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Chapter 60: Chapter 57: From Zero to One

Inside the examination hall of the Fudan University Physics Building.

Li Dong frowned as he studied the problem.

’Uniform system scan... transient temperature rise of the immersion liquid...’

He jotted down several key parameters for fluid dynamics and heat conduction on his scratch paper.

’So I have to solve the problem with the water!’

’What about using a microfluidic active cooling system?’

’I could build a high-frequency circulation channel above the wafer stage to carry the heat away instantly.’

Li Dong rejected the idea as soon as it came to him.

’No, that won’t work.’

’The wafer stage’s scanning speed is at least 1200 mm/s. The transient temperature rise in the immersion liquid from a single laser pulse is on the microsecond scale. A circulatory flow control system’s thermal response speed could never keep up.’

’Besides, the localized temperature rise has to be controlled to within 0.001℃ to meet the line width requirements.’

’A million probably wouldn’t cover it.’

Li Dong actually scratched his head and tried to approach the problem from a different angle.

’What about adaptive optics compensation, then? Adding a deformable mirror to the optical path, measuring the temperature in real time, and then canceling out the optical path difference?’

Li Dong shook his head, rejecting the idea once more.

’The problem states that the structure and parameters of the existing exposure optics system can’t be altered, so that’s out too.’

They were all dead ends.

Every viable theoretical solution Li Dong could think of was instantly wiped out the moment he applied the strict ”practical engineering constraints.”

Meanwhile, not far from Li Dong, beads of sweat had already begun to form on Qiao Fan’s forehead.

The scratch paper of this prized genius from Rongcheng Seventh Middle School was now covered in a dense scrawl of Navier-Stokes (N-S) equations and transient heat conduction partial differential equations.

Using the finite difference method, he had derived a three-dimensional numerical model for heat distribution in high-speed flowing pure water, pushing the theoretical calculation of the coupling between the flow and temperature fields to its absolute limit.

But... what then?

’So what if I’ve calculated it? To achieve this level of temperature control, the equipment’s size and cost...’

’Under a million? What a joke!’

”It’s impossible... Theoretically, it’s feasible, but from an engineering standpoint, it’s a dead end.”

Qiao Fan muttered to himself.

Meanwhile, the boy from Modu Middle School, sitting diagonally in front of Li Dong, looked deathly pale.

’Others might not grasp the true difficulty of this problem, but how could he not?’

His name was Lin Jian, the son of Lin Wei!

While others just saw it as an insanely difficult question that went beyond the curriculum, Lin Jian knew perfectly well this was the exact problem his own father had been trying to solve lately.

’Don’t tell me my dad wrote this question... Using it to test high school students?’

...

As time slowly passed, Li Dong forced himself to calm down.

’My approach must be wrong. If all those PhDs in optics at Huaxuan can’t solve it, what makes me think I can?’

Li Dong closed his eyes.

’What if...’

’If the great minds of the Cyan Dragon Study Group were looking at this problem, how would they approach it?’

Suddenly, he seemed to think of something.

A few days ago, while developing the refracting microscope, Newton had also gotten stuck in a similar dead end: ”how to polish a perfect lens medium to eliminate chromatic aberration.”

’How was it I ”advised” that god of physics back then?’

”Sir Newton, you are too focused on the perfection of the medium, causing you to overlook the properties of light itself.”

’The nature of light itself...’

Li Dong felt like he was onto something...

In his mind, Newton’s arrogant voice rang out.

’So, you’re His Excellency the Senior’s foolish nephew? Did you get your head slammed in the doors of the Royal Society?’

’How did you manage to get fixated on the ’medium’ again, after the very point your uncle made just a few days ago?’

”That’s right!”

Li Dong’s eyes snapped open.

’Everyone right now is exactly like Newton when he was trapped by ’lens chromatic aberration’! They’re all obsessing over the immersion liquid—the ’light propagation medium’!’

’They’re trying to reduce the medium’s temperature and refractive index fluctuations to zero!’

At that moment, the effect of Light and Shadow Insight activated.

The test paper vanished from Li Dong’s vision. All he saw was a beam of deep purple ArF excimer laser passing through an objective lens and into a pool of ultrapure water...

Li Dong could barely suppress a smile.

’I don’t need to brute-force control the medium’s refractive index (n) at all! I just have to control the light source’s vacuum wavelength (λ)!’

’Since the immersion liquid’s refractive index changes by Δn, I just need to simultaneously fine-tune the light source’s wavelength to give it a proportional compensation of Δλ.’

’As long as I ensure that (n+Δn)/(λ+Δλ) = n/λ always holds true, meaning λ and n maintain a strictly proportional relationship!’

’Then, substituting it into the phase formula, the new phase φ’ = 2π·(n+Δn)L/(λ+Δλ) = 2π·nL/λ = φ, which will always remain constant!’

More importantly, this equation was completely independent of the light ray’s angle of incidence and its optical path length.

Regardless of the light’s angle of incidence into the immersion liquid or its optical path length, as long as n and λ changed proportionally, the phase of every light ray would remain exactly the same. The principle of equal optical paths between the object and image would not be broken.

Even the change in the depth of focus caused by the shifting refractive index would be canceled out due to the extremely small amount of compensation.

Numerical aperture NA = n·sinθ, and depth of focus DOF = λ/NA².

Although the DOF could not be kept physically, absolutely constant when λ and n were fine-tuned proportionally, the resulting fluctuation in the depth of focus—from a wavelength tuning of just a few picometers (pm)—was only a thousandth of a nanometer. This was completely negligible within engineering tolerances!

He had found his angle of attack!

Li Dong immediately prepared to write down his ”real-time, synchronous, light-source wavelength-tuning compensation scheme” on the answer sheet.

However, just as he put pen to paper, he froze.

’Wait...’

’To fine-tune λ, the laser needs to receive a precise control command. Where would that command come from?’

’The problem requires a response time of ≤ 1ms and forbids any components from making direct contact with the wafer surface.’

’This means using a contact temperature sensor to measure the water temperature in real-time is completely unfeasible. The response time would be too slow.’

’The only way is... feedforward control!’

Li Dong’s well-rounded 0.2 attribute began to kick in.

’I have to establish a precise mapping model ahead of time.’

’All the core operating parameters for this system are already known.’

’I just need to run preliminary experiments to calibrate the transient temperature rise, ΔT, of the immersion liquid inside the exposure slit under different combinations of operating parameters. Then, using the temperature coefficient of the refractive index for ultrapure water, I can deduce the corresponding Δn. Finally, I can feed this pre-calibrated compensation value back to the laser, allowing it to complete the Δλ adjustment at the very instant the laser fires!’

But after figuring this part out, Li Dong was struck by a profound sense of powerlessness.

What was the key to establishing this zero-latency, predictive feedforward model?

It was the ability to accurately describe the coupled relationship between laser energy deposition, immersion liquid flow, and transient heat conduction under high-speed scanning boundary conditions.

Essentially, it meant solving the strongly coupled N-S and transient heat conduction equations to establish a precise numerical map between the operating parameters and Δn.

Li Dong looked at the fluid control equations he had listed on his scratch paper and gave a wry smile.

He understood wave optics, Classical Mechanics, and the basic laws of thermodynamics.

But he had never systematically studied computational fluid dynamics, nor had he mastered the numerical tools required to solve such a strongly nonlinear system of partial differential equations.

He could write down the form of the equations, but he couldn’t perform the practical model calibration and solve them.

’Teacher Chen was right. My knowledge really is just a series of isolated islands.’

’I need to build the bridges between them sooner... The ones Teacher Chen built for me are still too rudimentary.’

Li Dong put down his pen and sighed.

’I’ve found the angle of attack, the core optical principle is perfectly sound, and the path to an engineering solution is clear. But the final step for implementation—’constructing and solving the thermo-fluid coupled feedforward model’—is beyond my current abilities.’

He didn’t try to force it by fabricating mathematical formulas he didn’t understand.

Instead, he calmly wrote out his thought process in the answer section.

At the end, he added a final note:

[Engineering Implementation Bottleneck: To achieve precise compensation under all operating conditions, it is necessary to construct a strongly coupled thermo-fluid numerical model incorporating the Navier-Stokes and transient heat conduction equations. This model would be used to precisely calibrate and solve the mapping relationship.]

[Due to my current lack of the numerical solving ability for this type of nonlinear system of partial differential equations, as well as the necessary engineering calibration conditions, the specific control algorithm model cannot be fully constructed. Therefore, my solution to this problem ends here.]

Li Dong capped his pen and sighed.

’Guess I haven’t fleeced enough wool yet,’ he thought.

However, Li Dong didn’t realize just how critical the step he had just taken was. It was the leap from zero to one.