Linewidth Enhancement Factor
Author: the photonics expert Dr. Rüdiger Paschotta
Definition: a parameter quantifying the amplitude–phase coupling in a laser
Alternative term: Henry factor
Categories: laser devices and laser physics, physical foundations
Units: (dimensionless)
Formula symbol: <$\alpha$>
DOI: 10.61835/2c8 Cite the article: BibTex plain textHTML Link to this page LinkedIn
Any free-running (not stabilized) single-frequency laser has a certain finite linewidth, which is essentially due to noise from spontaneous emission into the resonator modes. For simple cases, that fundamental limit for the linewidth was calculated by Schawlow and Townes even before the first laser was experimentally demonstrated. Whereas this limit was later shown to be closely approached by a number of solid-state lasers, significantly higher linewidth values were measured for semiconductor lasers (laser diodes) even when the influence of technical noise was very low. It was then later found by Charles H. Henry [1] that the increased linewidths result from a coupling between intensity and phase noise, caused by a dependence of the refractive index on the carrier density in the semiconductor. Henry introduced the linewidth enhancement factor <$\alpha$> (also called Henry factor or alpha factor) to quantify that amplitude–phase coupling mechanism; essentially, <$\alpha$> is a proportionality factor relating phase changes to changes of the gain:
$$\Delta \varphi = \frac{\alpha }{2} \cdot \Delta g$$(The factor 1/2 serves to convert the change of power gain <$\Delta g$> to the change of amplitude gain.) Henry then found that the linewidth of the laser should be increased by the factor <$(1 + \alpha^2)$>, which turned out to be in reasonable agreement with experimental data.
Note that one may expect the linewidth enhancement factor to be the factor by which the linewidth is enhanced, but that factor is actually <$(1 + \alpha^2)$>.
Linewidth Enhancement Factor for Different Lasers
It is possible to calculate the <$\alpha$> factor of a semiconductor for a given carrier density from a band structure model, although this is not easy. For typical quantum wells, one often obtains values of the order to 2 to 5.
Quantum dot lasers are different from other semiconductor lasers (e.g. based on quantum wells) in terms of <$\alpha$> factor. Simple models suggest a very small <$\alpha$> factor, but different values are obtained experimentally – sometimes even negative values. This can be understood by taking into account the carriers not only of the quantum dots themselves, but also in the wetting layer. There are also various other subtle effects [6, 7].
Four-level solid-state lasers usually have a very small linewidth enhancement factor when operated near their gain maximum. Larger values are obtained when forcing operation at other wavelengths and for quasi-three-level laser gain media.
Additional Effects
Apart from increasing the laser linewidth in continuous-wave operation, a non-zero linewidth enhancement factor also causes a chirp when e.g. a laser is power modulated, or when an intense optical pulse passes an amplifier which it saturates.
Surprisingly, the amplitude–phase coupling related to the linewidth enhancement factor can under certain circumstances (with frequency-dependent loss) be used to reduce the linewidth even below the Schawlow–Townes limit [4, 5].
Measuring Linewidth Enhancement Factors
There are different methods for the measurement of the linewidth enhancement factor of a laser or a laser gain medium. Most common are those based on recording the optical spectrum of ASE for different excitation levels, on measuring amplitude and phase modulation caused by a modulated drive current, pump–probe measurements, and linewidth measurements.
More to Learn
Encyclopedia articles:
Suppliers
The RP Photonics Buyer's Guide contains 17 suppliers for linewidth measurement equipment. Among them:
HighFinesse
HighFinesse linewidth analyzers are the ultimate high-end instruments for measuring, analyzing and controlling frequency, linewidth and intensity noise of laser light sources. The product series covers a measurement range from 450 nm up to 1625 nm and enables the analysis of both very narrow laser lines down to 1 kHz as well as broader spectra up to 100 MHz. Thanks to the included software, the instrument delivers real-time analysis, with no need for further calculation. The analyzers feature an extremely high resolution and accuracy and are ideal for optimizing the stability of laser setups.
TOPTICA Photonics
The HighFinesse Linewidth Analyzers (LWAs) are very versatile, robust and compact instruments for measuring, analyzing and controlling frequency and intensity noise of lasers. This makes them ideal equipment for exact laser characterization in real time.
The HighFinesse wavelength meters are the unsurpassed high-end instruments for wave-length measurement of pulsed or continuous laser sources. They deliver the superb absolute and relative accuracy required by cutting edge scientific research, as well as industrial and medical applications.
Quantifi Photonics
Quantifi Photonics designs and manufactures a wide range photonic test solutions including lasers, optical spectrum analyzers, power meters and more. The company specialize in testing silicon photonics, co-packaged optics (CPO) and pluggable transceivers. Their solutions are used to test photonic integrated circuits (PICs), optical engines and pluggable optical transceivers, and are optimized for high density, high channel count applications in manufacturing environments. The company also offers unique solutions for coherent optical communications, photon Doppler velocimetry, and optical pulse analysis.
Bibliography
[1] | C. H. Henry, “Theory of the linewidth of semiconductor lasers”, IEEE J. Quantum Electron. 18 (2), 259 (1982); https://doi.org/10.1109/JQE.1982.1071522 |
[2] | C. H. Henry, “Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers”, J. Lightwave Technol. LT-4, 288 (1986) |
[3] | M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers – an overview”, IEEE J. Quantum Electron. 23 (1), 9 (1987); https://doi.org/10.1109/JQE.1987.1073204 |
[4] | A. Yariv et al., “Self-quenching of fundamental phase and amplitude noise in semiconductor lasers with dispersive loss”, Opt. Lett. 15 (23), 1359 (1990); https://doi.org/10.1364/OL.15.001359 |
[5] | Y. Shevy and H. Deng, “Frequency-stable and ultranarrow-linewidth semiconductor laser locked directly to an atom-cesium transition”, Opt. Lett. 23 (6), 472 (1998); https://doi.org/10.1364/OL.23.000472 |
[6] | S. Melnik et al., “The linewidth enhancement factor <$\alpha$> of quantum dot semiconductor lasers”, Opt. Express 14 (7), 2950 (2006); https://doi.org/10.1364/OE.14.002950 |
[7] | J. Molina Vázquez et al., “Linewidth enhancement factor of quantum-dot optical amplifiers”, IEEE J. Quantum Electron. 42 (10), 986 (2006); https://doi.org/10.1109/JQE.2006.881022 |
[8] | T. Fordell and A. M. Lindberg, “Experiments on the linewidth-enhancement factor for a vertical-cavity surface-emitting laser”, IEEE J. Quantum Electron. 43 (1), 6 (2007); https://doi.org/10.1109/JQE.2006.884583 |
[9] | R. P. Green et al., “Linewidth enhancement factor of terahertz quantum cascade lasers”, Appl. Phys. Lett. 92 (7), 071106 (2008); https://doi.org/10.1063/1.2883950 |
[10] | S. Gerhard et al., “Frequency-dependent linewidth enhancement factor of quantum-dot lasers”, IEEE Photon. Technol. Lett. 20 (20), 1736 (2008); https://doi.org/10.1109/LPT.2008.2004675 |
[11] | R. Pagano et al., “Above threshold estimation of alpha (Henry) parameter in stripe lasers using near- and far-field intensity measurements”, IEEE J. Quantum Electron. 47 (4), 439 (2011); https://doi.org/10.1109/JQE.2010.2091255 |
[12] | A. Consoli et al., “Self-validating technique for the measurement of the linewidth enhancement factor in semiconductor lasers”, Opt. Express 20 (5), 4979 (2012); https://doi.org/10.1364/OE.20.004979 |
[13] | B. Sinquin and M. Romanelli, “Determination of the linewidth enhancement factor of semiconductor lasers by complete optical field reconstruction”, Opt. Lett. 48 (4), 863 (2023); https://doi.org/10.1364/OL.483776 |
Questions and Comments from Users
Here you can submit questions and comments. As far as they get accepted by the author, they will appear above this paragraph together with the author’s answer. The author will decide on acceptance based on certain criteria. Essentially, the issue must be of sufficiently broad interest.
Please do not enter personal data here. (See also our privacy declaration.) If you wish to receive personal feedback or consultancy from the author, please contact him, e.g. via e-mail.
By submitting the information, you give your consent to the potential publication of your inputs on our website according to our rules. (If you later retract your consent, we will delete those inputs.) As your inputs are first reviewed by the author, they may be published with some delay.
2024-02-04
I would expect that the alpha factor to contribute only at low frequencies, with a cutoff frequency around the relaxation oscillation frequency. In such case, wouldn't the high frequency white noise level be the same as the ideal Schawlow–Townes limit?
The author's answer:
Yes, the linewidth enhancement factor is relevant only in frequency regions where the carrier density oscillates substantially.
For the linewidth, the low-frequency phase noise is relevant, so for that the factor fully counts, while very high-frequency noise is not affected.