PHYSICAL REVIEW A 84, 013838 (2011)

Measurement of birefringence inside a filament

Shuai Yuan,1,3,* Tie-Jun Wang,1 Olga Kosareva,2 Nikolay Panov,2 Vladimir Makarov,2 Heping Zeng,3 and See Leang Chin1

1Centre d’Optique, Photonique et Laser (COPL) et Departement de physique, de genie physique et d’optique,Universite Laval, Quebec, Quebec G1V 0A6, Canada

2International Laser Center and Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia3State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062, People’s Republic of China

(Received 22 March 2011; published 29 July 2011)

We quantified the ultrafast birefringence induced in the filament in an atomic gas by measuring the filament-induced polarization rotation of a probe pulse. Based on the dephasing of the probe’s orthogonal polarizationcomponents in argon, the experiment was done at 1 atm by copropagating a linearly polarized 400-nm probepulse with an 800-nm pump pulse which generated the filament. The probe’s elliptical polarization states wereshown under various initial pump-probe polarization schemes. These states were verified by comparing thefilament-induced probe polarization rotation angle and the ellipticity of the probe polarization.

DOI: 10.1103/PhysRevA.84.013838 PACS number(s): 42.65.Re, 52.38.Hb, 42.25.Lc

I. INTRODUCTION

Self-guided propagation of an ultrashort intense laser pulsein air was demonstrated to induce a filamentation chan-nel [1,2]. Many interesting phenomena have been observed,such as self-spatial filtering [3], self-compression [4], intensityclamping [5], molecular alignment [6–8], supercontinuumgeneration [9], and so on, providing unique capabilitiesfor applications such as remote sensing [10–13], lightningcontrol [14], or high harmonic [15], THz [16–20], and far-infrared pulse [21] generation.

A high-intensity laser field will induce birefringence inthe initially isotropic optical medium which can change thepolarization of a probe pulse. The linearly polarized probepulse becomes elliptically polarized; its major axis rotatesin the course of propagation [22]. Recently, it was provedby Bejot et al. [23] that filament-induced symmetry breakingin argon gas can act as a “gaseous half-wave plate” for theprobe. Later Marceau et al. [24], Chen et al. [25], and Calegariet al. [26] demonstrated that filament in molecular gases canspatially separate the orthogonal polarizations of a probe pulsewith the parallel-to-pump polarization component guidedalong the propagation axis, and the perpendicular-to-pumppolarization component defocused outside. More recently,based on the vectorial model [27], the key role of the probepulse cross focusing was shown to be the reason for thepolarization ellipse rotation.

In this paper, we provide a technique to quantify ultrafastbirefringence by measuring the rotation angle of the major axisof the probe’s elliptical polarization induced by cross-phasemodulation (XPM). Since no special time-resolved detectionis required, this simple technique can in principle be appliedto any wavelengths in transparent atomic gases and couldeven be extended to molecular gases. Unlike previous ultrafastbirefringence measurements done by measuring the spectralmodulation [28], or through a time domain treatment ofpump-probe experiments [29,30], by using this technique,we can directly give the probe’s elliptical polarization statesafter the filament. In terms of understanding the physics, we

*shuai.yuan.1@ulaval.ca, ye zoom@126.com

demonstrate that the filament-induced polarization rotationangle of the probe is related to the ellipticity of the probepolarization. That means at a certain probe polarizationrotation angle, it will have a fixed ellipticity of the probepolarization. This could happen both after and within thefilament region. It might be beneficial to verify the polarizationstates of the probe from their experimental results or theoreticalcalculations.

II. EXPERIMENT SETUP

Our experimental setup is shown in Fig. 1. The outputof a Ti:sapphire laser (10 Hz, 50 fs, 800 nm) was split intotwo parts by a beam splitter (BS) with 70% reflection and30% transmission. The reflected pulse with 1.1 mJ was usedas the pump and generated a 2-cm-long filament at the centerof an 80-cm-long gas cell filled with argon (purity >98%) at1 atm. Here the critical power for self-focusing is estimatedto be 10 GW [31], corresponding to 0.5 mJ for pulse energyin our laser output. The experiment was operated in a stableand single-filament condition. The transmitted beam passedthrough a 100-μm-thick KTP crystal for second harmonicgeneration and was reflected by two dichroic mirrors (DM)(with a high reflectivity of 99.5% at 400 nm and hightransmission of 90% at 800 nm) to filter out the remaining800 nm. The resulting pulse was set as probe (2.7 μJ, <100 fs,400 nm). The pump and probe beams were combined togetherby the second DM. Two half-wave plates (HWP) for 800and 400 nm were mounted on the pump and probe beampaths, respectively. Both beams were initially prepared inparallel linear polarizations, and focused by a plano-convexlens with a focal length of 50 cm. During the experiment,the angle between the polarizations of the pump and probewas varied according to the experimental design (see Sec. IV).After the filament, a dichroic mirror was used to block thepump beam while transmitting the probe. A white paperscreen was set ∼1 m after the filament. A CCD cameracovered by several neutral density (ND) filters and a bandpassfilter centered at 400 nm (bandwidth = 80 nm) was used toimage the transmitted probe pattern on the screen so as torecord the fluence distribution of the probe scattered on thescreen. A CCD camera was used in this experiment instead

013838-11050-2947/2011/84(1)/013838(6) ©2011 American Physical Society

SHUAI YUAN et al. PHYSICAL REVIEW A 84, 013838 (2011)

FIG. 1. (Color online) (a) Schematic diagram of our experimental setup. Time delay of the pump pulse (800 nm) can be changed by a delayline. DM: dichroic mirror; BS: beam splitter; HWP: half-wave plate; CCD: charge coupled device. (b) The polarization scheme for the probe.

of a photodiode since it could help us better understand theguiding and diffraction effect inside the filament by detectingthe spatial distribution of the probe. It may open furtherpossibilities in molecular gases as well. For example, thepolarization separator [25] in molecular gases at each revivalof molecules can be detected using this technique. In ourexperiment the transverse size of the probe beam, in whichthe polarization is rotated essentially, is similar to that of thefilament, since the probe has the same initial diameter as thepump, is focused by the same lens, and experiences nonlinearcross focusing induced by the pump. The bandwidth of thefilter was large enough to cover the full spectral width ofthe probe, including its spectral broadening due to XPM in thefilament [28]. The polarization of the probe beam was analyzedby a cube polarizer after the dichroic mirror. In order to adjustthe relative delay between the pump and probe pulses, weinstalled a motorized translation stage on the pump beam path.This experiment was done with a good temporal overlap forthe pump and probe, as described in Ref. [24]. We note that theprobe pulse probed the filament core of the pump because theywere both focused by the same lens and because the filamentguided the probe pulse [25,27]. Also during the experiment, wedid not observe any white light from our optics. This indicatesthat the effect of the optical components was minimized tothe extent that it did not affect our results significantly, unlikethe case of the self-compressed pulse propagation throughglass [32].

III. THEORETICAL ANALYSES

The mechanism of the measurement technique can beexplained by the difference in the nonlinear refractive indexes

generated by the driving laser field along its polarization axis[parallel or x axis; see Fig. 1(b)] and the orthogonal y axis.Integrated over the whole interaction length, the birefringenceinduces a dephasing between the probe beam’s componentsalong the fast and slow polarization axes [23]:

ε = 2π

∫�nXPM

probedz/λprobe, (1)

where �nXPMprobe is the XPM-induced index change between the

parallel and orthogonal components of the probe by consider-ing χ (3)

xxxx = 3χ (3)yyxx [22], and λprobe is the wavelength of the

probe pulse. The birefringence is quantitatively characterizedby the dephasing ε (phase difference between the parallel andorthogonal components of the probe pulse), which is relatedto the XPM-induced refractive index change, the pump andprobe interaction length, and the filament intensity.

We choose a laboratory coordinate system shown inFig. 1(b) to describe the two components of the probe fieldalong the fast (orthogonal, y) and slow (parallel, x) polarizationaxes:

E′2ωx = E2ωx cos(ωt − kz), (2)

E′2ωy = E2ωy cos(ωt − kz + ε), (3)

where E2ωx and E2ωy are the electric fields of the probe pulseprojected on the fast (y) and slow (x) polarization axes beforeinteracting with the pump while E′

2ωx and E′2ωyare the fields

after interaction with the pump inside the filament. By doingcoordinate rotation from the laboratory coordinate system (x,y)to the probe coordinate system (x′,y′), where x′ is parallel to

TABLE I. Measured the probe’s major axis rotation angle, the dephasing coefficient (ε), and the ellipticity square for the probe polarizationtheoretically (e2

sim) and experimentally (e2expt).

Initial probe polarization Experimental probe’s Calculated ellipticity Experimental ellipticity(ψprobe) rotation angle (major axis) Dephasing (ε) square (e2

sim) square (e2expt)

15◦ 16.4◦ ± 2◦ (0.527 ± 0.030)π 0.075 0.08530◦ 36.5◦ ± 2◦ (0.537 ± 0.025)π 0.321 0.33960◦ 35.7◦ ± 2◦ (0.542 ± 0.014)π 0.329 0.28275◦ 15.8◦ ± 2◦ (0.515 ± 0.039)π 0.071 0.055–30◦ 36.0◦ ± 2◦ (0.539 ± 0.011)π 0.328 0.298120◦ 37.1◦ ± 2◦ (0.547 ± 0.029)π 0.325 0.313

013838-2

MEASUREMENT OF BIREFRINGENCE INSIDE A FILAMENT PHYSICAL REVIEW A 84, 013838 (2011)

FIG. 2. (Color online) Polarization analysis of the transmitted probe at the end of filament for pairs of initial probe polarization anglesψprobe at 60◦ and 30◦ (a), 120◦ and –30◦ (b), 75◦ and 15◦ (c), with polarization scheme of the pump, probe and analyzer (d). Dashed straightlines and solid straight lines correspond to probe polarization after the analyzer when the pump is turned off (initial) and on (final), respectively.XPM-induced polarization rotations are marked by the blue arrows and characters.

the major axis of the rotated probe ellipse, we obtain [33]

cos ε = E22ωx − E2

2ωy

2E2ωxE2ωy

tan 2(ψprobe + φ), (4)

e2 =

(E2ωy

E2ωx

)2+ tan2(ψprobe + φ) − 2

(E2ωy

E2ωx

)tan(ψprobe + φ) cos ε

(E2ωy

E2ωx

)2tan2(ψprobe + φ) + 1 + 2

(E2ωy

E2ωx

)tan(ψprobe + φ) cos ε

, (5)

where ψprobe is defined as the initial angle between the linearprobe polarization and the x axis [Fig. 1(b)] and ϕ as theangle of rotation of the major axis of the elliptically polarizedprobe after propagating through the filament with respect to itsinitial polarization direction. With Eq. (4), the dephasing ε canbe obtained from the measured rotation angle ϕ, while usingEq. (5), the ellipticity e of the probe polarization, defined as

e = E′2ωy

E′2ωx

, can be evaluated by using the dephasing ε and therotation angle ϕ.

IV. EXPERIMENTAL RESULTS AND DISCUSSIONS

The experiment was carried out by rotating the transmis-sion axis of the analyzer with the pump and probe initiallinear polarization directions fixed. The probe polarization’stransformation after passing through the filament was studiedfor various pairs of the initial probe polarization angles +α

and –α, symmetric relative to the 45◦ line (with respect to

the x axis) in Fig. 2(d). The x axis (0◦) in this experimentwas defined parallel to the fixed pump polarization [seeFig. 2(d)]. The polarization states of the probe accumulatedalong the filament were measured by rotating the analyzer’stransmission axis. These states are shown in Figs. 2(a)–2(c),which correspond to the initial probe polarization anglesψprobe=45◦±15◦ = 60◦ and 30◦, ψprobe=45◦±75◦ = 120◦and –30◦, and ψprobe=45◦±30◦ = 75◦ and 15◦, respectively.After propagating through the filament, the probe polarizationintegrated over the whole beam became elliptical, with themajor axes rotated.

Table I summarizes the experimentally measured rotationangles and dephasing ε of different initial probe polarizations,and the corresponding calculated and measured ellipticitiessquared. The measured ellipticity is defined as the ratiobetween the length of the minor axis and that of the majoraxis on the Lorentz fit [solid curves in Figs. 2(a)–2(c)] ontothe experimentally measured angular transmission [squaresin Figs. 2(a)–2(c)]. The calculated ellipticity squared (e2

sim) is

013838-3

SHUAI YUAN et al. PHYSICAL REVIEW A 84, 013838 (2011)

FIG. 3. (Color online) (a) Polarization scheme of the pump, probe, and analyzer. (b) Probe energy, integrated over the whole transmittedprobe beam pattern is shown as a function of the probe polarization angle ψprobe, when pump was turned on (curve 1; black squares, solidcurve) and off (curve 2; orange triangles, solid curve). Their difference, which means the XPM-induced probe transmission, is shown in curve3 (red circles, solid curve, curve 1 minus curve 2) and fitted by cosine square.

derived from Eq. (5). The dephasing ε is obtained using Eq. (4)together with the measured probe polarization rotation angle.The calculated ellipticities (e2

sim) are in good agreement withthe directly measured ones (e2

expt). This not only convincinglyverifies the accuracy of the technique, but also indicates anunderlying physical connection between the probe polarizationrotation angle and the ellipticity of the probe polarization afterthe filament—because of the pump-induced birefringence.That means at a certain probe polarization rotation angle, it willhave the fixed ellipticity of the probe polarization. It happensboth after the pump-produced filament and inside the filamentwhen the probe pulse copropagates with the pump pulse. Herewe emphasize that even if we vary the angle between theinitial pump and probe’s polarizations, the probe’s dephasingε always yields roughly the same values, since the value ofdephasing ε depends only on the nonlinear refractive indexof argon and the integral of the pump clamped intensity overthe interaction length. Indeed, the clamped intensity variationalong the filament depends neither on the pump linear polariza-tion direction nor on the probe polarization. The former is dueto the initial isotropy of argon, the latter is due to the negligibleeffect of the probe on the pump during the propagation.That means if we replace argon by some unknown isotropicmolecular gas, we can retrieve the XPM-induced refractiveindex change �nXPM

probe in this gas by applying Eq. (1) to thedephasing ε and the filament length obtained in the experiment.

Figures 2(a)–2(c) illustrate that after interaction with thepump, for pairs of probe polarizations set at angles ψprobe =45◦ ± α [Fig. 2(d)], the resultant elliptical polarizations rotatethrough the same angle from its initial position towardsopposite directions. The symmetry of the rotation originatesfrom the constant dephasing (ε) between the two componentsof the probe polarization. This symmetry can be seen fromEq. (4) which gives

tan 2(ψprobe + ϕ) = 2|E2ωx ||E2ωy ||E2ωx |2 − |E2ωy |2 cos ε

= 2 sin ψprobe cos ψprobe

cos2 ψprobe − sin2 ψprobecos ε

= tan 2ψprobe cos ε. (6)

For angles such as ψprobe = 45◦ ± α, since ε stays constant,the term (tan 2ψprobe cos ε) on the right-hand side of Eq. (6)will have the same magnitude but the opposite sign; letus set (tan 2ψprobe cos ε = ±β). From Eq. (6), we get ϕ =−ψprobe ± tan−1 β

2 , which means the same polarization rotationangles towards opposite directions from the initial probedirection.

In order to have a more complete physical picture ofthe birefringence-induced rotation of the probe due to fila-ment guiding [25] and electronic XPM, two supplementaryexperiments were carried out by fixing the polarization ofthe pump and rotating the probe and vice versa whilemeasuring the probe transmission through the fixed analyzer.The first is to rotate the probe. In Fig. 3(a), we define theexperimentally fixed pump polarization as our x axis, whichis set at 45◦ to the fixed transmission axis of the analyzer. Byrotating the polarization of the probe, we measured the probetransmission after the analyzer, at various positions of α and−α symmetrical about the 45◦ line. Each point on Fig. 3(b)was obtained by integrating the whole probe beam pattern onthe paper screen imaged onto the CCD camera as the angle α

changes from –45◦ to 45◦. Since the transmission axis of theanalyzer was fixed, rotating the probe polarization will varythe transmitted probe energy. Curves 1 (total transmission)and 2 (background) in Fig. 3(b) were taken as the pumpwas turned on and off, respectively. The XPM-induced probetransmission as a function of angle α was depicted by curve 3in Fig. 3(b) by removing the background from the totaltransmission of the probe (curve 1 minus curve 2). In thismeasurement, the XPM-induced probe pulse transmissionreaches a maximum at α = 0◦ and shows equal values at thesymmetrical pairs of angles (α;−α) around the maximum.The equal values at the symmetrical pairs of angles (α,−α)matches well with that in Figs. 2(a)–2(c); i.e., the pairs ofprobe polarization angles, 45◦ ± α, rotate through the sameangles towards opposite directions after interacting with thepump.

The second experiment is to rotate the pump polarizationwith the probe and analyzer fixed. The scheme is shown inFig. 4(a). We define the experimentally fixed transmission

013838-4

MEASUREMENT OF BIREFRINGENCE INSIDE A FILAMENT PHYSICAL REVIEW A 84, 013838 (2011)

FIG. 4. (Color online) (a) Illustration of pump and probe polarization in the primed coordinate system. (b) The experimentally obtainedprobe transmission through the crossed analyzer for the pairs of angles θ and −θ . Here we use cos2 2θ to fit.

axis of the analyzer as our x axis (0◦) which is crossed bythe fixed probe polarization (90◦). θ is the angle between thepump polarization direction and the 45◦ line. By rotating thepump polarization about the 45◦ line symmetrically throughthe angle ±θ , the probe transmission through the analyzerintegrated over the whole beam pattern was recorded. Theresults are shown in Fig. 4(b). Similar to Fig. 3(b), thetransmitted probe after the analyzer also gives a maximumwhen the pump polarization is rotated to the 45◦ line. (θ = 0◦)and yields equal values at the symmetrical pairs of angles (θ ,–θ ). This effect can be simply understood by considering theprojection of the transmitted probe polarization on the x axisthrough XPM:

P(3)2ωx = χ (3)

xxxE2ωxEωxE∗ωx = χ (3)

xxyyE2ωxEωyE∗ωy

+χ (3)xyxyE2ωyEωxE

∗ωy + χ (3)

xyyxE2ωyEωyE∗ωx. (7)

While taking into account the experimental condition forthe initial probe polarization E2ωx = 0, we have

P(3)2ωx(2ω) = χ (3)

xyxyE2ωy(EωyE∗ωx + E∗

ωyEωx)

= χ (3)xyxyE2ωy |Eω|2 sin 2ψpump

= χ (3)xyxyE2ωy |Eω|2 cos(2θ ), (8)

since χ (3)xyxy = χ (3)

xyyx and ψpump = θ + 45◦. As shown inFig. 4(b), the experimental curve fits well with the curvey = η cos2 2θ , originating from Eq. (8) while η is a constantfor normalization.

V. DISCUSSION AND CONCLUSION

We experimentally demonstrated an optical gating tech-nique for quantitative ultrafast birefringence measurement inan atomic gas. By simply measuring the filament-inducedpolarization rotation of a probe pulse, we can easily and quan-titatively obtain the ultrafast birefringence caused by filamentguiding and cross-phase modulation. This technique can beused in atomic gases. In molecular gases, the same techniqueat proper delays could measure the birefringence as a resultof molecular rotation through Raman-type excitation as wellas the birefringence of the rotational wave packet at variousrevival times. This type of measurement could be applied toremote sensing of molecular gas targets because they possessnot only different revival times but also different and uniquebirefringence at proper delays, opening new perspectives forultrafast information processing and telecommunication.

ACKNOWLEDGMENTS

The authors thank C. Marceau for fruitful scien-tific discussions. This work was partially supported byNSERC, Canada Research Chair, the Canada Founda-tion for Innovation, the Canadian Institute for Photon-ics Innovation, le Fonds Quebecois pour la Recherchesur la Nature et les Technologies, Programme de boursesd’exemption des droits de scolarite supplementaires pourles etudiants etrangers, the Chinese Government Scholar-ships for Construction of High Level University graduateprogram, and the Russian Foundation for Basic Research(Grants No. 09-02-01200a, No. 09-02-01522a), RosnaukaNo. 02.740.11.0223.

[1] S. L. Chin, Femtosecond Laser Filamentation (Springer-Verlag,Berlin, 2010).

[2] A. Couairon and A. Mysyrowicz, Phys. Rep. 441, 47 (2007).[3] F. Theberge, N. Akozbek, W. W. Liu, A. Becker, and S. L. Chin,

Phys. Rev. Lett. 97, 023904 (2006).[4] L. Berge and S. Skupin, Phys. Rev. Lett. 100, 113902 (2008).[5] J. Kasparian, R. Sauerbrey, and S. L. Chin, Appl. Phys. B 71,

877 (2000).

[6] H. Cai, J. Wu, Y. Peng, and H. Zeng, Opt. Express 17, 5822(2009).

[7] S. Varma, Y.-H. Chen, and H. M. Milchberg, Phys. Rev. Lett.101, 205001 (2008).

[8] P. F. Lu, J. Liu, H. Li, H. F. Pan, J. Wu, and H.-P. Zeng, Appl.Phys. Lett. 97, 061101 (2010)

[9] R. R. Alfano, The Supercontinuum Laser Source (Springer-Verlag, New York, 1989).

013838-5

SHUAI YUAN et al. PHYSICAL REVIEW A 84, 013838 (2011)

[10] G. Heck, E. J. Judge, J. Odhner, M. Plewicki, and R. J. Levis,GomacTech Digest Conference, Orlando, FL, 2009 (unpub-lished), [http://www.temple.edu/capr/publications/PDFs/gomacLevisPaper-010709.pdf].

[11] J. Kasparian et al., Science 301, 61 (2003).[12] H. L. Xu et al., Appl. Phys. B 87, 151 (2007).[13] J. F. Daigle, Y. Kamali, M. Chateauneuf, G. Tremblay,

F. Theberge, J. Dubois, G. Roy, and S. L. Chin, Appl. Phys.B 97, 701 (2009).

[14] J. Kasparian et al., Opt. Express 16, 5757 (2008).[15] X. S. Zhang, A. Lytle, T. Popmintchev, X. B. Zhou, H. C.

Kapteyn, M. M. Murnane, and O. Cohen, Nat. Phys. 3, 270(2007).

[16] C. D’Amico, A. Houard, M. Franco, B. Prade, A. Mysyrowicz,A. Couairon, and V. T. Tikhonchuk, Phys. Rev. Lett. 98, 235002(2007).

[17] X. Xie, J.-M. Dai, and X.-C. Zhang, Phys. Rev. Lett. 96, 075005(2006).

[18] T.-J. Wang, S. Yuan, Y. Chen, J.-F. Daigle, C. Marceau,F. Theberge, M. Chateauneuf, J. Dubois, and S. L. Chin, Appl.Phys. Lett. 97, 111108 (2010).

[19] Y. Liu, A. Houard, B. Prade, S. Akturk, A. Mysyrowicz,and V. T. Tikhonchuk, Phys. Rev. Lett. 99, 135002 (2007).

[20] J. Liu, J.-M. Dai, S. L. Chin, and X.-C. Zhang, Nat. Photonics4, 627 (2010).

[21] F. Theberge, M. Chateauneuf, G. Roy, P. Mathieu, and J. Dubois,Phys. Rev. A 81, 033821 (2010).

[22] Y. R. Shen, The Principles of Nonlinear Optics, (John Wiley,New York, 1984), pp. 28, 286–301.

[23] P. Bejot, Y. Petit, L. Bonacina, J. Kasparian, M. Moret, and J.-P.Wolf, Opt. Express 16, 7564 (2008).

[24] C. Marceau, Y. Chen, F. Theberge, M. Chateauneuf, J. Dubois,and S. L. Chin, Opt. Lett. 34, 1419 (2009).

[25] Y. Chen, C. Marceau, F. Theberge, M. Chateauneuf, J. Dubois,and S. L. Chin, Opt. Lett. 33, 2731 (2008).

[26] F. Calegari, C. Vozzi, S. Gasilov, E. Benedetti, G. Sansone,M. Nisoli, S. De Silvestri, and S. Stagira, Phys. Rev. Lett. 100,123006 (2008).

[27] O. Kosareva et al., Opt. Lett. 35, 17 (2010).[28] C. Marceau, S. Ramakrishna, S. Genier, T. Wang, Y. Chen,

F. Theberge, M. Chateauneuf, J. Dubois, T. Seideman, and S. L.Chin, Opt. Commun. 283, 2732 (2010).

[29] A. C. Bernstein, M. McCormick, G. M. Dyer, J. C. Sanders, andT. Ditmire, Phys. Rev. Lett. 102, 123902 (2009).

[30] N. Tang and R. L. Sutherland, J. Opt. Soc. Am. B 14, 3412(1997).

[31] W. Liu and S. L. Chin, Opt. Express 13, 5750 (2005)[32] L. Berge, S. Skupin, and G. Steinmeyer, Phys. Rev. Lett. 101,

213901 (2008).[33] S. L. Chin, Fundamentals of Laser Optoelectronics (World

Scientific, Singapore, 1989), 1, pp. 170–171, Eqs. 7-20–7-22.(Note there are typographical errors in Eq. 7-20 (cosε instead ofcos2ε) and Eq. 7-22 (sin2α and cos2α in the denominator shouldbe interchanged).

013838-6