Microlens & CRA Optimization

This recipe shows how to sweep microlens parameters (height, radius, squareness) to maximize QE and study the effect of CRA (Chief Ray Angle) shift compensation on pixel performance.

Background

The microlens is the most optically significant element in a BSI pixel. Its shape determines how well light is focused onto the photodiode. Key parameters:

• Height ($H$): Controls the focusing power. Too tall and light over-focuses; too short and it under-focuses.
• Radius ($r_x,r_y$): Determines the lens aperture. Should be close to but less than half the pitch.
• Squareness ($n$): Superellipse parameter. $n=2$ is an ellipse; higher values create a more box-like lens with better fill factor.
• CRA shift: For pixels at the edge of a sensor, the microlens must be shifted to accept light at an angle. At the sensor edge, the chief ray arrives at an oblique angle. Without microlens shift, the focused spot misses the photodiode, causing QE loss and crosstalk. Modern image sensors offset the microlens toward the optical axis to compensate.

Setup

import numpy as np
import copy
import matplotlib.pyplot as plt
from compass.runners.single_run import SingleRunner
from compass.analysis.qe_calculator import QECalculator

base_config = {
    "pixel": {
        "pitch": 1.0,
        "unit_cell": [2, 2],
        "bayer_map": [["R", "G"], ["G", "B"]],
        "layers": {
            "air": {"thickness": 1.0, "material": "air"},
            "microlens": {
                "enabled": True, "height": 0.6,
                "radius_x": 0.48, "radius_y": 0.48,
                "material": "polymer_n1p56",
                "profile": {"type": "superellipse", "n": 2.5, "alpha": 1.0},
                "shift": {"mode": "none"},
            },
            "planarization": {"thickness": 0.3, "material": "sio2"},
            "color_filter": {
                "thickness": 0.6,
                "materials": {"R": "cf_red", "G": "cf_green", "B": "cf_blue"},
                "grid": {"enabled": True, "width": 0.05, "material": "tungsten"},
            },
            "barl": {"layers": [
                {"thickness": 0.010, "material": "sio2"},
                {"thickness": 0.025, "material": "hfo2"},
                {"thickness": 0.015, "material": "sio2"},
                {"thickness": 0.030, "material": "si3n4"},
            ]},
            "silicon": {
                "thickness": 3.0, "material": "silicon",
                "photodiode": {"position": [0, 0, 0.5], "size": [0.7, 0.7, 2.0]},
                "dti": {"enabled": True, "width": 0.1, "material": "sio2"},
            },
        },
    },
    "solver": {
        "name": "torcwa", "type": "rcwa",
        "params": {"fourier_order": [9, 9]},
        "stability": {"precision_strategy": "mixed", "fourier_factorization": "li_inverse"},
    },
    "source": {
        "wavelength": {"mode": "single", "value": 0.55},
        "polarization": "unpolarized",
    },
    "compute": {"backend": "auto"},
}

Microlens Staircase Approximation

Adjust the number of slices and lens squareness to see how the staircase approximation changes.

Number of slices: 20

Squareness (n): 2.5

Lens height:0.600 um

Number of slices:20

Slice thickness:0.0300 um

Estimated area error:2.33%

Interactive Pixel Stack Builder

Adjust the thickness of each layer in a BSI pixel cross-section. The visualization shows the vertical stack with refractive indices and a scale bar.

Microlens height: 0.60 um

Planarization: 0.30 um

Color filter: 0.60 um

BARL total: 0.080 um

Silicon: 3.0 um

Show DTI trenches

Show metal grid

Total stack height:4.58 um

Sweep 1: Microlens height

heights = np.arange(0.2, 1.01, 0.1)
avg_qe_vs_height = []

for h in heights:
    cfg = copy.deepcopy(base_config)
    cfg["pixel"]["layers"]["microlens"]["height"] = float(h)
    result = SingleRunner.run(cfg)

    # Average QE across all pixels
    all_qe = np.mean([qe[0] for qe in result.qe_per_pixel.values()])
    avg_qe_vs_height.append(all_qe)
    print(f"  height={h:.1f} um -> avg QE = {all_qe:.3f}")

plt.figure(figsize=(8, 5))
plt.plot(heights, avg_qe_vs_height, "o-", linewidth=2)
plt.xlabel("Microlens Height (um)")
plt.ylabel("Average QE at 550 nm")
plt.title("QE vs Microlens Height")
plt.grid(True, alpha=0.3)
plt.tight_layout()
plt.savefig("ml_height_sweep.png", dpi=150)

Sweep 2: Squareness parameter

n_values = [2.0, 2.5, 3.0, 4.0, 6.0, 10.0]
avg_qe_vs_n = []

for n in n_values:
    cfg = copy.deepcopy(base_config)
    cfg["pixel"]["layers"]["microlens"]["profile"]["n"] = n
    result = SingleRunner.run(cfg)

    all_qe = np.mean([qe[0] for qe in result.qe_per_pixel.values()])
    avg_qe_vs_n.append(all_qe)
    print(f"  n={n:.1f} -> avg QE = {all_qe:.3f}")

plt.figure(figsize=(8, 5))
plt.plot(n_values, avg_qe_vs_n, "s-", linewidth=2)
plt.xlabel("Superellipse Squareness (n)")
plt.ylabel("Average QE at 550 nm")
plt.title("QE vs Microlens Squareness")
plt.grid(True, alpha=0.3)
plt.tight_layout()
plt.savefig("ml_squareness_sweep.png", dpi=150)

Sweep 3: CRA Shift Optimization

The CRA (Chief Ray Angle) describes how far off-axis incident light arrives at pixels near the sensor edge. As the CRA increases, the focused spot drifts away from the photodiode center, reducing QE and increasing optical crosstalk. Compensating by shifting the microlens toward the optical axis is one of the most impactful optimizations for real sensor designs.

Interactive Cone Illumination Viewer

Visualize how Chief Ray Angle (CRA) and cone half-angle affect pixel illumination. The microlens shifts to compensate for oblique incidence.

CRA Angle: 10.0°

Cone half-angle: 14.5°

CRA:10.0°

Microlens shift:0.441 um

f/#:2.0

Solid angle:0.200 sr

Basic CRA sweep: no shift vs auto shift

Run each CRA angle twice -- once without microlens shift and once with automatic CRA-based shift:

cra_angles = np.arange(0, 31, 5)  # 0, 5, 10, ..., 30 degrees
qe_no_shift = []
qe_with_shift = []

for cra in cra_angles:
    # Without microlens shift
    cfg = copy.deepcopy(base_config)
    cfg["source"]["angle"] = {"theta_deg": float(cra), "phi_deg": 0.0}
    cfg["pixel"]["layers"]["microlens"]["shift"]["mode"] = "none"
    result = SingleRunner.run(cfg)
    avg_qe = np.mean([float(np.mean(qe)) for qe in result.qe_per_pixel.values()])
    qe_no_shift.append(avg_qe)

    # With auto CRA shift
    cfg2 = copy.deepcopy(cfg)
    cfg2["pixel"]["layers"]["microlens"]["shift"]["mode"] = "auto_cra"
    cfg2["pixel"]["layers"]["microlens"]["shift"]["cra_deg"] = float(cra)
    result2 = SingleRunner.run(cfg2)
    avg_qe2 = np.mean([float(np.mean(qe)) for qe in result2.qe_per_pixel.values()])
    qe_with_shift.append(avg_qe2)

    print(f"CRA={cra:2d} deg: no shift QE={avg_qe:.3f}, with shift QE={avg_qe2:.3f}")

Plot CRA response

fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(10, 8), height_ratios=[3, 1])

ax1.plot(cra_angles, qe_no_shift, "o-", label="No ML shift", linewidth=2, color="tab:red")
ax1.plot(cra_angles, qe_with_shift, "s-", label="Auto CRA shift", linewidth=2, color="tab:blue")
ax1.set_ylabel("Average QE at 550 nm")
ax1.set_title("CRA Response: Microlens Shift Compensation")
ax1.legend()
ax1.grid(True, alpha=0.3)
ax1.set_ylim(0, 1)

# QE improvement from shift
improvement = np.array(qe_with_shift) - np.array(qe_no_shift)
ax2.bar(cra_angles, improvement, width=3, color="tab:green", alpha=0.7)
ax2.set_xlabel("Chief Ray Angle (degrees)")
ax2.set_ylabel("QE improvement")
ax2.set_title("QE Gain from Microlens Shift")
ax2.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig("cra_shift_analysis.png", dpi=150)

Per-channel CRA analysis

Examine how each color channel is affected differently:

channels = {"R": "red", "G": "green", "B": "blue"}
channel_qe = {ch: {"no_shift": [], "with_shift": []} for ch in channels}

for cra in cra_angles:
    for mode, shift_mode in [("no_shift", "none"), ("with_shift", "auto_cra")]:
        cfg = copy.deepcopy(base_config)
        cfg["source"]["angle"] = {"theta_deg": float(cra), "phi_deg": 0.0}
        cfg["pixel"]["layers"]["microlens"]["shift"]["mode"] = shift_mode
        if shift_mode == "auto_cra":
            cfg["pixel"]["layers"]["microlens"]["shift"]["cra_deg"] = float(cra)
        result = SingleRunner.run(cfg)

        for ch in channels:
            ch_qe = [float(np.mean(qe)) for name, qe in result.qe_per_pixel.items()
                      if name.startswith(ch)]
            channel_qe[ch][mode].append(np.mean(ch_qe) if ch_qe else 0.0)

fig, axes = plt.subplots(1, 3, figsize=(16, 5))
for ax, (ch, color) in zip(axes, channels.items()):
    ax.plot(cra_angles, channel_qe[ch]["no_shift"], "o--",
            label="No shift", color=color, alpha=0.6)
    ax.plot(cra_angles, channel_qe[ch]["with_shift"], "s-",
            label="With shift", color=color)
    ax.set_xlabel("CRA (degrees)")
    ax.set_ylabel("QE at 550 nm")
    ax.set_title(f"{ch} Channel")
    ax.legend()
    ax.grid(True, alpha=0.3)
    ax.set_ylim(0, 1)

plt.suptitle("Per-Channel CRA Response", fontsize=14)
plt.tight_layout()
plt.savefig("cra_per_channel.png", dpi=150)

Spectral CRA analysis

For a complete picture, sweep both wavelength and CRA:

wavelengths = np.arange(0.40, 0.701, 0.02)  # 20 nm step for speed
cra_list = [0, 10, 20, 30]

fig, ax = plt.subplots(figsize=(10, 6))

for cra in cra_list:
    cfg = copy.deepcopy(base_config)
    cfg["source"] = {
        "wavelength": {"mode": "sweep", "sweep": {"start": 0.40, "stop": 0.70, "step": 0.02}},
        "angle": {"theta_deg": float(cra), "phi_deg": 0.0},
        "polarization": "unpolarized",
    }
    cfg["pixel"]["layers"]["microlens"]["shift"]["mode"] = "auto_cra"
    cfg["pixel"]["layers"]["microlens"]["shift"]["cra_deg"] = float(cra)

    result = SingleRunner.run(cfg)

    green_qe = np.mean([
        qe for name, qe in result.qe_per_pixel.items() if name.startswith("G")
    ], axis=0)
    ax.plot(result.wavelengths * 1000, green_qe, label=f"CRA={cra} deg", linewidth=2)

ax.set_xlabel("Wavelength (nm)")
ax.set_ylabel("Green QE")
ax.set_title("Green QE Spectrum at Various CRA (with ML shift)")
ax.legend()
ax.grid(True, alpha=0.3)
plt.tight_layout()
plt.savefig("cra_spectral.png", dpi=150)

Practical note on shift tables

Real sensor designs use a polynomial or lookup table for the microlens shift versus image height, calibrated during lens-sensor co-design. The auto_cra mode uses Snell's law ray tracing through all intermediate layers (planarization, color filter, BARL, silicon) to compute the shift, following the method described in Hwang & Kim, "A Numerical Method of Aligning the Optical Stacks for All Pixels," Sensors, vol. 23, no. 2, 702, 2023 (DOI: 10.3390/s23020702). For production designs, you can supply a custom shift table through the shift.table config option.

Sweep 4: 2D Optimization

For a more thorough study, sweep height and radius simultaneously:

heights = np.arange(0.3, 0.9, 0.1)
radii = np.arange(0.35, 0.50, 0.02)
qe_map = np.zeros((len(heights), len(radii)))

for i, h in enumerate(heights):
    for j, r in enumerate(radii):
        cfg = copy.deepcopy(base_config)
        cfg["pixel"]["layers"]["microlens"]["height"] = float(h)
        cfg["pixel"]["layers"]["microlens"]["radius_x"] = float(r)
        cfg["pixel"]["layers"]["microlens"]["radius_y"] = float(r)
        result = SingleRunner.run(cfg)
        qe_map[i, j] = np.mean([qe[0] for qe in result.qe_per_pixel.values()])

plt.figure(figsize=(8, 6))
plt.pcolormesh(radii, heights, qe_map, shading="auto", cmap="viridis")
plt.colorbar(label="Average QE")
plt.xlabel("Microlens Radius (um)")
plt.ylabel("Microlens Height (um)")
plt.title("QE vs Height and Radius")
plt.tight_layout()
plt.savefig("ml_2d_optimization.png", dpi=150)

Summary & Recommendations

• Height: QE peaks at an optimal height (typically 0.5-0.7 um for 1 um pitch). Too tall causes over-focusing; too short gives weak focusing.
• Squareness: Higher $n$ values (more square lens) generally improve QE because of better fill factor, but very high values can cause edge diffraction.
• CRA 0--10 deg: QE remains within 5% of normal incidence when ML shift is applied.
• CRA 10--20 deg: QE begins to drop. ML shift recovers 5--10% absolute QE compared to unshifted.
• CRA 20--30 deg: Significant QE loss even with shift. Short wavelengths (blue) are affected most because the small absorption depth makes them sensitive to focus offset.
• Blue channel is most sensitive to CRA because it absorbs near the surface where microlens focusing accuracy matters most.
• Red channel is least sensitive because photons penetrate deep into silicon regardless of focus quality.
• CRA shift is the single most impactful compensation for edge-of-sensor pixels. With auto CRA shift, the lens compensates and maintains QE up to 15-20 degrees before declining.