Laser Linewidth and Coherence
Linewidth Δν and coherence length Lc = c/(πΔν) determine interferometry and spectroscopy performance. Narrow linewidth enables long-path interferometers; transform-limited pulses satisfy Δν×Δt ≥ TBP.
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Gaussian line shape: k ≈ 0.66; Lorentzian: k ≈ 0.32. Schawlow-Townes limit sets fundamental linewidth floor. Femtosecond pulses have THz bandwidth—broad spectral coverage. Single-frequency lasers achieve coherence lengths > 1 km.
Ready to run the numbers?
Why: Coherence length limits interferometer path difference and holography depth. Narrow linewidth enables precision spectroscopy. Transform-limited pulses maximize peak power for given bandwidth.
How: Lc = c/(π×Δν×k) with shape factor k. Time-bandwidth product Δν×Δt ≥ TBP (0.44 Gaussian). Cavity Δν = FSR/F from finesse.
Run the calculator when you are ready.
Spectral Parameters
LINEWIDTH & COHERENCE ANALYSIS
Linewidth To Coherence
📝 Calculation Steps
Input Parameters
Center Wavelength: 1064.00 nm
Center Frequency: 2.82 × 10^14 Hz
Linewidth to Coherence Conversion
Linewidth: 2.65 × 10^11 Hz
Linewidth: 1.0000 nm
Line Shape: gaussian (factor: 0.664)
Formula: Lc = c / (π × Δν × k)
ext{Lc} = c / (\text{pi} imes \text{Delta} ν imes k)
Coherence Length: 0.54 mm→ 0.54 mm
Coherence Time: 1.81 × 10^-12 s
Quality Metrics
Quality Factor (Q): 1.06e+3
Spectral Purity: 9.40 × 10^-4
Visualizations
For educational and informational purposes only. Verify with a qualified professional.
🔬 Physics Facts
Coherence length limits maximum path difference in Michelson interferometers.
— RP Photonics
Schawlow-Townes limit sets quantum minimum linewidth for laser oscillators.
— NIST
Transform-limited pulses have minimum bandwidth for given duration.
— Ultrafast Optics
Fiber lasers can achieve sub-kHz linewidth for metrology applications.
— Thorlabs
📋 Key Takeaways
- • Laser linewidth (Δν or Δλ) is the spectral width measured as full-width at half-maximum (FWHM) and determines coherence properties
- • Coherence length is inversely proportional to linewidth: Lc = c / (π × Δν) for Lorentzian lineshapes
- • Narrower linewidth enables longer coherence lengths, essential for interferometry, LIDAR, and fiber sensing applications
- • The time-bandwidth product (TBP) sets the minimum bandwidth for transform-limited pulses: Δν × Δt ≥ TBP
- • Single-frequency lasers achieve sub-kHz linewidths with coherence lengths exceeding kilometers
🤔 Did You Know?
The Schawlow-Townes limit sets the fundamental quantum limit on laser linewidth, but real lasers typically operate 10-100× above this limit due to technical noise sources.
Source: Schawlow & Townes (1958), Physical Review
Frequency-stabilized HeNe lasers can achieve linewidths below 1 MHz, enabling coherence lengths over 100 meters for precision metrology applications.
Source: NIST Laser Metrology Standards
Ultra-narrow linewidth fiber lasers (<100 Hz) enable distributed fiber sensing over distances exceeding 100 km, revolutionizing structural health monitoring.
Source: OSA/Optica Photonics Technology Letters
💡 Expert Tips
- • For interferometry applications, ensure coherence length exceeds the maximum optical path difference by at least 3×
- • Use Pound-Drever-Hall locking or frequency offset locking to stabilize laser frequency and reduce linewidth
- • Higher cavity finesse and longer cavity length reduce linewidth according to Δν = FSR / F
- • Consider line shape (Gaussian vs Lorentzian) when calculating coherence length, as shape factors differ significantly
- • For transform-limited pulses, choose pulse shape carefully—Gaussian pulses have TBP = 0.441, while sech² pulses have TBP = 0.315
Laser linewidth (Δν or Δλ) is the spectral width of the laser emission, typically measured as the full-width at half-maximum (FWHM). It's inversely related to coherence length—narrower linewidth means longer coherence and better interferometric performance.
Key Relationships
Lc = c / (π × Δν)
Coherence length
Δν × Δt ≥ TBP
Time-bandwidth product
Typical Laser Linewidths
| Laser Type | Linewidth | Coherence Length | Applications |
|---|---|---|---|
| Stabilized HeNe | <1 MHz | >100 m | Metrology, interferometry |
| Single-freq Nd:YAG | 1-100 kHz | 1-100 km | LIDAR, holography |
| DFB Fiber Laser | 1-100 kHz | 1-100 km | Fiber sensing |
| ECDL (External Cavity) | 100 kHz-1 MHz | 100 m-3 km | Spectroscopy |
| Multi-mode diode | 1-3 nm | 0.1-0.3 mm | Pumping, illumination |
| Femtosecond Ti:Sapph | 10-100 nm | 3-30 μm | Ultrafast spectroscopy |
Time-Bandwidth Product
The time-bandwidth product (TBP) is the minimum product of pulse duration and spectral bandwidth, set by the Fourier transform relationship. Different pulse shapes have different TBP constants.
Gaussian
0.441
Sech²
0.315
Lorentzian
0.142
Rectangular
0.886
Coherence Requirements for Applications
| Application | Min. Coherence Length | Max. Linewidth | Notes |
|---|---|---|---|
| Fiber Gyroscope | >1 km | <100 kHz | Long fiber coil |
| LIDAR (long range) | >100 m | <1 MHz | Coherent detection |
| Holography | >1 m | <100 MHz | Scene depth dependent |
| Surface Profilometry | >1 cm | <10 GHz | Optical path difference |
| Fiber Sensing | >10 m | <10 MHz | Distributed sensing |
Linewidth Unit Conversions
Converting between frequency, wavelength, and wavenumber linewidths at center wavelength λ₀:
Δν ↔ Δλ
Δν ↔ Δσ
Δλ ↔ Δσ
❓ Frequently Asked Questions
Q: What is the difference between linewidth and bandwidth?
Linewidth typically refers to the spectral width of a continuous-wave laser, while bandwidth often describes the frequency spread of pulsed lasers. Both measure spectral spread but in different contexts.
Q: How does linewidth affect interferometry?
Narrower linewidth provides longer coherence length, allowing larger optical path differences before interference fringes disappear. For stable interference, coherence length should exceed the path difference by 3-5×.
Q: What causes laser linewidth broadening?
Technical noise (vibrations, temperature fluctuations, current noise), quantum noise (Schawlow-Townes limit), and cavity losses all contribute. Frequency stabilization techniques can reduce technical noise significantly.
Q: Can I measure linewidth directly?
Yes, using techniques like self-heterodyne detection, Fabry-Perot interferometry, or delayed self-heterodyne. For very narrow linewidths (<1 MHz), specialized equipment is required.
Q: What is the relationship between finesse and linewidth?
Higher cavity finesse (F) narrows the linewidth: Δν = FSR / F, where FSR is the free spectral range. This is why high-finesse cavities enable ultra-narrow linewidth lasers.
Q: How do I choose the right linewidth for my application?
Match coherence length to your application: LIDAR needs >100 m (linewidth <1 MHz), interferometry needs >1 m (<100 MHz), while pumping applications can tolerate much broader linewidths (>1 nm).
Q: What is transform-limited bandwidth?
The minimum spectral bandwidth for a given pulse duration, set by the Fourier transform relationship. Transform-limited pulses have the shortest duration possible for their bandwidth, with no excess chirp.
📚 Official Data Sources
⚠️ Disclaimer: This calculator provides theoretical estimates for educational purposes only. Actual laser linewidth depends on many factors including cavity design, stabilization techniques, environmental conditions, and technical noise sources. For precision applications, consult laser manufacturers and use calibrated measurement equipment. Always follow laser safety guidelines and regulations.
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