Depth of focus (DOF) is the total range of focal plane displacement along the optical axis over which a photolithographic system maintains critical dimension (CD), pattern profile, and sidewall angle within specified manufacturing tolerances — a fundamental metric governing scanner focus control budgets and yield stability in semiconductor volume production.

Optical Fundamentals and Defocus Physics

Rayleigh Depth of Focus Formulation:

• Rayleigh Equation: = k_2 \frac{\lambda}{NA^2}$, where $\lambda$ is exposure wavelength, $ is numerical aperture, and $ is a process-dependent factor (typically 0.4–0.8).
• Wavelength Dependencies: Advanced nodes transition from i-line (365 nm) to KrF (248 nm), ArF (193 nm dry/immersion), and EUV (13.5 nm), reducing absolute optical DOF at shorter wavelengths.
• NA Scaling Trade-off: Increasing numerical aperture enhances single-point resolution ( = k_1 \frac{\lambda}{NA}$) but quadratically degrades depth of focus, creating a severe focus window bottleneck in high-NA tools.
• **Process Factor *: Encompasses resist contrast, illumination coherence ($\sigma$), reticle enhancement techniques, and post-exposure bake diffusion limits.

Wavefront Phase Error under Defocus:

• Phase Shift Equation: The phase error introduced by axial defocus $\Delta z$ across pupil radius $\rho = r/R_{pupil}$ is expressed by Zernike defocus polynomial $:

578919\Delta \Phi(\rho) = \frac{2\pi}{\lambda} \cdot \Delta z \cdot \left[ 1 - \sqrt{1 - \left( NA \cdot \rho / n \right)^2} \right] \approx \frac{\pi}{\lambda} \Delta z \left( \frac{NA}{n} \right)^2 \rho^2578919

• Strehl Ratio Decay: Optical intensity peak at best focus degrades with RMS phase error according to \approx 1 - (2\pi \cdot W_{rms} / \lambda)^2$, causing image contrast loss as defocus exceeds $\lambda / (2 NA^2)$.
• Normalized Image Log-Slope (NILS): Defocus reduces contrast near feature edges; NILS drops below acceptable manufacturing thresholds ( < 2.0$), triggering pattern bridging or line collapse.

Rayleigh DOF versus Effective Process DOF

Rayleigh Criterion vs. Resist-Limited DOF:

• Optical DOF: Calculated purely from aerial image intensity distributions assuming ideal threshold photoresist response.
• Process DOF: Extracted experimentally from Bossung curves taking photoresist chemical amplification, acid diffusion length ( = 2\sqrt{D \cdot t_{PEB}}$), and etch bias into account.
• Resist Degradation Factor: Real process DOF is consistently 20–40% smaller than pure optical Rayleigh DOF due to finite resist contrast ($\gamma$) and top-loss/sidewall degradation.

Quantitative Contrast Metrics:

• Contrast Threshold: = \frac{I_{max} - I_{min}}{I_{max} + I_{min}} \ge C_{crit}$ (typically {crit} \ge 0.3$ for line/space patterns, $\ge 0.5$ for contact holes).
• Depth of Focus Extraction: Calculated as the focus range $\Delta z = z_{upper} - z_{lower}$ satisfying:

578919CD_{min} \le CD(z, E_{nom}) \le CD_{max} \quad \text{and} \quad \Theta_{sidewall}(z) \ge 85^\circ578919

Immersion Lithography and Refractive Index Scaling

Medium Refractive Index Impact:

• Immersion Medium: Replacing air (=1.0$) with ultra-pure deionized water ({H_2O} = 1.44$ at 193 nm) scales the effective wavelength in the fluid to $\lambda_0 / n$.
• Exact High-NA Immersion DOF Equation:

578919DOF_{immersion} = \frac{k_2 \cdot \lambda_0}{n \cdot \left( 1 - \sqrt{1 - (NA/n)^2} \right)}578919

• Hyper-NA Systems: Enables > 1.0$ (up to = 1.35$ in modern ArFi scanners), expanding focus latitude by a factor of \approx 1.44$ compared to an equivalent dry system operating at theoretical limits.

Polarization and Vector Optical Effects:

• TM-Polarization Loss: At high angles of incidence ($\theta > 45^\circ$ inside resist), TE-polarized light maintains interference contrast, whereas TM-polarized light interference drops as $\cos(2\theta)$, reducing focus window bounds.
• Azimuthal & Radial Polarization: Custom illuminator polarization states mitigate TM contrast loss, preserving DOF at dense line-space pitches below 40 nm.

Phase-Shift Masks and Optical Resolution Enhancement

Attenuated PSM (6% Att-PSM):

• Phase Interference: Absorber layer shifts background light by 80^\circ$ with 6% intensity transmission, sharpening edge transitions and broadening focus latitude by 15–25%.
• Side-Lobe Printing Risk: High transmission PSM (e.g., 18%) extends DOF further but risks unexposed background printing (side-lobe defects) near focus extremes.

Alternating PSM (Alt-PSM):

• Zero-Order Suppression: 80^\circ$ phase difference etched into alternating mask clear regions completely eliminates 0th order diffracted beam for equal lines and spaces.
• Two-Beam Interference Focus Invariance: Interference occurs strictly between $+1$ and hBc1$ diffracted orders, producing spatial intensity profiles that are inherently insensitive to defocus phase shifts to first order:

578919I(x, z) \propto \cos^2\left( \frac{2\pi x}{P} \right)578919

• DOF Gain: Expands effective focus latitude by $> 2.0\times$ relative to binary chrome masks, enabling extreme low-$ patterning.

Off-Axis Illumination (OAI) Interaction:

• Dipole / Quadrupole / Annular Source Profiles: Tilts incoming illumination vector by angle $\sin \theta_{ill} = \frac{\lambda}{2 P}$, causing 0th and $+1 diffracted orders to pass symmetrically through opposite sides of pupil.
• Optical Path Length Matching: Cancels 1st-order optical path difference under defocus, maximizing depth of focus for specific dense pitches at the expense of isolated feature DOF.

Aberration Coupling and Scanner Metrology

Zernike Lens Aberrations and Focal Plane Metrics:

• Spherical Aberration ( / Z_{16}$): Introduces focus shifts dependent on spatial frequency and illumination angle, causing focal plane tilt between dense and isolated patterns.
• Field Curvature ((x,y)$): Causes best focus position to vary across the exposure field, consuming part of the available scanner focus budget.
• Astigmatism ( / Z_6$): Shifts best focus independently for horizontal ($) and vertical ($) features (-V$ focus separation), restricting common horizontal/vertical process window.

Metrology and Sensor Calibration:

• Phase Grating Focus Sensors (FOCAL): Uses phase-shifting reticle marks to convert defocus directly into lateral alignment shifts measured by off-axis alignment scope.
• Diffraction-Based Overlay / Focus Metrology: Automated on-wafer target measurements using asymmetric target designs to map intra-field focus errors at high wafer throughput.

EUV Defocus and Advanced Node Limits

EUV Wavelength ($\lambda = 13.5\text{ nm}$) Transition:

• Single-Exposure EUV DOF: Extreme reduction in wavelength restores $ margins ( \approx 0.40$ at 28 nm pitch with =0.33$), yielding typical optical DOF of 80–120 nm.
• Anamorphic EUV (NA = 0.55): High-NA EUV employs \times / 8\times$ asymmetric magnification; DOF shrinks to $< 40\text{ nm}$, mandating sub-nanometer active scanner levelling compensation.

3D Mask Absorber & Stochastic Effects:

• Non-Telecentricity & Mask Shadowing: EUV reflective optics require ^\circ$ chief ray angle ($), causing phase mismatch across focus and non-symmetric Bossung curves.
• Stochastic Defectivity Limit: Near focus window boundaries, photon shot noise and local resist acid concentration fluctuations cause exponential increases in stochastic micro-bridging and line-breaking defects.

Focus Budget Allocation and Manufacturing Controls

Focus Budget Tree:

• Scanner Subsystems: Lens heating focus drift, laser spectral bandwidth variation ($\Delta \lambda_{E95}$ chromatic focus blur), reticle stage non-flatness, and optical sensor drift (typically 12–18 nm combined).
• Wafer & Process Contributors: Chemical mechanical planarization (CMP) topography variations, wafer chuck deformation, resist thermal expansion during PEB, and thin film interference non-uniformities (typically 15–25 nm combined).
• Total Focus Error Budget: Calculated via root-sum-square (RSS) summation:

5789193\sigma_{Focus\_Total} = \sqrt{\sum (3\sigma_{scanner})^2 + \sum (3\sigma_{wafer})^2 + \sum (3\sigma_{process})^2}578919

• Manufacturing Requirement: \sigma_{Focus\_Total}$ must remain strictly within the common overlapping process window depth of focus to guarantee zero defocus-induced yield loss.

Closed-Loop Run-to-Run (R2R) Focus Control:

• Advanced Process Control (APC): Integrates inline diffraction-based focus metrology (DBF) data to dynamically update scanner focus baseline offsets per lot and per exposure field.
• Intra-Field High-Order Compensation: Uses adaptive lens manipulator rings and active reticle stage tilting to correct field curvature and astigmatism dynamically during wafer exposure.

Summary and Engineering Best Practices

Focus Latitude Maximization Checklist:

• Illumination Optimization: Match source pupil shape (Dipole/Quadrupole/Annular) to target feature pitch and orientation to minimize zero-order path length differences.
• Reticle Design: Implement attenuated or alternating PSM and model-based SRAF placement to preserve aerial image slope across focus extremes.
• Material Engineering: Utilize high-contrast chemical amplification photoresists with optimized post-exposure bake thermal budgets to limit acid blur.
• Metrology Integration: Deploy inline diffraction-based focus monitoring to feed dynamic run-to-run scanner focus compensations and prevent intra-field focus drift.