MICROANGELO SCULPTING: MICROLENS ARRAY FABRICATION
6.3 Microlens Array Characterization
The characterization of the microlens arrays produced by thermocapillary replication began with inspection of the surface profile using coherence scanning interferometry (also called scanning white light interferometry). In this technique, white light incident on the sample surface is interfered with light which passed through a reference arm. The resulting interference pattern shows fringes modulated by an overall Gaussian envelope. The maximum of the envelope occurs when the length of the reference arm equals the length of the sample arm, so from this measurement the surface profile of the sample can be measured. Using a Zygo NewView 600 and a Zemetrics Zegage, we measured the surface profile of the fabricated microlens arrays and have presented four qualitatively distinct topologies in Fig. 6.4. The full list of fabrication parameters can be found in Table 6.1.
Figure 6.3: Diagram of the full MicroAngelo experimental setup Aluminium Chiller block
Sapphire Window
Molten PS Nanofilm Transparent Quartz Window
Thermal Paste
Aluminium Heater Block
RTDs
Fiberglass insulation
Spring-mounted riser plates
Motorized z-control Ceramic heater element
Vibration isolation table
SU-8 spacers
do −ho d1 SU-8 pin pattern
ho
∏
Diagram of the experimental setup used for MicroAngelo fabrication. Consistent with the geometries defined in previous chapters, the sample is heated from below and actively cooled from above. A full listing of the dimensions and parameters can be found in the text and in Table 6.1. Figure courtesy of Daniel Lim.
Fig. 6.4 exhibits four representative MLA topographies achieved through MicroAn- gelo fabrication, imaged using coherence scanning interferometry. The fabrication parameters and surface characteristics for each of the topographies are listed in Table 6.1. As seen in Fig. 6.4(a) and (b), we have successfully achieved both con- vex (converging) and concave (diverging) MLAs. Simple topologies are formed when the pin diameter, Dp, is much larger than the center-to-center pattern pitch, Π. WhenDpis around the size ofΠ, the concave ridges around convex microlenses overlap to form smaller interstitial lens arrays. This achieves a hierarchical MLA structure where a smaller array of lenses is formed in the interstitial region of the larger lens array, as seen in Fig. 6.4(c). Hierarchical MLAs exhibit two distinct length scales, corresponding to the vertical size of the two lens arrays. We also report the fabrication of a lens structure with a central depression at the vertex of each microlens, seen in Fig. 6.4(d), which we call the caldera-like structure. During the course of this project, we discovered that the caldera-like microlens structure bears a strong resemblance to the microdonut topology fabricated by Vespiniet al.
Figure 6.4: Topographies of fabricated microlens arrays (a) Convex
x (μm)
z (nm) z (nm) z (nm)z (nm)
x (μm) x (μm)
x (μm)
y (μm) y (μm)
y (μm) y (μm)
00
0
0
0
00
0 00
50 50
25
15 30 100
100 100
100 100
100 200 100
200
200 200
200
200
200 400 200 300
300
300
300 50
10
(b) Concave
(c) Hierarchical (d) Caldera-like
Surface topographies of fabricated microlens arrays imaged using coherence scanning interferometry.
Note that the vertical axes have units of nanometers and the horizontal axes have units of microns;
all microlenses shown here are actually wide and shallow. (a) Convex microlens array composed of converging lenses. (b) Concave microlens array composed of diverging lenses. (c) Hierarchical microlens array formed from two interdigitated arrays of different sizes. The vertical scale is logarithmically plotted to highlight the shorter secondary array between the main peaks. (d) Caldera- like microlens array with a central depression at the vertex of each lens. An additional array of smaller lenses is also visible in the interstitial region. Figure courtesy of Daniel Lim.
through spin-coating polymer onto a patterned pyroelectric substrate [57]. While Vespiniet al.attributes the central depression formation to a slump of material away from the protrusion vertex during spin-coating, we will show through first-principles computational simulation in Sec. 6.4 that our caldera-like arrays evolve from the bottom-up and outside-in. The technique investigated by Vespini et al. has only achieved convex caldera-like ("microdonut") structures, whereas MicroAngelo has achieved concave caldera-like topologies as well and hence has access to a larger variety of curved topologies.
Beyond the qualitative observations gleaned from observation of microlens array surface profiles, we have also quantitatively characterized the microlens array prop- erties using this surface profile data. In particular, we measured the lens diameter, fill factor, focal length, Fresnel number, asphericity, and surface roughness. Each of these measurements will be examined in the following sections.
6.3.1 Lens Diameter and Fill Factor
Since the lenses are formed with a continuous surface profile, there is no clear delineation between a lens and the neighboring interstitial regions. We defined a
Table 6.1: Parameter values for the fabricated microlens arrays Convex Concave Hierarchical Caldera-like
t (min) 30 60 15 45
Tcold(◦C) 60 60 60 60
Thot(◦C) 180 180 180 180
do−ho(nm) 1630±40 1430±50 1400±30 1410±20 d1(nm) 805±7 880±10 730±10 730±10
d2(nm) - 320±20 - -
Dp(µm) 20 50 50 50
Π(µm) 100 75 100 100
ho(nm) 228±2 288±4 288±4 288±4
Parameter values for the four microlens arrays imaged in Fig. 6.4. Uncertainties are one standard deviation unless otherwise stated.tis the fabrication time for which the heating elements were active.
d2is the depth of the photoresist depression in a block and is only applicable to concave microlens array fabrication, as in the experimental setup shown in Fig. 6.1(b).
lens as the region of all contiguous pixels of the same curvature. Note that for the caldera-like lenses we included the center region of opposite curvature. We chose to use 8-connected pixels to determine neighboring pixels. The calculation of mean curvature is very noisy when performed on raw interferometric data, so first we smoothed the raw data using a cubic smoothing spline in MATLAB [25]
(csapsroutine with smoothing parameter set to 10−4). From the fitted cubic spline, the mean curvature at all points was calculated and the points with the appropriate curvature were grouped together to form a lens. Note that the cubic spline was only used to find the points which were part of the lens. All calculations and derived values were performed on the raw and unsmoothed interferometric data. Since the resulting region was not strictly circular, we defined a characteristic diameter of the lens, Dlens, from the total area of the lens, Alens, by the relation
Dlens =2
rAlens
π . (6.1)
The fill factor of the lens array was calculated by taking Alens and dividing by the area of a unit cell. The computed lens diameters and fill factors can be found in Table 6.2.
6.3.2 Focal Length and Fresnel Number
With the lens domain determined by the sign of the curvature, the focal lengths of the fabricated microlenses were estimated by fitting the raw data within the lens domain to a paraboloid of the form
z(x,y)= zmax − (x0)2
2R1 − (y0)2
2R2, (6.2)
where zmax is the height of the lens at its vertex and R1 and R2 are the radii of curvature along the lateral principal axes, x0and y0. The principal axes of the lens are not guaranteed to coincide with the native coordinate system (x and y) of the interferometry data, so we used rotated coordinates
x0 y0
!
= cosθ −sinθ sinθ cosθ
! x−xo
y− yo
!
, (6.3)
whereθis the angle of rotation of the principal axes (x0, y0) to the raw data axes (x, y) and (xo,yo) are the coordinates of the lens vertex in the raw data coordinates. The use of two independent radii of curvature allows us to account for any astigmatism in the lens. The corresponding focal lengths, f1and f2, are then calculated from the lensmaker’s equation usingR1andR2
1
fi = n−1 Ri
, (6.4)
where we have assumed that the lens is thin and that the back side of the lens is planar, corresponding to an infinite radius of curvature. The larger of the two calculated focal lengths was defined to be f1 and the smaller was defined to be f2. Since these lenses were used with HeNe lasers with an optical wavelength of 632.8 nm, a refractive index of 1.580 was used for the PS. This value was measured in our lab using an Abbe refractometer; for more details on this instrument please see Appendix A.6.
With the calculated lens diameters and focal lengths, we can evaluate whether the lenses are operating in the near-field or far-field regime at the focal plane of the MLA using the Fresnel number. The Fresnel number, evaluated at the focal plane of the lens, is defined by
F = a2
λoptf, (6.5)
whereais a characteristic size of the aperture andλoptis the wavelength of light at which the lens is being used. If F is less than 1, the beam is in the far-field while if F is greater than 1, the beam is in the near-field. In our case, we can evaluate the Fresnel numbers for both individual lenses,Flens, and for the whole arrayFarray. These expressions are
Flens = D2lens
4λoptf, (6.6)
Farray= Π2
λoptf, (6.7)
where Π is the spatial period of the array. In all cases, the Fresnel numbers are small compared to unity, as can be seen in Table 6.2. This means that the lenses are operating in the Fraunhofer regime at the focal plane and diffraction is important.
6.3.3 Asphericity and Surface Roughness
To justify the selection of a paraboloidal geometry over a spherical geometry and quantify the degree of asphericity, we fit the lens cross section along its principal axes to an aspheric profile of the form
z(r)= zmax−
r2 (Dlens/2)
1+p
1−r2/(Dlens/2)2 +α4r4
, (6.8)
whereα4is the first aspheric coefficient and quantifies the degree of asphericity. The larger the value ofα4, the less spherical is the 1D lens profile. The lens profiles are displaced vertically so that the minimum (maximum) of the fitted convex (concave) lens lies at zero height and zmax corresponds to the height of the lens. To allow comparison of this asphericity over different lens sizes, we defined the asphericity ratio as the ratio of the α4 contribution to the surface profile relative to the lens height, evaluated one characteristic radius away from the lens vertex
AR≡
α4(Dlens/2)4 zmax
. (6.9)
The AR values calculated for the MLAs in Table 6.2 are less than one but still on the order of unity, indicating that the contribution due to the perturbing polynomial
is comparable to that of the spherical geometry. This result corroborates with the excellent fits obtained by the 2D paraboloidal surface in Eq. (6.2) over the lens surface. As a point of comparison for these reported values, a commercial spherical microlens array (Thorlabs MLA150-5C-M) evaluated using the same process yields an AR value of 0.04±0.03 and a commercial parabolic microlens array (Thorlabs MLA300-14AR-M) yields a larger AR value of 0.13±0.02.
The root mean square (RMS) residual of the 2D surface fit has two main contribu- tions: the non-conformity of the microlens geometry to the paraboloid shape and the high spatial frequency surface roughness contribution. The RMS residual of this fit therefore provides an upper bound to the surface roughness of the fabricated surfaces. The majority of the microlens fits achieve an RMS residual of less than 2 nm, which also provides an upper bound to the low surface roughness of the ultrasmooth microlenses.