Phase jump and intensity instabilities in a homogeneously broadened bidirectional ring laser with backscattering

Volume 82, number 1,2

OPTICS COMMUNICATIONS

1 April 1991

Phase jump and intensity instabilities in a homogeneously broadened bidirectional ring laser with backscattering F.C. C h e n g Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA Received 17 August 1990; revised manuscript received 7 December 1990

The equations of motion of a single frequency homogeneously broadened ring laser with optical backscattering have been investigated numerically by using Monte Carlo simulation. Studies of the coupled third order equations of motion with additive and multiplicative noise terms show that, under some conditions of backscattering, the laser has two regions of operation. When the magnitude of the backscattering coefficient R is less than a critical value, Re, which depends on the pump parameter, the laser exhibits the usual noise-induced spontaneous intensity switching. When IR I is larger than Pc, the two modes exhibit steady state anticorrelated oscillations. Jumps of _+7tin the phase difference of the two counter propagating waves have been demonstrated to occur in both the switching and oscillation regions. Noise-induced transitions from one region to another are also observed when [R [ is close to Re.

1. Introduction

It is well known that a homogeneously broadened ring laser, with two counter propagating modes, exhibits the effect o f r a n d o m intensity switching [ 14]. When light is reflected or scattered from one mode back into the other, the behaviour o f the laser can be drastically modified [ 5-14 ] and the mode switching effect may even be suppressed [ 5 ]. Studies have shown that the intensities of a homogeneously broadened bidirectional ring laser can be made to oscillate periodically if the backscattering effect is carefully controlled [ 5,7 ]. Investigations [ 13-15 ] of an inhomogeneously broadened HeNe laser with backscattering have also shown intensity oscillations. The third-order equations o f motion for that laser at line center with backscattering have been solved exactly [ 13,14 ] when additive noise terms a r e neglected. In addition to the anticorrelated intensity oscillation, Chyba [ 13 ] also found that the relative phase o f the two propagating waves shows sudden jumps o f _+z~ when one o f the intensities becomes zero. Similar effects o f 7t-phase j u m p have also been observed in both unidirectional and bidirectional farinfrared ring lasers, for which neither the atomic population nor the polarization can be adiabatically

eliminated [ 16,17 ]. Recently, Spreeuw et al. [ 15 ] used a mode coupling approach to solve the deterministic equations of a H e N e ring laser with backscattering. They interpreted the intensity oscillation as due to the frequency splitting in the mode structure o f the corresponding passive ring cavity and observed the frequency splitting experimentally. Using their approach, phase jumps can be seen as mode hops between the two standing-wave modes. Kuhlke and Jetschke [ 6 ] have studied the coupled equations o f motion for a homogeneously broadened ring laser with backscattering when both the additive and multiplicative noise terms were neglected. Their analysis o f the coupled deterministic equations using the method o f small perturbation showed the existence of a critical value Rc for the backscattering coefficient. For "off-phase" backscattering, when the magnitude o f the backscattering coefficient R is less than Re, they obtained a stable solution for the two intensities. Their results do not show r a n d o m intensity switching, because o f the absence o f the additive noise terms. When [Rt is larger than R~, they obtained solutions exhibiting self-sustained intensity oscillations. They calculated the field amplitudes to second order in the laser gain coefficient and showed that the oscillations are anharmonic in general and

0030-4018/91/$03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )

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become harmonic in the limit of large IRI. In the following, we wish to report the results of further numerical studies on the phase jumps and intensity oscillations of a homogeneously broadened bidirectional ring laser with backscattering. We include both additive and multiplicative noise terms in our simulations. Our results show that ~r-phase jumps can occur in both the switching and oscillation regions when either one of the intensities becomes zero.

values used for the dimensionless backscattering coefficient range between 0 and 500, and these are typical for a dye ring laser with an output coupler of 1-2% transmissivity and with one unit of dimensionless time approximately equal to 10- 5 s [ 19 ]. The backscattering is said to be "off phase" when 01 + 0 2 = ( 2 n + 1 )~, and "in phase" when 01 + 02=2n~ [ 12 ]. The expression for Re obtained from analyzing the deterministic coup!ed equations of motion for the case IR~l = IREI has been determined to be [6] Rc = a ( ~ - 1 ) / 2 ~ 1

2. The model

The dimensionless coupled equations of motion for a single-frequency homogeneously broadened ring laser form the starting point, with the introduction of additional terms for backscattering, additive noise, and multiplicative noise [4,9]

+Pgl +ql +REEl +pE2+q2

E1 = (al - IE112-2 lEE [2)El + R i g 2

,

/~2 = (a2 - [E212-2[E112)E2

,

(1)

(q*(t) qj(t'))=4Jod(t-t') ( i , j = 1, 2 ) , (p*(t)p(t'))=(Q/Tp)exp(-It-t'l/Tv), Ri=lRj[ exp(i0j), Ej=lEjl exP(i~oj), 0=1,2) ~0(t) =~02 -~ol.

(2) (3)

(4) (5)

Here El and E2 are the complex dimensionless amplitudes of the two counter-propagating modes, a~ and a2 are their corresponding real p u m p parameters, Rt and R2 are the corresponding dimensionless complex backscattering coefficients, q~ and q2 are the ~-correlated additive noise terms representing spontaneous emission fluctuations, p is the colored multiplicative noise term representing the pump fluctuations, Q is the strength of the pump noise, and TD is the correlation time associated with t h e p u m p fluctuations. We include the multiplicative noise in our equations so that the model is applicable to a dye laser in which ,pump fluctuations are significant. TypiCal values for Q and Tp in our simulations are 100 and 0.5, based on recent experimental measuremerits on a Rhodamine 6G dye ring laser pumped by the 514.5 nm line of an argon ion laser [18]. The 46

1 April 1991

),

(6)

where ~ is the cross coupling constant and a (=a~ =a2) is the pump parameter. For a homogeneously broadened laser, ~ is equal to 2 and Re is equal to a/2x/~.

3. Results and discussion

In fig. I a the intensities obtained from a computer solution of eqs. ( l ) - ( 5 ) show anticorrelated oscillations for "off phase" backscattering with 01+02=(2n+l)Tt. The pump parameter a ( = a l = a 2 ) is 100 and Re computed from eq. (6) is 20.4. The backscattering coefficient IR I ( = IRl I = IR21 ) is 500 and is larger than Re. The period of the harmonic oscillation has been determined to be 7r/IRI, just as for an inhomogenously broadened laser [ 13 ]. Fig. I a shows that the maxim u m value of the intensities is about 80, with a small fluctuation. In this case of large IR I and the laser far above threshold (large a), the effect of multiplicative noise is small and our results show good agreement with the deterministic results [6]. However, when the laser is not far above threshold (small a), the effect of multiplicative noise is significant. Although the period of the oscillations remains the same as zt/JR I, the maximum value of the intensities fluctuates in a much larger scale. Fig. lb displays the relative phase ~0 of the two intensities versus dimensionless time. It shows phase jumps of _+~ronly when one of the two intensities becomes zero. The results also show that the oscillations become anharmonic when [RI is decreased and becomes close to Re. In order to study the behaviour of the intensities and the relative phase when IR] is near the critical value Re, simulations have been carried out with both

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OPTICS COMMUNICATIONS

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Fig. 1. Plots of (a) dimensionless intensities and (b) relative phase ~(t) versus dimensionless time from numerical integration ofeqs. ( 1)-(4) with dimensionless iteration step size equal to 0.000 001, a~=az=a= 100, R~--R2----IRI mS00, Q= 100, and Tp=0.5. IRI is larger than Rc (=20.4). The two intensities exhibit anti-correlated oscillations. additive and multiplicative noise and with only additive noise. Figs. 2 and 3 show results with additive noise only. In fig. 2, the p u m p parameter a and the magnitude o f the backscattering coefficient ]R I are respectively 100 and 20. F r o m ¢q. (6), Rc is equal to 20.4 which is larger than IRI. In fig. 2a the two modes show the effect of r a n d o m switching. It is well known [ 1-4 ] that with no backscattering, when one mode switches on the other mode switches off. The mode is said to be " o f f " because the most probable intensity is zero. For our case with backscattering, when one mode is " o n " , the most probable intensity for the " o f f " mode is not exactly zero, but is very small compared to the intensity o f the " o n " mode. The magnitude o f this most probable intensity is found to increase with increasing strength o f backscattering. For this case with no p u m p fluctuations and IRI
-2/l;

= 0.0

I

I

I

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Fig. 2. Plots of (a) dimensionless intensities and (b) relative phase versus dimensionless time as in fig. l with iteration step size equal to 0.000 01, Qffi0,al =a2= 100, and Rl =R2= IRI =20. The two intensities exhibit random switching. sities o f the " o n " and " o f f " modes are similar to the ones obtained in the deterministic analysis [ 6 ]. Fig. 2b shows that the relative phase o f the two intensities exhibits jumps o f + 7z during mode switching, when one o f the intensities becomes zero. In fig. 3, a and [RI are respectively 100 and 21, so that this time IRI is larger than Rc ( = 2 0 . 4 ) . Fig. 3a shows that the intensities regularly turn " o n " and "off". By comparing fig. 2a and fig. 3a, we see clearly that when the magnitude o f the backscattering coefficient is close to the critical value Re, a small change in IRI may result in a substantial change in the dynamics o f the laser intensities. Fig. 3b shows phase jumps o f _+~ which are similar to fig. 1b. For fig. 4 and fig. 5, both additive and multiplicative noises terms are included in the simulations. 47

Volume 82, number 1,2 • (a/

120

l

96

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I

l

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Fig. 3. Plots of (a) dimensionless intensities and (b) relative phase versus dimensionless time as in fig. 2 with iteration step size=0.000 01, Q=0, a~ =a2= 100, and Ri =R2= IRI =21. The parameters used for figs. 4 and 5 are the same as for figs. 2 and 3. Q and Tp are respectively equal to 100 and 0.5. In fig, 4, IR I ( = 20) is less than R~ ( = 2 0 . 4 ) and in fig. 5, IR[ = (21) is larger than Re. Fig. 4a and 5a show that the two intensities exhibit both r a n d o m switching and periodic oscillations. When IR I is larger t h a n / ~ , the laser tends to operate more often in the oscillation region than in the switching region. Conversely when IRI is less than Re, it tends to operate more often in the switching region than in the oscillation region. In eq. ( 1 ), the effect of p u m p fluctuations on backscattering can be interpreted as the replacement of the constant p u m p parameter a by a r a n d o m p u m p parameter (7)

where a ( = a t = a 2 ) is the constant p u m p parameter 48

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(b)

1 April 1991

-2/i; t J i o t 0.0 3.2 6.4 9.6 dimensionless time t

I

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Fig. 4. Plots of (a) dimensionless intensities and (b) relative phase versus dimensionless time as in fig. 1 with iteration step size equal to 0.000 01, Q= 100, Tp=0.5, al=a2=100, and Ri =R2= IRI =20. and p is the multiplicative noise term. Eq. (6) then has to be modified ~--l R~ = 2x/2(~ + 1 ) A = 2

~--1 ~

(a+v).

(8)

Rc can be viewed as having a r a n d o m component which is determined by the p u m p fluctuations. Whether the laser operates in the switching region or the oscillation region depends on whether Rc is smaller or larger than the backscattering coefficient IRt. We can interpret figs. 4a and 5a to mean that the laser is being indirectly driven by the multiplicative noise to operate between the r a n d o m switching region and the periodic oscillation region. In all these cases (figs. 2b, 3b, 4b, and 5b) n-phase j u m p s

Volume 82, number 1,2

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1 April 1991

-0.4

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Fig. 6. Plots of the normalizedcross-correlationfunctionA~ versus the backscattering coefficient IRI for a= 100, Q= 100, and Tp= 0.5. The change ofA 12across R¢ is smooth.

IRI > Rc, Q = 100

-7~

•

"1

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Fig. 5. Plots of (a) dimensionless intensities and (b) relative phase versus dimensionlesstime as in fig. 4 with iteration step size=0.000 01, Q=100, Tp=0.5, al=a2=100, and RI= R=-- IRI =21. are f o u n d to occur in b o t h switching and oscillation

regions, when either one of the intensities becomes zero. Fig. 6 shows a plot of the normalized cross-correlation function 212 [4 ] for the two intensities versus [R I in the " o f f " phase case. The pump parameter is 100 and Rc is again 20.4. When IRI is larger than Re, the two intensities show anticorrelated oscillations and 212 increases first rapidly and then slowly toward - 0.5 with increasing IR I. The value of 212 decreases sharply toward - 1 . 0 when [RI is decreased below Re, but the change of 212 across Re is smooth. The numerical simulations also indicate that the change of 212 at IRI =Re is smooth even when the pump fluctuations are neglected, probably due to the contribution of additive noise. Including

p u m p fluctuations in the simulations causes the transition at [RI =Re to become even smoother. It has been known that the effect of pump fluctuations on the intensity cross correlation of a homogeneously broadened laser without backscattering is significant only when the laser is close to threshold [ 4 ]. When the laser is far above threshold (large a) and [RI =0, the theory with and without pump fluctuations [ 4,1 ] both predict a value of - 1 for the intensity cross correlation. This is consistent with our simulation results shown in fig. 6. When the laser is close to threshold (small a), the pump fluctuations dominate the contribution to the intensity cross correlation and the transition between the two regions cannot be observed. For an inhomogeneously broadened laser tuned to line center (¢= 1 ), with equal pump parameters and with " o f f phase" backscattering, 212 is independent of [R[ [9, 12]. The dependence of 212 on [R [ for a homogeneously broadened laser with "off phase" backscattering can be attributed to the strong coupling between the modes (¢=2). When the phase of the backscattering is slightly detuned from the "off phase" case, numerical simulations also show 7t-phase jumps whenever one of the intensities is driven to zero by the noise. The probability of observing the jumps in this case is much less than for the exactly "offphase" case. These conditions are similar to those obtained for an inhomogeneously broadened laser [ 13 ].

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4. Conclusions In conclusion, our numerical simulations show that it-phasejumps can occur in a homogeneously broadcncd ring laserwhen the backscatteringis "off phase" or close to "off phase'. The resultsdemonstrate that the laserwith "off phase" backscattering can operate in two regions, a switching region or an oscillation region, depending on the magnitude of the backscatteringcoefficient.Noise-induced transitionsfrom one region to another have been observed numerically.For the case of"in phase" backscattcring, ncithcr 7r-phase jumps nor intensity oscillations have bccn found in our simulations. The anti-correlated intensity oscillations have been observed experimentally in a dye ring laser with backscattering [5 ]. However, to our knowledge, 7t-phasejumps have not yet bccn observed in any homogeneously broadened laser. W e suggest that a typical dye ring laser with carefully adjusted backscattering should bc able to exhibit it-phasejumps and confirm our results.

Acknowledgement The author is grateful to Dr. L. Mandel for his guidance and support. The author is much indebted to Dr. T.H. Chyba for fruitful discussions and for providing the computer programs. The author thanks B.E. Magill for modifying one of the programs. This work was supported by grants from the National Sci-

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1 April 1991

ence Foundation and from the Office of Naval Research.

References [1] Surendra Singh and L. Mandel, Phys. Rev. A 20 (1979) 2459. [2 ] Rajarshi Roy and L. Mandel, Optics Comm. 34 (1980) 133. [ 3 ] P. Lett, W. Christian, Surendra Singh and L. Mandel, Phys. Rev. Lett. 47 ( 1981 ) 1892. [ 4 ] P. Lett and L. Mandel, J. Opt. Soc. Am. B 2 ( 1985 ) 1615. [ 5 ] S. Schmter and D. Kuhlke, Opt. Quant. Elect. 13 ( 1981 ) 247. [6] D. Kuhlke and G. Jetschke, Physica C 106 ( 1981 ) 287. [ 7 ] D. Kuhlke and R. Horak, Physica C 111 ( 1981 ) 111. [ 8 ] Rajarshi Roy and L. Mandel, Optics Comm. 35 (1980) 247. [9]W.R. Christian and L. Mandel, Phys. Rev. A 34 (1986) 3932. [ l0 ] W.R. Christian and L. Mandel, }. Soc. Opt. Am. B 5 (1988) 1406. [ l 1] W.R. Christian, T.H. Chyba, E.C. Gage and L. Mandel, Optics Comm. 66 (1988) 238. [ 12] L. Pesquera, R. Blanco and M.A. Rodriguez, Phys. Rev. A 39 (1989) 5777; L. Pesquera and R. Blanco, Optics Comm. 74 (1989) 102. [13] T.H. Chyba, Phys. Rev. A40 (1989) 6327. [ 14] T.H. Chyba, Optics Comm. 76 (1990) 395. [ 15 ] R.J.C. Spmeuw, R. Centeno Neelen, N.J. van Druten, E.R. Eliel and J.P. Woerdman, Phys. Rev. A 42 (1990) 4315. [ 16 ] C.O. Weiss, N.B. Abraham and U. Hubner, Phys. Rev. Lett. 61 (1988) 1587. [17] L.M. Hoffer, G.L. Lippi, N.B. Abraham and P. Mandel, Optics Comm. 66 (1988) 219. [ 18] E.C. Gage and L. Mandel, Phys. Rev. A 38 (1988) 5166. [ 19 ] E.C. Gage and L. Mandel, J. Soc. Opt. Am. B 6 (1989) 287.