- Email: [email protected]
Spontaneous breaking of cylindrical symmetry in an optically pumped laser
Volume 78, number 3,4
OPTICS COMMUNICATIONS
1 September 1990
Spontaneous breaking of cylindrical symmetry in an optically pumped laser Chr. Tamm and C.O. Weiss
1
Labor 4.42, Physikalisch-Technische Bundesanstalt, Bundesallee 100, D-3300 Braunschweig, Fed. Rep. Germany
Received 6 February 1990
We describe and discuss an experiment on the formation of stationary transverse patterns of the optical field in a laser. The gain medium of an optically pumped, traveling-wave laser is in resonance with degenerate Gauss-Laguerre TEMpo and TEMo2transverse modes of the laser resonator. For low pump powers close to the laser threshold, the laser oscillates in the cylindrically symmetricTEMIomode. A field pattern of broken cylindrical symmetry emerges corttinuously above a critical pump power level. The experimental observations confirm the theoretical prediction of Lugiato et al. on the spontaneous breaking of the cylindrical symmetry in lasers [L.A. Lugiato, F. Prati, L.M. Narducci and G.-L. Oppo, Optics Comm. 69 (1989) 387].
The f o r m a t i o n o f spatial patterns in nonequilibrium systems with spatial degrees o f freedom is currently investigated in different fields o f n o n l i n e a r physics [1]. Recent theoretical studies on instabilities o f the transverse field d i s t r i b u t i o n in n o n l i n e a r optical systems have analyzed the spontaneous breaking o f spatial s y m m e t r y a n d the f o r m a t i o n o f transverse radiation patterns in passive n o n l i n e a r optical resonators [2] and lasers [3,4]. In this paper, we describe an e x p e r i m e n t that d e m onstrates a laser instability connected with a breaking o f the spatial symmetry. We show the spontaneous breaking o f cylindrical s y m m e t r y in the r a d i a t i o n pattern o f a travelling-wave laser where both the gain m e d i u m a n d the t r a n s v e r s e - m o d e structure o f the laser resonator are cylindrically symmetric. The transition to the state o f b r o k e n symmetry occurs when the p u m p bias level o f the laser is increased above a critical value, i.e., for a sufficiently strong d e p a r t u r e o f the system from thermal equilibrium. This e x p e r i m e n t a l d e m o n s t r a t i o n o f p a t t e r n f o r m a t i o n in a q u a n t u m - o p t i c a l system is significant since it permits direct c o m p a r i s o n with the predictions o f an analytical m o d e l [4]. It m a y also be relevant to a m o r e general u n d e r s t a n d i n g o f Present address: Physics Department, University of Queensland, St. Lucia, Brisbane, Australia 4067.
the effects o f transverse-mode c o m p e t i t i o n in highpower solid state lasers [ 5 ], a n d to the d e v e l o p m e n t o f optical switching and computing devices that make use o f the f o r m a t i o n o f transverse r a d i a t i o n patterns in nonlinear optical resonators [6]. The experimental situation is as follows. The active m e d i u m o f an optically p u m p e d laser is in resonance with the eigenfrequencies o f TEMloq and TEMo2q G a u s s - L a g u e r r e transverse optical modes [ 7 ] o f the laser resonator for one value o f the longitudinal m o d e index q. We recall that the transverse field d i s t r i b u t i o n o f the TEMpo m o d e is cylindrically s y m m e t r i c with one circular node line while the TEMo2 m o d e is not cylindrically s y m m e t r i c as its field distribution has two radial node lines. In the ideal case o f a cylindrically s y m m e t r i c optical reson a t o r without diffraction losses, both m o d e s would have exactly equal resonance frequencies since under these conditions, G a u s s - L a g u e r r e m o d e s TEMp~ o f equal transverse o r d e r 2 p + 1 are frequency degenerate [7]. In our experiment, the difference o f the eigenfrequencies o f the TEM10 and TEMo2 modes caused by diffraction at apertures a n d by deviations o f the laser resonator from cylindrical symmetry, is much smaller than the cavity linewidth. F o r modes o f different longitudinal or transverse order, the gain is below lasing threshold as a result o f their larger diffraction losses or largely different resonance fre-
0030-4018/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
253
Volume 78, number 3,4
OPTICS COMMUNICATIONS
quencies. The transverse d i s t r i b u t i o n o f p u m p intensity is circularly s y m m e t r i c and has its m a x i m u m at the optical axis o f the resonator. The laser oscillates under conditions where the gain resonance line o f the active m e d i u m is essentially homogeneously b r o a d e n e d . The atomic variables can adiabatically follow the temporal evolution o f the laser field (good cavity l i m i t ) , and the longitudinal variation o f the field envelope in the gain m e d i u m is negligible (limit o f longitudinally u n i f o r m field [ 3 ] ). U n d e r these conditions, it is expected [4] that the laser oscillates in a cylindrically s y m m e t r i c G a u s s Laguerre T E M m m o d e p a t t e r n for p u m p intensities close to the lasing threshold, and that above a critical p u m p intensity level which m a i n l y d e p e n d s on the width o f the transverse gain profile, a stationary field pattern o f b r o k e n cylindrical s y m m e t r y develops whose shape is d e t e r m i n e d by the interference o f phase-locked T E M m and TEMo2 modes. This symmetry breaking p h e n o m e n o n can be u n d e r s t o o d by taking into account that in single-transverse-mode TEMpo oscillation, the gain of the laser m e d i u m is saturated only incompletely in the region a r o u n d the node line o f the TEMpo field distribution. F o r p u m p intensities above the critical level, the residual unsaturated gain is sufficiently large to render pure TEMpo oscillation unstable against the growth o f a TEMo2 c o m p o n e n t in the transverse field distribution o f the laser [4], The fact that the resulting intensity pattern is stationary and that self-focusing or defocusing effects which could arise in off-resonant optical excitation are absent, leads to a particularly simple experimental situation. This distinguishes our e x p e r i m e n t from earlier observations o f s y m m e t r y breaking and pattern f o r m a t i o n in nonlinear optical systems with phase-conjugating mirrors [8] where adequate theoretical models seem to be missing. In some previous observations o f s y m m e t r y breaking in lasers, only o n e - d i m e n s i o n a l scans o f b e a m profiles were reported, so that the shapes o f the complete intensity pattern were not known [ 9 ]. It may be noted that stationary laser oscillation in phase-locked G a u s s - H e r m i t e TEMpo and TEMm transverse laser m o d e s has recently been shown to lead to field patterns o f broken space-inversion s y m m e t r y [ 10], a n d to bistability o f the corresponding transverse optical phase pattern [6]. The e x p e r i m e n t (see fig. l a ) uses a s o d i u m d i m e r 254
1 September 1990
(b)
Fig. 1. Experimental setup. (a) Na2 laser system. The intensity of the pump beam PB is varied by means of the aeoustooptic modulator AOM; FL: focusing lens. The laser ring resonator is formed by plane outcoupling mirror M 1 and concave (r= 0.713 m ) mirror M2; Pr 1: Brewster prism; diameter of aperture Ap ~ 1.2 ram. The Brewster prism Pr2 separates the laser output from the transmitted part of the pump field. Detector D, lock-in circuit Ll and piezoelement PZT form a servo loop stabilizing the optical path length of the laser resonator (see text). BEx is a beam expansion lens. (b) Spatially resolving photodetector. The shaded areas correspond to a partly masked quadrant photodetector; the symmetry center of the detector is aligned to the center of the output pattern of the laser. The output currents of the individual elements are added and subtracted in the way shown to obtain the average intensity signal I~ and the intensity difference signal Is. For the detector position shown, and for an angular position of the intensity pattern as in fig. 2c, d, one obtains a nearly optimal response of ld to the angular variation of intensity in the pattern. (Na2) v a p o r ring laser, collinearly p u m p e d by the collimated TEMoo beam o f a single-frequency 2 = 488 nm argon-ion laser that is tuned into resonance with the center o f the D o p p l e r - b r o a d e n e d (6, 43 ) X ~E + (3, 4 3 ) B l I l u transition o f Na2. In the 40 m m long Na2 v a p o r zone, the p u m p b e a m radius for an 1/e 2 d r o p o f the p u m p intensity is given by wp ~ 250 gin; this value is d e t e r m i n e d by the ( m e a s u r e d ) radius o f the u n c o l l i m a t e d p u m p b e a m and the focal length and the position of the collimating lens. Within an e s t i m a t e d accuracy o f _+ 15%, the p u m p b e a m radius wv matches the radius o f the gaussian envelope o f the resonator field (i.e., the radius for an 1/e2-drop of the intensity o f the TEMoo field) in the active me-
Volume 78, number 3,4
OPTICS COMMUNICATIONS
dium, WL. At a temperature of the sodium heat-pipe cell of ~ 670 K and a helium buffer gas pressure of l 0 mbar, approximately 50% of the pump laser power is absorbed in the Na2 vapor. As a lasing transition with relatively high gain at low p u m p intensities, the 2 = 525 nm (Q 13 ) transition from the upper level of the pump transition is selected by an intracavity prism. As a result of the velocity-selective optical excitation of the Na2 vapor by the pump laser, the Na2 oscillates in a traveling wave copropagating with the pump field [ 11 ], and in a single longitudinal mode. For the pump intensities used ( < 20 W / m m 2), and with absorption and outcoupling losses of the resonator field of ~ 5% which are sufficiently small to obtain comparable intracavity intensities of pump and laser field, the gain resonance line is approximately lorentzian and the gain linewidth of ~ 50 MHz (hwhm) is determined mainly by the collisional relaxation of the lower levels of the pump and laser transitions [ l 1 ]. The optical ring resonator of the Na2 laser consists of two mirrors and a Brewster prism that is also used to couple the pumping radiation into the resonator (see fig. l ). The optical roundtrip path d in the resonator is equal to the radius of curvature r of the concave resonator mirror. The free spectral range of the resonator is ~ 420 MHz. With respect to the location of the resonances of higher-order transverse modes, the resonator is equivalent to a linear cavity with one concave and one flat mirror spaced by d = r~ 2. The eigenfrequencies of Gauss-Laguerre modes TEMpi with 2 p + l = 4 k + 2 ( k = 0 , l, 2, ...) thus lie exactly halfway between subsequent TEMoo resonances [ 7 ]. In the experiment, only the TEMm and TEMo2 modes exceed the lasing threshold since the corresponding cavity resonance is kept in coincidence with the peak of the gain profile, and modes with k > 0 are suppressed by a circular intracavity aperture (see fig. 1 ). The polarization of the laser field is constant and uniform over the beam cross section as a result of the Brewster-angle optical surfaces inside the cavity. The Brewster windows of the Na2 cell and the oblique reflection at the concave resonator mirror introduce astigmatic distortions of the intracavity field. In general, such distortions destroy the cylindrical symmetry of the resonator, thereby leading to oscillation in Gauss-Hermite rather than Gauss-La-
1 September 1990
guerre modes [ 12 ]. In this case, generally beat components are observed in the laser output. They arise from the interference of nearly degenerate transverse modes that are not phase-locked and oscillate at slightly different optical frequencies. In our case, the astigmatic distortions of mirror and Brewster windows compensate each other to some extent; the residual astigmatism is compensated by rotating the intracavity Brewster prism by a small amount ( < 1 o ) out of the position of minimum deviation. The criterion for a sufficient compensation of astigmatism here is the absence of mode-beating oscillations in the laser output; for an uncompensated setup, transverse-mode beating frequencies in the range of 0.5...5 MHz are observed. The photographs of fig. 2a-d show the output intensity pattern of the laser for different values of optical p u m p power; all other experimental parameters are left unchanged. The intensity patterns shown in fig. 2a and b have a closed, nearly circular node line, indicating that the laser oscillates in a Gauss-Laguerre TEMIo optical-resonator mode at low pump power levels. The difference between the intensity patterns shown in fig. 2b and fig. 2c is given by the transition from a closed node line to a quadrate of node points. This modification of the output intensity pattern of the laser with pump power is predicted in ref. [4] as a result of the instability of the cyliridrically symmetric Gauss-Laguerre TEMm mode pattern in a laser above a critical value of the pump parameter. Comparison with the theoretical results of ref. [ 4 ] permits the conclusion that the intensity patterns shown in fig. 2c and d arise from a field configuration which can be described as a superposition of Gauss-Laguerre TEMm and TEMo2 modes. The observation that the superposition pattern shows node points (as opposed to the node line of the TEMm field distribution) can be related to the fact that the two mode fields combine in phase quadrature. The emergence of four peripheral local maxima of the intensity at a higher pump level (see fig. 2d) indicates a relative increase of the TEMo2 component in the field pattern when the pump power is increased further above the instability threshold. Fig. 3 shows, as a function of the pump power, the output signals of a photodetector arrangement that is sensitive to the angular variation of intensity in the output pattern of the laser (see fig. lb). The 255
Volume 78, number 3,4
OPTICS COMMUNICATIONS
l September 1990
Fig. 2. Photographs of the output intensity pattern of the laser for various values of optical pump power P. (a) P= 0.1 W; (b) P= 0.15 W: (c) P= 0.2 W; (d) P= 0.3 W. The photographs show the intensity distribution in the output patterns without correctly reflecting the change of the average output intensity of the laser with pump power. s w e e p rate o f t h e p u m p p o w e r was negligibly slow w i t h respect to the t i m e c o n s t a n t s o f laser r e s o n a t o r a n d gain m e d i u m . Fig. 3 d e m o n s t r a t e s t h a t there is 256
a w e l l - d e f i n e d t h r e s h o l d p u m p p o w e r for the breaking o f the c i r c u l a r s y m m e t r y o f the i n t e n s i t y pattern, a n d that the a n g u l a r m o d u l a t i o n o f i n t e n s i t y in the
Volume 78, number 3,4
scale) [linear
0"5 t
ID
0
OPTICS C O M M U N I C A T I O N S
J
~
-o.~-I
I)
' [Matt]
022
Fig. 3. Average intensity signal Is and intensity difference signal I d as a function of optical pump power P. The two broken lines denote the pump power for laser threshold, Pth, and the pump power at the instability threshold, Pi, respectively. The data collection time was 1 ms. output pattern increases approximately linearly with p u m p power. The ratio o f the p u m p powers at the instability threshold, Pi, and at lasing threshold, Pth, is found to be Pi/Pth.~ 1.5, for the case o f approximately equal gaussian envelope diameters o f p u m p and laser beam, Wp~ WE (see above). Assuming equal shapes for the transverse profiles o f p u m p intensity and gain, and neglecting the partial absorption o f the p u m p light in the Na2 vapor, the predicted position o f the instability threshold is given by Pi/Pth = 2.6 for Wp=WL [4]. It is clear that the angular position o f the intensity pattern that emerges above the instability threshold is only marginally stable for an optical system o f perfect cylindrical symmetry. Experimentally, one thus can expect that the angular position o f the pattern is determined by small deviations from cylindrical symmetry in the optical system resulting from, e.g., misalignment of optical elements and air turbulence. For a highly symmetric system, the position might be sensitive also to the spatial and temporal fluctuations o f the spontaneous emission in the gain medium. It is found in our case that the angular position o f the pattern in fact fluctuates strongly over time intervals larger than a few milliseconds if the optical setup and the resonator mode structure has a nearly perfect effective cylindrical symmetry (as judged from the symmetry o f the laser output pattern below the instability threshold). One typical example for
1 September 1990
the observed fluctuations is shown in fig. 4. Air currents were identified as the main cause of the fluctuations. For the photographs shown in fig. 2 and for the measurement of the instability threshold (fig. 3), the fluctuations o f the pattern position were reduced by slightly displacing the p u m p beam from the position of m a x i m u m symmetry. The experiment was also carried out for smaller and larger values of the p u m p beam radius Wp. The experimental observations remain qualitatively unchanged as Wp is varied in the range 0.5 WL< Wp< 3WL. In agreement with theory, the instability threshold is found to approach the lasing threshold for increasing p u m p beam diameter. As for wp.~ WL, the observed values o f Pi/Pth are smaller than predicted. Considering the predicted [ 4 ] strong influence of the width o f the transverse gain distribution on the position of the instability threshold, it is reasonable to assume that the origin o f the discrepancies between calculated and experimentally observed instability thresholds lies in slightly differing shapes of the transverse profiles o f p u m p intensity and laser gain. One possible source o f such deviations is the thermal diffusion o f Na2 molecules across the p u m p beam. With
i S10t 0 (linear
scal--e~t/ }TD TiNe [r~s] Fig. 4. Temporal variation of average intensity signal Is and intensity difference signal Id under conditions where strong fluctuations of the angular position of the pattern appear (see text ). The pump power is varied in a triangular waveform between laser threshold and a maximum value of .~0.5 W; two cyclesare shown. The arrows and dotted lines denote the time intervals where the pump power is above the instability threshold. The irregular behavior of the difference signal Id can be attributed to fluctuations of the angular position of the pattern. 257
Volume 78, number 3,4
OPTICS COMMUNICATIONS
an i n d e p e n d e n t m e a s u r e m e n t of the shape of the transverse small-signal gain distribution in the laser m e d i u m , e.g. by means of a separate probe laser, one should be able to clearly identify the origin of the observed reduction of the instability thresholds. For large p u m p b e a m diameters, % > 3 WL, it is difficult tO observe the breaking of the cylindrical symmetry in the intensity pattern of the laser since the transition is accompanied by the strong relative fluctuations of output power which occur near the laser threshold. In agreement with recent theoretical predictions, field patterns of a different type are observed for higher p u m p powers [ 13 ]. For wp<0.5 WL, the intensity of the p u m p field is sufficient to locally saturate the gain m e d i u m . U n d e r these conditions, it is found that the output power of the laser approaches a constant value if the optical p u m p power is increased, and that the intensity pattern remains cylindrically symmetric also for the highest p u m p power available ( ~ 0 . 5 W ) . It may be noted that the experimental observations discussed above were made u n d e r conditions where the laser output power varies approximately linearly with the p u m p power in the whole range above the laser threshold (see fig. 3). Here one may expect that p u m p saturation effects play no role. In conclusion, we have studied the formation of a field pattern of broken cylindrical symmetry in an optically p u m p e d ring laser whose geometry is effectively cylindrically symmetric. For p u m p powers near to the lasing threshold, the laser in our case oscillates in the cylindrically symmetric G a u s s - L a guerre TEMpo mode of the optical resonator. U n d e r otherwise unchanged experimental conditions, a stationary field pattern of broken cylindrical symmetry develops if the p u m p power is increased above a threshold value. One characteristic feature of this transverse pattern is the quadratic arrangement of node points where the optical field amplitude goes to zero. An e x a m i n a t i o n of the transverse variation of the optical phase in this pattern shows that in the vicinity of the node points, the equiphase surface of the optical field has a helical structure, the helices of neighbouring node points being mirror images of each other. This implies that the field pattern considered here not only corresponds to a state of broken cylindrical symmetry, but also to a state of broken space258
1 September 1990
inversion symmetry. It may be noted that a helical equiphase surface is characteristic also of lasers oscillating in a stationary T E M ~ hybrid mode pattern [ 6,14 ], and of the "optical vortex" pattern which has been studied recently as a stable field configuration of laser resonators with large Fresnel n u m b e r s [ 151The authors gratefully acknowledge the assistance of W. Dekker in the preparation of the experiment. This work has been supported by the Deutsche Forschungsgemeinschaft a n d by an ESPRIT grant of the C o m m i s s i o n of the European C o m m u n i t i e s .
References [ 1] See, e.g., H. Haken, Advancedsynergetics (Springer,Berlin, Heidelberg, 1987) p. 1. [ 2 ] J.V. Moloney,in: Instabilities and chaos in quantum optics, eds. F.T. Arecchi and R.G. Harrison (Springer, Berlin, Heidelberg, 1987) p. 139; L.A. Lugiato and R. Lefever, Phys. Rev. Lett. 58 (1987) 2209. [3] L.A. Lugiato, C. Oldano and L.M. Narducci, J. Opt. Soc. Am. B 5 (1988) 879; feature issue on Nonlinear dynamics of lasers, eds. D.K. Bandy,J.R. Tredicceand A.N. Oraevsky. [4] L.A. Lugiato, F. Prati, L.M. Narducci and G.-L. Oppo, Optics Comm. 69 (1989) 387. [5 ] R. Hauck, F. Hollinger and H. Weber, Optics Comm. 47 (1983) 141. [6 ] Chr. Tamm and C.O. Weiss, 1990J. Opt. Soc. Am. B, feature issue on Transverseeffects in nonlinear-opticalsystems, eds. N.B. Abraham and W.J. Firth. [7 ] H. Kogelnikand T. Li, Appl. Optics 5 ( 1966) 1550. [8] G. Grynberg, E. Le Bihan and P. Verkerk, Optics Comm. 67 (1988) 363; G. Giusfredi, J.F. Valley, R. Pon, G. Khitrova and H.M. Gibbs, J. Opt. Soc. Am. B 5 (1988) 1188, feature issue on Nonlineardynamicsof lasers, eds. D.K. Bandy,J.R. Tredicce and A.N. Oraevsky. [9] D.J. Biswas,Vas Dev and U.K. Chatterjee, Phys. Rev. A 38 (1988) 555; J.R. Tredicce, E.J. Quel, A.M. Ghazzawi, C. Green, M.A. Pernigo, L.M. Narducci and L.A. Lugiato, Phys. Rev. Len. 62 (1989) 1274. [ 10] Chr. Tamm, Phys. Rev. A 38 (1988) 5960. [11 ] B. Wellegehausen, IEEE J. Quantum Electron. QE-15 (1979) 1108. [ 12] See, e.g., A.E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986) pp. 647-648. [ 13] M. Brambilla, L.A. Lugiato, F. Prati, Chr. Tamm and C.O. Weiss, to be presented at 1990 IQEC conference,Anaheim, California. [ 14] J.M. Vaughan and D.V. Willets, Optics Comm. 30 (1979) 263. [ 15] P. Coullet, L. Gil and F. Rocca, Optics Comm. 73 (1989) 403.
OPTICS COMMUNICATIONS
1 September 1990
Spontaneous breaking of cylindrical symmetry in an optically pumped laser Chr. Tamm and C.O. Weiss
1
Labor 4.42, Physikalisch-Technische Bundesanstalt, Bundesallee 100, D-3300 Braunschweig, Fed. Rep. Germany
Received 6 February 1990
We describe and discuss an experiment on the formation of stationary transverse patterns of the optical field in a laser. The gain medium of an optically pumped, traveling-wave laser is in resonance with degenerate Gauss-Laguerre TEMpo and TEMo2transverse modes of the laser resonator. For low pump powers close to the laser threshold, the laser oscillates in the cylindrically symmetricTEMIomode. A field pattern of broken cylindrical symmetry emerges corttinuously above a critical pump power level. The experimental observations confirm the theoretical prediction of Lugiato et al. on the spontaneous breaking of the cylindrical symmetry in lasers [L.A. Lugiato, F. Prati, L.M. Narducci and G.-L. Oppo, Optics Comm. 69 (1989) 387].
The f o r m a t i o n o f spatial patterns in nonequilibrium systems with spatial degrees o f freedom is currently investigated in different fields o f n o n l i n e a r physics [1]. Recent theoretical studies on instabilities o f the transverse field d i s t r i b u t i o n in n o n l i n e a r optical systems have analyzed the spontaneous breaking o f spatial s y m m e t r y a n d the f o r m a t i o n o f transverse radiation patterns in passive n o n l i n e a r optical resonators [2] and lasers [3,4]. In this paper, we describe an e x p e r i m e n t that d e m onstrates a laser instability connected with a breaking o f the spatial symmetry. We show the spontaneous breaking o f cylindrical s y m m e t r y in the r a d i a t i o n pattern o f a travelling-wave laser where both the gain m e d i u m a n d the t r a n s v e r s e - m o d e structure o f the laser resonator are cylindrically symmetric. The transition to the state o f b r o k e n symmetry occurs when the p u m p bias level o f the laser is increased above a critical value, i.e., for a sufficiently strong d e p a r t u r e o f the system from thermal equilibrium. This e x p e r i m e n t a l d e m o n s t r a t i o n o f p a t t e r n f o r m a t i o n in a q u a n t u m - o p t i c a l system is significant since it permits direct c o m p a r i s o n with the predictions o f an analytical m o d e l [4]. It m a y also be relevant to a m o r e general u n d e r s t a n d i n g o f Present address: Physics Department, University of Queensland, St. Lucia, Brisbane, Australia 4067.
the effects o f transverse-mode c o m p e t i t i o n in highpower solid state lasers [ 5 ], a n d to the d e v e l o p m e n t o f optical switching and computing devices that make use o f the f o r m a t i o n o f transverse r a d i a t i o n patterns in nonlinear optical resonators [6]. The experimental situation is as follows. The active m e d i u m o f an optically p u m p e d laser is in resonance with the eigenfrequencies o f TEMloq and TEMo2q G a u s s - L a g u e r r e transverse optical modes [ 7 ] o f the laser resonator for one value o f the longitudinal m o d e index q. We recall that the transverse field d i s t r i b u t i o n o f the TEMpo m o d e is cylindrically s y m m e t r i c with one circular node line while the TEMo2 m o d e is not cylindrically s y m m e t r i c as its field distribution has two radial node lines. In the ideal case o f a cylindrically s y m m e t r i c optical reson a t o r without diffraction losses, both m o d e s would have exactly equal resonance frequencies since under these conditions, G a u s s - L a g u e r r e m o d e s TEMp~ o f equal transverse o r d e r 2 p + 1 are frequency degenerate [7]. In our experiment, the difference o f the eigenfrequencies o f the TEM10 and TEMo2 modes caused by diffraction at apertures a n d by deviations o f the laser resonator from cylindrical symmetry, is much smaller than the cavity linewidth. F o r modes o f different longitudinal or transverse order, the gain is below lasing threshold as a result o f their larger diffraction losses or largely different resonance fre-
0030-4018/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
253
Volume 78, number 3,4
OPTICS COMMUNICATIONS
quencies. The transverse d i s t r i b u t i o n o f p u m p intensity is circularly s y m m e t r i c and has its m a x i m u m at the optical axis o f the resonator. The laser oscillates under conditions where the gain resonance line o f the active m e d i u m is essentially homogeneously b r o a d e n e d . The atomic variables can adiabatically follow the temporal evolution o f the laser field (good cavity l i m i t ) , and the longitudinal variation o f the field envelope in the gain m e d i u m is negligible (limit o f longitudinally u n i f o r m field [ 3 ] ). U n d e r these conditions, it is expected [4] that the laser oscillates in a cylindrically s y m m e t r i c G a u s s Laguerre T E M m m o d e p a t t e r n for p u m p intensities close to the lasing threshold, and that above a critical p u m p intensity level which m a i n l y d e p e n d s on the width o f the transverse gain profile, a stationary field pattern o f b r o k e n cylindrical s y m m e t r y develops whose shape is d e t e r m i n e d by the interference o f phase-locked T E M m and TEMo2 modes. This symmetry breaking p h e n o m e n o n can be u n d e r s t o o d by taking into account that in single-transverse-mode TEMpo oscillation, the gain of the laser m e d i u m is saturated only incompletely in the region a r o u n d the node line o f the TEMpo field distribution. F o r p u m p intensities above the critical level, the residual unsaturated gain is sufficiently large to render pure TEMpo oscillation unstable against the growth o f a TEMo2 c o m p o n e n t in the transverse field distribution o f the laser [4], The fact that the resulting intensity pattern is stationary and that self-focusing or defocusing effects which could arise in off-resonant optical excitation are absent, leads to a particularly simple experimental situation. This distinguishes our e x p e r i m e n t from earlier observations o f s y m m e t r y breaking and pattern f o r m a t i o n in nonlinear optical systems with phase-conjugating mirrors [8] where adequate theoretical models seem to be missing. In some previous observations o f s y m m e t r y breaking in lasers, only o n e - d i m e n s i o n a l scans o f b e a m profiles were reported, so that the shapes o f the complete intensity pattern were not known [ 9 ]. It may be noted that stationary laser oscillation in phase-locked G a u s s - H e r m i t e TEMpo and TEMm transverse laser m o d e s has recently been shown to lead to field patterns o f broken space-inversion s y m m e t r y [ 10], a n d to bistability o f the corresponding transverse optical phase pattern [6]. The e x p e r i m e n t (see fig. l a ) uses a s o d i u m d i m e r 254
1 September 1990
(b)
Fig. 1. Experimental setup. (a) Na2 laser system. The intensity of the pump beam PB is varied by means of the aeoustooptic modulator AOM; FL: focusing lens. The laser ring resonator is formed by plane outcoupling mirror M 1 and concave (r= 0.713 m ) mirror M2; Pr 1: Brewster prism; diameter of aperture Ap ~ 1.2 ram. The Brewster prism Pr2 separates the laser output from the transmitted part of the pump field. Detector D, lock-in circuit Ll and piezoelement PZT form a servo loop stabilizing the optical path length of the laser resonator (see text). BEx is a beam expansion lens. (b) Spatially resolving photodetector. The shaded areas correspond to a partly masked quadrant photodetector; the symmetry center of the detector is aligned to the center of the output pattern of the laser. The output currents of the individual elements are added and subtracted in the way shown to obtain the average intensity signal I~ and the intensity difference signal Is. For the detector position shown, and for an angular position of the intensity pattern as in fig. 2c, d, one obtains a nearly optimal response of ld to the angular variation of intensity in the pattern. (Na2) v a p o r ring laser, collinearly p u m p e d by the collimated TEMoo beam o f a single-frequency 2 = 488 nm argon-ion laser that is tuned into resonance with the center o f the D o p p l e r - b r o a d e n e d (6, 43 ) X ~E + (3, 4 3 ) B l I l u transition o f Na2. In the 40 m m long Na2 v a p o r zone, the p u m p b e a m radius for an 1/e 2 d r o p o f the p u m p intensity is given by wp ~ 250 gin; this value is d e t e r m i n e d by the ( m e a s u r e d ) radius o f the u n c o l l i m a t e d p u m p b e a m and the focal length and the position of the collimating lens. Within an e s t i m a t e d accuracy o f _+ 15%, the p u m p b e a m radius wv matches the radius o f the gaussian envelope o f the resonator field (i.e., the radius for an 1/e2-drop of the intensity o f the TEMoo field) in the active me-
Volume 78, number 3,4
OPTICS COMMUNICATIONS
dium, WL. At a temperature of the sodium heat-pipe cell of ~ 670 K and a helium buffer gas pressure of l 0 mbar, approximately 50% of the pump laser power is absorbed in the Na2 vapor. As a lasing transition with relatively high gain at low p u m p intensities, the 2 = 525 nm (Q 13 ) transition from the upper level of the pump transition is selected by an intracavity prism. As a result of the velocity-selective optical excitation of the Na2 vapor by the pump laser, the Na2 oscillates in a traveling wave copropagating with the pump field [ 11 ], and in a single longitudinal mode. For the pump intensities used ( < 20 W / m m 2), and with absorption and outcoupling losses of the resonator field of ~ 5% which are sufficiently small to obtain comparable intracavity intensities of pump and laser field, the gain resonance line is approximately lorentzian and the gain linewidth of ~ 50 MHz (hwhm) is determined mainly by the collisional relaxation of the lower levels of the pump and laser transitions [ l 1 ]. The optical ring resonator of the Na2 laser consists of two mirrors and a Brewster prism that is also used to couple the pumping radiation into the resonator (see fig. l ). The optical roundtrip path d in the resonator is equal to the radius of curvature r of the concave resonator mirror. The free spectral range of the resonator is ~ 420 MHz. With respect to the location of the resonances of higher-order transverse modes, the resonator is equivalent to a linear cavity with one concave and one flat mirror spaced by d = r~ 2. The eigenfrequencies of Gauss-Laguerre modes TEMpi with 2 p + l = 4 k + 2 ( k = 0 , l, 2, ...) thus lie exactly halfway between subsequent TEMoo resonances [ 7 ]. In the experiment, only the TEMm and TEMo2 modes exceed the lasing threshold since the corresponding cavity resonance is kept in coincidence with the peak of the gain profile, and modes with k > 0 are suppressed by a circular intracavity aperture (see fig. 1 ). The polarization of the laser field is constant and uniform over the beam cross section as a result of the Brewster-angle optical surfaces inside the cavity. The Brewster windows of the Na2 cell and the oblique reflection at the concave resonator mirror introduce astigmatic distortions of the intracavity field. In general, such distortions destroy the cylindrical symmetry of the resonator, thereby leading to oscillation in Gauss-Hermite rather than Gauss-La-
1 September 1990
guerre modes [ 12 ]. In this case, generally beat components are observed in the laser output. They arise from the interference of nearly degenerate transverse modes that are not phase-locked and oscillate at slightly different optical frequencies. In our case, the astigmatic distortions of mirror and Brewster windows compensate each other to some extent; the residual astigmatism is compensated by rotating the intracavity Brewster prism by a small amount ( < 1 o ) out of the position of minimum deviation. The criterion for a sufficient compensation of astigmatism here is the absence of mode-beating oscillations in the laser output; for an uncompensated setup, transverse-mode beating frequencies in the range of 0.5...5 MHz are observed. The photographs of fig. 2a-d show the output intensity pattern of the laser for different values of optical p u m p power; all other experimental parameters are left unchanged. The intensity patterns shown in fig. 2a and b have a closed, nearly circular node line, indicating that the laser oscillates in a Gauss-Laguerre TEMIo optical-resonator mode at low pump power levels. The difference between the intensity patterns shown in fig. 2b and fig. 2c is given by the transition from a closed node line to a quadrate of node points. This modification of the output intensity pattern of the laser with pump power is predicted in ref. [4] as a result of the instability of the cyliridrically symmetric Gauss-Laguerre TEMm mode pattern in a laser above a critical value of the pump parameter. Comparison with the theoretical results of ref. [ 4 ] permits the conclusion that the intensity patterns shown in fig. 2c and d arise from a field configuration which can be described as a superposition of Gauss-Laguerre TEMm and TEMo2 modes. The observation that the superposition pattern shows node points (as opposed to the node line of the TEMm field distribution) can be related to the fact that the two mode fields combine in phase quadrature. The emergence of four peripheral local maxima of the intensity at a higher pump level (see fig. 2d) indicates a relative increase of the TEMo2 component in the field pattern when the pump power is increased further above the instability threshold. Fig. 3 shows, as a function of the pump power, the output signals of a photodetector arrangement that is sensitive to the angular variation of intensity in the output pattern of the laser (see fig. lb). The 255
Volume 78, number 3,4
OPTICS COMMUNICATIONS
l September 1990
Fig. 2. Photographs of the output intensity pattern of the laser for various values of optical pump power P. (a) P= 0.1 W; (b) P= 0.15 W: (c) P= 0.2 W; (d) P= 0.3 W. The photographs show the intensity distribution in the output patterns without correctly reflecting the change of the average output intensity of the laser with pump power. s w e e p rate o f t h e p u m p p o w e r was negligibly slow w i t h respect to the t i m e c o n s t a n t s o f laser r e s o n a t o r a n d gain m e d i u m . Fig. 3 d e m o n s t r a t e s t h a t there is 256
a w e l l - d e f i n e d t h r e s h o l d p u m p p o w e r for the breaking o f the c i r c u l a r s y m m e t r y o f the i n t e n s i t y pattern, a n d that the a n g u l a r m o d u l a t i o n o f i n t e n s i t y in the
Volume 78, number 3,4
scale) [linear
0"5 t
ID
0
OPTICS C O M M U N I C A T I O N S
J
~
-o.~-I
I)
' [Matt]
022
Fig. 3. Average intensity signal Is and intensity difference signal I d as a function of optical pump power P. The two broken lines denote the pump power for laser threshold, Pth, and the pump power at the instability threshold, Pi, respectively. The data collection time was 1 ms. output pattern increases approximately linearly with p u m p power. The ratio o f the p u m p powers at the instability threshold, Pi, and at lasing threshold, Pth, is found to be Pi/Pth.~ 1.5, for the case o f approximately equal gaussian envelope diameters o f p u m p and laser beam, Wp~ WE (see above). Assuming equal shapes for the transverse profiles o f p u m p intensity and gain, and neglecting the partial absorption o f the p u m p light in the Na2 vapor, the predicted position o f the instability threshold is given by Pi/Pth = 2.6 for Wp=WL [4]. It is clear that the angular position o f the intensity pattern that emerges above the instability threshold is only marginally stable for an optical system o f perfect cylindrical symmetry. Experimentally, one thus can expect that the angular position o f the pattern is determined by small deviations from cylindrical symmetry in the optical system resulting from, e.g., misalignment of optical elements and air turbulence. For a highly symmetric system, the position might be sensitive also to the spatial and temporal fluctuations o f the spontaneous emission in the gain medium. It is found in our case that the angular position o f the pattern in fact fluctuates strongly over time intervals larger than a few milliseconds if the optical setup and the resonator mode structure has a nearly perfect effective cylindrical symmetry (as judged from the symmetry o f the laser output pattern below the instability threshold). One typical example for
1 September 1990
the observed fluctuations is shown in fig. 4. Air currents were identified as the main cause of the fluctuations. For the photographs shown in fig. 2 and for the measurement of the instability threshold (fig. 3), the fluctuations o f the pattern position were reduced by slightly displacing the p u m p beam from the position of m a x i m u m symmetry. The experiment was also carried out for smaller and larger values of the p u m p beam radius Wp. The experimental observations remain qualitatively unchanged as Wp is varied in the range 0.5 WL< Wp< 3WL. In agreement with theory, the instability threshold is found to approach the lasing threshold for increasing p u m p beam diameter. As for wp.~ WL, the observed values o f Pi/Pth are smaller than predicted. Considering the predicted [ 4 ] strong influence of the width o f the transverse gain distribution on the position of the instability threshold, it is reasonable to assume that the origin o f the discrepancies between calculated and experimentally observed instability thresholds lies in slightly differing shapes of the transverse profiles o f p u m p intensity and laser gain. One possible source o f such deviations is the thermal diffusion o f Na2 molecules across the p u m p beam. With
i S10t 0 (linear
scal--e~t/ }TD TiNe [r~s] Fig. 4. Temporal variation of average intensity signal Is and intensity difference signal Id under conditions where strong fluctuations of the angular position of the pattern appear (see text ). The pump power is varied in a triangular waveform between laser threshold and a maximum value of .~0.5 W; two cyclesare shown. The arrows and dotted lines denote the time intervals where the pump power is above the instability threshold. The irregular behavior of the difference signal Id can be attributed to fluctuations of the angular position of the pattern. 257
Volume 78, number 3,4
OPTICS COMMUNICATIONS
an i n d e p e n d e n t m e a s u r e m e n t of the shape of the transverse small-signal gain distribution in the laser m e d i u m , e.g. by means of a separate probe laser, one should be able to clearly identify the origin of the observed reduction of the instability thresholds. For large p u m p b e a m diameters, % > 3 WL, it is difficult tO observe the breaking of the cylindrical symmetry in the intensity pattern of the laser since the transition is accompanied by the strong relative fluctuations of output power which occur near the laser threshold. In agreement with recent theoretical predictions, field patterns of a different type are observed for higher p u m p powers [ 13 ]. For wp<0.5 WL, the intensity of the p u m p field is sufficient to locally saturate the gain m e d i u m . U n d e r these conditions, it is found that the output power of the laser approaches a constant value if the optical p u m p power is increased, and that the intensity pattern remains cylindrically symmetric also for the highest p u m p power available ( ~ 0 . 5 W ) . It may be noted that the experimental observations discussed above were made u n d e r conditions where the laser output power varies approximately linearly with the p u m p power in the whole range above the laser threshold (see fig. 3). Here one may expect that p u m p saturation effects play no role. In conclusion, we have studied the formation of a field pattern of broken cylindrical symmetry in an optically p u m p e d ring laser whose geometry is effectively cylindrically symmetric. For p u m p powers near to the lasing threshold, the laser in our case oscillates in the cylindrically symmetric G a u s s - L a guerre TEMpo mode of the optical resonator. U n d e r otherwise unchanged experimental conditions, a stationary field pattern of broken cylindrical symmetry develops if the p u m p power is increased above a threshold value. One characteristic feature of this transverse pattern is the quadratic arrangement of node points where the optical field amplitude goes to zero. An e x a m i n a t i o n of the transverse variation of the optical phase in this pattern shows that in the vicinity of the node points, the equiphase surface of the optical field has a helical structure, the helices of neighbouring node points being mirror images of each other. This implies that the field pattern considered here not only corresponds to a state of broken cylindrical symmetry, but also to a state of broken space258
1 September 1990
inversion symmetry. It may be noted that a helical equiphase surface is characteristic also of lasers oscillating in a stationary T E M ~ hybrid mode pattern [ 6,14 ], and of the "optical vortex" pattern which has been studied recently as a stable field configuration of laser resonators with large Fresnel n u m b e r s [ 151The authors gratefully acknowledge the assistance of W. Dekker in the preparation of the experiment. This work has been supported by the Deutsche Forschungsgemeinschaft a n d by an ESPRIT grant of the C o m m i s s i o n of the European C o m m u n i t i e s .
References [ 1] See, e.g., H. Haken, Advancedsynergetics (Springer,Berlin, Heidelberg, 1987) p. 1. [ 2 ] J.V. Moloney,in: Instabilities and chaos in quantum optics, eds. F.T. Arecchi and R.G. Harrison (Springer, Berlin, Heidelberg, 1987) p. 139; L.A. Lugiato and R. Lefever, Phys. Rev. Lett. 58 (1987) 2209. [3] L.A. Lugiato, C. Oldano and L.M. Narducci, J. Opt. Soc. Am. B 5 (1988) 879; feature issue on Nonlinear dynamics of lasers, eds. D.K. Bandy,J.R. Tredicceand A.N. Oraevsky. [4] L.A. Lugiato, F. Prati, L.M. Narducci and G.-L. Oppo, Optics Comm. 69 (1989) 387. [5 ] R. Hauck, F. Hollinger and H. Weber, Optics Comm. 47 (1983) 141. [6 ] Chr. Tamm and C.O. Weiss, 1990J. Opt. Soc. Am. B, feature issue on Transverseeffects in nonlinear-opticalsystems, eds. N.B. Abraham and W.J. Firth. [7 ] H. Kogelnikand T. Li, Appl. Optics 5 ( 1966) 1550. [8] G. Grynberg, E. Le Bihan and P. Verkerk, Optics Comm. 67 (1988) 363; G. Giusfredi, J.F. Valley, R. Pon, G. Khitrova and H.M. Gibbs, J. Opt. Soc. Am. B 5 (1988) 1188, feature issue on Nonlineardynamicsof lasers, eds. D.K. Bandy,J.R. Tredicce and A.N. Oraevsky. [9] D.J. Biswas,Vas Dev and U.K. Chatterjee, Phys. Rev. A 38 (1988) 555; J.R. Tredicce, E.J. Quel, A.M. Ghazzawi, C. Green, M.A. Pernigo, L.M. Narducci and L.A. Lugiato, Phys. Rev. Len. 62 (1989) 1274. [ 10] Chr. Tamm, Phys. Rev. A 38 (1988) 5960. [11 ] B. Wellegehausen, IEEE J. Quantum Electron. QE-15 (1979) 1108. [ 12] See, e.g., A.E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986) pp. 647-648. [ 13] M. Brambilla, L.A. Lugiato, F. Prati, Chr. Tamm and C.O. Weiss, to be presented at 1990 IQEC conference,Anaheim, California. [ 14] J.M. Vaughan and D.V. Willets, Optics Comm. 30 (1979) 263. [ 15] P. Coullet, L. Gil and F. Rocca, Optics Comm. 73 (1989) 403.