On this page
Laser Systems
1. What defines a laser system?
For experimental work, the most useful specifications are usually:
| Quantity | Why it matters |
|---|---|
| Optical frequency $\nu$ / wavelength $\lambda$ | determines the addressed transition |
| Linewidth $\Delta\nu$ | determines spectral purity and coherence |
| Frequency noise $S_\nu(f)$ | shows where the linewidth/noise originates |
| Relative intensity noise (RIN) | produces amplitude noise and AC-Stark fluctuations |
| Output power | sets available Rabi frequency and nonlinear-conversion efficiency |
| Beam quality $M^2$ | determines focusing and mode matching |
| Polarization | determines optical selection rules and nonlinear-crystal efficiency |
| Pointing stability | affects coupling into cavities, fibers and atomic samples |
A narrow quoted linewidth does not guarantee good long-term frequency accuracy. A laser can have excellent short-term coherence while drifting by many MHz over minutes or hours.
2. What is laser linewidth?
An ideal monochromatic field would be
\[E(t)=E_0 e^{i2\pi\nu_0 t}.\]A real laser has amplitude and phase fluctuations:
\[E(t)=A(t)e^{i[2\pi\nu_0 t+\phi(t)]}.\]The instantaneous frequency is
\[\nu(t)=\nu_0+\frac{1}{2\pi}\frac{d\phi}{dt}.\]The optical spectrum therefore has a finite width. The term laser linewidth normally refers to the full width at half maximum (FWHM) of a specified optical line shape, but the result depends on the noise process and the observation time.
Lorentzian and Gaussian limits
White frequency noise produces a Lorentzian-like central line. Slow technical frequency fluctuations often produce Gaussian or non-Lorentzian broadening.
For a Lorentzian line with FWHM $\Delta\nu$, a useful coherence-time scale is
\[\tau_c\approx\frac{1}{\pi\Delta\nu},\]and the corresponding vacuum coherence length is approximately
\[L_c\approx c\tau_c.\]Thus a 1 MHz Lorentzian linewidth corresponds to a coherence time of roughly $0.32\ \mu\text{s}$, while a 1 kHz linewidth corresponds to roughly $0.32$ ms.
3. Fundamental and technical linewidth contributions
The ideal quantum-limited linewidth follows the Schawlow-Townes scaling. A representative form is
\[\Delta\nu_{ST}\propto \frac{h\nu}{P_{out}}\,(\Delta\nu_{cav})^2,\]with additional factors determined by gain, spontaneous emission and laser architecture. Semiconductor lasers also contain amplitude-phase coupling through the Henry factor $\alpha_H$, which increases the linewidth roughly through a factor $1+\alpha_H^2$ in the simplest treatment.
In most laboratory systems, the measured linewidth is dominated by technical noise rather than the fundamental Schawlow-Townes limit. Important sources include:
- current-driver noise;
- cavity-length fluctuations inside the laser;
- temperature drift;
- acoustic and vibration noise;
- optical feedback;
- pump-power noise;
- air-path fluctuations;
- electronics noise in the servo system.
4. Frequency-noise spectral density
A more complete description than a single linewidth is the frequency-noise power spectral density
\[S_\nu(f)\quad [\text{Hz}^2/\text{Hz}].\]For white frequency noise with one-sided PSD level $S_\nu^0$, the Lorentzian linewidth is commonly written, under the corresponding PSD convention,
\[\Delta\nu_L\approx \pi S_\nu^0.\]Real lasers rarely contain only white frequency noise. The relation between $S_\nu(f)$ and optical line shape is discussed quantitatively by Di Domenico, Schilt and Thomann in their frequency-noise/linewidth analysis.
5. Approximate linewidth scales
The following are orders of magnitude, not specifications. Actual values vary greatly with design and measurement bandwidth.
| Laser architecture | Representative free-running linewidth scale |
|---|---|
| simple semiconductor diode / DFB | sub-MHz to many MHz |
| external-cavity diode laser (ECDL) | tens of kHz to MHz class |
| narrow-linewidth fiber / solid-state laser | Hz to tens-of-kHz class |
| cavity-stabilized research laser | Hz to sub-Hz class |
Modern cavity-stabilized systems can reach linewidths near or below 1 Hz. NIST reports compact ultrastable reference systems targeting approximately 1 Hz linewidth and has demonstrated cavity-stabilized lasers with linewidths well below 1 Hz in optimized systems.
6. Measuring laser linewidth
Heterodyne beat measurement
Two independent lasers at nearby optical frequencies are combined on a fast photodiode. The electrical beat frequency is
\[\nu_b=|\nu_1-\nu_2|.\]For two independent Lorentzian lasers,
\[\Delta\nu_b=\Delta\nu_1+\Delta\nu_2.\]If the two lasers have equal Lorentzian linewidth,
\[\Delta\nu_{laser}\approx\frac{\Delta\nu_b}{2}.\]For equal Gaussian linewidths, variances add instead, giving
\[\Delta\nu_{laser}\approx\frac{\Delta\nu_b}{\sqrt{2}}.\]This distinction is important when interpreting beat-note spectra.
Delayed self-heterodyne
The laser is split into two paths. One path passes through a long fiber delay and often an AOM frequency shift before recombination. When the delay is much longer than the coherence time, the two arms become effectively decorrelated. For a Lorentzian source, the measured long-delay self-heterodyne FWHM is approximately twice the original laser linewidth.
Frequency discriminator / cavity method
A steep cavity resonance converts frequency fluctuations into intensity fluctuations. After calibration of the cavity slope, a photodetector and spectrum analyzer can recover $S_\nu(f)$.
Optical frequency comb
For ultranarrow lasers, beating against a comb referenced to a known ultrastable oscillator provides absolute and relative frequency-noise information over very large dynamic ranges.
Cavity-stabilized lasers
7. Fabry-Perot reference cavity
For a two-mirror cavity of optical length $nL$, the free spectral range is
\[\mathrm{FSR}=\frac{c}{2nL}.\]The cavity finesse is
\[\mathcal F=\frac{\mathrm{FSR}}{\Delta\nu_{cav}},\]where $\Delta\nu_{cav}$ is the FWHM resonance width of the passive cavity. Thus
\[\Delta\nu_{cav}=\frac{\mathrm{FSR}}{\mathcal F}.\]The resonance frequencies approximately obey
\[\nu_m=\frac{mc}{2nL}.\]For small length fluctuations,
\[\frac{\delta\nu}{\nu}\approx-\frac{\delta L}{L}\]when refractive-index fluctuations are neglected.
Worked cavity example
For $L=10$ cm in vacuum,
\[\mathrm{FSR}=\frac{c}{2L}\approx1.50\ \text{GHz}.\]For finesse $\mathcal F=3\times10^5$,
\[\Delta\nu_{cav}\approx5\ \text{kHz}.\]This does not mean the locked laser linewidth must be 5 kHz.
8. Why a cavity can produce a laser linewidth far below the cavity linewidth
The cavity acts as a high-slope frequency discriminator. A feedback loop continuously keeps the laser near the center of the resonance. The achievable frequency error can therefore be a tiny fraction of the cavity resonance width.
A useful conceptual chain is
\[\boxed{\text{stable cavity length}\rightarrow\text{stable resonance frequency}\rightarrow\text{error signal}\rightarrow\text{feedback}\rightarrow\text{narrow laser}.}\]The final locked-laser noise depends on
- cavity thermal/Brownian noise;
- cavity vibration sensitivity;
- cavity temperature drift;
- residual amplitude modulation;
- shot noise and detector noise;
- servo gain and servo bandwidth;
- actuator noise;
- free-running laser noise outside the loop bandwidth.
NIST reports compact Fabry-Perot reference cavities with cavity-stabilized laser linewidths well below 1 Hz, showing directly that the stabilized laser linewidth can be much narrower than the passive cavity resonance width.
9. Pound-Drever-Hall (PDH) locking
PDH locking is one of the standard techniques for stabilizing a laser to a high-finesse cavity.
The EOM creates optical sidebands at
\[\nu_0\pm f_m.\]The reflected carrier and sidebands interfere at the photodetector. Demodulation at $f_m$ produces a bipolar error signal approximately linear near the cavity resonance.
The servo sends this error to one or more laser actuators:
- diode current — fast but limited range;
- intracavity or external EOM — very fast, small range;
- laser-cavity PZT — moderate bandwidth, larger range;
- temperature — very slow, large tuning range.
A practical high-performance loop often uses fast + slow actuators simultaneously.
10. What linewidth should be expected after cavity locking?
There is no universal number. The result depends on the cavity and servo.
A useful hierarchy is
| System | Typical limiting mechanism |
|---|---|
| ordinary free-running laser | laser cavity, current, temperature, acoustic noise |
| laser locked to moderate-finesse cavity | servo residual + cavity drift |
| laser locked to ultrastable cavity | cavity Brownian/thermal noise, vibration and residual servo noise |
| state-of-the-art ultrastable system | reference-cavity thermal noise and technical noise below that level |
NIST has demonstrated a chip-based laser with an integrated 1 s linewidth of about 1.1 Hz and frequency instability below $10^{-14}$ from 1 ms to 1 s using an ultrastable cavity reference.
Length stability required for a 1 Hz optical reference
At $\lambda=852$ nm,
\[\nu\approx\frac{c}{\lambda}\approx3.52\times10^{14}\ \text{Hz}.\]A 1 Hz frequency fluctuation corresponds to a fractional instability
\[\frac{\delta\nu}{\nu}\approx2.8\times10^{-15}.\]For a 10 cm cavity this corresponds to an effective length fluctuation of order
\[|\delta L|\sim L\frac{|\delta\nu|}{\nu}\approx2.8\times10^{-16}\ \text{m}.\]This illustrates why ultrastable cavities require careful vibration isolation, thermal engineering and low-noise mirror coatings.
Second-harmonic-generation (SHG) laser systems
11. What is SHG?
Second-harmonic generation converts photons at angular frequency $\omega$ into coherent radiation at $2\omega$ through the second-order nonlinear susceptibility $\chi^{(2)}$:
\[P_i^{(2)}=\epsilon_0\sum_{jk}\chi^{(2)}_{ijk}E_jE_k.\]For a monochromatic pump
\[E_\omega(t)\propto\cos(\omega t),\]the quadratic polarization contains a term oscillating at $2\omega$.
The output wavelength is
\[\lambda_{2\omega}=\frac{\lambda_\omega}{2}.\]Examples:
- 1064 nm $\rightarrow$ 532 nm;
- 1156 nm $\rightarrow$ 578 nm;
- 1020 nm $\rightarrow$ 510 nm;
- 1018 nm $\rightarrow$ 509 nm.
The last two ranges are particularly relevant to cesium Rydberg excitation.
12. SHG conversion efficiency
In the undepleted-pump limit, the generated second-harmonic power scales approximately as
\[P_{2\omega}\propto d_{eff}^2 L^2 P_\omega^2\,\mathrm{sinc}^2\left(\frac{\Delta kL}{2}\right),\]where
\[\Delta k=k_{2\omega}-2k_\omega\]for birefringent phase matching. For quasi-phase matching,
\[\Delta k=k_{2\omega}-2k_\omega-\frac{2\pi}{\Lambda},\]where $\Lambda$ is the poling period.
Common nonlinear media include
- PPLN / MgO:PPLN;
- PPKTP;
- LBO;
- BBO;
- KTP.
Single-pass SHG
Simple, robust and broad in servo behavior, but conversion efficiency may be limited by available peak intensity and interaction length.
Cavity-enhanced SHG
The fundamental field is resonantly enhanced in an external cavity containing the nonlinear crystal. This can dramatically increase conversion efficiency, but introduces another cavity-locking problem.
13. What happens to linewidth and frequency noise after SHG?
This is subtle and is often oversimplified.
Write the fundamental field as
\[E_\omega(t)=A e^{i[\omega t+\phi(t)]}.\]In ideal SHG,
\[E_{2\omega}(t)\propto E_\omega^2(t) \propto A^2e^{i[2\omega t+2\phi(t)]}.\]Therefore
\[\boxed{\phi_{2\omega}(t)=2\phi_\omega(t).}\]The instantaneous absolute frequency fluctuation becomes
\[\boxed{\delta\nu_{2\omega}=2\delta\nu_\omega.}\]The frequency-noise PSD therefore becomes
\[\boxed{S_{\nu,2\omega}(f)=4S_{\nu,\omega}(f),}\]which is a 6 dB increase in absolute frequency-noise PSD.
However, the fractional frequency noise remains unchanged:
\[\boxed{\frac{\delta\nu_{2\omega}}{\nu_{2\omega}}= \frac{\delta\nu_\omega}{\nu_\omega}.}\]Does SHG double the linewidth or quadruple it?
There is no universal answer without specifying the linewidth definition and noise statistics.
- If linewidth is dominated by slowly varying Gaussian frequency jitter, doubling the absolute frequency excursions tends to double the Gaussian spectral width.
- If the spectrum is Lorentzian and dominated by white frequency noise, $S_\nu$ increases by 4, and the Lorentzian linewidth associated with that white-noise floor can increase by approximately 4.
- Real lasers contain mixed noise, so the safest procedure is to propagate or measure the full frequency-noise spectrum rather than apply a single rule.
A 2023 frequency-conversion experiment directly observed that SHG doubles the temporal phase fluctuation and produces a fourfold frequency-noise PSD increase. In another experiment developed specifically for cesium Rydberg excitation, a 1018 nm source measured at about 20 kHz linewidth produced a 509 nm SHG source estimated at about 40 kHz, illustrating a different linewidth convention/noise regime.
14. Cavity locking combined with SHG
Three architectures are common.
A. Lock the fundamental laser first, then frequency-double
This is conceptually clean. The SHG output inherits the stabilized phase, with the SHG phase/frequency-noise scaling described above.
B. Generate SHG and use the harmonic as the frequency reference
This is useful when the best reference exists at the final wavelength. A frequency correction $\delta\nu$ at the harmonic corresponds to approximately $\delta\nu/2$ at the fundamental.
C. Lock the SHG enhancement cavity to the laser
This maximizes nonlinear-conversion efficiency but does not, by itself, make the laser an ultrastable frequency reference. The enhancement cavity follows the laser unless a separate absolute-frequency stabilization loop is present.
15. Practical SHG linewidth example for cesium Rydberg systems
A useful architecture for green/blue Rydberg coupling lasers is
\[\boxed{ \text{ECDL seed} \rightarrow \text{fiber amplifier} \rightarrow \text{SHG crystal} \rightarrow \text{visible coupling laser} }\]Qian et al. demonstrated 1018 nm $\rightarrow$ 509 nm single-pass SHG using an ECDL-seeded Yb fiber amplifier and MgO:PPLN, obtaining about 2 W at 509 nm. The 1018 nm linewidth was measured near 20 kHz and the 509 nm linewidth was estimated near 40 kHz. The system was developed for cesium Rydberg-atom experiments.
For narrow Rydberg EIT, the visible coupling-laser linewidth contributes directly to the optical coherence budget. A useful experimental model is
\[\gamma_{gr}^{eff}\approx\gamma_{gr}^{intrinsic}+\gamma_{probe}+\gamma_{coupling}+\gamma_{technical},\]where the exact combination depends on the noise statistics and density-matrix convention.
16. Laser linewidth versus atomic linewidth
For resonant spectroscopy, compare the laser linewidth to the relevant atomic coherence width.
If
\[\Delta\nu_{laser}\ll\Gamma_{atom},\]the laser usually contributes little additional broadening.
If
\[\Delta\nu_{laser}\sim\Gamma_{atom},\]laser frequency noise can noticeably broaden and distort the spectrum.
If
\[\Delta\nu_{laser}\gg\Gamma_{atom},\]spectroscopic resolution is fundamentally limited by the laser unless a narrow coherent component remains within the atomic response bandwidth.
For EIT and Raman-type coherences, the relevant quantity is often the relative two-photon phase/frequency noise of the two lasers rather than each laser linewidth independently.
17. A practical laser-stabilization hierarchy
The sequence is not a learning prescription; it represents the layers of physical noise control present in many real laser systems.
18. What should be recorded when characterizing a laser?
For reproducible experimental reporting, record at least:
- center wavelength/frequency;
- free-running linewidth and measurement method;
- locked linewidth and integration/observation time;
- frequency-noise PSD if available;
- reference-cavity FSR, finesse and cavity linewidth;
- lock method and servo bandwidth;
- optical power before and after SHG;
- nonlinear crystal and phase-matching method;
- SHG conversion efficiency;
- RIN;
- beam waist and $M^2$;
- polarization;
- long-term drift or Allan deviation.
19. Selected references
- R. W. P. Drever et al., “Laser phase and frequency stabilization using an optical resonator,” Applied Physics B 31, 97–105 (1983). DOI
- E. D. Black, “An introduction to Pound-Drever-Hall laser frequency stabilization,” American Journal of Physics 69, 79–87 (2001). DOI
- G. Di Domenico, S. Schilt and P. Thomann, “Simple approach to the relation between laser frequency noise and laser line shape,” Applied Optics 49, 4801–4807 (2010). DOI
- J. Guo et al., “Chip-based laser with 1-hertz integrated linewidth,” Science Advances (2022). NIST
- C. McLemore et al., “Thermal Noise-Limited Laser Stabilization to an 8 mL Volume Fabry-Perot Reference Cavity with Microfabricated Mirrors,” APL Photonics (2022). NIST
- C. Ling et al., “Self-Injection Locked Frequency Conversion Laser,” Laser & Photonics Reviews (2023). DOI
- J. Qian et al., “2 W single-frequency, low-noise 509 nm laser via single-pass frequency doubling of an ECDL-seeded Yb fiber amplifier,” Applied Optics 57, 8733–8737 (2018). DOI
- J. Feng et al., “Noise suppression, linewidth narrowing of a master oscillator power amplifier at 1.56 μm and the second harmonic generation output at 780 nm,” Optics Express 16, 11871–11877 (2008). DOI
Related: Optics & Photonics · Quantum Technologies · Rydberg Semiclassical Optics · Measurements & Instruments