Laser line width

The laser line width is the spectral line width of a laser beam. This article presents the latest findings on the development of spectral coherence and line width of a laser.

theory

History: First derivation of the laser line width

The first man-made coherent light source was a " burl ". The acronym MASER means " Microwave Amplification by Stimulated Emission of Radiation " (in German: Microwave amplification by stimulated emission of radiation). More precisely, it was the ammonia maser, which emits at a wavelength of 12.5 mm, which was first operated in 1954 by Gordon , Zeiger and Townes . A year later, the same authors theoretically inferred the burl line width by making the reasonable approximations that their ammonia burl

(i) works in continuous-wave mode (English: " continuous-wave ", cw),

(ii) has an ideal four-level power scheme, and

(iii) does not suffer any intrinsic resonator losses, but only decoupling losses through the mirrors.

Remarkably, this derivation is based on purely semi-classical assumptions. It describes the ammonia molecules as quantum emitters, while classical electromagnetic fields are assumed (on the other hand no quantized fields or quantum fluctuations ). This results in half the half -width (“ half-width-at-half-maximum ”, HWHM) of the Maser emission

${\displaystyle \Delta \nu _{\rm {M}}^{*}={\frac {4\pi k_{\rm {B}}T(\Delta \nu _{\rm {c}}^{*})^{2}}{P_{\rm {aus}}}}\Leftrightarrow \Delta \nu _{\rm {M}}={\frac {2\pi k_{\rm {B}}T(\Delta \nu _{\rm {c}})^{2}}{P_{\rm {aus}}}},}$

which is marked here with an asterisk and is converted to the full-width-at-half-maximum (FWHM) of the Maser emission. is the Boltzmann constant, is the temperature, is the output power of the laser, and and are the passive HWHM and FWHM line widths of the microwave resonator used. ${\displaystyle \Delta \nu _{\rm {M}}=2\Delta \nu _{\rm {M}}^{*}}$ ${\displaystyle k_{\rm {B}}}$ ${\displaystyle T}$ ${\displaystyle P_{\rm {aus}}}$ ${\displaystyle \Delta \nu _{\rm {c}}^{*}}$ ${\displaystyle \Delta \nu _{\rm {c}}=2\Delta \nu _{\rm {c}}^{*}}$

In 1958, two years before Maiman (which was initially as "optical maser") is built the first laser, transferred Schawlow and Townes the maser line width in the visible and near infrared spectral region by the thermal energy of the photon energy replaced . is Planck's quantum of action and is the frequency of the laser light. This transfer approximates that ${\displaystyle k_{\rm {B}}T}$ ${\displaystyle h\nu _{\rm {L}}}$ ${\displaystyle h}$ ${\displaystyle \nu _{\rm {L}}}$

(iv) a photon is coupled into the laser mode by spontaneous emission during the resonator decay time, ${\displaystyle \tau _{\rm {c}}}$

and leads to the Schawlow-Townes line width :

${\displaystyle \Delta \nu _{\rm {L,ST}}^{*}={\frac {4\pi h\nu _{\rm {L}}(\Delta \nu _{\rm {c}}^{*})^{2}}{P_{\rm {aus}}}}\Leftrightarrow \Delta \nu _{\rm {L,ST}}={\frac {2\pi h\nu _{\rm {L}}(\Delta \nu _{\rm {c}})^{2}}{P_{\rm {aus}}}}.}$

The transfer from the microwave to the optical spectral range is also based on a purely semi-classical basis, without the assumption of quantized fields or quantum fluctuations. Therefore, the original Schawlow-Townes equation is based completely on semi-classical physical assumptions and is a four-fold approximation of a more general laser line width, which is derived in the following.

Passive resonator mode: resonator decay time

A Fabry-Perot resonator of geometric length , consisting of two mirrors, is considered , which is homogeneously filled with an active laser medium that has a refractive index . As a reference situation, we define the passive resonator mode, the active laser medium of which is transparent, i.e. that is, it does not act to amplify or absorb light. ${\displaystyle \ell }$ ${\displaystyle n}$

Light travels in the resonator with speed , the speed of light being in a vacuum. Thus, the resonator round trip time and the free spectral range are given by ${\displaystyle c=c_{0}/n}$ ${\displaystyle c_{0}}$ ${\displaystyle t_{\rm {RT}}}$ ${\displaystyle \Delta \nu _{\rm {FSR}}}$

${\displaystyle t_{\rm {RT}}={\frac {1}{\Delta \nu _{\rm {FSR}}}}={\frac {2\ell }{c}}.}$

Light in the longitudinal resonator mode of interest to us oscillates at the qth resonance frequency

${\displaystyle \nu _{L}={\frac {q}{t_{\rm {RT}}}}=q\Delta \nu _{\rm {FSR}}.}$

The exponential extraction time and the corresponding rate constant resulting from the specular reflections of the two resonator mirror to ${\displaystyle \tau _{\rm {aus}}}$ ${\displaystyle 1/\tau _{\rm {aus}}}$ ${\displaystyle R_{i}}$ ${\displaystyle i=1,2}$

${\displaystyle R_{1}R_{2}=e^{-t_{\rm {RT}}/\tau _{\rm {aus}}}\Rightarrow {\frac {1}{\tau _{\rm {aus}}}}={\frac {-\ln {(R_{1}R_{2})}}{t_{\rm {RT}}}}.}$

The exponential intrinsic loss time and the corresponding rate constant result from the intrinsic resonator losses per resonator revolution ${\displaystyle \tau _{\rm {verl}}}$ ${\displaystyle 1/\tau _{\rm {verl}}}$ ${\displaystyle L_{\rm {RT}}}$

${\displaystyle 1-L_{\rm {RT}}=e^{-t_{\rm {RT}}/\tau _{\rm {verl}}}\Rightarrow {\frac {1}{\tau _{\rm {verl}}}}={\frac {-\ln {(1-L_{\rm {RT}})}}{t_{\rm {RT}}}}.}$

The resulting exponential resonator decay time and the corresponding rate constant are thus ${\displaystyle \tau _{c}}$ ${\displaystyle 1/\tau _{\rm {c}}}$

${\displaystyle {\frac {1}{\tau _{\rm {c}}}}={\frac {1}{\tau _{\rm {aus}}}}+{\frac {1}{\tau _{\rm {verl}}}}={\frac {-\ln {[R_{1}R_{2}(1-L_{\rm {RT}})]}}{t_{\rm {RT}}}}.}$

All three exponential decay times average over the resonator cycle time In the following we assume that , , , and , and therefore , and to not change significantly within the spectral range that is for laser linewidth of interest. ${\displaystyle t_{\rm {RT}}.}$ ${\displaystyle \ell }$ ${\displaystyle n}$ ${\displaystyle R_{1}}$ ${\displaystyle R_{2}}$ ${\displaystyle L_{\rm {RT}}}$ ${\displaystyle \tau _{\rm {aus}}}$ ${\displaystyle \tau _{\rm {verl}}}$ ${\displaystyle \tau _{\rm {c}}}$

Passive resonator mode: Lorentz line width, Q factor, coherence time and length

In addition to the resonator decay time , the spectral coherence properties of the passive resonator mode can be expressed by the following parameters. The FWHM-Lorentz line width of the passive resonator mode, which occurs in the Schawlow-Townes equation, is obtained by Fourier transformation of the exponential resonator decay time into the frequency space, ${\displaystyle \tau _{\rm {c}}}$ ${\displaystyle \Delta \nu _{\rm {c}}}$ ${\displaystyle \tau _{\rm {c}}}$

${\displaystyle \Delta \nu _{\rm {c}}={\frac {1}{2\pi \tau _{\rm {c}}}}.}$

The quality factor ( Q factor) is defined as the energy stored in the resonator divided by the energy lost per oscillation cycle of the light, ${\displaystyle Q_{\rm {c}}}$ ${\displaystyle W_{\rm {speich}}}$ ${\displaystyle W_{\rm {lost}}}$

${\displaystyle Q_{\rm {c}}=2\pi {\frac {W_{\rm {speich}}(t)}{W_{\rm {lost}}(t)}}=2\pi {\frac {\varphi (t)}{-{\frac {1}{\nu _{L}}}{\frac {d}{dt}}\varphi (t)}}=2\pi \nu _{L}\tau _{\rm {c}}={\frac {\nu _{L}}{\Delta \nu _{\rm {c}}}},}$

where the number of photons is in the resonator mode. The coherence time and the coherence length of the light emitted from the resonator mode are given by the equation ${\displaystyle \varphi =W_{\rm {speich}}/h\nu _{L}}$ ${\displaystyle \tau _{\rm {c}}^{\rm {coh}}}$ ${\displaystyle \ell _{\rm {c}}^{\rm {coh}}}$

${\displaystyle \tau _{\rm {c}}^{\rm {coh}}={\frac {1}{c}}\ell _{\rm {c}}^{\rm {coh}}=2\tau _{\rm {c}}.}$

Active resonator mode: gain, resonator decay time, Lorentz line width, Q factor, coherence time and length

Taking into account the population densities and the upper and lower laser levels and the effective cross-sections and the stimulated emission and absorption at the resonance frequency , the gain per unit length in the active laser medium at the resonance frequency results ${\displaystyle N_{2}}$ ${\displaystyle N_{1}}$ ${\displaystyle \sigma _{\rm {e}}}$ ${\displaystyle \sigma _{\rm {a}}}$ ${\displaystyle \nu _{L}}$ ${\displaystyle \nu _{L}}$

${\displaystyle g=\sigma _{\rm {e}}N_{2}-\sigma _{\rm {a}}N_{1}.}$

One value means gain at the resonance frequency , while one value means absorption. This results in a correspondingly lengthened or shortened resonator decay time with which the photons move away from the active resonator mode: ${\displaystyle g>0}$ ${\displaystyle \nu _{L}}$ ${\displaystyle g<0}$ ${\displaystyle \tau _{\rm {L}}}$

${\displaystyle {\frac {1}{\tau _{\rm {L}}}}={\frac {1}{\tau _{\rm {c}}}}-cg.}$

The other four properties of the spectral coherence of the active resonator mode are obtained in the same way as for the passive resonator mode. The Lorentz line width results from Fourier transformation into the frequency space,

${\displaystyle \Delta \nu _{\rm {L}}={\frac {1}{2\pi \tau _{\rm {L}}}}.}$

A value leads to a gain reduction of the line width, while a value leads to a broadening of the absorption. The Q factor is ${\displaystyle g>0}$ ${\displaystyle g<0}$

${\displaystyle Q_{\rm {L}}=2\pi {\frac {W_{\rm {speich}}(t)}{W_{\rm {lost}}(t)}}=2\pi {\frac {\varphi (t)}{-{\frac {1}{\nu _{L}}}{\frac {d}{dt}}\varphi (t)}}=2\pi \nu _{L}\tau _{\rm {L}}={\frac {\nu _{L}}{\Delta \nu _{\rm {L}}}}.}$

The coherence time and length are

${\displaystyle \tau _{\rm {L}}^{\rm {coh}}={\frac {1}{c}}\ell _{\rm {L}}^{\rm {coh}}=2\tau _{\rm {L}}.}$

Spectral coherence factor

The factor by which the resonator decay time is lengthened by amplification or shortened by absorption is introduced here as a so-called spectral coherence factor : ${\displaystyle \Lambda }$

${\displaystyle \Lambda :={\frac {1}{1-cg\tau _{\rm {c}}}}.}$

All five parameters introduced above, which describe the spectral coherence equally, scale with the same spectral coherence factor : ${\displaystyle \Lambda }$

${\displaystyle \tau _{\rm {L}}=\Lambda \tau _{\rm {c}},(\Delta \nu _{\rm {L}})^{-1}=\Lambda (\Delta \nu _{\rm {c}})^{-1},Q_{\rm {L}}=\Lambda Q_{\rm {c}},\tau _{\rm {L}}^{\rm {coh}}=\Lambda \tau _{\rm {c}}^{\rm {coh}},\ell _{\rm {L}}^{\rm {coh}}=\Lambda \ell _{\rm {c}}^{\rm {coh}}.}$

Lasende Resonator-Mode: Fundamental laser line width

For a given number of photons in the lasing resonator mode, the stimulated emission rate and resonator decay rate are: ${\displaystyle \varphi }$

${\displaystyle R_{\rm {stim}}=cg\varphi ,}$

${\displaystyle R_{\rm {abkling}}={\frac {1}{\tau _{\rm {c}}}}\varphi .}$

The spectral coherence factor is thus

${\displaystyle \Lambda ={\frac {R_{\rm {abkling}}}{R_{\rm {abkling}}-R_{\rm {stim}}}}.}$

The resonator decay time of the lasing resonator mode is

${\displaystyle \tau _{\rm {L}}=\Lambda \tau _{\rm {c}}={\frac {R_{\rm {abkling}}}{R_{\rm {abkling}}-R_{\rm {stim}}}}\tau _{\rm {c}}.}$

The fundamental laser line width is

${\displaystyle \Delta \nu _{\rm {L}}={\frac {1}{\Lambda }}\Delta \nu _{\rm {c}}={\frac {R_{\rm {abkling}}-R_{\rm {stim}}}{R_{\rm {abkling}}}}\Delta \nu _{\rm {c}}.}$

This fundamental line width applies to lasers with any energy level scheme (four or three level scheme or any situation between these two extremes), in operation below, at, and above the laser threshold, a gain that is smaller, equal to or greater than the losses, and in a CW or transient laser regime.

From this derivation it becomes clear that the fundamental line width of a CW laser is based on the semi-classical effect that the gain increases the resonator decay time.

Continuous wave laser: The gain is smaller than the loss

The spontaneous emission rate into the lasing resonator mode is given by

${\displaystyle R_{\rm {spon}}=c\sigma _{\rm {e}}N_{2}.}$

It should be noted that the rate is basically positive, because with every single emission process an atomic excitation is converted into a photon in the laser mode. This rate is the source term of the laser radiation and must by no means be misinterpreted as "noise". The photon rate equation for a single laser mode is given as ${\displaystyle R_{\rm {spon}}}$

${\displaystyle {\frac {d}{dt}}\varphi =R_{\rm {spon}}+R_{\rm {stim}}-R_{\rm {abkling}}=c\sigma _{\rm {e}}N_{2}+cg\varphi -{\frac {1}{\tau _{\rm {c}}}}\varphi .}$

A CW laser is characterized by a temporally constant number of photons in the laser mode. Hence is . In a CW laser, the stimulated and spontaneous emission rates together compensate for the resonator decay rate. thats why ${\displaystyle d\varphi /dt=0}$

${\displaystyle R_{\rm {stim}}-R_{\rm {abkling}}=-R_{\rm {spon}}<0.}$

The stimulated emission rate is less than the resonator decay rate. In other words: the gain is smaller than the losses. This fact has been known for decades and has been used to quantitatively describe the behavior of semiconductor lasers at the laser threshold. Even well above the laser threshold, the gain is still minimally smaller than the losses. It is precisely this extremely small difference that creates the finite line width of a CW laser.

From this derivation it becomes clear that a laser is basically an amplifier of spontaneous emission and the CW laser line width is caused by the semi-classical effect that the gain is smaller than the losses. In quantum-optical descriptions of the laser line width, which are based on the main density operator equation, it can be shown that the gain is smaller than the losses.

Schawlow-Townes approximation

See also : Schawlow-Townes limit

As mentioned above, it is clear from its historical derivation that the original Schawlow-Townes equation is a four-fold approximation of the fundamental laser line width. Starting from the fundamental laser line width , which was derived above, one obtains exactly the original Schawlow-Townes equation by explicitly using the four approximations (i) - (iv): ${\displaystyle \Delta \nu _{\rm {L}}}$

(i) It is a pure CW laser. The same applies accordingly

${\displaystyle R_{\rm {abkling}}-R_{\rm {stim}}=R_{\rm {spon}}\Rightarrow }$

${\displaystyle \Delta \nu _{\rm {L}}={\frac {1}{\Lambda }}\Delta \nu _{\rm {c}}={\frac {R_{\rm {abkling}}-R_{\rm {stim}}}{R_{\rm {abkling}}}}\Delta \nu _{\rm {c}}={\frac {R_{\rm {spon}}}{R_{\rm {abkling}}}}\Delta \nu _{\rm {c}}.}$

(ii) It is an ideal four level laser. The same applies accordingly

${\displaystyle N_{1}=0\Rightarrow cg=c(\sigma _{\rm {e}}N_{2}-\sigma _{\rm {a}}N_{1})=c\sigma _{\rm {e}}N_{2}=R_{\rm {spon}}\Rightarrow }$

${\displaystyle \Delta \nu _{\rm {L}}={\frac {R_{\rm {spon}}}{R_{\rm {abkling}}}}\Delta \nu _{\rm {c}}={\frac {cg}{{\frac {1}{\tau _{\rm {c}}}}\varphi }}\Delta \nu _{\rm {c}}.}$

(iii) Intrinsic resonator losses are negligible. The same applies accordingly

${\displaystyle {\frac {1}{\tau _{\rm {verl}}}}=0\Rightarrow {\frac {1}{\tau _{\rm {c}}}}={\frac {1}{\tau _{\rm {aus}}}}\Rightarrow P_{\rm {aus}}=h\nu _{\rm {L}}{\frac {1}{\tau _{\rm {aus}}}}\varphi =h\nu _{\rm {L}}{\frac {1}{\tau _{\rm {c}}}}\varphi \Rightarrow }$

${\displaystyle \Delta \nu _{\rm {L}}={\frac {cg}{{\frac {1}{\tau _{\rm {c}}}}\varphi }}\Delta \nu _{\rm {c}}={\frac {cgh\nu _{\rm {L}}}{P_{\rm {aus}}}}\Delta \nu _{\rm {c}}.}$

(iv) Exactly one photon is coupled into the laser mode by spontaneous emission during the resonator decay time. In an ideal four-level CW laser, this would happen exactly at the unreachable point at which the spectral coherence factor , the number of photons in the resonator mode and the output power become infinitely large, i.e. at the point at which the amplification equal to the losses. The same applies accordingly ${\displaystyle \tau _{\rm {c}}}$ ${\displaystyle \Lambda }$ ${\displaystyle \varphi }$ ${\displaystyle P_{\rm {aus}}}$

${\displaystyle R_{\rm {stim}}=R_{\rm {abkling}}\Rightarrow R_{\rm {spon}}=cg={\frac {1}{\tau _{\rm {c}}}}=2\pi \Delta \nu _{\rm {c}}\Rightarrow }$

${\displaystyle \Delta \nu _{\rm {L}}={\frac {cgh\nu _{\rm {L}}}{P_{\rm {aus}}}}\Delta \nu _{\rm {c}}={\frac {2\pi h\nu _{\rm {L}}(\Delta \nu _{\rm {c}})^{2}}{P_{\rm {aus}}}}=\Delta \nu _{\rm {L,ST}}.}$

This means that if one applies the same four approximations (i) - (iv), which were already used in the first derivation, to the fundamental laser line width , one consequently obtains the original Schawlow-Townes equation. ${\displaystyle \Delta \nu _{\rm {L}}}$

In summary, this is the fundamental laser line width

${\displaystyle \Delta \nu _{\rm {L}}={\frac {1}{\Lambda }}\Delta \nu _{\rm {c}}={\frac {R_{\rm {abkling}}-R_{\rm {stim}}}{R_{\rm {abkling}}}}\Delta \nu _{\rm {c}}=(1-cg\tau _{\rm {c}})\Delta \nu _{\rm {c}}=\Delta \nu _{\rm {c}}-{\frac {cg}{2\pi }},}$

while the original Schawlow-Townes equation is a four-fold approximation of this fundamental laser line width and is therefore primarily of historical significance.

Additional mechanisms of line broadening and narrowing

After its publication in 1958, the original Schawlow-Townes equation was expanded in a number of ways. These extended equations are often referred to by the same name, namely the Schawlow-Townes line width. This has resulted in confusion in the published literature on laser line widths, since it is often unclear which extension of the original Schawlow-Townes equation the respective authors are considering or using.

Some semi-classical extensions aimed to eliminate one or more of the above approximations (i) - (iv), whereby these extended equations became more similar in their physical content to the fundamental laser line width derived above.

The following possible extensions to the fundamental laser line width have been proposed:

1. Hempstead and Lax , as well as independently Hermann Haken , predicted quantum mechanically that near the laser threshold an additional reduction of the laser line width by a factor of two would occur. However, such a reduction has only been observed experimentally in very few cases.
2. In a semi-classical way, Petermann explained a line broadening previously observed experimentally in semiconductor lasers waveguiding through optical amplification instead of through a difference in refractive index. Siegman later showed that this effect is caused by the non- orthogonality of transverse modes. Woerdman and co-workers extended this idea to longitudinal modes and polarization modes. Therefore, sometimes the so-called "Petermann K-factor" is added to the equation for the laser line width.
3. Henry predicted an additional line broadening in quantum mechanics, which occurs because variations in the refractive index due to electron-hole pair excitations in semiconductor lasers (but of course in all other excitation processes too) cause phase variations in the resonator orbit. Therefore, sometimes the so-called “Henry factor” is added to the equation for the laser line width. ${\displaystyle \alpha }$

Measurement of the laser line width

One of the first methods of measuring the spectral coherence of a laser was interferometry. A typical method for measuring the laser line width is self- heterodyne interferometry .

Continuous wave laser

The laser line width of a typical single-mode He-Ne laser at a wavelength of 632.8 nm, in the absence of further reducing optics in the resonator, is typically on the order of 1 GHz. Rare-earth-doped dielectric or semiconductor-based lasers with distributed feedback through integrated Bragg mirrors have typical line widths on the order of 1 kHz. The laser line width of stabilized CW lasers can be far less than 1 kHz. Observed line widths are larger than the fundamental laser line width because technical influences occur (e.g. temporal fluctuations in the optical or electrical pump power, mechanical vibrations, changes in the refractive index and length due to temperature fluctuations, etc.).

Individual evidence

1. ^ JP Gordon, HJ Zeiger, CH Townes: Molecular microwave oscillator and new hyperfine structure in the microwave spectrum of NH3 . In: Physical Review . 95, No. 1, 1954, pp. 282-284. doi : 10.1103 / PhysRev.95.282 .
2. ↑ ^{a b c d e f g h} J. P. Gordon, HJ Zeiger, CH Townes: The maser − New type of microwave amplifier, frequency standard, and spectrometer . In: Physical Review . 99, No. 4, 1955, pp. 1264-1274. doi : 10.1103 / PhysRev.99.1264 .
3. ↑ TH Maiman: Stimulated optical radiation in Ruby . In: Nature . 187, No. 4736, 1960, pp. 493-494. doi : 10.1038 / 187493a0 .
4. ↑ ^{a b c d e f} A. L. Schawlow, CH Townes: Infrared and optical masers . In: Physical Review . 112, No. 6, 1958, pp. 1940-1949. doi : 10.1103 / PhysRev.112.1940 .
5. ↑ ^{a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae af ag ah ai aj ak al am} M. Pollnau, M. Eichhorn: Spectral coherence , Part I: Passive resonator linewidth, fundamental laser linewidth, and Schawlow-Townes approximation . In: Progress in Quantum Electronics . In press, Nr. Journal Pre-proof, 2020, p. 100255. doi : 10.1016 / j.pquantelec.2020.100255 .
6. ↑ ^{a b c d e} N. Ismail, CC Kores, D. Geskus, M. Pollnau: Fabry-Pérot resonator: spectral line shapes, generic and related Airy distributions, linewidths, finesses, and performance at low or frequency-dependent reflectivity . In: Optics Express . 24, No. 15, 2016, pp. 16366–16389. bibcode : 2016OExpr..2416366I . doi : 10.1364 / OE.24.016366 . PMID 27464090 .
7. ↑ M. Pollnau: Phase aspect in photon emission and absorption . In: Optica . 5, No. 4, 2018, pp. 465–474. doi : 10.1364 / OPTICA.5.000465 .
8. ↑ HS Sommers: Spontaneous power and the coherent state of injection lasers . In: Journal of Applied Physics . 45, No. 4, 1974, pp. 1787-1793. doi : 10.1063 / 1.1663491 .
9. ↑ HS Sommers: Threshold and oscillation of injection lasers: a critical review of laser theory . In: Solid-State Electronics . 25, No. 1, 1982, pp. 25-44. doi : 10.1016 / 0038-1101 (82) 90091-0 .
10. ^ Siegman, AE (1986) Lasers , University Science Books, Mill Valley, California, ch. 13, pp. 510-524.
11. ↑ G. Björk, Y. Yamamoto: Analysis of semiconductor microcavity lasers using rate equations . In: IEEE Journal of Quantum Electronics . 27, No. 11, 1991, pp. 2386-2396. doi : 10.1109 / 3.100877 .
12. ^ Sargent III, M .; Scully, MO; Lamb, Jr., WE (1993) "Laser Physics", 6th edition, Westview Press, Ch. 17.
13. ^ RD Hempstead, M. Lax: Classical noise. VI. Noise in self-sustained oscillators near threshold . In: Physical Review . 161, No. 2, 1967, pp. 350-366. doi : 10.1103 / PhysRev.161.350 .
14. ↑ Haken, H. (1970) “Laser Theory”, Vol. XXV / 2c of Encyclopedia of Physics, Springer.
15. ↑ K. Petermann: Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding . In: IEEE Journal of Quantum Electronics . QE-15, No. 7, 1979, pp. 566-570. doi : 10.1109 / JQE.1979.1070064 .
16. ^ AE Siegman: Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers . In: Physical Review A . 39, No. 3, 1989, pp. 1253-1263. doi : 10.1103 / PhysRevA.39.1253 . PMID 9901361 .
17. ^ AE Siegman: Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators . In: Physical Review A . 39, No. 3, 1989, pp. 1264-1268. doi : 10.1103 / PhysRevA.39.1264 . PMID 9901362 .
18. ↑ WA Hamel, JP Woerdman: Nonorthogonality of the longitudinal eigenmodes of a laser . In: Physical Review A . 40, No. 5, 1989, pp. 2785-2787. doi : 10.1103 / PhysRevA.40.2785 . PMID 9902474 .
19. ↑ AM van der Lee, NJ van Druten, AL Mieremet, MA van Eijkelenborg, Å. M. Lindberg, MP van Exter, JP Woerdman: Excess quantum noise due to nonorthogonal polarization modes . In: Physical Review Letters . 79, No. 5, 1989, pp. 4357-4360. doi : 10.1103 / PhysRevA.40.2785 . PMID 9902474 .
20. ^ CH Henry: Theory of the line width of semiconductor lasers . In: IEEE Journal of Quantum Electronics . 18, No. 2, 1982, pp. 259-264. doi : 10.1109 / JQE.1982.1071522 .
21. ^ OS Heavens, Optical Masers (Wiley, New York, 1963).
22. ↑ T. Okoshi, K. Kikuchi, A. Nakayama: Novel method for high resolution measurement of laser output spectrum . In: Electronics Letters . 16, No. 16, 1980, pp. 630-631. doi : 10.1049 / el: 19800437 .
23. ↑ JW Dawson, N. Park, KJ Vahala: An improved delayed self-heterodyne interferometer for linewidth measurements . In: IEEE Photonics Technology Letters . 4, No. 9, 1992, pp. 1063-1066. doi : 10.1109 / 68.157150 .
24. ↑ EH Bernhardi, HAGM van Wolferen, L. Agazzi, MRH Khan, CGH Roeloffzen, K. Wörhoff, M. Pollnau, RM de Ridder: Ultra-narrow-linewidth, single-frequency distributed feedback waveguide laser in Al2O3: Er3 + on silicon . In: Optics Letters . 35, No. 14, 2010, pp. 2394-2396. doi : 10.1364 / OL.35.002394 . PMID 20634841 .
25. ↑ CT Santis, ST Steger, Y. Vilenchik, A. Vasilyev, A. Yariv: High-coherence semiconductor lasers based on integral high-Q resonators in hybrid Si / III-V platforms . In: Proceedings of the National Academy of Sciences of the United States of America . 111, No. 8, 2014, pp. 2879–2884. doi : 10.1073 / pnas.1400184111 . PMID 24516134 . PMC 3939879 (free full text).
26. ^ LW Hollberg, CW dye lasers, in Dye Laser Principles , FJ Duarte and LW Hillman (eds.) (Academic, New York, 1990) Chapter 5.