We show that the quantum-limited linewidth of a laser can be enhanced when the polarization eigenmodes of the laser resonator are nonorthogonal. For the theoretical description of this phenomenon we introduce a simple coupled two-mode model. Experimentally, we observed an enhancement of the quantum noise by a factor of 60 in a He-Xe gas laser

Lindberg, Å.M.

Mieremet, A.L.

van der Lee, A.M.

van Druten, N.J.

van Eijkelenborg, M.A.

van Exter, M.P.

Woerdman, J.P.

International Migration, Integration and Social Cohesion online publications

UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)Excess quantum noise due to nonorthogonal polarization modesvan der Lee, A.M.; van Druten, N.J.; Mieremet, A.L.; van Eijkelenborg, M.A.; Lindberg, Å.M.;van Exter, M.P.; Woerdman, J.P.DOI10.1103/PhysRevLett.79.4357Publication date1997Published inPhysical Review LettersLink to publicationCitation for published version (APA):van der Lee, A. M., van Druten, N. J., Mieremet, A. L., van Eijkelenborg, M. A., Lindberg, Å.M., van Exter, M. P., & Woerdman, J. P. (1997). Excess quantum noise due to nonorthogonalpolarization modes. Physical Review Letters, 79(22), 4357-4360.https://doi.org/10.1103/PhysRevLett.79.4357General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:08 Mar 2023VOLUME 79, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 1 DECEMBER 1997ionenontheExcess Quantum Noise due to Nonorthogonal Polarization ModesA. M. van der Lee, N. J. van Druten, A. L. Mieremet, M. A. van Eijkelenborg, Å. M. Lindberg,M. P. van Exter, and J. P. WoerdmanHuygens Laboratory, Leiden University, P.O. Box 9504, Leiden, The Netherlands(Received 7 August 1997)We show that the quantum-limited linewidth of a laser can be enhanced when the polarizateigenmodes of the laser resonator are nonorthogonal. For the theoretical description of this phenomwe introduce a simple coupled two-mode model. Experimentally, we observed an enhancement ofquantum noise by a factor of 60 in a He-Xe gas laser. [S0031-9007(97)04703-0]PACS numbers: 42.50.Lc, 42.55.Lt, 42.60.Daedinstmfrceiseenionedu-0].rmsli-er-e-deydi-mtthen-orsit-itThe coherence of a laser is fundamentally limited bspontaneous emission into the lasing mode. Since thetroduction of the fundamental limit to the laser linewidthby Schawlow and Townes [1] several refinements and etensions have been found [2]. One of the most intriguinmodifications to the original Schawlow-Townes formula ithe so-called excess-noise factor, denoted byK, first pro-posed by Petermann [3] in 1979 for gain-guided laseSiegman [4] showed that excess noise is a general featof open resonators, directly related to the nonorthogonalof the laser eigenmodes in such resonators. More speccally, it was shown that in unstable resonators theK factorcan increase the effect of the spontaneous-emission noin the laser mode by orders of magnitude. Only recentsome of the key issues have received a firm experimenbasis. In particular, a large valuesø330d of K in a hard-edged unstable resonator [5] and resonant behavior ofKas a function of mirror size [6] have been observed. Inrecent paper [7], some of us have speculated that for velarge K factors all spontaneous emission would be chaneled into the laser mode, thus presenting an alternativeproach towards the long-sought thresholdless single-molaser. Therefore, experimental realization of a laser wia very largeK factor is of great importance.So far, only the spatial nonorthogonality of theresonator eigenmodes (transverse and longitudinal) hbeen considered as a source of excess quantum noIn this Letter we consider thepolarization degrees offreedom (in a stable-cavity laser). We show, both theretically and experimentally, that nonorthogonal polarization eigenmodes can lead to a substantial excess-nofactor, which we denote byKpol. Since large polarizationexcess noise arises in a simple two-mode descriptionmay provide further insight into the physical origin oexcess noise. Also, by combining spatial and polarizatiomode nonorthogonality one may hope to obtain evelarger K factors. It is to be expected that within theparaxial approximation the overallK factor of a laserfactorizes in a spatialK and a polarizationK . Thusour work opens the way to even largerK factors, byexploring near-degeneracy of polarization eigenmodesunstable-cavity lasers.0031-9007y97y79(22)y4357(4)$10.00yin-x-gsrs.ureityifi-iselytalaryn-ap-dethasise.o--ise, itfnninThe polarization state of a laser is in general determinby the anisotropies of the passive cavity and of the gamedium. The linear part of these anisotropies is moconveniently described using the Jones vector formalis[8], in which the polarization dynamics takes the form oa coupled two-mode problem:ddtµE1E2∂­ iMµE1E2∂1µL1L2∂, (1)where E1 and E2 are the (complex) amplitudes of theorthogonal basis vectors which are here chosen to bes1ands2 for concreteness, andM is a 2 3 2 Jones matrixdescribing the laser resonator. A Langevin noise souL has been added to take into account the quantum nodue to spontaneous emission.Excess noise now appears in a natural way whEq. (1) is transformed from the (orthogonal)s1, s2basis to the eigenmode basis of the resonator matrixM.If the eigenmodes are nonorthogonal, the transformatleads to enhancement of the strength of the transformLangevin noise into the separate eigenmodes, and to mtual correlations between these two noise sources [9,1The enhanced influence of the noise is expressed in teof the increased diffusion rate of the eigenmode amptudes, by a factorK . As in the case of the transvers[4] and longitudinal [11] excess-noise factor, the polaization excess-noise factorKpol of an eigenmodejeil ofM can generally be expressed in terms of the overlap btween this eigenmode and the corresponding adjoint mojyil, as Ki ­ jkyijeilj22. Here we have employed theDirac bra-ket notation to bring out the complete analogwith the usual expressions for transverse and longitunal excess-noise factors (we have takenj il and jyil tobe normalized to unity, cf. [12]). For a two-mode systethe adjoint modesjyild are easily constructed: the adjoinmode belonging to the first eigenmode corresponds topolarization state that is orthogonal to the second eigemode and vice versa. In fact, the excess-noise factof the two modes are identical, and can be easily rewrten in terms of the eigenmodes only (i.e., without explicreference to the adjoint modes) as© 1997 The American Physical Society 4357VOLUME 79, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 1 DECEMBER 1997tllyh-hearesingwereddesneeftore-a-ss-,i-ryK ­11 2 jke1je2lj2. (2)This equation shows explicitly that excess noise only apears when the eigenmodes are nonorthogonal (otherwke1je2l ­ 0 and K ­ 1), and thatK is highest when thetwo modes are nearly identicalske1je2l ø 1d.A simple way to create nearly identical polarizationeigenmodes is to haveM take the form [13]M ­µV iAiA 2V∂, (3)where 2V is the frequency splitting of thes1 ands2 modes andiA represents a (dissipative) couplingbetween the two. The frequencies and damping ratesthe eigenmodes are given, respectively, by the real aimaginary parts of the eigenvaluesl6 of the matrixM,l6 ­ 6pV2 2 A2 ; 6vy2 . (4)These are shown in Fig. 1. Two regimes can be ditinguished: (i) the lock bandjVj , jAj, where theeigenmodes are linearly polarized, and have the safrequency but different damping, (ii) outside the locband jVj . jAj, where the eigenmodes are ellipticallypolarized and have different frequencies but equal lossWhen, for instance,jVj # jAj, outside the lock band, theeigenmodes become more and more identical; they evofrom s1, s2 (when jVj ¿ jAj) towards the same linearpolarization (whenjVj ­ jAj). As the overlap of theeigenmodes is given byjke1je2lj ­ jAyVj, outside thelock band the polarization excess-noise factor as givenEq. (2) becomesFIG. 1. (a) Frequencies (solid lines) and net gains (dashlines) of the eigenmodes, as a function ofVyA. Thepolarization states of the eigenmodes are also indicated.The polarization excess-noise factorKpol, as derived from thenonorthogonality of the polarization eigenmodes.4358p-iseofnds-mekes.lvebyed(b)Kpol ­V2V2 2 A2. (5)Inside the lock bandKpol is obtained from Eq. (5) by ex-changing the role ofV andA. The resulting polarizationexcess-noise factor is shown in Fig. 1(b). The key poinis that asV ! A the polarization eigenmodes becomenearly identical, and the excess-noise factor diverges.The linear model considered above describes essentiathe situation at threshold; to describe a laser above thresold, the linear analysis needs to be supplemented with tnonlinearities introduced by the gain medium. Althoughthe essential features of the analysis discussed abovepreserved, it turns out that saturation somewhat modifiethe excess-noise characteristics of the laser, dependon the specific case considered. For concreteness,limit the discussion below to the case that is relevant foour experiments; namely, that of a gas laser where ths1 and thes2 modes are weakly coupled by the gainsaturation, leading to linear polarization of the laser fiel[14]. We also assume that the laser is operated outsithe lock bandsjVj . jAjd. The dynamics reduces then tothe evolution of the angle of linear polarizationu, whichis governed by the Adler equation [15]:Ùu ­ V 2 A sins2ud . (6)The polarization angleu will rotate at a frequencyvy2[Eq. (4)]. For jVj ¿ jAj the rotation will be harmonic,but this rotation will become increasingly anharmonic athe lock band is approached. Including the Langevinoise terms of Eq. (1) leads to a Langevin noise sourcin Eq. (6), with a strength~I21y2. This is the sourceof the well-known inverse-power dependence of thdiffusion rate of the phase, that is at the heart othe Schawlow-Townes formula [1,2]. Straightforwardanalysis shows that the excess noise that arises duethe nonorthogonality of the eigenmodes in the lineamodel [Eq. (2)], is mapped onto an enhancement of thsensitivity ofu to noise, due to the anharmonicity impliedby Eq. (6) [16]. We find that the diffusion rate of theangleu is enhanced by a factorKpol ­V2 1 A2y2V2 2 A2. (7)The problem of diffusion of the phase in the Adler equation due to a Langevin noise term has in fact been anlyzed in great detail in a series of papers by Cresseret al.[17], in the context of quantum noise in laser gyros. Inthe limit of weak noise the same result as in Eq. (7) wafound. The difference between Eq. (7) and the excesnoise factor from the linear model, as given by Eq. (5)is a factor1 1 A2y2V2, which is in the order of unity(for the casejVj . jAj which we consider here), so theexcess-noise factor is expected to be only slightly modfied by saturation.We have experimentally verified the polarizationexcess-noise factor predicted above. The setup was veVOLUME 79, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 1 DECEMBER 1997g-ticda-hein-e toisofedetionhealctrahsimilar to that used previously to determine quantumlimited linewidths [15,18]. A He-Xe gas laser wasoperated on line center of the 3.51-mm Xe transition, thecapillary being filled with a He pressure of 1.2 kPa andXe pressure of a few Pa. The discharge was maintainby rf excitation. The capillary was terminated by twoquartz plates with a single-pass transmission of 92each. The 8.6-cm long stable-cavity laser consisteda concave dielectric mirror (32% reflection, radius ocurvature 30 cm) and a flat, highly reflective gold mirroplaced around the capillary. Coils around the laser caviproduced an axial magnetic field of&1 mT, inducing acircular birefringence through the Faraday effect in thgain medium. This controls the detuning of the twomodessVd. The dissipative couplingsiAd of the s1 ands2 modes was produced by an intracavity 2-mm-thicCaF2 plate, which introduces linear dichroism. This platewith its normal at roughly 35± from the resonator axis,created a loss difference between thes- and p-polarizedlight (ø15% per round trip). The output of the laser wassent through a polarizer, detected on a cryogenic InSphoto diode with a bandwidth of 12 MHz, and monitoredon an rf-spectrum analyzer.As mentioned above, the output of the laser is linearlpolarized, due to weak coupling of thes1 ands2 modesby the gain saturation. When operated outside the lobandsjVj . jAjd, the angle of polarization will rotate asdescribed by Eq. (6) at a fundamental frequencyv 2.In the experiment this is detected after the polarizeas an intensity modulation at a fundamental frequencv, leading to spectra as in Fig. 2. For large magnetfields sV ¿ Ad the rotation is harmonic and the spectrumconsists of a single Lorentzian [Fig. 2(a)], while close tothe lock pointsV * Ad the rotation becomes anharmonic[Fig. 2(b)], and higher harmonics appear in the spectrumVery close to the lock point all the harmonics meltogether [Fig. 2(c)] and no information about the firsharmonic in the spectrum can be obtained.In Fig. 3 the fundamental rotation frequencyv asdetermined from the rf spectra is plotted as a functioof the current generating the axial magnetic field. Thsolid line is a fit to the data, from which all the relevanparameters under operating conditions can be determinnamely, the value of the dichroism present (A, typically afew MHz), the proportionality between the applied currenand the Faraday rotation, and an offset in the magnefield caused by stray magnetic fields.The widths of the peaks in the spectra are proportionto the quantum-noise strength [6,7,15,18]. We havmeasured the fundamental laser linewidth (determinefrom the width of the first harmonic in the spectra) aa function of the axial magnetic field [19]. ForV ¿A (orthogonal polarization eigenmodes,Kpol ­ 1) werecover the usual Schawlow-Townes linewidth [2,15,18By dividing all the measured linewidths by the asymptotilinewidth sV ¿ Ad we find the polarization excess-noise-aed%offrtyek,byckryic.ttneted,tticaleds].cFIG. 2. Measured rf polarization-rotation spectra as the manetic field is progressively reduced. (a) For high magnefields the spectrum contains a Lorentzian peak at the funmental frequencyv, with no harmonics visible. (b) As themagnetic field is lowered and the lock point is approached, trotation becomes increasingly anharmonic, and the widthscrease, due to the enhanced quantum noise. (c) Very closthe lock point the harmonics are not separated anymore.factor, Kpol. This is plotted in Fig. 4 as function of theFaraday rotationV. The experimental data clearly showa dramatic increase in the linewidth as the lock pointapproached (an enhancement of more than a factor60), in good agreement with both the linear theory bason mode nonorthogonality [solid line, Eq. (5)], and thenhancement expected on the basis of the Adler equa[dashed line, Eq. (7)]. We cannot distinguish between tquality of the two predictions. Note that the theoreticFIG. 3. Fundamental rotation frequencyv of the polarizationas a function of the currentsIBd generating the axial magneticfield. Points: measured frequencies, as determined from spesuch as in Fig. 2. The solid line is a fit to the data from whicthe linear dichroismsAd and the Faraday rotationsVd can bedetermined (see text).4359VOLUME 79, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 1 DECEMBER 1997n,v..edoineoflir],,.dsinnFIG. 4. Measurement of the polarization excess-noise factowhich shows strong enhancement of the quantum noise cloto the lock point. Points: measuredKpol as a function ofthe Faraday rotationV. The solid curve is the theoreticalexpectation from mode nonorthogonality theory [Eq. (5)], thdashed curve displays the expected enhancement based onAdler equation; see Eq. (7).curves were drawn without using any fit parameter, athe information needed for calculatingKpol was obtainedfrom the fit in Fig. 3.It is surprising that the mode-nonorthogonality theorbased only on the linear model [Eq. (1)], gives suchgood quantitative description of the excess quantum noisdespite the fact that in our experiment the nonlineariimplied by the gain saturation plays an important role ithe polarization behavior.In conclusion, we demonstrated the existencea polarization excess-noise factor, by exploring thnear-degeneracy of polarization eigenstates defined bycircular birefringence and a linear dichroism. Aquantum-linewidth enhancement of up to a factor of 6was observed. It was found that outside the lock banthe saturation only slightly modifies the excess-noisbehavior. It is interesting to note that, as the lock pointapproached from outside the lock band, the excess-nofactor increases dramatically, while the losses of theigenmodes remain constant [see Fig. 1(a)]. This shoexplicitly that there is no direct, simple relation betweeeigenmode losses and excess noise.We acknowledge support of the Stichting voor Fundamenteel Onderzoek der Materie (FOM) which isupported by NWO. We also acknowledge suppofrom the European Union under the TMR ContracNo. ERB4061PL95-1021 (Microlasers and Cavity QEDand the ESPRIT Contract No. 20029 (ACQUIRE).4360r,seethellyae,tynofea0deisiseewsn-srtt)[1] A. L. Schawlow and C. H. Townes, Phys. Rev.112, 1940(1958).[2] M. P. van Exter, S. J. M. Kuppens, and J. P. WoerdmaPhys. Rev. A51, 809 (1995), and references therein.[3] K. Petermann, IEEE J. Quant. Electron.15, 566 (1979).[4] A. E. Siegman, Phys. Rev. A39, 1253 (1989);39, 1264(1989).[5] Y.-J. Cheng, C. G. Fanning, and A. E. Siegman, Phys. ReLett. 77, 627 (1996).[6] M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen,and J. P. Woerdman, Phys. Rev. Lett.77, 4314 (1996).[7] M. A. van Eijkelenborg, M. P. van Exter, and J. P.Woerdman, Phys. Rev. A (to be published).[8] E. Hecht and A. Zajac,Optics(Addison-Wesley, Reading,MA, 1974).[9] H. A. Haus and S. Kawakami, IEEE J. Quant. Electron21, 63 (1985).[10] In the experiment the excess noise factor is determinfrom the spectral width of the beat note between twsimultaneously oscillating modes. One might worryabout the mutual correlations between the Langevnoise sources driving the two modes. However, thescorrelations exist only on short time scales. Becausethe frequency difference of the two modes the mutuacorrelations will die out on time scales larger than thereciprocal frequency difference (see discussion in [4p. 1267).[11] W. A. Hamel and J. P. Woerdman, Phys. Rev. Lett.64,1506 (1990).[12] G. H. C. New, J. Mod. Opt.42, 799 (1995).[13] J. P. Woerdman and R. J. C. Spreeuw, inAnalo-gies in Optics and Micro-Electronics,edited by W.van Haeringen and D. Lenstra (Kluwer AcademicDordrecht, The Netherlands, 1990), pp. 135–150.[14] M. Sargent III, M. O. Scully, and W. E. Lamb, Jr.,LaserPhysics(Addison-Wesley, Reading, MA, 1974).[15] M. A. van Eijkelenborg, C. A. Schrama, and J. PWoerdman, Opt. Commun.119, 97 (1995).[16] M. P. van Exter, N. J. van Druten, A. M. van der Lee, anJ. P. Woerdman (to be published).[17] J. D. Cresseret al., Phys. Rev. A25, 2214 (1982); Phys.Rev. A 25, 2226 (1982); J. D. Cresser, Phys. Rev. A26,398 (1982).[18] Å. M. Lindberg, M. A. van Eijkelenborg, and J. P.Woerdman, IEEE J. Quant. Electron.33, 1767 (1997).[19] In the experiment, the output power of the laser wapurposely decreased asV was increased, using theinverse-power dependence of the quantum linewidth,order to keep the quantum linewidth above the detectiolimit of 10 kHz (set by technical noise). Relatively smallcorrections due to the power-dependences ofNsp [20] andof the Faraday rotation have been accounted for.[20] S. J. M. Kuppenset al., IEEE J. Quant. Electron.32, 383(1996).

English

Excess quantum noise due to nonorthogonal polarization modes

UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)Excess quantum noise due to nonorthogonal polarization modesvan der Lee, A.M.; van Druten, N.J.; Mieremet, A.L.; van Eijkelenborg, M.A.; Lindberg, Å.M.;van Exter, M.P.; Woerdman, J.P.DOI10.1103/PhysRevLett.79.4357Publication date1997Published inPhysical Review LettersLink to publicationCitation for published version (APA):van der Lee, A. M., van Druten, N. J., Mieremet, A. L., van Eijkelenborg, M. A., Lindberg, Å.M., van Exter, M. P., & Woerdman, J. P. (1997). Excess quantum noise due to nonorthogonalpolarization modes. Physical Review Letters, 79(22), 4357-4360.https://doi.org/10.1103/PhysRevLett.79.4357General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:15 Apr 2023VOLUME 79, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 1 DECEMBER 1997ionenontheExcess Quantum Noise due to Nonorthogonal Polarization ModesA. M. van der Lee, N. J. van Druten, A. L. Mieremet, M. A. van Eijkelenborg, Å. M. Lindberg,M. P. van Exter, and J. P. WoerdmanHuygens Laboratory, Leiden University, P.O. Box 9504, Leiden, The Netherlands(Received 7 August 1997)We show that the quantum-limited linewidth of a laser can be enhanced when the polarizateigenmodes of the laser resonator are nonorthogonal. For the theoretical description of this phenomwe introduce a simple coupled two-mode model. Experimentally, we observed an enhancement ofquantum noise by a factor of 60 in a He-Xe gas laser. [S0031-9007(97)04703-0]PACS numbers: 42.50.Lc, 42.55.Lt, 42.60.Daedinstmfrceiseenionedu-0].rmsli-er-e-deydi-mtthen-orsit-itThe coherence of a laser is fundamentally limited bspontaneous emission into the lasing mode. Since thetroduction of the fundamental limit to the laser linewidthby Schawlow and Townes [1] several refinements and etensions have been found [2]. One of the most intriguinmodifications to the original Schawlow-Townes formula ithe so-called excess-noise factor, denoted byK, first pro-posed by Petermann [3] in 1979 for gain-guided laseSiegman [4] showed that excess noise is a general featof open resonators, directly related to the nonorthogonalof the laser eigenmodes in such resonators. More speccally, it was shown that in unstable resonators theK factorcan increase the effect of the spontaneous-emission noin the laser mode by orders of magnitude. Only recentsome of the key issues have received a firm experimenbasis. In particular, a large valuesø330d of K in a hard-edged unstable resonator [5] and resonant behavior ofKas a function of mirror size [6] have been observed. Inrecent paper [7], some of us have speculated that for velarge K factors all spontaneous emission would be chaneled into the laser mode, thus presenting an alternativeproach towards the long-sought thresholdless single-molaser. Therefore, experimental realization of a laser wia very largeK factor is of great importance.So far, only the spatial nonorthogonality of theresonator eigenmodes (transverse and longitudinal) hbeen considered as a source of excess quantum noIn this Letter we consider thepolarization degrees offreedom (in a stable-cavity laser). We show, both theretically and experimentally, that nonorthogonal polarization eigenmodes can lead to a substantial excess-nofactor, which we denote byKpol. Since large polarizationexcess noise arises in a simple two-mode descriptionmay provide further insight into the physical origin oexcess noise. Also, by combining spatial and polarizatiomode nonorthogonality one may hope to obtain evelarger K factors. It is to be expected that within theparaxial approximation the overallK factor of a laserfactorizes in a spatialK and a polarizationK . Thusour work opens the way to even largerK factors, byexploring near-degeneracy of polarization eigenmodesunstable-cavity lasers.0031-9007y97y79(22)y4357(4)$10.00yin-x-gsrs.ureityifi-iselytalaryn-ap-dethasise.o--ise, itfnninThe polarization state of a laser is in general determinby the anisotropies of the passive cavity and of the gamedium. The linear part of these anisotropies is moconveniently described using the Jones vector formalis[8], in which the polarization dynamics takes the form oa coupled two-mode problem:ddtµE1E2∂­ iMµE1E2∂1µL1L2∂, (1)where E1 and E2 are the (complex) amplitudes of theorthogonal basis vectors which are here chosen to bes1ands2 for concreteness, andM is a 2 3 2 Jones matrixdescribing the laser resonator. A Langevin noise souL has been added to take into account the quantum nodue to spontaneous emission.Excess noise now appears in a natural way whEq. (1) is transformed from the (orthogonal)s1, s2basis to the eigenmode basis of the resonator matrixM.If the eigenmodes are nonorthogonal, the transformatleads to enhancement of the strength of the transformLangevin noise into the separate eigenmodes, and to mtual correlations between these two noise sources [9,1The enhanced influence of the noise is expressed in teof the increased diffusion rate of the eigenmode amptudes, by a factorK . As in the case of the transvers[4] and longitudinal [11] excess-noise factor, the polaization excess-noise factorKpol of an eigenmodejeil ofM can generally be expressed in terms of the overlap btween this eigenmode and the corresponding adjoint mojyil, as Ki ­ jkyijeilj22. Here we have employed theDirac bra-ket notation to bring out the complete analogwith the usual expressions for transverse and longitunal excess-noise factors (we have takenj il and jyil tobe normalized to unity, cf. [12]). For a two-mode systethe adjoint modesjyild are easily constructed: the adjoinmode belonging to the first eigenmode corresponds topolarization state that is orthogonal to the second eigemode and vice versa. In fact, the excess-noise factof the two modes are identical, and can be easily rewrten in terms of the eigenmodes only (i.e., without explicreference to the adjoint modes) as© 1997 The American Physical Society 4357VOLUME 79, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 1 DECEMBER 1997tllyh-hearesingwereddesneeftore-a-ss-,i-ryK ­11 2 jke1je2lj2. (2)This equation shows explicitly that excess noise only apears when the eigenmodes are nonorthogonal (otherwke1je2l ­ 0 and K ­ 1), and thatK is highest when thetwo modes are nearly identicalske1je2l ø 1d.A simple way to create nearly identical polarizationeigenmodes is to haveM take the form [13]M ­µV iAiA 2V∂, (3)where 2V is the frequency splitting of thes1 ands2 modes andiA represents a (dissipative) couplingbetween the two. The frequencies and damping ratesthe eigenmodes are given, respectively, by the real aimaginary parts of the eigenvaluesl6 of the matrixM,l6 ­ 6pV2 2 A2 ; 6vy2 . (4)These are shown in Fig. 1. Two regimes can be ditinguished: (i) the lock bandjVj , jAj, where theeigenmodes are linearly polarized, and have the safrequency but different damping, (ii) outside the locband jVj . jAj, where the eigenmodes are ellipticallypolarized and have different frequencies but equal lossWhen, for instance,jVj # jAj, outside the lock band, theeigenmodes become more and more identical; they evofrom s1, s2 (when jVj ¿ jAj) towards the same linearpolarization (whenjVj ­ jAj). As the overlap of theeigenmodes is given byjke1je2lj ­ jAyVj, outside thelock band the polarization excess-noise factor as givenEq. (2) becomesFIG. 1. (a) Frequencies (solid lines) and net gains (dashlines) of the eigenmodes, as a function ofVyA. Thepolarization states of the eigenmodes are also indicated.The polarization excess-noise factorKpol, as derived from thenonorthogonality of the polarization eigenmodes.4358p-iseofnds-mekes.lvebyed(b)Kpol ­V2V2 2 A2. (5)Inside the lock bandKpol is obtained from Eq. (5) by ex-changing the role ofV andA. The resulting polarizationexcess-noise factor is shown in Fig. 1(b). The key poinis that asV ! A the polarization eigenmodes becomenearly identical, and the excess-noise factor diverges.The linear model considered above describes essentiathe situation at threshold; to describe a laser above thresold, the linear analysis needs to be supplemented with tnonlinearities introduced by the gain medium. Althoughthe essential features of the analysis discussed abovepreserved, it turns out that saturation somewhat modifiethe excess-noise characteristics of the laser, dependon the specific case considered. For concreteness,limit the discussion below to the case that is relevant foour experiments; namely, that of a gas laser where ths1 and thes2 modes are weakly coupled by the gainsaturation, leading to linear polarization of the laser fiel[14]. We also assume that the laser is operated outsithe lock bandsjVj . jAjd. The dynamics reduces then tothe evolution of the angle of linear polarizationu, whichis governed by the Adler equation [15]:Ùu ­ V 2 A sins2ud . (6)The polarization angleu will rotate at a frequencyvy2[Eq. (4)]. For jVj ¿ jAj the rotation will be harmonic,but this rotation will become increasingly anharmonic athe lock band is approached. Including the Langevinoise terms of Eq. (1) leads to a Langevin noise sourcin Eq. (6), with a strength~I21y2. This is the sourceof the well-known inverse-power dependence of thdiffusion rate of the phase, that is at the heart othe Schawlow-Townes formula [1,2]. Straightforwardanalysis shows that the excess noise that arises duethe nonorthogonality of the eigenmodes in the lineamodel [Eq. (2)], is mapped onto an enhancement of thsensitivity ofu to noise, due to the anharmonicity impliedby Eq. (6) [16]. We find that the diffusion rate of theangleu is enhanced by a factorKpol ­V2 1 A2y2V2 2 A2. (7)The problem of diffusion of the phase in the Adler equation due to a Langevin noise term has in fact been anlyzed in great detail in a series of papers by Cresseret al.[17], in the context of quantum noise in laser gyros. Inthe limit of weak noise the same result as in Eq. (7) wafound. The difference between Eq. (7) and the excesnoise factor from the linear model, as given by Eq. (5)is a factor1 1 A2y2V2, which is in the order of unity(for the casejVj . jAj which we consider here), so theexcess-noise factor is expected to be only slightly modfied by saturation.We have experimentally verified the polarizationexcess-noise factor predicted above. The setup was veVOLUME 79, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 1 DECEMBER 1997g-ticda-hein-e toisofedetionhealctrahsimilar to that used previously to determine quantumlimited linewidths [15,18]. A He-Xe gas laser wasoperated on line center of the 3.51-mm Xe transition, thecapillary being filled with a He pressure of 1.2 kPa andXe pressure of a few Pa. The discharge was maintainby rf excitation. The capillary was terminated by twoquartz plates with a single-pass transmission of 92each. The 8.6-cm long stable-cavity laser consisteda concave dielectric mirror (32% reflection, radius ocurvature 30 cm) and a flat, highly reflective gold mirroplaced around the capillary. Coils around the laser caviproduced an axial magnetic field of&1 mT, inducing acircular birefringence through the Faraday effect in thgain medium. This controls the detuning of the twomodessVd. The dissipative couplingsiAd of the s1 ands2 modes was produced by an intracavity 2-mm-thicCaF2 plate, which introduces linear dichroism. This platewith its normal at roughly 35± from the resonator axis,created a loss difference between thes- and p-polarizedlight (ø15% per round trip). The output of the laser wassent through a polarizer, detected on a cryogenic InSphoto diode with a bandwidth of 12 MHz, and monitoredon an rf-spectrum analyzer.As mentioned above, the output of the laser is linearlpolarized, due to weak coupling of thes1 ands2 modesby the gain saturation. When operated outside the lobandsjVj . jAjd, the angle of polarization will rotate asdescribed by Eq. (6) at a fundamental frequencyv 2.In the experiment this is detected after the polarizeas an intensity modulation at a fundamental frequencv, leading to spectra as in Fig. 2. For large magnetfields sV ¿ Ad the rotation is harmonic and the spectrumconsists of a single Lorentzian [Fig. 2(a)], while close tothe lock pointsV * Ad the rotation becomes anharmonic[Fig. 2(b)], and higher harmonics appear in the spectrumVery close to the lock point all the harmonics meltogether [Fig. 2(c)] and no information about the firsharmonic in the spectrum can be obtained.In Fig. 3 the fundamental rotation frequencyv asdetermined from the rf spectra is plotted as a functioof the current generating the axial magnetic field. Thsolid line is a fit to the data, from which all the relevanparameters under operating conditions can be determinnamely, the value of the dichroism present (A, typically afew MHz), the proportionality between the applied currenand the Faraday rotation, and an offset in the magnefield caused by stray magnetic fields.The widths of the peaks in the spectra are proportionto the quantum-noise strength [6,7,15,18]. We havmeasured the fundamental laser linewidth (determinefrom the width of the first harmonic in the spectra) aa function of the axial magnetic field [19]. ForV ¿A (orthogonal polarization eigenmodes,Kpol ­ 1) werecover the usual Schawlow-Townes linewidth [2,15,18By dividing all the measured linewidths by the asymptotilinewidth sV ¿ Ad we find the polarization excess-noise-aed%offrtyek,byckryic.ttneted,tticaleds].cFIG. 2. Measured rf polarization-rotation spectra as the manetic field is progressively reduced. (a) For high magnefields the spectrum contains a Lorentzian peak at the funmental frequencyv, with no harmonics visible. (b) As themagnetic field is lowered and the lock point is approached, trotation becomes increasingly anharmonic, and the widthscrease, due to the enhanced quantum noise. (c) Very closthe lock point the harmonics are not separated anymore.factor, Kpol. This is plotted in Fig. 4 as function of theFaraday rotationV. The experimental data clearly showa dramatic increase in the linewidth as the lock pointapproached (an enhancement of more than a factor60), in good agreement with both the linear theory bason mode nonorthogonality [solid line, Eq. (5)], and thenhancement expected on the basis of the Adler equa[dashed line, Eq. (7)]. We cannot distinguish between tquality of the two predictions. Note that the theoreticFIG. 3. Fundamental rotation frequencyv of the polarizationas a function of the currentsIBd generating the axial magneticfield. Points: measured frequencies, as determined from spesuch as in Fig. 2. The solid line is a fit to the data from whicthe linear dichroismsAd and the Faraday rotationsVd can bedetermined (see text).4359VOLUME 79, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 1 DECEMBER 1997n,v..edoineoflir],,.dsinnFIG. 4. Measurement of the polarization excess-noise factowhich shows strong enhancement of the quantum noise cloto the lock point. Points: measuredKpol as a function ofthe Faraday rotationV. The solid curve is the theoreticalexpectation from mode nonorthogonality theory [Eq. (5)], thdashed curve displays the expected enhancement based onAdler equation; see Eq. (7).curves were drawn without using any fit parameter, athe information needed for calculatingKpol was obtainedfrom the fit in Fig. 3.It is surprising that the mode-nonorthogonality theorbased only on the linear model [Eq. (1)], gives suchgood quantitative description of the excess quantum noisdespite the fact that in our experiment the nonlineariimplied by the gain saturation plays an important role ithe polarization behavior.In conclusion, we demonstrated the existencea polarization excess-noise factor, by exploring thnear-degeneracy of polarization eigenstates defined bycircular birefringence and a linear dichroism. Aquantum-linewidth enhancement of up to a factor of 6was observed. It was found that outside the lock banthe saturation only slightly modifies the excess-noisbehavior. It is interesting to note that, as the lock pointapproached from outside the lock band, the excess-nofactor increases dramatically, while the losses of theigenmodes remain constant [see Fig. 1(a)]. This shoexplicitly that there is no direct, simple relation betweeeigenmode losses and excess noise.We acknowledge support of the Stichting voor Fundamenteel Onderzoek der Materie (FOM) which isupported by NWO. We also acknowledge suppofrom the European Union under the TMR ContracNo. ERB4061PL95-1021 (Microlasers and Cavity QEDand the ESPRIT Contract No. 20029 (ACQUIRE).4360r,seethellyae,tynofea0deisiseewsn-srtt)[1] A. L. Schawlow and C. H. Townes, Phys. Rev.112, 1940(1958).[2] M. P. van Exter, S. J. M. Kuppens, and J. P. WoerdmaPhys. Rev. A51, 809 (1995), and references therein.[3] K. Petermann, IEEE J. Quant. Electron.15, 566 (1979).[4] A. E. Siegman, Phys. Rev. A39, 1253 (1989);39, 1264(1989).[5] Y.-J. Cheng, C. G. Fanning, and A. E. Siegman, Phys. ReLett. 77, 627 (1996).[6] M. A. van Eijkelenborg, Å. M. Lindberg, M. S. Thijssen,and J. P. Woerdman, Phys. Rev. Lett.77, 4314 (1996).[7] M. A. van Eijkelenborg, M. P. van Exter, and J. P.Woerdman, Phys. Rev. A (to be published).[8] E. Hecht and A. Zajac,Optics(Addison-Wesley, Reading,MA, 1974).[9] H. A. Haus and S. Kawakami, IEEE J. Quant. Electron21, 63 (1985).[10] In the experiment the excess noise factor is determinfrom the spectral width of the beat note between twsimultaneously oscillating modes. One might worryabout the mutual correlations between the Langevnoise sources driving the two modes. However, thescorrelations exist only on short time scales. Becausethe frequency difference of the two modes the mutuacorrelations will die out on time scales larger than thereciprocal frequency difference (see discussion in [4p. 1267).[11] W. A. Hamel and J. P. Woerdman, Phys. Rev. Lett.64,1506 (1990).[12] G. H. C. New, J. Mod. Opt.42, 799 (1995).[13] J. P. Woerdman and R. J. C. Spreeuw, inAnalo-gies in Optics and Micro-Electronics,edited by W.van Haeringen and D. Lenstra (Kluwer AcademicDordrecht, The Netherlands, 1990), pp. 135–150.[14] M. Sargent III, M. O. Scully, and W. E. Lamb, Jr.,LaserPhysics(Addison-Wesley, Reading, MA, 1974).[15] M. A. van Eijkelenborg, C. A. Schrama, and J. PWoerdman, Opt. Commun.119, 97 (1995).[16] M. P. van Exter, N. J. van Druten, A. M. van der Lee, anJ. P. Woerdman (to be published).[17] J. D. Cresseret al., Phys. Rev. A25, 2214 (1982); Phys.Rev. A 25, 2226 (1982); J. D. Cresser, Phys. Rev. A26,398 (1982).[18] Å. M. Lindberg, M. A. van Eijkelenborg, and J. P.Woerdman, IEEE J. Quant. Electron.33, 1767 (1997).[19] In the experiment, the output power of the laser wapurposely decreased asV was increased, using theinverse-power dependence of the quantum linewidth,order to keep the quantum linewidth above the detectiolimit of 10 kHz (set by technical noise). Relatively smallcorrections due to the power-dependences ofNsp [20] andof the Faraday rotation have been accounted for.[20] S. J. M. Kuppenset al., IEEE J. Quant. Electron.32, 383(1996).

Search4Dev

UvA-DARE

NARCIS 

Excess quantum noise due to nonorthogonal polarization modes

Abstract

Similar works

Full text

Available Versions

International Migration, Integration and Social Cohesion online publications

International Migration, Integration and Social Cohesion online publications

Search4Dev

UvA-DARE

NARCIS