How Aerial Image Simulation Works (Hopkins Model)

What Is the Aerial Image?

The aerial image is the intensity distribution of light just above the wafer surface, before it enters the photoresist. It is the optical system's output — a direct consequence of the mask pattern, the illumination source shape, and the lens aberrations. Simulating it accurately is the foundation of all OPC work.

The Hopkins Formulation

In 1953, H.H. Hopkins derived a compact formulation for partially coherent imaging that remains the basis of all modern lithography simulators. The key insight: a partially coherent source can be decomposed into incoherent point sources. Each point source illuminates the mask coherently, producing a coherent aerial image. The total intensity is the incoherent sum over all source points.

The formulation involves three key quantities:

• S(f,g) — the source intensity distribution (the illumination shape in spatial frequency coordinates)
• M(f,g) — the Fourier transform of the mask transmission function
• P(f,g) — the pupil function, equal to 1 inside the NA circle and 0 outside (for an ideal aberration-free lens)

For each source point, the coherent image amplitude is computed by multiplying the shifted mask spectrum with the pupil function and inverse-Fourier-transforming back to the spatial domain. The total intensity is the incoherent sum of the squared amplitudes across all source points.

Transmission Cross-Coefficients (TCCs)

Direct evaluation of the Hopkins integral is expensive for every new mask. In practice, simulators precompute the Transmission Cross-Coefficient (TCC) matrix, which captures how pairs of mask spatial frequency components interfere at the image plane.

The TCC is a property of the optical system only — it depends on the source shape, pupil function, and NA, but not on the mask. Once computed for a given optical configuration, it can be reused for any mask pattern on that tool.

For efficiency, the TCC is decomposed into eigenvectors, called coherent decomposition kernels. A small number of dominant kernels — typically 8 to 20 — captures more than 99% of the optical response. Each kernel is convolved with the mask independently (a coherent image), and the resulting intensities are summed. This kernel-based approach is what makes real-time aerial image simulation feasible.

Resist Threshold Model

The aerial image alone does not predict the printed pattern. After the aerial image reaches the resist, a threshold model determines where the resist clears. In the simplest model, pixels with intensity above a threshold Ith clear the resist (define the bright features). More accurate models (Dill, Mack) account for the chemical amplification and diffusion in the resist film.

The predicted contour — the iso-intensity line at the threshold — becomes the basis for measuring EPE and driving OPC iterations.

What litopc Computes

litopc runs the Abbe form of this calculation directly rather than through a precomputed TCC: the source (a σ = 0.7 disk by default, or an annulus on Pro) is sampled into point sources on equal-area rings, each point images the mask coherently through a circular NA/λ pupil with a physical defocus phase, and the intensities are summed. The engine is scalar and thin-mask, with no aberrations, and the resist is a single intensity threshold. Because the mask is re-imaged for every evaluation, there is no kernel truncation; the cost is kept small by forming the image on a grid that over-samples the (1 + σ)·NA/λ passband and interpolating exactly back to the raster.

That is enough to show the effects this article is about — corner rounding, line-end pull-back, iso/dense bias, the loss of contrast through focus — and to drive a model-based OPC loop on them, but it is not calibrated to any process. The enabled presets are DUV dry (193 nm, NA 0.93) and EUV Low-NA (13.5 nm, NA 0.33). The exact equations, with their limits, are in the model guide, and the validation page checks them against analytic expectations.

Try it yourself in the litopc simulator — no installation required. Open Simulator →