Simulation of a Micro-lens Array - Breault...Using ASAP and FDTD Solutions . This technical...

ASAP Technical Publication bropn1175 (Sept. 28, 2005)

Simulation of a Micro-lens Array Using ASAP and FDTD Solutions

This technical publication describes finite-difference, time-domain (FDTD) simulations via field distribution import from FDTD Solutions™ by Lumerical, and the process for effectively using them in the Advanced Systems Analysis Program (ASAP®) from Breault Research Organization (BRO). It also calculates and stores detailed history information for every ray that is traced, allowing more detailed analysis of problem paths through your system. If you are interested in taking advantage of this integrated approach, you must own licensed copies of both ASAP and FDTD Solutions. The example presented in this document illustrates an end-to-end simulation of a digital imaging system that uses micro-lenses above each CMOS pixel. The macroscopic optical lens system is modeled with ASAP, while the microscopic lens array and CMOS backplane is modeled by FDTD Solutions. The goal of the simulation is to calculate the pixel cross-talk when the system is illuminated by (plane waves near normal incidence and 20º incidence) a plane wave near normal incidence and near 20° incidence. Notes

• Before working through this example, you must be familiar with both ASAP and FDTD Solutions. • The six simulations that need to be run with FDTD Solutions take several hours on a single computer. • The following files are provided with this documentation: digital_camera.inr, microlens_lowres.fsp,

microlens.fsp, and microlens.lsf.

Breault Research Organization, Inc.

Copyright 2005-2015 All rights reserved 6400 East Grant Road, Suite 350 Tucson, Arizona 85715 USA

www.Breault.com [email protected] 800.882.5085 USA | Canada 1.520.721.0500 Worldwide 1.520.721.9630 Fax

Background The camera lens used in this example comes from U.S. Patent 5,706,141, issued in January 1988 and published in Optics and Photonics News in March 1998 (page 45). It is a digital still camera lens with a short focal length of 5.26 mm, F/2.8, and the total Field Of View (FOV) is 49.2°. The CMOS detector array has a pixel spacing of 4 µm. A micro-lens with a radius of curvature of 4µm, in a dielectric of refractive index n = 1.8, is positioned approximately 16 µm above each detector. A schematic view of the macroscopic and microscopic optical system is shown in Figure 1, and a schematic diagram of the entire micro-lens array is shown in Figure 2.

Figure 1. Macroscopic optical system to be modeled with ASAP and the micro-lens array to be modeled with FDTD Solutions.


Copyright 2005-2014 All rights reserved Simulation of a Micro-lens Array

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Figure 2. Schematic diagram of the entire micro-lens array. The two points (A and B) where the fields are calculated by ASAP are labeled by black squares. The detailed micro-lens pattern within the 12x12 µm2 black squares is also shown. The approximate alignments of the focused spots at each site are shown by red stars.

The camera is illuminated at normal incidence and at 20° incidence. The focused spot illuminates the detector array at sites A and B as shown in Figure 2. At each site, two misalignments of 0.02176° (along the y-axis and at 45° between the x and y axes) are introduced, which cause the focused spot to misalign with the detector array. In total, we have six different alignments to calculate, as shown in Table 1 below.

Table 1. Six alignments that will be calculated

Simulation site angle of incidence Misalignment 1 A 0° None 2 A 0° 0.02176° towards y axis 3 A 0° 0.02176° at 45° between x and y axes 4 B 20° None 5 B 20° 0.02176° towards y axis 6 B 20° 0.02176° at 45° between x and y axes

For each of the six cases, we will calculate the pixel crosstalk by integrating the total electric field amplitude intensity (intensity is Watts/steradian and used mainly for point sources) at each detector in the backplane.



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Solution The solution is separated into two steps, as described in Table 2. Each step is then described in detail.

Table 2. Solution Steps

Step Purpose Product 1 Model the macroscopic optical system that delivers the image of a point

source to the micro-lens array. Calculate the irradiance both on-axis near the center of the array and near the edge of the micro-lens array.

ASAP

2 Model the interaction of the resulting beam when it is aligned perfectly with a pixel and misaligned with a pixel. Calculate the pixel cross talk.

FDTD Solutions



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Step 1―Macroscopic beam delivery to the micro-lens surface The macroscopic optical system, shown in Figure 1, is a digital camera lens. The script that generates this optical system in ASAP is digital_camera.inr. After setting up the optical system model in ASAP, the source is traced to a dummy plane located in close proximity to the micro-lens surface as shown in Figure 3. Note that no micro-lens surface features are included in the ASAP model. A FIELD calculation stores the complex vector electric field in microdata_1.dat, microdata_2.dat,…, microdata_6.dat files, which is then exported to the *.fld file format using the CVF command. The files are saved as microdata_1.fld, microdata_2.fld,…, microdata_6.fld. The energy distribution at the dummy plane 50 nm above the micro-lens surface is shown in Figure 4. Note that WINDOW dimensions, PIXELS setting, and location of the dummy plane may require iteration to arrive at the conditions suitable for an accurate FDTD simulation. In this case, the WINDOW is chosen to include the center pixel and its nearest neighbors (12µm × 12µm), which helps ensure that all the focused energy is captured within the window. A choice of PIXELS 151 provides spatial resolution necessary to avoid phase ambiguities. After completing the export operation of six files in ASAP, we switch to FDTD Solutions to continue the simulation. Figure 3. Ray trace of for six source conditions. Three are close to normal incidence and focus to the center of the micro-lens array. Three others are incident at 20° and focus to the corner of the micro-lens array. (See Figure 2 for locations where the fields are calculated schematically.)



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a) site A, source angle 0° b) site A, source angle 0.02176° c) site A, diagonal source angle

0.02176°

d) site B, source angle 20° e) site B, source angle 20.02176° f) site B, diagonal source angle

20.02176°

Figure 4. Detailed field distributions. First column is at site A and second is at site B, as shown in Figure 2.



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Step 2―Modeling the sub-wavelength features of the micro-lens surface Open FDTD Solutions and open the example file, microlens.fsp included with this example. The geometry consists of an array of micro-lenses and is shown in Figure 5. The detailed dimensions are shown in Figure 6.

Figure 5. Micro-lens array, drawn by FDTD Solutions. The ASAP source (shown by the gray line and purple arrow) launches the fields calculated by ASAP onto the micro-lens array. The yellow line below the array collects the transmitted field information that can be propagated through the homogeneous n-1.8 materials to the backplane, 16 µm behind the front surface of the micro-lens array.



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Figure 6. Dimensions of Micro-lens array

The following steps show how to setup, run and analyze the simulations of the micro-lens array, as well as calculate the pixel cross-talk at the backplane. Step 2a―Run a low-resolution calculation We generally recommend running a quick, low-resolution simulation to assure that simulation, sources, and monitors are all set up appropriately. Check that the simulation has run long enough to extract accurate data from the frequency monitors, which perform Fourier transforms of the time signal. We will try a simulation at a mesh size of dx=60 nm. The wavelength of the source is λ0=587.562 nm. The number of points per wavelength is λ0/(n dx)≅5.4. In FDTD, we typically want a minimum of ten points per wavelength in the highest index material so we can expect to get only qualitative results from this simulation. In later steps, we will use dx=40 nm or 8.2 points per wavelength, from which we can extract quantitative results.

1. Open the file, microlens_lowres.fsp, included with this example. 2. Select the Simulation Area and edit the properties. Set the following properties:

Table 3. Properties for Simulation Area

property value dx 60 nm dy 60 nm dz 60 nm simulation time 100 fs

3. Select the ASAP Source, and load the field data microdata_1.fsp. When you plot the field data, it should

look like Figure 7 below. Check the Frequency/Wavelength settings, and make sure that the wavelength is 587.562 nm.



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Figure 7. Irradiance of the ASAP source in FDTD Solutions, read from the file microdata_1.fld.

4. Check the memory requirements: from the SIMULATE menu, select Check memory requirements. The

total memory is 197 MB. Your computer should have more memory than this to run the simulation. 5. Run the simulation. Note that after the meshing is complete, you will receive a warning that you do not have

10 points-per-wavelength in air. Click OK to proceed, ignoring this warning with the simulation. 6. View the movie microlens.mpg that was created with this simulation. From this movie, you can see that

the simulation was run for enough time for the fields to move through the structure. 7. More quantitatively, plot the Ex as a function of time for the time monitors called below, diagonal1 and

diagonal2, shown Figure 8 below. You can see that the main pulse is well past by 80 fs, and the remaining field is approximately three orders of magnitude smaller than the initial pulse. Based on this, we can run our full simulation for 80 fs.

a) below b) diagonal1 c) diagonal2

Figure 8: Ex component of the electric field as a function of time at three points below the micro-lens array (monitors below, diagonal1 and diagonal2). Based on these results, we can reduce our simulation time to 80 fs.



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Step 2b―Run a simulation for each source field created in ASAP

1. Open the file microlens.fsp provided with this example. 2. Edit the Simulation Area and verify that the simulation time is set to 80 fs. 3. The script file microlens.lsf is provided with this example. This script file breaks the simulation into

three sub-steps, which can each be done independently: • Load the template file microlens.fsp, provided with this example, and import the data from each of the

six fld files (microdata_i.fld) created by ASAP. It will save a different fsp file called microdata_i.fsp corresponding to each fld file.

• Run the six fsp files microdata_i.fsp created in the previous step. Each of these files can take 30 minutes to several hours to run, depending on the speed of your computer.

• Load each of the microdata_i.fsp files with completed simulation results and perform a cross-talk analysis of the pixels at the backplane 16 µm behind the micro-lens array. The projection step can several hours to run, depending on the speed of your computer.

Run the script file microlens.lsf. You will see this dialog box displayed:

You can choose which of the three steps you to run. The first step will take only a few seconds, but remember that running all six fsp files will take several hours and requires 498 MB of memory. You may want to leave this calculation running overnight. The analysis step performed by micro-lens.lsf is described in detail below. Step 2c―Analysis steps The following steps are all performed automatically by microlens.lsf when you click the third check box in the Micro lens calculation wizard. This stage of the analysis can take a few hours, depending on the speed of your computer.

1. Project the field to a depth of 16 µm in the x-z plane. At this distance from the focusing optics of the lens, we cannot use a far field projection that makes assumptions about being in the very far field. Instead, we use the built-in scripting function farfieldexact3d, which can project onto any rectangular surface defined by vectors x, y and z, as long as the projection is further than approximately one wavelength. The difficulty with this projection is that it is very slow, but is still faster than running a very large FDTD Simulation.



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An optimized version of farfieldexact3d will be available in future releases of FDTD Solutions. To save time, the resolution is relatively coarse, but you will see the two plots in Figure 9 for both the on-axis field (microdata_1.fsp) and the 20-degree field (microdata_4.fsp).

a) on-axis b) 20°

Figure 9: Projected irradiance from the micro-lens array to depth of 16 µm for a) on axis and b) 20° incidence beams. The spot is focal length is approximately 5 µm.

2. The second part of the analysis routine is to perform a cross-talk analysis at the back plane. We will assume for

simplicity that the backplane is made up of square detectors that are aligned with each micro-lens, as shown in Figure 10, and we will ignore any dead space between the detectors. The fields are then projected to the backplane and the total intensity at each detector is integrated. Finally, the electric field intensity is plotted over 5x5 pixels for each of the six different source fields in Figure 11 and Figure 12.

a) front side b) back side detectors

Figure 10: A schematic view of a) the front side micro-lens array and b) the detector array at the backplane.



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0.02176°


20.02176°

Figure 11: 5x5 pixel responses, showing the cross-talk on a linear scale



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0.02176°


20.02176°

Figure 12: 5x5 pixel responses, showing the cross-talk on a log scale

Summary A method for using ASAP and FDTD Solutions to calculate pixel cross-talk in a digital camera with a micro-lens array has been demonstrated. The method can be easily extended to consider more complicated micro-lens designs, for example, ones that include multilayer dielectric stacks.



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