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Journal of the Optical Society of America B

Laser isolas

John C. Englund, William C. Schieve

Published Jan 1, 1985• 17 citations
Publisher

Abstract

When the strong Doppler limit is applicable, a tuned laser containing Doppler-broadened gain and absorption media may produce a steady-state standing-wave field that is both frequency pushed and isolated from the branch of on-resonance steady states. The locus of the isolated states plotted as a function of the linear gain is a pair of closed loops, called isolas. Similar running-wave modes are not found in this limit. A connection is indicated between the isolated steady states and both the mode-pushed states discovered by Bennett [ The Physics of Gas Lasers ( Gordon and Breach, New York, 1977)] and the instability found by Casperson [ Phys. Rev. A21, 911 ( 1980)].

DOI
10.1364/josab.2.000081

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I presented those results at the Paris Quantum Electronics Conference in February 1963, but the volume containing my paper describing those equations, wasn’t published until a year later. 4 4. W. R. Bennett Jr, in Quantum Electronics: Proceedings of the Third International Congress, Paris, vol. 1, P. Grivet, N. Bloembergen, eds., Columbia U. Press, New York (1964), pp. 441, 450. That work was discussed in much more detail in an article based on lectures that I gave at Brandeis University in the summer of 1969.One final aspect of the role of hole-burning effects in laser spectroscopy that was omitted in the otherwise fine article by Brewer and coauthors is the narrow absorption peaks initially observed in 1968 by Veniamin Chebotayev and his student V. N. Lisitzin at Novosibirsk, Russia, in a laser containing a saturable absorber. 5 5. V. N. Lisitzin, “Experimental Investigations of Gas Lasers with Saturable Absorbers as Resonators,” research thesis, AcademGorodok, Novosibirsk, Russia (1968). The extremely narrow resonant peaks they found were due to hole-burning effects in the Doppler-broadened absorber. Chebotayev and Lisitzin also extended that work to include tuning dips from strong-wave–weak-wave interactions in external absorption cells, a process that anticipated the later, more elegant work by Theodor Hänsch, Marc Levenson, and Arthur Schawlow. 6 6. T. W. Hänsch, M. D. Levenson, A. L. Schawlow, Phys. Rev. Lett. 26, 945 (1971). In those experiments, the strong wave burns a hole in the line that is probed by a weak wave traveling in the opposite direction. Since the absorber can involve a ground-state atom or molecule, much narrower natural widths can be obtained in the absorber experiments than in those with the amplifying medium.REFERENCESSection:ChooseTop of pageREFERENCES <<1. W. R. Bennett Jr, Phys. Rev. 126, 580 (1962). https://doi.org/10.1103/PhysRev.126.580 , Google ScholarCrossref2. W. R. Bennett Jr, in Applied Optics Supplement on Optical Masers, Optical Society of America, Washington, DC (1962), pp. 24, 57, 58. Google Scholar3. R. A. McFarlane, W. R. Bennett Jr, W. E. Lamb Jr, Appl. Phys. Lett. 2, 189 (1963). https://doi.org/10.1063/1.1753727 , Google ScholarCrossref, CAS4. W. R. Bennett Jr, in Quantum Electronics: Proceedings of the Third International Congress, Paris, vol. 1, P. Grivet, N. Bloembergen, eds., Columbia U. Press, New York (1964), pp. 441, 450. Google Scholar5. V. N. Lisitzin, “Experimental Investigations of Gas Lasers with Saturable Absorbers as Resonators,” research thesis, AcademGorodok, Novosibirsk, Russia (1968). Google Scholar6. T. W. Hänsch, M. D. Levenson, A. L. Schawlow, Phys. Rev. Lett. 26, 945 (1971). Google ScholarCrossref© 2008 American Institute of Physics.

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When the strong Doppler limit is applicable, a tuned laser containing Doppler-broadened gain and absorption media may produce a steady-state standing-wave field that is both frequency pushed and isolated from the branch of on-resonance steady states. The locus of the isolated states plotted as a function of the linear gain is a pair of closed loops, called isolas. Similar running-wave modes are not found in this limit. A connection is indicated between the isolated steady states and both the mode-pushed states discovered by Bennett [ The Physics of Gas Lasers ( Gordon and Breach, New York, 1977)] and the instability found by Casperson [ Phys. Rev. A21, 911 ( 1980)].
In Chaps. 8 and 9, we considered a laser medium consisting of homo­ geneously broadened, stationary atoms. When the lasing atoms move as in a gas, they see an electric field with shifted frequency due to the Doppler effect, as shown in Fig. 10-1. One might argue that it is merely necessary to average the complex polarization or the susceptibility Xn over the frequency range corresponding to the velocity distribution, that is, calculate a new ^ n , for which Here the frequency co = coo + Kv, where coo is the frequency at atomic line (1) center, v is the component of velocity along the laser (z) axis, and the frequen­ cy distribution is determined by a Maxwell-Boltzmann velocity distribution W(y) = ( w)-1 exp [ - (v/w)2]. (2) Here u is the most probable speed of the atom, and K is the wave number. The simple recipe (1) is, in fact, valid for an inhomogeneously broadened mediumt consisting of stationary atoms such as ruby at low temperatures. However, in the standing-wave laser, the atoms not only see Doppler-shifted frequencies, but also move through the standing-wave electric field, effectively seeing an amplitude-modulated field. Equivalently, each atom sees two frequencies, for the standing-wave field is the sum of oppositely directed running waves, one of which appears to be Doppler upshifted, one down­ shifted. Hence it is necessary to consider both the frequency shift and the time dependence of the atomic z coordinate when calculating the polarization of a gaseous medium.
Richard Brewer, Aram Mooradian, and Boris Stoicheff, in their review “The Early Days of Precision Laser Spectroscopy” (Physics Today, Physics Today 0031-9228 60 1 2007 49 https://doi.org/10.1063/1.2709559. January 2007, page 49 ), implied that Willis Lamb conceived of the notion that his dip at line center could be explained through hole-burning effects. In fact, Lamb had no physical interpretation of the tuning dip when he first showed by purely mathematical analysis that the dip might exist. After reading his original manuscript, I suggested hole burning to him in a letter dated 7 July 1962 (available at the online version of this letter).The subject had come up in a lecture I gave, attended by Gordon Gould, on my previous research on hole-burning effects. 1 1. W. R. Bennett Jr, Phys. Rev. 126, 580 (1962). https://doi.org/10.1103/PhysRev.126.580 I had only discussed the effects of running waves in one direction, and Gordon asked, “What about the other running wave in the standing wave?” After drawing a diagram on the blackboard, I realized, as did Gordon, that the presence of the second running wave would result in a dip as the laser was tuned through line center.These holes in the Doppler line approach each other and overlap at the middle. Because the power stimulated in each hole falls within about a natural width of the atoms’ Doppler-shifted frequency, the number of atoms stimulated to emit at line center can decrease by a factor of two. After showing that interpretation to Lamb, I published it later that year in a review paper on gaseous optical masers. 2 2. W. R. Bennett Jr, in Applied Optics Supplement on Optical Masers, Optical Society of America, Washington, DC (1962), pp. 24, 57, 58. During my time at Bell Laboratories, Ross McFarlane and I started an experiment to look for the dip in a helium–neon laser using a much shorter length than the one I had described in reference 1. I was in the process of moving from Bell Labs to Yale University, and McFarlane had to finish the experiment alone. He was the first to observe the tuning dip experimentally, but for several months we withheld publication of a paper reporting the results 3 3. R. A. McFarlane, W. R. Bennett Jr, W. E. Lamb Jr, Appl. Phys. Lett. 2, 189 (1963). https://doi.org/10.1063/1.1753727 until Abraham Szoke and Ali Javan had had a chance to verify the effect at MIT.While waiting for their experiment, I showed that the “mirror-image” hole in the Doppler profile was responsible for the “mode-pushing” effect described in Lamb’s theory and that the hole-burning interpretation led to the same single-mode equations as his third-order theory (with different notation). I presented those results at the Paris Quantum Electronics Conference in February 1963, but the volume containing my paper describing those equations, wasn’t published until a year later. 4 4. W. R. Bennett Jr, in Quantum Electronics: Proceedings of the Third International Congress, Paris, vol. 1, P. Grivet, N. Bloembergen, eds., Columbia U. Press, New York (1964), pp. 441, 450. That work was discussed in much more detail in an article based on lectures that I gave at Brandeis University in the summer of 1969.One final aspect of the role of hole-burning effects in laser spectroscopy that was omitted in the otherwise fine article by Brewer and coauthors is the narrow absorption peaks initially observed in 1968 by Veniamin Chebotayev and his student V. N. Lisitzin at Novosibirsk, Russia, in a laser containing a saturable absorber. 5 5. V. N. Lisitzin, “Experimental Investigations of Gas Lasers with Saturable Absorbers as Resonators,” research thesis, AcademGorodok, Novosibirsk, Russia (1968). The extremely narrow resonant peaks they found were due to hole-burning effects in the Doppler-broadened absorber. Chebotayev and Lisitzin also extended that work to include tuning dips from strong-wave–weak-wave interactions in external absorption cells, a process that anticipated the later, more elegant work by Theodor Hänsch, Marc Levenson, and Arthur Schawlow. 6 6. T. W. Hänsch, M. D. Levenson, A. L. Schawlow, Phys. Rev. Lett. 26, 945 (1971). In those experiments, the strong wave burns a hole in the line that is probed by a weak wave traveling in the opposite direction. Since the absorber can involve a ground-state atom or molecule, much narrower natural widths can be obtained in the absorber experiments than in those with the amplifying medium.REFERENCESSection:ChooseTop of pageREFERENCES <<1. W. R. Bennett Jr, Phys. Rev. 126, 580 (1962). https://doi.org/10.1103/PhysRev.126.580 , Google ScholarCrossref2. W. R. Bennett Jr, in Applied Optics Supplement on Optical Masers, Optical Society of America, Washington, DC (1962), pp. 24, 57, 58. Google Scholar3. R. A. McFarlane, W. R. Bennett Jr, W. E. Lamb Jr, Appl. Phys. Lett. 2, 189 (1963). https://doi.org/10.1063/1.1753727 , Google ScholarCrossref, CAS4. W. R. Bennett Jr, in Quantum Electronics: Proceedings of the Third International Congress, Paris, vol. 1, P. Grivet, N. Bloembergen, eds., Columbia U. Press, New York (1964), pp. 441, 450. Google Scholar5. V. N. Lisitzin, “Experimental Investigations of Gas Lasers with Saturable Absorbers as Resonators,” research thesis, AcademGorodok, Novosibirsk, Russia (1968). Google Scholar6. T. W. Hänsch, M. D. Levenson, A. L. Schawlow, Phys. Rev. Lett. 26, 945 (1971). Google ScholarCrossref© 2008 American Institute of Physics.
When a tuned standing-wave gas laser operates on-resonance, neighboring frequencies experience gain in accordance with the Lamb dip. Bennett1 showed that it is possible for certain of these frequencies also to satisfy the phase-matching condition required by the dispersion relation. In such cases, gas lasers possess three possible steady states—one on-resonance state and two frequency-pushed ones. Bennett suggested that this effect could give rise to bistability and hysteresis in the frequency vs detuning characteristics.
Population pulsations are fluctuations in the population difference (of a two level system) due to the presence of two or more coherent waves interfering in the medium. In this work, we show that population pulsations generated by three waves, a central wave and two mode-locked sidebands, are responsible for both the multiwavelength and the single-wavelength instabilities of single-mode lasers containing homogeneously-broadened media. The role of the population pulsations in establishing these instabilities, however, diminish as the central mode is detuned away from the atomic resonance frequency. For homogeneously-broadened lasers, we find two regions of single-wavelength instability. The first is at line center, for which population pulsations are solely responsible, and the second is off line center where the unsaturated medium provides the required gain and anomalous dispersion. For the case of inhomogeneously-broadened lasers, we show that population pulsations significantly increase the instability range over that predicted by Casperson for single-mode bad-cavity lasers. Both the unidirectional ring and the standing-wave cavities are treated. The Fourier expansion technique, used in this work, for treating three-frequency operation in saturation spectroscopy is shown to be equivalent (in appropriate limits) to the linear stability analysis in laser theory and optical bistability. We also show, in single-sideband saturation spectroscopy, that for long interaction lengths propagation effects can significantly influence the absorption and dispersion coefficients of the medium. Finally, we show that under certain conditions the pronounced splittind effects of the population pulsations develop into regions of intense absorption.