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Unidirectionality of an optically pumped far infrared ring laser
15 June 1995
OPTICS COMMUNICATIONS Optics Communications
ELSEVIER
117 ( 1995) 462-468
Unidirectionality of an optically pumped far infrared ring laser Kyoji Matsushima a, Noriyoshi Higashidab,
Noburu Sokabe b, Tomio Ariyasu a
a Department of Electrical Engineering, Kansai University, 3-3-35 Yamate-cho. Suita 564, Japan b Department of Applied Physics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558, Japan Received
21 July 1994; revised version received 21 March 1995
Abstract
An experimental and theoretical investigation has been made on the unidirectional operation of an optically pumped far infrared ring laser. A ring laser operating on the 119 ,um line of CH30H experiences reversal of output direction in either case of (a) the pump frequency being tuned across the line center of the infrared pump transition or (b) the fir cavity being tuned across the far infrared line center. A model based on two-mode laser theory predicts the output directionality of the optically pumped fir ring laser.
1. Introduction
Optically pumped far infrared lasers (OPFRL) have attracted considerable attention owing to such interesting behavior as temporal chaos and an unidirectional operation. Lorenz chaos [l] was report on a unidirectional NH3 ring laser for the first time [ 2-41. Unidirectionality of the output is another peculiarity of OPFRLs. Experimental and theoretical investigations have been devoted to temporal chaos in relation to instability of lasers [ 5,6], and unidirectional operation of OPFRL has also attracted much attention. It was already suggested that unidirectional operation has its origin in gain anisotropy as well as in competition between the two counter-propagating fir fields [ 71. The gain anisotropy due to the asymmetric velocity distribution of the molecules optically pumped by a pump laser has been investigated extensively. More detailed studies on the mode competition due to the nonlinearity in lasers as a self-sustained oscillator, however, is necessary to understand the mechanism of the output directionality of OPFRL.
0030-4018/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIOO30-4018(95)00189-l
A recent theoretical investigation on output directionality of an OPFRL [ 81 has revealed a new type of anisotropy which gives rise to stationary bi-directional output. The bi-directional output has been observed experimentally on an NH3 optically pumped ring laser operating in mid infrared region [ 91. Matrix continued fractions expansion [lo] was applied to calculate fir gains and stationary fir intensities in an OPFRL model of Doppler broadened three-level system including Raman type process. However, stability of the stationary solutions therein reported has never been examined. In this paper, experimental as well as theoretical studies on a ring laser operating on the 119 ,um line of CH30H are reported. A model of a three-level system interacting with the pump wave and the two counterpropagating fir waves is analysed. The intensitydetermining equations are solved to third order of perturbation. The competition between two fir modes emitting in forward and backward directions is shown to play an important role in output directionality.
K. Mats&ha
et al./ Optics Communications
117 (1995) 462468
PA SIGNAL
463
FORWARD I
I I
1
I
1
Fig. 1. Experimental set-up of an optically pumped FIR ring laser. PM: plane mirror, pyroelectric detector, MIC: microphone, PC: personal computer.
CM: concave
SIGNAL /
mirror, W: NaCl window,
b(2)
:
2. Experimental
A scheme of the experimental setup for an optically pumped molecular ring laser is shown in Fig. 1. The ring cavity was installed inside a vacuum vessel. The cavity consists of two concave mirrors (Ml, M2) and a plane mirror (M3). Each of the concave mirrors has a 1.5mm bore central coupling hole. The infrared pump light comes into the ring cavity through the hole on Ml. The forward fir output is coupled out of the cavity through the hole on M2, while the backward output through the hole on Ml. The mirror M2 is mounted on a precision mechanical translation stage which facilitates frequency tuning of the ring cavity. The fir outputs were detected with pyroelectric detectors (D1 and D2) and recorded on a personal computer for further processing the signal. The pump laser is a conventional cw CO2 laser delivering a maximum output power of approximately 10 W on the 9P(36) line which produces CH30H 119 ,ccm fir laser line. Photoacoustic absorption signal was monitored with a microphone installed inside the vessel. A typical recording of the 119 ,um output as obtained with the pump frequency scanned over the entire FSR of the pump laser is shown in Fig. 2 for
IOfiHz
_
Pump Detuning
+
Fig. 2. 119 ,um output power versus pump detuning. CH30H vapour pressure was 17 Pa and the maximum pump power was 8.5 W. The fir cavity detuning is positive.
CHsOH pressure of 17 Pa. For large negative pump detunings, the ring laser emits its output only in backward direction. The output direction reversed, regardless of the ring cavity detuning from the fir resonance, when the pump laser frequency was tuned across the
K. Matsushima et al./ Optics Communications I1 7 (1995) 462-468
464
Fig. 4. A model for an optically
-
> Pump Detuning
pumped
ring fir laser.
Rabi frequencies of the fir waves are denoted by fK and 2ru”’ (S = + or -), respectively. The positive sign is for the fir wave copropagating with the pump wave, and the negative sign for the counterpropagating one. Diagonal elements pii of density matrix satisfy
+
Fig. 3. Forward (dotted line curve) and backward (solid line curve) output power versus pump detuning. Fir cavity detuning (A > 0) decreases from (a) to (c). CH3OH vapour pressure was 17 Pa.
line center of the pump transition as shown in Fig. 3. The fir cavity detuning in Fig. 3 is positive and decreases gradually from (a) to (c) . The forward output power takes a maximum at different pump frequency depending on fir detuning. The ring laser reverses its output direction at a definite pump frequency, which is in close coincidence with the center frequency of the infrared pump transition of the 119 ,ccm line, regardless of the fir detuning. The pump offset of this line has been reported to be +23 MHz [ 111.
3. Analysis
dpW/dt _ ia dpllldt
= -ya(pa~~ - p&) - ia’+‘(ab+l’ - ug’*) (o-6;’ _ o-b;’ *) = -Yl(PII
-
= -y2(~22
-
(1)
cG2),
py,>
+ia’+‘(ab+l’-ab+l’*) dPz/dt
iP(~02
+ict+)(a~;’
- ~$2)
+ iP(c02
--CT&)*),
(2)
-
(3)
a$2>,
where yi (i = 0, 1,2) are energy relaxation constants of the state i. CT;?are slowly varying amplitudes of the off-diagonal elements pij defined as pat E cr&‘exp [-i(f2t +a&’
exp l-i(0t
~02 = mexp
[-i(W
- Kz)] + Kz)l -
,
KPZ)] ,
P12--~exp[-i[(f2~-f2)t-(Kp-K)~]] +c+i;‘exp[--i[(@--Wt-(Kp+K)zll.
3.1. A model and basic equations Fig. 4 shows a three-level scheme for optically pumped FIR lasers. The pump transition 2-O and the fir transition O-l share common upper level 0. Both transitions are assumed to be Doppler broadened. The laser medium uniformly fills the ring cavity. The infrared pump wave with an angular frequency of L& and an wavenumber of Kp couples with the 2-O transition with Rabi frequency 2& while co- and counterpropagating fir waves with the same frequency D couple with the O-l transition. The wavenumbers and
In rotating wave approximation, d@/dt
= -yera~~’
+ i(AF
+’
satisfy
Kv)a~~’
- ia’*‘(poo - ~11) + ipa\:)*, daon/dt
(4)
= -yo2~02 + i( Ap - KPV)C~O~
- i/3(p00- ~22) + icy’+‘ui;~+ ia'-'cri;'
,
(5)
dcr;;’ /dt = - y12a’,;’ + i [(AP - A) - (KP F K)vl c$’ + ia’ l ) ~702 - iflab:’ * ,
(6)
K. Matsushima Table 1 Definitions
of the quantities
et al./ Optics Communications
1 I7 (1995) 462468
465
used in the present analysis
00 g(f)
3 7s
Ll(A 7 Kv)
B(Ap
-
s
Ku)~B(A~
KPU)~(U)
du -
YC
I
--m M
p
--
EYOl YO
Ll(d~
Yo+Yl -
- Kp)
- EB(Ap
2Yl
- Kpu)
f(v)dv I
YO +YI --
Ll(A+Ku)L,(A-Ku)B(Ap-Kpu) [
zB(Ap
- Kpu)
a1
f(“) _exp [-(“Iu)21
B(n) 2YO2(YO + Y2yz) P2
I(*)
_
=2
f(u)
du
DO(u) EdOf(u)
ufi
f2=d2+
1
Y02P2
f
ro(x2+~)
4(cu’*‘)2
YOY2
701
where A (= 0 - w) and Ap (= 0p - wp) are the fir and the pump detunings, respectively, and yij is the relaxation constant of coherence between the states i and j. We, herein, assume that 712 > (yet, ym), i.e. Raman-type two photon process is ignored. The stationary inversion density N(u) du ( = (~00 - ptt ) du) of the molecules with the velocity component of u along the optical axis is given by
where Li (x) (i = 1,2), I’*) and Zp are the dimensionless Lorentzian function, dimensionless intensities of fir and ir radiation, respectively. Table 1 summarizes the notation. Since the amplitude of the induced polarization oscillating at the fir frequency 0 leads to PC*] = Nt JY” PO1 PO1 (*) du, the fir intensity determining equations [ 121 lead to dZ’*‘/dt = -ycZ’*’
N(v) dv =
dv = f, + _f2[LI (A - Ku)l’+’ + L1 (A + Ku)Z(-)] ]fso”(4
+ fd%Gl
1
2EOYOI Ji
(7) whereI@ dv (- (p&,-p:,) dv) andDO dv (= ( p!&.- p&) dv) are the inversion density between the states 0 and 1 and the population difference between the states 2 and 0 in the absence of ir and fir fields in the cavity, respectively. ft, fz and fs are defined as
co
Yw$I’
J -CO
Ll (A F Ku) N(u) du,
where Nt and yc are the totalmolecular density and the cavity decay constant for the fir power, respectively. To the third order approximation in perturbation, Eq. (7) leads to N(v) dv = f3Do(v> 1 --
x
(
fl
f’[& f:
(A
-
Ku)z’+’
+ L1 (A + Kv)Z’-‘1
dv, >
f3
=
$2(Ap
- Kpv)Ip,
(10)
(11)
(12)
where it has been assumed that A@(v) = 0 at room temperature. Accordingly, the intensity determining equations for the forward and the backward fir waves are given respectively by
K. Matsushima et al./ Optics Communications I I7 (I 995) 462-468
466 Table 2 Stationary
solutions
of Eq. (13) and its stability Stationary
solutions
Conditions
for stability
(i)
I’+’ = 0,
I’-’
= 0
g’+’ 5 0, g’-’
(ii)
I(+) = g’+‘/s’+’
I’-’
zz 0
{g’+’ > 0, g’-’
< - 0) or {G(+) > 0, G(-)
5 0)
(iii)
I’+’ = 0,
I’-’
= g’-‘/s’-’
(g’+’ 5 0, g(-)
> 0} or {G(+) 5 0, G’-)
> 0}
(iv)
I’+’ = G’+‘/(r’+‘@,
I’-’
= G’-‘/(r’-‘0)
G(+) > 0, G’-’
> 0
G’*’ =
-_g
(*)
_qg'F,/s'W
Table 3 Cavity and molecular
parameters
and
5 0
0 E 1 - q2/(s(+)s’-‘)
used for numerical
calculations.
Cavity decay constant Molecule FIR wavelength Pump wavelength Relaxation constants
Dephasing
is ignored,
i.e. ‘yo2 = (70 + yz)/2 1 x 10-Z 7s
YC
%HsOH A AP -YOl ? YO 3 YI Y2 YOZ
Dipole matrix element for the FlR transition Dipole matrix element for the IR transition Most probable speed of molecules
PO1
/-Wz
0.69 1.57 1.13 2.2 4.0
U
119pm 9.7 pm x lo6 (s Pa)-’ x lo6 (s. Pa)-’ x lo6 (s. Pa)-* x 10W30 C. m x 10p31 C . m 395 m/s
(*) (*) (*) (*) (*)
* Ref. 1151. &‘+‘/dr
= I’+)(g(+)
df’-‘/&
= f’-‘@-’
_ S’+‘~‘+’ _ @-‘), _ $-‘I’-’
_ @‘+‘),
(1W (13b)
where g’*j are the net gains, s(*) are self-saturation coefficients, and q is cross-saturation coefficient. The coefficients g(*), s(*) and q are defined in Table 1.
experimental conditions. Output directionality of the 119 pm ring laser is determined by the values of g’“‘, s(*) and q. The stable stationary solutions for fir intensities are calculated from the parameters summarized in Table 3.
3.3. Directionality 3.2. Stability of the solutions Eqs. (13) are similar to those appeared in the twomode laser model [ 131. In steady-state regime, stability of the solutions to the intensity determining equations ( 13) is examined just in the same way as that for the two-mode laser. For the coupling constant (5 = q2/s(+)s(-‘) smaller than unity, i.e. for “weak coupling”, the solutions and their stability are summarized in Table 2. Solution (i) is trivial. Solution (iv) corresponds to bi-directional operation which is realized in conventional laser-gyroscopes [ 141. Solutions (ii) and (iii) correspond to unidirectional operation due to mode competition which suppresses simultaneous oscillation of the forward and the backward waves. For the CH30H 119 pm line, 5 < 1 under the present
The threshold condition plotted on the parameter plane spanned by cavity and pump detunings is shown in Fig. 5. The net gain is 0 alone the outer contour. In region B wherein solution (iii) is stable, the backward output is preferred, while in region F the forward output is expected. Bi-directional output is expected in region B & F wherein solution (iv) is stable. No lasing is expected outside the regions denoted by B, F andB&F. Calculated fir intensities as a function of dimensionless pump detuning are plotted in Fig. 6 for different cavity detunings. Fig. 6 predicts the reversal of output direction which is in agreement with the experimental results. It also predicts that the reversal of the output direction occurs when the pump frequency is tuned
K. Matsushima
lp=l.O
et al/Optics
Communications
, A/r n,
II7 (1995) 462-468
-16
-8
467
0
AP’T
8
16
02
Fig. 6. Forward (I’-‘) and backward (I(+)) fir intensities calculated from Eq. (13) versus pump dehming for different fir cavity detunings. Ip = 0.5.
Fig. 5. Calculated output directionality on (Ap, A) plane for different pump intensities. F: Forwardly unidirectional output, B: Backwardly unidirectional output, FL? B: bi-directional output.
across the resonance (dp = 0). Bi-directional output is expected for pump detuning around the resonance and the range of pump detuning for which bi-directional output is obtained decreases with a decrease in pump intensity. As the dimensionless pump intensity (Zp) is estimated to be 0.3 to 0.5 in our experiments, a narrow bi-directional region of pump detuning is expected. However, bi-directional output has not been detected, possibly owing to rather poor resolution of cavity scanning of the pump laser. 4. Discussion The 119 ,um CHsOH OPFRL operates unidirectionally due to the mode competition between the counterpropagating fir waves. In ordinary ring gas lasers, co- and counter-propagating waves in the cavity do not strongly couple with each other except for central cavity tuning. For central cavity tuning and a low rotation rate, the two travelling waves strongly cou-
ple with each other. Therefore, mode competition and frequency locking due to back scattering may occur between the two waves [ 161. In optically pumped farinfrared ring lasers, two fir waves in the cavity strongly couple with each other regardless of fir cavity detuning. This may be due to the fact that the velocity distribution of the optically pumped molecules is limited in a narrow region characterized by the homogeneous width of the pump transition. The CHsOH OPFRL reverses its output direction when the pump frequency is scanned across the ir line center. The reversal occurs irrespective of ring cavity detuning. Therefore, the reversal of output direction may find its application in frequency reference in the ir region of spectrum. Several narrow absorption lines have been utilized as absolute frequency references for frequency stabilized lasers. Usually, the laser frequency is weakly modulated in order to detect the center of the absorption line and to get discrimination signal for frequency stabilization. However, it is preferable to avoid frequency modulation for some metrological purposes [ 171. We may be able to detect the sign of pump frequency deviation from the line center of pump transition by monitoring the output direction of an OPFRL.
468
K. Matsushima et al./ Optics Communications 117 (i995) 442468
The present analysis has show that the direction reversal occurs at the line center of the pump transition. It is noted that Raman type process is ignored in this analysis. Though the contribution of Raman type process is small because of a large ratio of Kp/K [ 81, it gives rise to a slight shift in pump frequency, for which the output direction reverses, from the ir line center [ 18,191.
5. Conclusion An experimental and theoretical investigation has been made on an optically pumped CHsOH fir ring laser operating on the 119 pm line. The ring laser operated unidirectionally and experienced abrupt reversal of the output direction when the pump frequency was tuned across the line center of the ir molecular resonance. Intensity determining equations have been derived for a model of Doppler-broadened three-level optically pumped ring laser and stability of the stationary solutions has been examined. Mode competition has been shown to be responsible for the output unidirectionality under off resonance pumping. The present model also predicts that the direction preference is dependent on the pump as well as the cavity detunings. An application to absolute frequency reference for ir lasers has been suggested.
Acknowledgements The authors would like to thank T. Matsumoto his assistance in the computer work.
for
References [ 11 H. Haken, Phys. L&t. A 53 (1975) 77. [2] CO. Weiss and W. Klishe, Optics Conuu. 51 (1984) 47. 131 C.O. Weiss, W. Klische, P.S. Ering and M. Cooper, Optics Comm. 52 (1985) 15. [4] CO. Weiss and J. Brock, Phys. Rev. Lett. 57 (1986) 2804. [5] N.B. Abraham, L.A. Lugiato and L.M. Narducci, J. Opt. Sot. Am. B 2 (1985) 7. [6] D.J. Biswas, R.G. Harrison, CO. Weiss, W. Klische, D. Dangoisse, P. Glorieux and N.M. Lawandy, in: Instabilities and Chaos in Quantum Optics, eds. FT. Arecchi and R.G. Harrison (Springer, Berlin, 1987) p. 109. [7] J. Heppner and C.O. Weiss, Appl. Phys. Lett. 33 (1978) 590. [S] R. Vilaseca, J.Martomll, RCorbalan and P Laguarta, Optics Comm. 70 (1989) 131. [9] W. Wazen and G.L. Bourdet, IEEE J. Quantum Electron. QE27 (1991) 152. [lo] R. Vilaseca, G. Orriols and L. Roso, Appl. Phys. B 34 (1984) 73. [ 1 l] M. Inguscio, E Stmmia and J.O. Henningsen, in: Reviews of Infrared and Millimeter Waves, Vol. 2, eds. K.J. Button, M. Inguscio and F. Strumia (Plenum Press, New York, 1984) p. 119. [ 121 M. Sargent III, M.O. Scully and Jr. W.E. Lamb, Laser Physics (Addison-Wesley, Masachusetts, 1974) p. 100. [ 131 M. Sargent III, M.O. Scully and Jr. W.E. Lamb, Laser Physics (Addison-Wesley, Masachusetts, 1974) p. 120. [ 141 W.M. Macek and Jr. D.T. Davis, Appl. Phys. Lett. 2 (1963) 63. [ 151 J. Heppner, C.O. Weiss, II. Htibner and G. S&inn, IEEE J. Quantum Electron. QE-16 (1980) 392. ] 161 F. Aronowitz and R.J. Collins, Appl. Phys. I&t. 9 (1966) 55. [ 171 A. Onae, T. Kurosawa, Y. Miki and E. Sakuma, Optics Comm. 88 (1992) 110. [ 181 N. Skribanowitz, MS. Feld,’ R.E. Francke, M.J. Kelly and A. Javan, Appl. Phys. Lett. 19 (1971) 161. [ 191 N. Skribanowitz, 1.P Herman, Jr. R.M. Osgood, M.S. Feld and A. Javan, Appl. Phys. Lett. 20 (1972) 428.
OPTICS COMMUNICATIONS Optics Communications
ELSEVIER
117 ( 1995) 462-468
Unidirectionality of an optically pumped far infrared ring laser Kyoji Matsushima a, Noriyoshi Higashidab,
Noburu Sokabe b, Tomio Ariyasu a
a Department of Electrical Engineering, Kansai University, 3-3-35 Yamate-cho. Suita 564, Japan b Department of Applied Physics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558, Japan Received
21 July 1994; revised version received 21 March 1995
Abstract
An experimental and theoretical investigation has been made on the unidirectional operation of an optically pumped far infrared ring laser. A ring laser operating on the 119 ,um line of CH30H experiences reversal of output direction in either case of (a) the pump frequency being tuned across the line center of the infrared pump transition or (b) the fir cavity being tuned across the far infrared line center. A model based on two-mode laser theory predicts the output directionality of the optically pumped fir ring laser.
1. Introduction
Optically pumped far infrared lasers (OPFRL) have attracted considerable attention owing to such interesting behavior as temporal chaos and an unidirectional operation. Lorenz chaos [l] was report on a unidirectional NH3 ring laser for the first time [ 2-41. Unidirectionality of the output is another peculiarity of OPFRLs. Experimental and theoretical investigations have been devoted to temporal chaos in relation to instability of lasers [ 5,6], and unidirectional operation of OPFRL has also attracted much attention. It was already suggested that unidirectional operation has its origin in gain anisotropy as well as in competition between the two counter-propagating fir fields [ 71. The gain anisotropy due to the asymmetric velocity distribution of the molecules optically pumped by a pump laser has been investigated extensively. More detailed studies on the mode competition due to the nonlinearity in lasers as a self-sustained oscillator, however, is necessary to understand the mechanism of the output directionality of OPFRL.
0030-4018/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIOO30-4018(95)00189-l
A recent theoretical investigation on output directionality of an OPFRL [ 81 has revealed a new type of anisotropy which gives rise to stationary bi-directional output. The bi-directional output has been observed experimentally on an NH3 optically pumped ring laser operating in mid infrared region [ 91. Matrix continued fractions expansion [lo] was applied to calculate fir gains and stationary fir intensities in an OPFRL model of Doppler broadened three-level system including Raman type process. However, stability of the stationary solutions therein reported has never been examined. In this paper, experimental as well as theoretical studies on a ring laser operating on the 119 ,um line of CH30H are reported. A model of a three-level system interacting with the pump wave and the two counterpropagating fir waves is analysed. The intensitydetermining equations are solved to third order of perturbation. The competition between two fir modes emitting in forward and backward directions is shown to play an important role in output directionality.
K. Mats&ha
et al./ Optics Communications
117 (1995) 462468
PA SIGNAL
463
FORWARD I
I I
1
I
1
Fig. 1. Experimental set-up of an optically pumped FIR ring laser. PM: plane mirror, pyroelectric detector, MIC: microphone, PC: personal computer.
CM: concave
SIGNAL /
mirror, W: NaCl window,
b(2)
:
2. Experimental
A scheme of the experimental setup for an optically pumped molecular ring laser is shown in Fig. 1. The ring cavity was installed inside a vacuum vessel. The cavity consists of two concave mirrors (Ml, M2) and a plane mirror (M3). Each of the concave mirrors has a 1.5mm bore central coupling hole. The infrared pump light comes into the ring cavity through the hole on Ml. The forward fir output is coupled out of the cavity through the hole on M2, while the backward output through the hole on Ml. The mirror M2 is mounted on a precision mechanical translation stage which facilitates frequency tuning of the ring cavity. The fir outputs were detected with pyroelectric detectors (D1 and D2) and recorded on a personal computer for further processing the signal. The pump laser is a conventional cw CO2 laser delivering a maximum output power of approximately 10 W on the 9P(36) line which produces CH30H 119 ,ccm fir laser line. Photoacoustic absorption signal was monitored with a microphone installed inside the vessel. A typical recording of the 119 ,um output as obtained with the pump frequency scanned over the entire FSR of the pump laser is shown in Fig. 2 for
IOfiHz
_
Pump Detuning
+
Fig. 2. 119 ,um output power versus pump detuning. CH30H vapour pressure was 17 Pa and the maximum pump power was 8.5 W. The fir cavity detuning is positive.
CHsOH pressure of 17 Pa. For large negative pump detunings, the ring laser emits its output only in backward direction. The output direction reversed, regardless of the ring cavity detuning from the fir resonance, when the pump laser frequency was tuned across the
K. Matsushima et al./ Optics Communications I1 7 (1995) 462-468
464
Fig. 4. A model for an optically
-
> Pump Detuning
pumped
ring fir laser.
Rabi frequencies of the fir waves are denoted by fK and 2ru”’ (S = + or -), respectively. The positive sign is for the fir wave copropagating with the pump wave, and the negative sign for the counterpropagating one. Diagonal elements pii of density matrix satisfy
+
Fig. 3. Forward (dotted line curve) and backward (solid line curve) output power versus pump detuning. Fir cavity detuning (A > 0) decreases from (a) to (c). CH3OH vapour pressure was 17 Pa.
line center of the pump transition as shown in Fig. 3. The fir cavity detuning in Fig. 3 is positive and decreases gradually from (a) to (c) . The forward output power takes a maximum at different pump frequency depending on fir detuning. The ring laser reverses its output direction at a definite pump frequency, which is in close coincidence with the center frequency of the infrared pump transition of the 119 ,ccm line, regardless of the fir detuning. The pump offset of this line has been reported to be +23 MHz [ 111.
3. Analysis
dpW/dt _ ia dpllldt
= -ya(pa~~ - p&) - ia’+‘(ab+l’ - ug’*) (o-6;’ _ o-b;’ *) = -Yl(PII
-
= -y2(~22
-
(1)
cG2),
py,>
+ia’+‘(ab+l’-ab+l’*) dPz/dt
iP(~02
+ict+)(a~;’
- ~$2)
+ iP(c02
--CT&)*),
(2)
-
(3)
a$2>,
where yi (i = 0, 1,2) are energy relaxation constants of the state i. CT;?are slowly varying amplitudes of the off-diagonal elements pij defined as pat E cr&‘exp [-i(f2t +a&’
exp l-i(0t
~02 = mexp
[-i(W
- Kz)] + Kz)l -
,
KPZ)] ,
P12--~exp[-i[(f2~-f2)t-(Kp-K)~]] +c+i;‘exp[--i[(@--Wt-(Kp+K)zll.
3.1. A model and basic equations Fig. 4 shows a three-level scheme for optically pumped FIR lasers. The pump transition 2-O and the fir transition O-l share common upper level 0. Both transitions are assumed to be Doppler broadened. The laser medium uniformly fills the ring cavity. The infrared pump wave with an angular frequency of L& and an wavenumber of Kp couples with the 2-O transition with Rabi frequency 2& while co- and counterpropagating fir waves with the same frequency D couple with the O-l transition. The wavenumbers and
In rotating wave approximation, d@/dt
= -yera~~’
+ i(AF
+’
satisfy
Kv)a~~’
- ia’*‘(poo - ~11) + ipa\:)*, daon/dt
(4)
= -yo2~02 + i( Ap - KPV)C~O~
- i/3(p00- ~22) + icy’+‘ui;~+ ia'-'cri;'
,
(5)
dcr;;’ /dt = - y12a’,;’ + i [(AP - A) - (KP F K)vl c$’ + ia’ l ) ~702 - iflab:’ * ,
(6)
K. Matsushima Table 1 Definitions
of the quantities
et al./ Optics Communications
1 I7 (1995) 462468
465
used in the present analysis
00 g(f)
3 7s
Ll(A 7 Kv)
B(Ap
-
s
Ku)~B(A~
KPU)~(U)
du -
YC
I
--m M
p
--
EYOl YO
Ll(d~
Yo+Yl -
- Kp)
- EB(Ap
2Yl
- Kpu)
f(v)dv I
YO +YI --
Ll(A+Ku)L,(A-Ku)B(Ap-Kpu) [
zB(Ap
- Kpu)
a1
f(“) _exp [-(“Iu)21
B(n) 2YO2(YO + Y2yz) P2
I(*)
_
=2
f(u)
du
DO(u) EdOf(u)
ufi
f2=d2+
1
Y02P2
f
ro(x2+~)
4(cu’*‘)2
YOY2
701
where A (= 0 - w) and Ap (= 0p - wp) are the fir and the pump detunings, respectively, and yij is the relaxation constant of coherence between the states i and j. We, herein, assume that 712 > (yet, ym), i.e. Raman-type two photon process is ignored. The stationary inversion density N(u) du ( = (~00 - ptt ) du) of the molecules with the velocity component of u along the optical axis is given by
where Li (x) (i = 1,2), I’*) and Zp are the dimensionless Lorentzian function, dimensionless intensities of fir and ir radiation, respectively. Table 1 summarizes the notation. Since the amplitude of the induced polarization oscillating at the fir frequency 0 leads to PC*] = Nt JY” PO1 PO1 (*) du, the fir intensity determining equations [ 121 lead to dZ’*‘/dt = -ycZ’*’
N(v) dv =
dv = f, + _f2[LI (A - Ku)l’+’ + L1 (A + Ku)Z(-)] ]fso”(4
+ fd%Gl
1
2EOYOI Ji
(7) whereI@ dv (- (p&,-p:,) dv) andDO dv (= ( p!&.- p&) dv) are the inversion density between the states 0 and 1 and the population difference between the states 2 and 0 in the absence of ir and fir fields in the cavity, respectively. ft, fz and fs are defined as
co
Yw$I’
J -CO
Ll (A F Ku) N(u) du,
where Nt and yc are the totalmolecular density and the cavity decay constant for the fir power, respectively. To the third order approximation in perturbation, Eq. (7) leads to N(v) dv = f3Do(v> 1 --
x
(
fl
f’[& f:
(A
-
Ku)z’+’
+ L1 (A + Kv)Z’-‘1
dv, >
f3
=
$2(Ap
- Kpv)Ip,
(10)
(11)
(12)
where it has been assumed that A@(v) = 0 at room temperature. Accordingly, the intensity determining equations for the forward and the backward fir waves are given respectively by
K. Matsushima et al./ Optics Communications I I7 (I 995) 462-468
466 Table 2 Stationary
solutions
of Eq. (13) and its stability Stationary
solutions
Conditions
for stability
(i)
I’+’ = 0,
I’-’
= 0
g’+’ 5 0, g’-’
(ii)
I(+) = g’+‘/s’+’
I’-’
zz 0
{g’+’ > 0, g’-’
< - 0) or {G(+) > 0, G(-)
5 0)
(iii)
I’+’ = 0,
I’-’
= g’-‘/s’-’
(g’+’ 5 0, g(-)
> 0} or {G(+) 5 0, G’-)
> 0}
(iv)
I’+’ = G’+‘/(r’+‘@,
I’-’
= G’-‘/(r’-‘0)
G(+) > 0, G’-’
> 0
G’*’ =
-_g
(*)
_qg'F,/s'W
Table 3 Cavity and molecular
parameters
and
5 0
0 E 1 - q2/(s(+)s’-‘)
used for numerical
calculations.
Cavity decay constant Molecule FIR wavelength Pump wavelength Relaxation constants
Dephasing
is ignored,
i.e. ‘yo2 = (70 + yz)/2 1 x 10-Z 7s
YC
%HsOH A AP -YOl ? YO 3 YI Y2 YOZ
Dipole matrix element for the FlR transition Dipole matrix element for the IR transition Most probable speed of molecules
PO1
/-Wz
0.69 1.57 1.13 2.2 4.0
U
119pm 9.7 pm x lo6 (s Pa)-’ x lo6 (s. Pa)-’ x lo6 (s. Pa)-* x 10W30 C. m x 10p31 C . m 395 m/s
(*) (*) (*) (*) (*)
* Ref. 1151. &‘+‘/dr
= I’+)(g(+)
df’-‘/&
= f’-‘@-’
_ S’+‘~‘+’ _ @-‘), _ $-‘I’-’
_ @‘+‘),
(1W (13b)
where g’*j are the net gains, s(*) are self-saturation coefficients, and q is cross-saturation coefficient. The coefficients g(*), s(*) and q are defined in Table 1.
experimental conditions. Output directionality of the 119 pm ring laser is determined by the values of g’“‘, s(*) and q. The stable stationary solutions for fir intensities are calculated from the parameters summarized in Table 3.
3.3. Directionality 3.2. Stability of the solutions Eqs. (13) are similar to those appeared in the twomode laser model [ 131. In steady-state regime, stability of the solutions to the intensity determining equations ( 13) is examined just in the same way as that for the two-mode laser. For the coupling constant (5 = q2/s(+)s(-‘) smaller than unity, i.e. for “weak coupling”, the solutions and their stability are summarized in Table 2. Solution (i) is trivial. Solution (iv) corresponds to bi-directional operation which is realized in conventional laser-gyroscopes [ 141. Solutions (ii) and (iii) correspond to unidirectional operation due to mode competition which suppresses simultaneous oscillation of the forward and the backward waves. For the CH30H 119 pm line, 5 < 1 under the present
The threshold condition plotted on the parameter plane spanned by cavity and pump detunings is shown in Fig. 5. The net gain is 0 alone the outer contour. In region B wherein solution (iii) is stable, the backward output is preferred, while in region F the forward output is expected. Bi-directional output is expected in region B & F wherein solution (iv) is stable. No lasing is expected outside the regions denoted by B, F andB&F. Calculated fir intensities as a function of dimensionless pump detuning are plotted in Fig. 6 for different cavity detunings. Fig. 6 predicts the reversal of output direction which is in agreement with the experimental results. It also predicts that the reversal of the output direction occurs when the pump frequency is tuned
K. Matsushima
lp=l.O
et al/Optics
Communications
, A/r n,
II7 (1995) 462-468
-16
-8
467
0
AP’T
8
16
02
Fig. 6. Forward (I’-‘) and backward (I(+)) fir intensities calculated from Eq. (13) versus pump dehming for different fir cavity detunings. Ip = 0.5.
Fig. 5. Calculated output directionality on (Ap, A) plane for different pump intensities. F: Forwardly unidirectional output, B: Backwardly unidirectional output, FL? B: bi-directional output.
across the resonance (dp = 0). Bi-directional output is expected for pump detuning around the resonance and the range of pump detuning for which bi-directional output is obtained decreases with a decrease in pump intensity. As the dimensionless pump intensity (Zp) is estimated to be 0.3 to 0.5 in our experiments, a narrow bi-directional region of pump detuning is expected. However, bi-directional output has not been detected, possibly owing to rather poor resolution of cavity scanning of the pump laser. 4. Discussion The 119 ,um CHsOH OPFRL operates unidirectionally due to the mode competition between the counterpropagating fir waves. In ordinary ring gas lasers, co- and counter-propagating waves in the cavity do not strongly couple with each other except for central cavity tuning. For central cavity tuning and a low rotation rate, the two travelling waves strongly cou-
ple with each other. Therefore, mode competition and frequency locking due to back scattering may occur between the two waves [ 161. In optically pumped farinfrared ring lasers, two fir waves in the cavity strongly couple with each other regardless of fir cavity detuning. This may be due to the fact that the velocity distribution of the optically pumped molecules is limited in a narrow region characterized by the homogeneous width of the pump transition. The CHsOH OPFRL reverses its output direction when the pump frequency is scanned across the ir line center. The reversal occurs irrespective of ring cavity detuning. Therefore, the reversal of output direction may find its application in frequency reference in the ir region of spectrum. Several narrow absorption lines have been utilized as absolute frequency references for frequency stabilized lasers. Usually, the laser frequency is weakly modulated in order to detect the center of the absorption line and to get discrimination signal for frequency stabilization. However, it is preferable to avoid frequency modulation for some metrological purposes [ 171. We may be able to detect the sign of pump frequency deviation from the line center of pump transition by monitoring the output direction of an OPFRL.
468
K. Matsushima et al./ Optics Communications 117 (i995) 442468
The present analysis has show that the direction reversal occurs at the line center of the pump transition. It is noted that Raman type process is ignored in this analysis. Though the contribution of Raman type process is small because of a large ratio of Kp/K [ 81, it gives rise to a slight shift in pump frequency, for which the output direction reverses, from the ir line center [ 18,191.
5. Conclusion An experimental and theoretical investigation has been made on an optically pumped CHsOH fir ring laser operating on the 119 pm line. The ring laser operated unidirectionally and experienced abrupt reversal of the output direction when the pump frequency was tuned across the line center of the ir molecular resonance. Intensity determining equations have been derived for a model of Doppler-broadened three-level optically pumped ring laser and stability of the stationary solutions has been examined. Mode competition has been shown to be responsible for the output unidirectionality under off resonance pumping. The present model also predicts that the direction preference is dependent on the pump as well as the cavity detunings. An application to absolute frequency reference for ir lasers has been suggested.
Acknowledgements The authors would like to thank T. Matsumoto his assistance in the computer work.
for
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