS platform thanks to the possibility of inserting a gold mirror beneath the antenna [17].
For the shallow etched SWG, the sensitivity to the etch depth variations has been studied in the case of F Fx = 0.8 and
F Fy= 0.83, which corresponds to an etch area of 100 nm × 100 nm for a free-space wavelength of 1.55 µm and an emission
angle of 10◦. The results in Fig. 9 are compared with the sensitivity to critical dimension loss, showing that the coupling strength is a linear function of the etch depth with a slope of 0.04 mm−1nm−1.
These results confirm that SWG, compared to the other types of grating antennas, offers a much wider range of coupling strengths that can be used to shape the radiation pattern according to equation (6). However, their fabrication can be negatively impacted by etching deviations. In the worst case, when the holes are 100 nm per side, a critical dimension loss of 17 nm would cause the beam divergence to change of 50% from its design value. The same would happen to shallow etched SWG when the etch depth is 140 nm rather than 120 nm. While the tolerance to critical dimension loss can be taken into account during the design process through careful choice of the fill factors, the etch depth has to be controlled at the nanometer level during the fabrication process. GAs based on shallow SWGs are more tolerant than deep SWGs, as they offer finer etch depth control and are less sensitive to critical dimension loss. The simulation results and sensitivity to process variations will be compared to experimental results from several GAs with varying fill factors and shallow or deep etch that are currently being fabricated.
nm
m
m
1
Fig. 9. Attenuation constant α of a shallow etched SWG grating antenna with (F Fx, F Fy) = (0.8, 0.83) as a function of the deviation of the etch depth
III. 2D BEAMFORMING
The previous section focused on grating antennas that can produce an arbitrary beam shape in the plane ψ = 0 (see Fig. 1). As derived in the Appendix, the shape of the beam in the planes θ = constant is proportional to
|E (const, ψ)| ∝ cos k0W 2 sin ψ k0W 2 sin ψ 2 −π 2 2
Consequently, it is possible to change this shape the same way it has been done for the shape in the plane ψ = 0, i.e. by changing the coupling strength along y according to
2α(y) = |A(y)|
2
cos2πy We
(9)
where A(y) is the wanted amplitude of the electrical near field (i.e. the Fourier transform of the desired shape of the beam) and the denominator is the amplitude of the electrical field of the fundamental quasi-TE mode in a waveguide with effective width We. However, modulation of the coupling strength in the y direction requires changing the fill factors of the SWG GA
along y and this causes the propagating mode to cross regions with different effective indices. In other words, the SWG acts as a lens for the light inside the GA, since a lower fill factor means lower refractive index and a higher fill factor means higher refractive index (see Fig. 10). This can cause phase errors that would impair the formation of a collimated beam in the far field.
Fig. 10. Equivalence between an SWG and a lens. The color represents the local refractive index of the material, with white the lowest value and black the highest.
For this reason, beam shaping in the planes θ = constant can be accomplished through an optical phased array (OPA). This consists of an array of antennas fed by a distribution network and a phase-shifting section. The distribution network is made of a cascade of Mach-Zender Modulators (MZM) that can partition the input power among the two outputs with an arbitrary ratio. One phase-shifter per antenna is, then, used to correct for the phase mismatch introduced by the MZMs and to electronically steer the beam in ψ direction. When only beam shaping is needed, the phase-shifting section can be removed by placing MZMs with two phase-shifters in the two branches at the last stage of the distribution network, as shown in Fig. 11. In the standard IMOS platform, phase-shifting is currently obtain through the thermo-optic effect. Nevertheless, power consumption and footprint reduction is achievable through MEMS twin waveguides [29].
Fig. 11. OPA with a distribution network of MZMs for beam shaping.
The far field of a uniform linear array is given by [21]:
EOP A(θ, ψ) = E(θ, ψ)AF (ψ) (10)
= E(θ, ψ)sin(
N χ 2 )
N sin(χ2) (11)
where E(θ, ψ) is the far field of a single antenna, AF (ψ) is the array factor, N is the number of antennas in the array and χ = k0d sin ψ − ϕ, being k0 the free-space wavenumber, d the pitch of the array and ϕ the phase difference between
consecutive antennas of the array. The maxima occurs for χ = ±2πm, m = 0, 1, . . . and, for a phase shift φ = k0d sin ψ0,
the array beam can be steered in the direction ψ0. The −3 dB points of the array factor occur when χN = ±2.782 rad. For
example, if d = 5 µm, the FWHM would be 0.12◦ for N = 128 and 0.06◦ for N = 256.
Hence, the FWHM of the beam can be decreased or increased by respectively increasing or decreasing N , which is achieved by using the MZMs as switches. Moreover, the distribution network can also split the light unevenly, allowing for AF (ψ) with lower sidelobes. In fact, uniform linear arrays, in which the light is evenly split among the N antennas, exhibit the highest sidelobe levels, above −13.46 dB, and the smallest FWHM for a given array aperture N d. Distributing more power in the central part of the array and less at the edges generates an array factor with lower sidelobes and larger FWHMs. This way an OPA made of N SWG GAs can achieve the beam shaping functionality also in the ψ direction.
IV. FUTUREWORK
A. Vertically emitting GA
While periodic leaky-wave antennas offer the possibility to scan the beam by sweeping the wavelength, they are not inherently suited to scan through broadside (0◦) because of the stopband effect, i.e. the GA becomes a second-order Bragg reflector that couples the forward-propagating fundamental mode to the back-propagating mode (see Fig. 12a). Nonetheless, for the alignment sensor, GAs that can couple light at normal or at angles very close to the normal would reduce the circuit footprint and remove the need for beam steering in the ψ direction, so that a standard GA or an OPA without phase-shifting section can be used.
Here, this problem is addressed by designing an SWG GA able to mitigate the stopband effect by inserting two identical etched regions per period, separated by λ0/4nef f to ensure destructive interference between subsequent reflections [30] (see
Fig. 12b).
Fig. 13 shows the percentage of power reflected in the feeding waveguide for a GA with single SWG (green line) and for the
Fig. 12. Structures of GA for broadside emission: (a) standard GA, (b) double SWG per period, (c) double-sided processed nitride GA
GA with double SWG (orange). These lines show that the structure can obtain a noticeable reduction of the stopband region. However, the reflection coefficient still presents a sharp peak when the grating period equals exactly the Bragg condition.
To completely suppress this peak it is necessary to break the symmetry of the grating period with respect to the middle
m
R
SiN
R
double
R
single
Fig. 13. Comparison between the reflection of three different GAs emitting at 0◦. SWG1 is a SWG GA with one etched rectangular area per period as in Fig. 12a. SWG2 refers to the SWG GA as in Fig. 12b ans SiN to the structure in Fig. 12c.
of the SWGs. A structure as in Fig. 12c, where the grating is defined in a nitride layer above and below the InP membrane, accomplishes this by taking advantage of the double-sided processing available in IMOS. The thickness and the refractive index of the nitride layer are chosen to create constructive interference for the upwards diffracted light and destructive interference for that downwards diffracted, as shown in the picture, consequently achieving broadside emission and, theoretically, 100% radiation efficiency, regardless of the BCB thickness [31] (for a silicon-rich nitride with index 2.4, the required thickness is ≈ 250 nm). The simulated reflection for this structure is also reported in blue in Fig. 13, for comparison with the previous structures. Although, this structure requires a dedicated process flow (e.g. LPCVD of low-stress silicon-rich nitride and four
Fig. 14. Geometry of the alignment sensor with θt6= 0. (a) and (b) illustrate views rotated of 90◦with respect to each other.
lithography steps, two for nitride wet etching everywhere except on the GA and two for dry etching of the gratings) and it is sensitive to alignment errors since destructive interference only occurs when the trench above the InP membrane is ≈ 140 nm apart from the trench below (an error of 50 nm corresponds to an interference amplitude of 0.3, instead of 0).
In conclusion, among these GAs, the double-sided processed SiN-based antenna is the only one that can efficiently couple the light at 0◦. However, it requires four additional lithography steps and process optimization for the deposition of low-stress nitride films. Acceptable performance can also be obtain by the GA with two SWGs per period, which is process compatible, but it would require an optical isolator.
B. Alignment Sensor
1) Concept: The geometry of the problem is illustrated in Fig. 14. The alignment marker is in the origin and the PIC is placed at a distance h from the wafer plane, where the marker is situated. A transmitting antenna (Tx) illuminates the marker that diffracts the light in the three orders 0, +1 and −1. These are collected by receiving antennas (Rx0, Rx+1, Rx−1) and
the information carried by the phase of the received signals is processed in the chip.
There are two possible design scenarios: θt6= 0 and θt= 0. The second case makes use of vertically emitting antennas and
it has the advantage of lowering the footprint of the sensor and the number of controls required to drive it.
When θt6= 0, the PIC concept is shown in Fig. 15. A coupler splits the light coming from a laser diode into two paths, one
feeding the transmitting antenna (Tx) and the other goes to a second coupler. The other input of this second coupler is the 0th order diffracted light, collected by (Rx0). Such configuration acts as a Mach-Zender interferometer, in which one arm includes
the two antennas and the marker. Detecting the signal at the output of the interferometer, it is possible to determine the distance of the wafer from the PIC. Two other antennas, (Rx±1), collect the first diffraction orders and a multimode interference coupler
makes them interfere. As the marker is moved below the PIC, the interference generates a sinusoidal function with a period of half the pitch of the marker, as shown in Fig. 16. Measuring multiple locations and two markers per location with slightly different pitch gives sub-nanometer accuracy through the nonius principle.
2) Circuit requirements: When the height h is 1 cm and the marker is a square with area A2 = 30 µm × 30 µm, a transmitting antenna emitting at an angle θt = 10◦ from normal has to be positioned at xt = −h tan θt = −1.76 mm. To
Fig. 15. Circuit of the alignment sensor with θt6= 0. Positions relative to the marker are in mm in brackets. Length and width also in mm.
uniformly illuminate the area of the marker, the FWHM has to be arctan(xt/h + A/2h) − arctan(xt/h − A/2h) = 0.167◦
in the plane xz. From equation (5), the required coupling strength is found to be 5.8 mm−1, corresponding to holes of 125 nm × 125 nm and a length of 540 µm for a GA with constant coupling strength. As shown in Fig. 3a, by modulating α, it is possible to obtain almost the same FWHM but decreasing the SLR to −30 dB. For this antenna’s length, Fig. 3b returns a range of required coupling strength that goes from 0 to 17 mm−1. The FWHM in the ψ direction is ≈ 0.17◦ and, from (3), the width of the antenna has to be 620 µm. The same specifications apply to the antenna receiving the 0th order. The marker diffracts the light according to the grating equation
km= kinc+ mK m = . . . , −2, −1, 0, 1, 2, . . .
m
Fig. 16. Expected detected signal as function of marker offset position.
TABLE II
PARAMETERS FOR THE ANTENNAS OF THE ALIGNMENT SENSOR
θ (◦) ψ (◦) FWHMθ(◦) FWHMψ(◦) Length (mm) Width (mm) OPA
AlignmentSensor 1: θt6= 0
Tx/Rx0 10 0 0.167 0.170 0.54 0.62 No
Rx±1 10.36 15 0.166 0.160 0.54 0.64 128 GAs 4µm wide and 5µm apart
AlignmentSensor 2: θt= 0
Tx 0 0 0.17 0.17 0.52 0.62 No
with km the wavevector of the m-th diffraction order, kinc the wavevector of the incident wave and K the wavevector of
the grating. Thus, if the marker has a pitch of P = 6µm along y, i.e. K = 2π/P y, the positions and coupling angles of the antennas (Rx±1) receiving the ±1 diffraction orders are
x±1= h k0sin θt kz ≈ 1.8 mm y±1= h ±K kz ≈ ±2.7 mm θ±1= arctan k0sin θt kz ≈ 10.36◦ ψ±1= arcsin y±1 q x2 ±1+ y±12 + h2 ≈ ±15 ◦ with kz= q k2 0− k20sin θt2− K2.
This can be achieved with an OPA able to steer the beam at ψ±1, avoiding interference from the side lobes and with a
FWHM in ψ of arctan(y±1/h + A/2h) − arctan(y±1/h − A/2h) = 0.16◦. The first condition imposes antennas for the array
with a width smaller than 4 µm. The second requirement can be obtained pushing the sidelobes further than the FWHM of the single antenna, which is set to ψ±1. Thus, the pitch of the array has to be smaller than 5.8 µm. The third specification
gives the number of required antennas. For a pitch d of 5µm, the number of array elements N must be larger than 102. With N = 128, the total array width is 640 µm. Such PIC requires, at least, 258 control signals, twice 128 for the OPAs receiving the ±1 diffraction orders and at least one for the laser diode and one for the photodetector. (Note that it is not possible to rotate the receiving antennas instead of steering the beam in ψ with an OPA because the polarization of the diffracted light and that of the antenna would not match.)
3) Vertical antennas: The situation is much simpler when vertical emitting antennas are available. In this case, the trans-mitting antenna (Tx) is placed exactly above the marker location, while the (Rx±1) collecting the diffraction orders are also
standard antennas, not OPAs, placed along the same direction of the transmitting antenna as shown in Fig. 17.
For the same marker pitch and distance from the wafer as before, the transmitting antenna is located in x = 0, y = 0 and has an FWHM of 0.17◦ both in θ and in ψ. Thus, the coupling strength and the antenna dimensions are similar to the previous case. The receiving antennas are placed in x±1= ±2.67 mm, y = 0, with coupling angles θ±1= 14.97◦, ψ = 0 and FWHM
of 0.16◦ in θ and 0.17◦ in ψ, requiring a slightly longer GA with lower coupling strength, around 5.5 mm−1. However, this approach needs an optical isolator that prevents the 0th diffraction order from coupling back into the transmitting antenna and the Bragg reflections from interfering with the laser.
These requirements are calculated under the hypothesis of being in the far field, however the focusing plane is at 1 cm. All the previous derivation remains correct when using focusing antennas, in which the grooves are ellipses and not straight lines, as in focusing fiber couplers. The comparison between the two circuits is provided in Table II.
V. CONCLUSION
Previous works on lidar systems demonstrated how PICs can produce narrow beams and wide FoV for detection and ranging at distances of hundreds of meters. However, these devices are not well-suited for the alignment sensor, which requires control
Fig. 17. (a) Illustration of the sensor. (b) Circuit of the alignment sensor with θt= 0.
over the shape of the beam to optimize the overlap with the phase-grating marker. In this work, the analytical analysis of the grating antennas provided design rules to achieve an arbitrary shape of the beam in the θ direction. Mainly, (6) provides the required coupling strength to obtain the desired beam shape. Furthermore, simulations of GAs based on SWGs showed that these nanostructures could provide the range of coupling strengths required for the alignment sensor, as shown in Fig. 7 and discussed in Section IV. These findings provide a different approach to obtain optical integrated antennas, complementary to what has already been done in silicon platforms [5]–[12] and in IMOS [17]. Additionally, SWG-based GAs can be defined in the same lithography steps already used in the standard IMOS fabrication process and a linear array of these can be used to shape the beam in the ψ direction.
Most notably, this is the first time to the author’s knowledge that a PIC for laser-based phase-grating alignment is proposed. Successful demonstration of the circuits proposed here could result in increased throughput of the lithography tool thanks to the faster mapping of the wafer surface provided by multiple PICs measuring the entire wafer at once. However, SWG GAs are expected to be sensitive to etching variations. Thus, future work should focus on the characterization of these antennas and on process optimization.
APPENDIX
FORMULATION
In periodic structures, a single mode propagating along the positive x-axis can be written as the linear combination of partial waves or Floquet modes [23]:
E(x, y, z) =
∞
X
n=−∞
En(y, z)e−jkx,nx (12)
where bold symbols denote vectors, p is the period of the grating, kx,n= kx,0+2πnp and kx,0 is the complex wavenumber
of the zeroth-order mode, whose real part is approximately the real propagation constant of the fundamental mode of the waveguide without the grating. Hence, the zeroth-order mode is a slow wave (i.e. it’s phase velocity is slower than the speed
of light) and, consequently, does not contribute to radiating leakage. Radiation occurs when the period p is chosen such that at least one mode is a fast wave (usually only the −1 order). From (12), immediately follows that all the Floquet modes decay at the same rate α = αx,0 along the antenna, in fact kx,n= βx,0+2πnp − jαx,0, by separating the wavenumber kx,0 in its real
part βx,0and imaginary part αx,0. This attenuation constant incorporates all the loss mechanisms, and it is characteristic of the
entire structure, including the reflective plane below the perturbed waveguide. Thus, the sought radiative loss can be written as ηrα, where 0 ≤ ηr≤ 1 affects the fraction of power actually radiated but not the normalized radiation pattern. Assuming
that the width W of the GA is much larger than its thickness and the zeroth order mode is the fundamental quasi-TE mode, (12) can be rewritten as E(x, y, z) =y Y (y)b ∞ X n=−∞ Zn(z)e−jkx,nx =y cosb πy We ∞ X n=−∞ Zn(z)e−jkx,nx (13)
wherey is the unit vector directed along the positive y-axis, Eb n(y, z) has been factorize in the product of Y (y)Zn(z) because
of the assumption on the geometry of the antenna and Y (y) = cos(πy/We) because of the assumption on the fundamental
mode being quasi-TE, with We the effective width of the mode. From equivalence principle and image theory [32], the far
field of this structure in the region z > 0 is approximately that of a free-space radiating x-directed strip of magnetic density current, lying in the plane z = 0 and flowing along the x-axis. Denoting the free-space propagation constant with k0 and
assuming radiation only from the −1 order mode, the far field is then found to be
Eθ(θ, ψ) = jk0
e−jk0r
4πr (Ixcos ψ + Iysin ψ sin θ) (14)
Eψ(θ, ψ) = jk0 e−jk0r 4πr Iycos θ (15) with I(θ, ψ) = Z Z
E−1(y, 0)e−jkx,−1xejk·rdxdy
=yb Z W/2 −W/2 Z L/2 −L/2 Z−1(0) cos πy W e−αx−jβ−1x
×ejk0y sin ψ+jk0x sin θ cos ψdxdy (16)
=y (−j)b cos k0W 2 sin ψ k0W 2 sin ψ 2 −π 2 2 × sinc L 2 (k0cos ψ sin θ − kx,−1)
where, without loss of generality, Z−1(0) has been used as normalization constant. Finally, substituting (16) into (14) and (15),
we obtain the two polarizations of electric far field
Eθ(θ, ψ, r) = k0e−jk0r 4πr cos k0W 2 sin ψ k0W 2 sin ψ 2 −π 2 2
× sin ψ sin θ sinc L
2 (k0cos ψ sin θ − kx,−1) (17) Eψ(θ, ψ, r) = k0e−jk0r 4πr cos k0W 2 sin ψ k0W 2 sin ψ 2 −π 2 2 × cos θ sinc L 2 (k0cos ψ sin θ − kx,−1) (18)
showing that the field is strongly polarized along bψ for small values of ψ and θ. Neglecting the radial dependence, in the plane ψ = 0, the field amplitude is
|E(θ, 0)| ∝ cos θ sinc L 2 (k0sin θ − kx,−1) (19)
having the maximum when
k0sin θmax= <{k−1} = β−1, (20)
This result is in agreement with the well-known grating equation β0− k0sin θmax= mK with m = ±1, ±2, . . . and K = 2πp .
Under the hypothesis that ∆θ3dB θmax, the FWHM of (19) is [21]
∆θ3dB ≈
1 L/λ0cos θmax
. (21)
However, if the GA is much longer than the decay length 1/α, the grating acts as if it were infinitely long. Taking the limit of (16) for L going to +∞, yields
|E(θ, 0)| ∝ s cos θ α k0 2 + sin θ −β−1 k0 2 (22) ∆θ3dB = 2α k0cos θmax . (23)
In any plane θ = const, the magnitude of the field is proportional to
|E (const, ψ)| ∝ cos k0W 2 sin ψ k0W 2 sin ψ 2 −π 2 2 (24)
with the maximum emission at ψ = 0 and an FWHM of
∆ψ3dB = 2 arcsin 1.86762 λ πW . (25)
ACKNOWLEDGMENT
I would like to thank my supervisor, Prof. Yuquing Jiao, and my co-supervisor, Dr. Vadim Pogoretskiy, for their invaluable help and guidance throughout this project, always supporting me in pursuing my ideas. I sincerely appreciated it.
Thanks to Desalegn, Federico, Yi, Jorn and all who in one way or another contributed in the completion of this thesis. I would also like to ackwnowledge Prof. Castoldi and Prof. Calabretta, who made this Erasmus program possible.
Finally, my heartfelt gratitude to Giorgia, my parents and my sisters who have always been there to listen and encourage me. Grazie!
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