Can amplified spontaneous emission produce intense laser guide stars for adaptive optics?
aa r X i v : . [ a s t r o - ph . I M ] D ec Can amplified spontaneous emission produceintense laser guide stars for adaptive optics? P AUL H ICKSON , J OSCHUA H ELLEMEIER , AND R UI Y ANG Department of Physics and Astronomy, University of British Columbia, 6224 Agriculture Road,Vancouver BC, V6T 1Z1,Canada School of Physics and Astronomy, Yunnan University, South Section, East Outer Ring Road, ChenggongDistrict, Kunming, 650500, China * [email protected] Abstract:
Adaptive optics (AO) is a key technology for ground-based optical and infraredastronomy, providing high angular resolution and sensitivity. AO systems employing laserguide stars (LGS) can achieve high sky coverage, but their performance is limited by LGSreturn flux. We examine the potential of two new approaches that might produce high-intensityatmospheric laser beacons. Amplified spontaneous emission could potentially boost the intensityof beacons produced by conventional resonant excitation of atomic or molecular species in theupper atmosphere. This requires the production of a population inversion in an electronictransition that is optically-thick to stimulated emission. Potential excitation mechanisms includecontinuous wave pumping, pulsed excitation and plasma generation. Alternatively, a high-powerfemtosecond pulsed laser could produce a white-light supercontinuum high in the atmosphere.The broad-band emission from such a source could also facilitate the sensing of the tilt componentof atmospheric turbulence. © 2020 Optical Society of America
1. Introduction
Adaptive optics (AO) is an important technology employed by many ground-based telescopes tocompensate atmospheric turbulence, thereby greatly improving the resolution of astronomicalimages [1]. Many AO systems employ laser guide star (LGS) beacons that allow sensing of theturbulence in any desired direction, increasing sky coverage [2]. Multiple beacons can be used toimprove performance and increase the size of the corrected field of view [3–6]. Large telescopestypically employ resonant LGS systems in which mesospheric atoms are excited, typically the 𝐷 transition of neutral sodium [7].One factor limiting the performance of such AO systems is the low brightness of the LGS.Much research has been devoted to optimizing the return flux of sodium LGS [8, 9], and itappears that the limit of this technology is being approached. This motivates us to look foralternative ways to create LGS.One interesting possibility is to induce amplified spontaneous emission (ASE) [10] in orderto produce very bright LGS. Laser pumping of an abundant atmospheric atomic or molecularspecies could in principle produce a population inversion that would amplify radiation generatedby spontaneous emission, as it propagates back towards the telescope. ASE in sodium vapourhas been demonstrated in laboratory experiments [11].A second possible new approach is to use a high-power femtosecond pulsed laser to generatewhite-light supercontinuum (SC) emission high in the atmosphere [12, 13]. This might helpaddress a limitation of conventional monochromatic laser AO in which the tilt component ofatmospheric turbulence cannot be sensed because the deflection of downward propagating lightfrom the LGS is equal and opposite to the deflection of the upward propagating radiation thatcreates the beacon. This degeneracy is normally broken by observations of stars in the field ofview, but such stars are generally very faint, limiting the accuracy that can be achieved. Thereation of a polychromatic LGS, which emits light at two or more widely-separated wavelengths,has been proposed to solve this problem [14–16], but has not been demonstrated in practice.In this paper we examine the physical processes of ASE and SC with the aim of assessingthe feasibility of applying these effects in astronomical adaptive optics, and determining theconditions that would be required to produce useful AO beacons.
2. Amplified spontaneous emission
The essential physics of ASE can be captured by consideration of an atomic or molecular speciesthat has a two-level transition, placed in a radiation field of specific energy density 𝜌 𝜈 ( 𝜈 ) . Let 𝜈 be the central frequency of the transition. If non-radiative excitation and de-excitation can beignored, the average fraction 𝑥 of atoms in the upper states is described by the rate equation 𝑑𝑥𝑑𝑡 = − 𝑥 𝐴 − 𝜌 𝜈 [ 𝑥𝐵 − ( − 𝑥 ) 𝐵 ] = − 𝐴 (cid:26) 𝑥 + 𝜆 𝜌 𝜈 𝜋ℎ (cid:20) 𝑥 − 𝑔 𝑔 ( − 𝑥 ) (cid:21) (cid:27) , (1)where 𝐴 , 𝐵 and 𝐵 are the Einstein coefficients for spontaneous and induced emission andabsorption [10]. In obtaining the second line of Eq (1), we have used the usual relations betweenthe Einstein coefficient, 𝑔 𝐵 = 𝑔 𝐵 and 𝐵 = 𝐴 𝜆 / 𝜋ℎ , where 𝜆 = 𝑐 / 𝜈 is the wavelengthand 𝑔 and 𝑔 are the statistical weights of the lower and upper states respectively.In the limit of high intensity illumination, the excitation fraction reaches an equilibrium value 𝑥 𝑒 , that can be found by equating the LHS of Eq. (1) to zero, 𝑥 𝑒 = 𝑔 /( 𝑔 + 𝑔 ) . (2)The specific intensity 𝐼 𝜈 of radiation propagating through the medium is described by theequation of radiative transfer [17], 𝑑𝐼 𝜈 𝑑𝑠 = − 𝜅𝐼 𝜈 + 𝑗 𝜈 . (3)Here 𝜅 is the absorption coefficient, 𝑗 𝜈 is the emission coefficient, and 𝑠 is distance measuredalong the propagation path. These coefficients are related to the atomic number density 𝑛 ,excitation fraction 𝑥 , and the Einstein coefficients by 𝑗 𝜈 = ℎ𝜈 𝜋 𝐴 𝑥𝑛𝜑, (4) 𝜅 = ℎ𝜈𝑐 [( − 𝑥 ) 𝐵 − 𝑥𝐵 ] 𝑛𝜑 = 𝜆 𝜋 𝑦 𝐴 𝑛𝜑. (5)Here 𝜑 ( 𝜈 ) is the line profile, normalized to have unit integral, and 𝑦 = 𝑥 𝑒 − 𝑥 − 𝑥 𝑒 = 𝑔 𝑔 ( − 𝑥 ) − 𝑥. (6)If the excitation fraction can be increased above the equilibrium value, ( 𝑥 > 𝑥 𝑒 ), producinga population inversion, the absorption coefficient becomes negative and the intensity of propa-gating radiation grows until the radiation escapes from the medium. If the excited region is along narrow cylinder, as for excitation by a pump laser launched from a telescope, forward andbackward propagating beams would develop. The intensity of these beams could in principle bevery large, ultimately limited only by the power absorbed by the medium from the pump laser. . LGS return flux The flux of radiation returning to the telescope depends on the atomic parameters, columndensity, and the nature of the excitation. For emission generated within the medium, Eq (3) hasthe solution 𝐼 𝜈 ( 𝜏 ) = 𝑒 − 𝜏 ∫ 𝜏 𝑆 𝜈 ( 𝑢 ) 𝑒 𝑢 𝑑𝑢, (7)where 𝜏 = ∫ 𝜅𝑑𝑠, (8)is the optical depth and 𝑆 𝜈 = 𝑗 𝜈 𝜅 = − ℎ𝜈 𝑐 𝑥𝑦 . (9)is the source function. If the excitation fraction 𝑥 is constant within the emitting region, 𝑆 𝜈 isconstant and can be taken outside the integral, which is then easily evaluated. The result is 𝐼 𝜈 = ℎ𝜈 𝑐 𝑥𝑦 ( − 𝑒 − 𝜏 ) . (10)The optical depth becomes 𝜏 = 𝜆 𝜋 𝑦𝑁 𝐴 𝜑, (11)where 𝑁 = ∫ ∞ 𝑛𝑑𝑠 (12)is the column density of the atomic or molecular species.Photons arriving at the telescope will be confined to a solid angle Ω ≃ A/ 𝑧 , where A isthe transverse area of the emitting region and 𝑧 is the line-of-sight distance to its center. Thephoton flux (photons s − m − ) is found by dividing the intensity by the photon energy ℎ𝜈 andintegrating over solid angle and frequency. Thus, Φ ≃ Ω ℎ𝜈 ∫ ∞ 𝐼 𝜈 𝑑𝜈 ≃ A ℎ𝜈𝑧 ∫ ∞ 𝐼 𝜈 𝑑𝜈. (13)Substituting Eqs (10) and (11) into Eq. (13) and setting 𝑥 =
1, we obtain
Φ = − Ω 𝑐 𝑥𝑦 ∫ ∞ (cid:26) exp (cid:20) − 𝑐 𝜋𝜈 𝑦𝑁 𝐴 𝜑 ( 𝜈 ) (cid:21) − (cid:27) 𝜈 𝑑𝜈. (14)The line profile is a symmetric function that is sharply peaked about the transition frequency 𝜈 ,so little error is introduced by replacing 𝜈 by 𝜈 in the coefficient of 𝜑 . We can also change thevariable of integration to 𝑞 = 𝜈 − 𝜈 and make use of use the symmetry to obtain Φ = − Ω 𝜆 𝑥𝑦 ∫ ∞ (cid:2) 𝑒 − 𝜏 𝜁 − (cid:3) 𝑑𝑞. (15)where 𝜁 ( 𝑞 ) = 𝜑 ( 𝑞 )/ 𝜑 , 𝜑 ≡ 𝜑 ( 𝜈 ) and 𝜏 = 𝜆 𝜋 𝑦𝑁 𝐴 𝜑 (16)is the optical depth at the line center.e start by expanding the exponential and integrating term by term, Φ = − Ω 𝜆 𝑥𝑦 ∞ Õ 𝑘 = (− 𝜏 ) 𝑘 𝑘 ! ∫ ∞ 𝜁 𝑘 𝑑𝑞. (17)For a Lorentzian line profile, of frequency half-width 𝑤 (half-width at half maximum intensity), 𝜁 𝐿 = [ + ( 𝑞 / 𝑤 ) ] − , (18) 𝜑 = /( 𝜋𝑤 ) . (19)Substituting Eq. (18) into Eq. (17), we obtain Φ = − Ω 𝜆 𝑥𝑦 ∞ Õ 𝑘 = (− 𝜏 ) 𝑘 𝑘 ! ∫ ∞ [ + ( 𝑞 / 𝑤 ) ] − 𝑘 𝑑𝑞. (20)The integral is a Mellin transform and can be expressed in terms of gamma functions Γ ( 𝑥 ) , Φ = − Ω 𝜆 𝑥𝑦 ∞ Õ 𝑘 = (− 𝜏 ) 𝑘 𝑘 ! Γ ( / ) Γ ( 𝑘 − / ) Γ ( 𝑘 ) . (21)Recalling that Γ ( / ) = √ 𝜋 and 𝑘 ! = Γ ( 𝑘 + ) , we can express this in terms of a generalizedhypergeometric function 𝑝 𝐹 𝑞 . Making the substitution 𝑘 → 𝑘 + Φ = √ 𝜋𝜏 𝑤 Ω 𝜆 𝑥𝑦 ∞ Õ 𝑘 = Γ ( 𝑘 + / ) Γ ( 𝑘 + ) (− 𝜏 ) 𝑘 𝑘 ! = 𝜋𝜏 𝑤 Ω 𝜆 𝑥𝑦 𝐹 ( /
2; 2; − 𝜏 ) . (22)Substituting from Eqs. (16) and (19) and simplifying, we obtain Φ = − A 𝜋𝑧 𝑥𝑁 𝐴
21 1 𝐹 ( /
2; 2; − 𝜏 ) , (23)This extends previous work by Milonni and Eberly, who give an approximate solution [10].For the case of complete excitation (all atoms in the upper state), 𝑦 = −
1, and there is anear-exponential dependence of the flux on the magnitude of the optical depth. The first fewterms of the hypergeometric series are, (setting 𝑦 = − Φ = 𝑥𝑁 A 𝐴 𝜋𝑧 (cid:20) − 𝜏 + 𝜏 + · · · (cid:21) . (24)The numerator of the fraction is the total number of spontaneous transitions per second from allexcited atoms. The first term in the series can thus be recognized as the flux from spontaneousemission alone, and subsequent terms generate the exponential growth with optical depth. It isclear from this that if the transition is optically-thin ( | 𝜏 | ≪ ) there is essentially no ASE gain .It is convenient to express the optical depth in terms of the cross section 𝜎 for stimulatedemission, defined by 𝜎 ( 𝜈 ) = ℎ𝜈𝑐 𝐵 𝜑 ( 𝜈 ) = 𝜆 𝜋 𝐴 𝜑 ( 𝜈 ) . (25)With this definition, the optical depth at the line centre becomes 𝜏 = 𝑦𝑁𝜎 ( 𝜈 ) , (26)e see that the critical parameters that determine the optical depth are the excitation fraction,column density and transition cross section .A similar analysis can be done for a Gaussian line profile, 𝜁 = exp [− ln 2 ( 𝑞 / 𝑤 ) ] , (27) 𝜑 = p ln 2 /( 𝜋𝑤 ) (28)In that case, the flux is given by Φ = Ω 𝜋 𝑥𝑁 𝐴 𝜉 (− 𝜏 ) , (29) ≃ 𝑥𝑁 A 𝐴 𝜋𝑧 (cid:20) − √ 𝜏 + √ 𝜏 + · · · (cid:21) , (30)The function 𝜉 ( 𝑡 ) ≡ ∞ Õ 𝑘 = 𝑡 𝑘 ( 𝑘 + ) / ( 𝑘 + ) ! (31)approaches an exponential as 𝑡 → ∞ .
4. Excitation mechanisms
A prerequisite to achieving ASE is a mechanism that generates a population inversion in themedium.
A possible method of excitation is to use a continuous wave (CW) laser to pump atoms to theupper state. In order to produce a population inversion, one needs at least three transitions thatinclude a metastable level. This level is populated by spontaneous transitions from a higherlevel that is pumped by the laser. A difficulty with this approach is that the metastable level willnecessarily have a comparatively-low cross section, as shown by Eq. (25). Thus, the opticaldepth will be small, unless the column density is very large.A second problem is the relatively-narrow linewidth of CW lasers, typically on the order of10 Hz. This is roughly 1% of the Doppler width of ∼ A second possible approach is coherent excitation in which a pulsed laser is used to induceRabi oscillations in the atomic transition. For a 𝜋 -pulse, the product of the laser irradiance 𝐹 and pulse length 𝜏 𝑝 is such that one half of a Rabi oscillation occurs, inverting the population.Atoms that are in the lower state are transferred to the upper state and vice versa. The conditionfor this is [18] 𝐹𝜏 𝑝 = ℏ 𝑐𝜋 /( 𝜆 𝐴 ) . (32)For an optical transition of a typical strong resonance line, and a 100 ps pulse length, anirradiance on the order of 10 W 𝑚 − is required. For a beam area of 1 m , this corresponds toa pulse energy of 1 mJ, which is readily achievable.A pulsed laser delivering a train of 𝜋 -pulses could repeatedly excite the atomic population,creating a population inversion. However, to support ASE, this population inversion must bemaintained for a time comparable to the propagation time through the mesosphere, on the orderof 10 − s. This puts an upper limit on the transition rate of 𝐴 . . That results in a very-lowoptical depth for all mesospheric metals [19] .3. High-power pulsed lasers A third approach is offered by high-power femtosecond pulsed lasers. Laser pulses can induceionization of atmospheric atoms and molecules if the electric field in the pulse is comparable tothe Coulomb field of the nucleus at the atomic radius [20, 21]. This occurs at an irradiance of 𝐹 ∼ W m − . If the laser pulse is pre-chirped (time-dependent frequency) it can be intensifiedby atmospheric dispersion as it propagates, producing a plasma at a chosen distance [12]. Thedesired excitation state might then be achieved during recombination.
5. Supercontinuum emission
The advent of terrawatt femtosecond lasers offers intriguing new possibilities for adaptive optics.A ubiquitous feature of the interaction of very intense pulses with continuous media, includinggases, is the generation of white-light “supercontinuum” (SC) emission [22, 23]. The exactnature of this radiation is not well understood, but likely arises from a combination of nonlineareffects [24]. In the atmosphere, filaments of emission have been generated that can extend forseveral km [12,25,26]. Emission from these filaments is found to be highly directional, peakingin the forward and backscattered direction. The intensity of the backward-propagating beam isfound to be enhanced compared to the predicted linear Rayleigh-Mie scattering theory. Thisis presumed to be a result of non-linear reflection arising from longitudinal index-of-refractionvariations produced by Kerr and plasma effects [27].Remote SC filaments can be generated by chirping the transmitted pulses. Simulationsindicate that pulses of sufficient intensity and coherence can be formed at altitudes as high as 20km [13, 28, 29].The white-light emission observed in SC radiation is believed to result from self-phasemodulation (SPM) due to nonlinear interaction with the medium. SPM occurs when the pulseinduces a change in the index of refraction in the medium due to the optical Kerr effect.This induces a chirp in the pulse, broadening its frequency spectrum. The degree of spectralbroadening increases with increasing laser power. Spectra of atmospheric SC filaments showemission that extends over the entire visible range from the ultraviolet to the infrared [12, 28].
6. Discussion and conclusions
We have estimated the photon flux that could in principle be provided by ASE, if a suitable atomicor molecular population, transition, and excitation mechanism can be found. We conclude thatno significant ASE gain can arise if the transition is optically-thin. For ASE, the magnitude ofthe optical depth must exceed unity, ideally by an order of magnitude. The optical depth dependsonly on the excitation fraction, the column density and the transition cross section for stimulatedemission. The metallic species present in the mesosphere do not have sufficient column densityto support ASE [19].High-power pulsed lasers that can create a plasma filament and superradiance in the atmo-sphere may provide an opportunity to realize high-intensity white-light laser guide stars. Iffeasible, this would allow sensing of atmospheric tilt, in addition to higher-order wavefrontdistortion. A potential drawback is the relatively low altitude of the filaments that have so farbeen produced, which increases the “cone effect”. However, this can be mitigated by the use ofmultiple laser guide stars.
Acknowledgements
PH acknowledges financial support from the Natural Sciences and Engineering Research Coun-cil of Canada. He thanks NAOC for hospitality during a sabbatical visit made possible byfinancial support from the Chinese Academy of Sciences, via the CAS Presidents Internationalellowship Initiative, 2017VMA0013. RY acknowledges financial support by Yunnan ProvincialDepartment of Education for her visiting scholar position at UBC.
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