Autonomous Sensing, Photonics & LiDAR Systems
Pulsed ToF LiDAR Power & Unambiguous Range Formula
Calculate Laser Time-of-Flight Round-Trip Delay (ns), Extended-Target Lambertian Received Power P_rx = P_pk * rho * (D^2 / (4 * R^2)) * eta, and Maximum Unambiguous PRF Range.
Extended-target automotive LiDAR scales as 1 / R^2—not monostatic radar’s 1 / R^4!
Engineers crossing over from microwave radar to automotive and airborne pulsed Time-of-Flight (ToF) LiDAR frequently apply a 1 / R^4 range law by mistake! Because a collimated milliradian laser spot is SMALLER than a vehicle bumper, wall, or road surface, 100% of the transmitted laser beam intercepts the diffuse Lambertian target (eliminating the outbound 1 / R^2 spreading loss!) and scatters into pi steradians, so clear-air Received Power scales inversely with R^2: P_rx_nW = 10^9 * P_peak_W * rho_target * (D_rx_m^2 / (4 * R_m^2)) * eta_opt! Simultaneously, Round-Trip Flight Time is t_ToF_ns = (2 * R_m) / 0.30 (at c = 0.30 m/ns), and Pulse Repetition Frequency PRF_kHz caps Maximum Unambiguous Range at R_unamb_m = 150,000 / PRF_kHz before second-time-around echo aliasing creates phantom close-in obstacles!
Verify on Peak_Laser_Power P_pk = 100 W (B2), Target_Lambertian_Reflectance rho = 0.20 (20%, B3), Receiver_Lens_Diameter D_rx = 0.06 m (60 mm, B4), Target_Range R = 150 m (B5 -> Round-Trip ToF = 300 / 0.30 = 1,000.00 ns in E2), Optical_Efficiency eta_opt = 0.50 (B6), and Laser_PRF = 500 kHz (B7): Received Optical Power is 10^9 * 100 * 0.20 * (0.0036 / 90,000) * 0.50 = 400.00 nW (E3); Max Unambiguous Range is 150,000 / 500 = 300.00 m (E4); confirming "Unambiguous Echo (R <= R_unamb)" (E5)!
| Pulsed ToF LiDAR Metric (Ppk=100W, rho=20%, Drx=60mm, R=150m, eta=50%, PRF=500kHz) | Formula | Verified Output |
|---|---|---|
| Round-Trip Flight Time (ns) | (2 * R_m) / 0.30 | 1,000.00 ns (1.00 us) |
| Received Return Power (nW) | 10^9 * Ppk * rho * (Drx^2/(4*R^2)) * eta | 400.00 nW (100.00 nW @ R=300m) |
| Max Unambiguous Range (m) | 150000 / PRF_kHz | 300.00 m (125.00 m @ 1,200 kHz) |
| Pulse-Aliasing Safety Gate | R_m <= R_unambiguous | Unambiguous Echo (R <= R_unamb) |
Extended-Target Lambertian LiDAR Received Power (nW) Formula
Validate P_peak > 0, 0 < rho <= 1, D_rx > 0, R > 0, 0 < eta_opt <= 1, and PRF_kHz > 0!
=LET(p_pk, B2, rh_t, B3, d_rx, B4, rn_m, B5, et_o, B6, pr_f, B7, IF(OR(p_pk <= 0, rh_t <= 0, rh_t > 1, d_rx <= 0, rn_m <= 0, et_o <= 0, et_o > 1, pr_f <= 0), "Invalid LiDAR Optical Budget", ROUND(1000000000 * p_pk * rh_t * ((d_rx ^ 2) / (4 * (rn_m ^ 2))) * et_o, 2)))Inverse-Square Check (R = 300 m) & Ghost-Echo Aliasing Test (PRF = 1,200 kHz)
Double range to 300 m: P_rx quarters to 100.00 nW! Crank PRF to 1,200 kHz at R = 150 m: R_unamb shrinks to 125.00 m, triggering "Aliased Ghost Target (R > R_unamb)" unless pulse coding/jitter is active!
=IF(ISTEXT(E3), E3, IF(B5 <= E4, "Unambiguous Echo (R <= R_unamb)", "Aliased Ghost Target (R > R_unamb)"))Try the example
Use the original values or change them to test your understanding.