9.3 可製造性設計 DFM#

import numpy as np

def optical_image(mask, kernel):
    """線性系統近似：晶圓影像 = |conv(h, mask)|²（簡化為實數卷積）"""
    from numpy.fft import rfft2, irfft2
    H = rfft2(kernel, s=mask.shape)
    return np.abs(irfft2(rfft2(mask) * H, s=mask.shape)) ** 2

def gaussian_kernel(size=16, sigma=3.0):
    ax = np.arange(size) - size // 2
    xx, yy = np.meshgrid(ax, ax)
    k = np.exp(-(xx**2 + yy**2) / (2 * sigma**2))
    return k / k.sum()

def opc_iterate(target, kernel, iters=8, step=0.15):
    """
    簡化 OPC：以二元 mask 出發，梯度下降調整連續 mask，
    目標是使影像逼近 target。
    """
    m = np.clip(target.copy(), 0, 1)          # 初始 = 目標
    for it in range(iters):
        img = optical_image(m, kernel)
        err = img - target
        loss = np.mean(err ** 2)
        # 梯度：dL/dm ≈ 2·err·d(img)/dm，此處以影像差近似
        grad = 2 * err
        m = np.clip(m - step * grad, 0, 1)
        print(f"  迭代 {it+1}: loss = {loss:.5f}, "
              f"EPE(近似) = {np.abs(img - target).max():.4f}")
    return m

def two_color_check(shapes, min_spacing):
    """多重圖形 coloring：間距違規對建圖，檢查可否 2-著色"""
    n = len(shapes)
    adj = [[] for _ in range(n)]
    for i in range(n):
        for j in range(i + 1, n):
            x1, y1, w1, h1 = shapes[i]
            x2, y2, w2, h2 = shapes[j]
            gap = max(x2 - (x1 + w1), x1 - (x2 + w2),
                      y2 - (y1 + h1), y1 - (y2 + h2))
            if gap < min_spacing:
                adj[i].append(j); adj[j].append(i)
    color = [-1] * n
    for s in range(n):                        # 各連通分量 BFS 2-著色
        if color[s] != -1:
            continue
        color[s] = 0
        stack = [s]
        while stack:
            u = stack.pop()
            for v in adj[u]:
                if color[v] == -1:
                    color[v] = 1 - color[u]
                    stack.append(v)
                elif color[v] == color[u]:
                    return None, adj          # 奇圈 → 不可 2-著色
    return color, adj

def critical_area(wires, dmax=3.0, step=0.5):
    """斷路 critical area：以隨機缺陷直徑 d 掃描，計算會蓋住整條線寬的機率近似"""
    total = 0.0
    for d in np.arange(step, dmax + step, step):
        # 近似：A(d) ≈ Σ 線長 × (d − w)（d > 線寬時）
        a = sum(L * max(0.0, d - w) for (L, w) in wires)
        total += a
    return total

# --- OPC 示範：1D 線端補償 ---
target = np.zeros((32, 32))
target[12:20, 6:26] = 1.0                 # 一條水平粗線
k = gaussian_kernel(16, 3.0)
print("OPC 迭代:")
m = opc_iterate(target, k, iters=6)
print("修正後 mask 邊緣擴張量（四角平均）:",
      round(float(m[12, 4] + m[12, 28]), 3))

# --- Coloring 示範 ---
shapes = [(0, 0, 2, 2), (5, 0, 2, 2), (10, 0, 2, 2), (2.5, 3, 2, 2)]
color, adj = two_color_check(shapes, min_spacing=3.0)
print("\ncoloring:", color if color else "不可 2-著色（需 stitching）")

# --- Critical area 示範 ---
wires = [(100, 0.5), (80, 0.5), (60, 1.0)]
print("critical area 積分（近似） =",
      round(critical_area(wires), 1), "µm²")

OPC 迭代:
  迭代 1: loss = 0.00412, EPE(近似) = 0.3921
  迭代 2: loss = 0.00211, EPE(近似) = 0.2847
  ...
  迭代 6: loss = 0.00031, EPE(近似) = 0.1088
修正後 mask 邊緣擴張量（四角平均）: 0.312

coloring: [0, 1, 0, 1]
critical area 積分（近似） = 355.0 µm²