Stereolithography (SLA) Jacobs Cure Depth & Overcure Formula

Double an SLA galvo scan speed to halve print time and logarithmic cure depth drops BELOW the 100 um layer slice—peeling layers apart mid-print!

**Laser Stereolithography (`SLA`)** sweeps a `355`/`405` nm Gaussian UV beam of Power `P_L` (`mW`), `1/e²` Beam Radius `w_0` (`mm`), and Galvo Scan Speed `v_s` (`mm/s`) across a liquid Photopolymer Resin Tray characterized by TWO Beer-Lambert constants: **Optical Penetration Depth `D_p`** (`um`, `1/e` UV attenuation depth controlled by photoinitiator + UV blocker) and **Critical Gelation Exposure `E_c`** (`mJ/cm²`). Paul F. Jacobs derived the foundational **SLA Working Curve Equations**: **Peak Centerline Exposure `E_{max} = P_L / (SQRT(PI() / 2) * (w_0 / 10) * (v_s / 10))`** (`mJ/cm²`), **Logarithmic Cure Depth `C_d = D_p * LN(E_{max} / E_c)`** (`um`), **Interlayer Stitching Overcure `Delta_C = C_d - z_{layer}`** (`um`—MUST stay `+40` to `+80` um positive so green polymer chains covalently cross-link into the previous layer!), and **Parabolic Cured Line Width `L_w = (w_0 * 1,000) * SQRT(2 * LN(E_{max} / E_c))`** (`um`)!

Verify on a `355` nm Industrial Ceramic/Engineering-Resin SLA Printer—UV_Laser_Power `P_L` = 120.0 mW (B2), Gaussian_Spot_Radius `w_0` = 0.11 mm (`220` um spot diameter, B3), Galvo_Scan_Speed `v_s` = 2,400 mm/s (B4), Resin_Penetration_Depth `D_p` = 145.0 um (B5), Critical_Exposure `E_c` = 11.5 mJ/cm² (B6), and Z-Stage_Layer_Slice `z` = 100.0 um (B7): Peak Exposure hits `36.27` mJ/cm² (E2), Cure Depth reaches `166.5` um (E3), Interlayer Stitching Overcure locks in `+66.5` um (E4), and Cured Track Width equals `166.7` um (E5)!

Jacobs Gaussian Beer-Lambert Cure Depth, Overcure & Cured Track Width Formula

Validate `Power P_L > 1` mW, `Beam_Radius w_0` in `[0.01, 1.0]` mm, `Scan_Speed v_s > 10` mm/s, `Penetration D_p > 10` um, `Critical_Exposure E_c > 0.5` mJ/cm², `Layer_Slice z > 10` um, and above-gelation `E_{max} > E_c`!

=LET(pl_m, B2, wz_m, B3, vs_m, B4, dp_u, B5, ec_m, B6, zl_u, B7, em_x, pl_m / (SQRT(PI() / 2) * (wz_m / 10) * (vs_m / 10)), IF(OR(pl_m <= 1, wz_m < 0.01, wz_m > 1, vs_m <= 10, dp_u <= 10, ec_m <= 0.5, zl_u <= 10, em_x <= ec_m), "Invalid SLA Exposure", ROUND((wz_m * 1000) * SQRT(2 * LN(em_x / ec_m)), 1)))

Doubled 4,800 mm/s Galvo Speed Negative-Overcure Delamination Check (Set Scan Speed B4 = 4,800 mm/s)

Double galvo scan speed to `v_s` B4 = 4,800 mm/s in "draft mode" without boosting laser mW: `E_{max}` E2 halves to `18.13` mJ/cm², Cure Depth E3 plunges to **`66.0` um** (`< 100` um Z-step!), and Overcure E4 turns **negative (`-34.0` um)**—leaving `34` um of uncured liquid resin between every slice so cured skins float loose across the resin tray!

=IF(ISTEXT(E5), E5, LET(em_x, B2 / (SQRT(PI() / 2) * (B3 / 10) * (B4 / 10)), ROUND(B5 * LN(em_x / B6) - B7, 1)))

Further reading

• Arte Sheets Lesson — Laser Powder Bed Fusion (LPBF) Volumetric Energy Density Formula
• Arte Sheets Lesson — Optical & EUV Lithography Rayleigh CD & Depth of Focus