FMCW LiDAR: Frequency Modulated Continuous Wave

• FMCW LiDAR is a coherent optical ranging method that employs a linearly chirped continuous-wave laser to directly map round-trip delay to RF frequency.
• The technique achieves sub-100μm range resolution and supports multidimensional metrology through trilateration with multiple fiber-coupled emitters.
• Advanced digital signal processing, including fractional Fourier transforms, extracts precise beat frequencies while mitigating speckle noise for improved accuracy.

A frequency modulated continuous wave (FMCW) lidar scheme is a coherent optical ranging method in which the instantaneous frequency of a continuous-wave laser is linearly chirped over a fixed bandwidth. Distance to a target is encoded in the radio-frequency (RF) beat note generated by mixing the returned and local oscillator waves. Modern FMCW lidar systems can attain sub-100 μm range resolution, support multi-dimensional coordinate recovery via trilateration, and achieve extended, stand-off, non-contact metrology. The FMCW approach is fundamentally distinct from pulsed time-of-flight or classical interferometry in its direct mapping of round-trip delay to measurable RF frequency. Innovations in ultrabroadband lasers, high-coherence chirp sources, fiber-based distribution, and advanced digital signal processing have established FMCW lidar schemes as powerful tools for precision length measurement, surface profiling, and three-dimensional spatial reconstruction (Mateo et al., 2015).

1. Fundamental Principles of FMCW Ranging

A key characteristic of the FMCW lidar scheme is the linear frequency chirp of a continuous-wave laser:

$\nu(t) = \nu_0 + K t,$

where $\nu_0$ is the starting optical frequency, $K = B/T$ is the chirp rate with sweep bandwidth $B$ over sweep duration $T$. The chirped beam is split, with one path sent to the target and back (incurring a round-trip delay $\tau = 2 R / c$), and the other reserved as a local oscillator (LO).

On coherent heterodyne mixing, a beat frequency

$f_b = K \tau = \frac{2 K R}{c},$

emerges. Solving for range yields the fundamental relation

$R = \frac{c\, f_b}{2 K}.$

This mapping reduces the ranging task to precision RF frequency measurement, inherently immune to absolute optical frequency drift, provided the chirp rate is precisely known. The finite optical bandwidth $B$ sets the Fourier-limited range resolution:

$\Delta R = \frac{c}{2B}.$

This scheme supports both static and dynamic targets, as Doppler shifts arising from target velocity can be separated from the beat frequency by appropriate dual-ramp (up/down) chirp and analysis.

2. System Architecture and Multidimensional Measurement

A canonical high-resolution FMCW lidar system distributes the chirped light across multiple, fiber-coupled transmitters. In a demonstrated configuration, a narrow-linewidth, ultrabroadband chirped laser—actively linearized and amplified—is split into distinct branches via optical fiber couplers (Mateo et al., 2015):

• 10% is reserved as the local oscillator (LO).
• 90% is divided among transmitters:
  • Two high-power branches ("Emitter B" and "Emitter C", ~220 mW) flood-illuminate the target.
  • One low-power branch ("Emitter/Receiver A", ~44 μW) is collimated and focused to a ~400 μm spot, scanned via a galvo mirror.

All back-scattered and reflected light is collected into the receiver A and heterodyned against the LO in an auto-balanced detector. After analog-to-digital conversion, a fractional Fourier transform (FRFT) yields the full beat-frequency spectrum, with peaks corresponding to various round-trip delay paths.

For multidimensional (2D/3D) metrology, the system performs trilateration: algebraic differences of specific beat-frequency peaks provide absolute free-space distances between optical emitters and the scan point on the target, while eliminating internal fiber delay contributions. In 2D, the intersection of distance circles determined by known emitter coordinates and measured ranges recovers the target coordinate. In 3D, a third transmitter yields the intersection of spheres.

3. Range Resolution and Experimental Metrics

The achievable range resolution is determined by the chirp bandwidth $B$, with

$\Delta R = \frac{c}{2 B}.$

For a bandwidth $B \approx 2\,\text{THz}$, the theoretical range resolution is ~75 μm; in practice, with $B_\mathrm{eff} = 1.6\,\text{THz}$, experimental resolution was $\Delta R \approx 94 μm$ (Mateo et al., 2015).

Key experimental parameters:

• Laser chirp: B ≈ 1.6 THz, T ≈ 8 ms (30 Hz repetition), K ≈ 2.0×10¹⁴ Hz/s
• Single-point (static) range precision: ≲ 10 μm (high SNR, diffuse target)
• Transverse coordinate RMS precision at 1.5 m stand-off: ≈ 100–200 μm
• Limiting factor: speckle-induced range fluctuations due to diffuse scattering, mitigated by windowed averaging and FRFT, at the cost of halved range resolution

Measured results demonstrate the capacity for millimeter to sub-100 μm ranging precision. The method accommodates both specular and highly diffuse (scattering) targets.

4. Self-Calibration and Flexibility

Unlike coordinate metrology schemes demanding precise surveying of transmitter positions prior to operation, the FMCW trilateration approach admits in situ self-calibration. In a typical implementation, one emitter (e.g., C) is mounted on a motorized translation stage monitored by an interferometric length gauge. By holding the scanning point fixed and stepping C to known positions, measured ranges to the scan point allow determination of all relevant coordinate origins by algebraic inversion. Repetition at multiple scan points refines the spatial calibration of all emitters to millimeter-level accuracy. This relaxes laborious surveying requirements and supports flexible, reconfigurable field setups.

5. Digital Signal Processing and Coordinate Extraction

Signal acquisition begins with digitization of the detector output. The inference workflow comprises:

• Windowing each time record into overlapping segments
• Application of a raised-cosine weighting function
• Fractional Fourier transforms (FRFT) producing the beat-frequency spectrum
• Peak identification for all expected delay features (e.g., Dₐₐ, D_ba, D_ca)
• Algebraic combination (e.g., computing $b = (D_{ba} - D_{aa}) \cdot c/(2K)$) to recover physical free-space ranges
• Geometric inversion (circle/sphere intersection) to obtain 2D/3D surface coordinates
• Profile accumulation across a scanned range to reconstruct surface topography

Speckle noise is mitigated by incoherent averaging of FFTs/FRFTs over overlapping time windows; this approach halves the range resolution but improves coordinate stability under diffuse-scattering conditions.

6. Applications, Performance Limits, and Future Directions

The high-coherence FMCW lidar with trilateration is suited for non-contact surface metrology and multi-dimensional spatial surveying at stand-off distances. Achieved performance includes ≤ 10 μm single-point precision under high SNR, and ≈ 100–200 μm coordinate precision for scanned 2D profiles with 1.5 m stand-off—currently limited by speckle effects in the presence of diffuse scattering (Mateo et al., 2015). The scheme inherently scales to larger volumes and longer ranges due to range independence from stand-off and suitability for field self-calibration.

The architecture enables self-calibrating, large field-of-regard precision metrology without requiring absolute alignment or point-by-point mapping of transmitters, and paves the way for further integration with advanced scanning optics and automated calibration methodologies.

Reference:

Multi-Dimensional, Non-Contact Metrology using Trilateration and High Resolution FMCW Ladar (Mateo et al., 2015)