Deep beneath XianGuang Research Institute, the ultra‑clean laboratory resembled an eternal battlefield, only now the enemy was no longer visible mechanical failures or material defects, but invisible barriers set by physical laws themselves, hidden in the subtlest interactions of light and matter. The High NA EUV prototype stood like a silent behemoth, containing immense power to carve future chips. Yet Xiuxiu and her team now faced a specter magnified dramatically by rising resolution—**optical proximity effect**.
In common imagination, lithography might resemble using an extremely fine pen to directly "draw" circuits on silicon wafers. Reality is far more complex. When EUV light passes through a mask bearing circuit patterns, trying to project perfect design shapes onto silicon, a series of unavoidable physical phenomena occur: light diffraction blurs pattern edges; adjacent patterns interfere, causing unintended extra exposure or shadows; under the larger‑angle light incidence brought by High NA, these effects become more significant and unpredictable. The result: the final shapes appearing on the silicon wafer exhibit annoying deviations from the original design on the mask—lines may widen, narrow, end‑shorten, corners round, or unwanted auxiliary shapes appear. This is optical proximity effect—like a mischievous, unruly photocopier always adding its unwelcome "strokes" when copying the finest drawings.
At previous technology nodes, such deviations could still be compensated by process adjustments or simple design rules. But at the 2nm and smaller scales targeted by High NA, optical proximity effect is no longer tolerable error, but a critical obstacle deciding chip‑function success or failure. Attempting to eliminate it completely by infinitely improving optical‑system physical precision has hit limits of cost and physical laws—like trying to correct water‑surface ripples with a hammer.
Facing this high wall built by fundamental physics, Xiuxiu's gaze turned toward a solution in another dimension—**computational lithography**.
This wasn't further polishing optical hardware, but using algorithmic power to "trick" physical laws, performing compensation for physical distortion in the virtual world beforehand. Its core idea: transforming chip manufacturing from a linear "design‑fabricate" flow into an intelligent, continuously iterating feedback loop of "design‑simulate‑optimize‑fabricate."
In the newly established "Computational Lithography Center" adjacent to the lab, huge screens no longer showed real‑time monitoring of vacuum chambers and robotic arms; instead, they were filled with complex geometry, wave‑equation visualizations, and cascading data. Here was the frontline where algorithms clashed with physical laws.
Xiuxiu stood before the main console, explaining to the core team the directions they'd fully attack, especially the most challenging yet promising branch—**inverse lithography technology**.
"Traditional 'forward' lithography thinking is: 'I have a mask design, I expect to get corresponding shapes on silicon,'" Xiuxiu's voice was clear and forceful in the room humming with equipment. "But inverse lithography technology turns this problem completely upside‑down."
She pulled up a schematic. Left side: the perfect target shapes chip designers want on silicon—straight lines, sharp corners, uniform spacing. Right side: the **actual** likely results after simulating optical and photoresist‑chemical‑physical processes—filled with various optical‑proximity‑effect distortions, unsatisfactory.
"The problem ILT solves," Xiuxiu's fingertip traced across the screen connecting target and distorted shapes, "given our desired target shapes, and our known optical‑system and process models, **infer**: What kind of seemingly 'distorted' or 'deformed' patterns should we place on the mask, such that when this 'distorted' mask pattern undergoes real‑physical‑process 'distortion,' it exactly yields the initially desired perfect target shapes on silicon?"
This line of thought itself brimmed with dialectical wisdom. It didn't try to eliminate distortion, but **actively, precisely introduced a controllable, oppositely‑directed 'pre‑distortion'**, so that the two distortions cancel each other out physically, achieving "negative‑times‑negative‑equals‑positive." It's like aiming not directly at the bullseye to hit it under strong crosswind, but calculating an aiming point offset based on wind force and arrow speed.
"That sounds like magic," a young algorithm engineer murmured. "But theoretically…"
"Theoretically, it's an extremely complex, large‑scale, non‑convex mathematical optimization problem," Xiuxiu continued, tone grave. "We need to build an extremely accurate **full‑flow physical model** simulating everything from EUV light emission, modulation by mask, projection through High NA complex optical system, chemical reaction with photoresist, until final shape formation on silicon. This model must include electromagnetic‑field propagation of light (often requiring solving complex Maxwell's equations), photoresist exposure kinetics, development process, even effects of subsequent etching processes."
She paused, letting the task's enormity sink in. "Then, with target shapes as constraints, mask patterns as variables, through iterative optimization algorithms (maybe gradient descent, genetic algorithms, or more advanced global optimization methods), search for that mask design minimizing difference between simulation results and target shapes. This search space is astronomical—a chip design contains billions or tens of billions of pattern elements; each tiny adjustment affects the whole."
The screen began showing examples of mask patterns ILT might generate. No longer the neat rectangles and polygons designers recognized, but a complex, abstract‑art‑like image filled with strange protrusions, depressions, jagged edges, and auxiliary dots. Every detail of these seemingly random patterns is "computed" by the algorithm to precisely compensate optical proximity effects at specific locations under specific conditions.
"This is a mask generated by inverse lithography technology," Xiuxiu pointed at the dazzling pattern. "It looks 'ugly,' doesn't match human design intuition, but it's the optimal or near‑optimal solution under physical‑law constraints to achieve target shapes. Its complexity directly reflects the complexity of the physical process."
Introducing ILT means a fundamental change in chip‑manufacturing flow. Design‑team‑submitted target layouts first enter this "computational lithography" loop. Here, super‑computer clusters run for days or weeks, performing massive physical simulation and optimization calculations, generating that "pre‑distorted" mask file. This file is used to fabricate the actual mask, which then enters the lithography machine for wafer processing. Processed results can feed back into the model via metrology data, further calibrating and optimizing this virtual "pre‑distortion" process. Manufacturing is no longer simple execution of design, but continuous dialogue and collaboration between design and physical reality, with computation as the bridge.
This was a grand blueprint, but Xiuxiu knew its core bottleneck well. The accuracy of that full‑flow physical model directly determines ILT compensation's effectiveness. Building such a complex multi‑physics model and achieving its efficient, stable inverse optimization required mathematical tools and computational theory far beyond traditional engineering. It involved partial‑differential‑equation numerical solving, large‑scale nonlinear optimization, uncertainty quantification, even new mathematical frameworks to handle such high‑dimensional, strongly‑constrained inverse problems.
She thought of Yue'er. Of that mind capable of discerning abstract mathematical structures, skilled at building rigorous theoretical systems. Of her robustness thinking demonstrated when responding to B‑so‑lay's critique—incorporating "flexibility" and "uncertainty" into mathematical frameworks. The core difficulty ILT faced—how to stably solve for that optimal "pre‑distortion" pattern in physical models full of uncertainties and complex constraints—wasn't that a profoundly mathematical problem?
Engineering practice had touched theoretical boundaries. She needed thinking like Yue'er's to provide the firmest mathematical foundation for this algorithm‑built bridge fighting physical distortion.
Without hesitation, Xiuxiu directly called Yue'er's video communication.
Yue'er's face appeared on screen, background her study piled with books and blackboards covered in formulas. She seemed deep in thought; seeing Xiuxiu, a gentle smile appeared.
"Xiuxiu, how's it going? Any ideas on High NA metrology limits?" Yue'er asked with concern.
"Metrology's another hard nut, still cracking," Xiuxiu waved, cutting to the chase. "Sister Yue'er, I've encountered a problem that likely needs your help more. We're fully attacking computational lithography, especially inverse lithography technology."
She concisely explained the challenge of optical proximity effect, the core idea of ILT "pre‑distortion" compensation, emphasizing the near‑hopeless mathematical complexity faced in full‑flow physical modeling and inverse optimization.
"…So, we need not merely an algorithm that runs, but a theoretically rigorous, efficient enough framework capable of handling such ultra‑large‑scale inverse problems." Xiuxiu's eyes held expectation and earnestness. "Our engineering practice has reached a crossroads needing cutting‑edge mathematical‑theory guidance. Sister Yue'er, I formally, solemnly invite you to join our computational lithography project as **Chief Science Advisor**. We need your wisdom to help us build this algorithmic bridge connecting ideal design with physical reality."
On screen, after listening, Yue'er's eyes gradually lit with familiar radiance—the flame kindled in a mathematician's eyes facing challenging intellectual problems. The idea of transforming chip manufacturing into a grand mathematical optimization problem greatly attracted her. Especially the involved inverse‑problem solving, optimization under uncertainty, seemed deeply linked with her recent thoughts on "boundaries" and robustness theory.
She didn't answer immediately, but contemplated a moment, then slowly said: "Transforming shape generation under physical constraints into an optimization problem in high‑dimensional space… that's very interesting. Especially your mentioned 'pre‑distortion' concept—that essentially seeks an 'inverse image' or 'near‑inverse image' under a physical mapping. It reminds me of some concepts in function spaces, and how to stably solve such problems in the presence of noise and model error. This indeed is a profound mathematical problem."
She looked up at Xiuxiu, a firm, interested smile appearing: "Xiuxiu, I'm honored by your invitation. This is precisely an excellent opportunity for deep integration between theoretical mathematics and cutting‑edge engineering practice. I accept. Let's see together if we can use mathematical keys to open this gate toward higher‑precision manufacturing."
Xiuxiu exhaled a long breath, a big stone lifted from her heart, replaced by huge excitement and anticipation. She knew Yue'er's joining wasn't merely adding a top‑tier intellectual resource; it marked the beginning of true **deep integration** within "XianGuang"—the two formerly relatively independent worlds of "theory" and "engineering" converging.
Mathematics would no longer remain merely symbols and theorems on paper; it would directly participate in great engineering fighting physical limits, shaping the material world. And the complex difficulties in engineering practice would provide the freshest, most challenging research material for mathematical theory, pushing it toward new frontiers.
"Wonderful! Sister Yue'er!" Xiuxiu said excitedly. "I'll have someone grant you highest access immediately, send all related models, data, algorithm documents. Our mathematical journey is about to start from this micro‑world of chip design!"
Ending the call, Xiuxiu felt new strength infusing her heart. The road ahead remained thorny; High NA's metrology limits stood like a blocking tiger; ILT's mathematical abyss awaited crossing. But now, she wasn't fighting alone. Beside her, Mozi's unlimited capital support served as backing; now, Yue'er's profound mathematical wisdom provided guidance.
Capital, technology, theory—this solid iron triangle, on the new battlefield of computational lithography, again demonstrated its unparalleled synergistic power. Xiuxiu turned, facing her team, voice resolute and forceful:
"Everyone, our Chief Science Advisor, Professor Yue'er, has officially joined! Next, let's fight this hard battle using algorithms against physical effects—together with a mathematician! Goal: achieve perfect shape replication under High NA!"
The laboratory echoed with hopeful, spirited responses. A joint expedition driven by engineering needs, led by mathematical theory, delving into micro‑world mysteries, thus set sail. The epic of lithography was about to enter a new era interwoven with computing power and wisdom.