- Email: [email protected]
Raman amplification of coherent Stokes wave at biharmonic laser pumping of single-mode silica fibers
~ ~t~1
I January 1995
ELSEVIER
Optics Communications 113 (1995) 498-504
OPTICS COMMUNICATIONS
UJ5
Raman amplification of coherent Stokes wave at biharmonic laser pumping of single-mode silica fibers A.Ya. Karasik a, T. Tsuboi b • General Physics Institute, Vavilov St. 38. Moscow 11 7333. Russian Federation b Faculty ofEngineering, Kyoto Sangyo University, Kam igamo, Kyoto 603. Japan Received 3 May 1994; revision received 24 August 1994
Abstract
We have studied a nonlinear process in the generation of narrow-band tunabl e Stokes radiation of frequency v. = 2 V2 - VI under a biharmonic tunable pulsed laser pumped with VI and V2 frequencies ( VI > vz) in single-mode silica fibers . The Stokes radiation is generated at a high conversion efficiency up to 40% when the pumping frequency difference is VI - V2 < 1200 em -I which corresponds to the region of vibronic resonances of fused silica, while its power is decreased by several orders at 1200 em - I < VI - V2 < 3000 ern - I. The process is explained by Raman amplification of the coherent probe Stokes wave which is generated by the four-photon mixing.
1. Introduction
detected by varying
VI - Vz
continuously, i.e. by tun-
ing in ac cordance with the law ofFPM under the IPM
The nonlinear four-photon mixing (FPM) in multimode silica fibers, which is caused by a monochromatic laser pumping, leads to an efficient Stokes and anti-Stokes generation [1,2]. Such a coherent light scattering is due to an intermode phase matching (IPM) over a large fiber length. In this case, waves of different frequencies propagate in different waveguide modes, and the modes of more than one with different effective refractive indexes take part in the nonlinear process. The IPM conditions are achieved when the phase mismatch due to frequenc y (v) dispersion of refractive index n (v) is compensated by the intermode dispersion. By the biharmonic pumping of the multi-mode fibers, where two laser frequencies VI and Vz (VI> vz) are used, the excitation of coherent molecular vibrations with frequenc y Vph (= V I - vz) generates an efficient anti-Stokes wave with frequency Va (=2Vl- V2)' Experimentally the anti-Stokes wave is
conditions ka =Lk, -k2 for mode wave-vectors [3,4]. The tunable range is determined by the spectrum of a parametric amplification [5] . The bandwidth of the spectrum depends on the fiber imperfections and pumping power [5,6]. As a result, the continuous tuning range of anti-Stokes radiation is relatively broad and reaches to 260 ern - I [6] . In addition to the coherent ant i-Stokes Raman scattering (CARS), an intense narrow band Stokes radiation of frequenc y Vs ( = 2 Vz - VI) was found to be generated by the biharmonic pumping in not only multi-mode fibers but also single-mode fibers [4] . The high efficiency of this process at collinear interaction of waves and the broad range of frequency tuning (vz- Vs~ 1000 cm- I ) indicate an absence of the IPM conditions. The reason of this nonlinear process was discussed briefly in Ref. [4] , but the mechan ism has not been clarified . The present work was undertaken to elucidate the
0030-4018/95/$ 09.50 e 1995 Elsevier Science B.V. AlI rights reserved. SSDl 0030·4018 (94 )00508-7
A.Yo. Karasik, T. Tsuboi I Optics Communications 113 (J 995) 498-504
mechanism of the nonlinear process caused by the biharmonic pumping in silica fibers. We show that the mechanism is determined by the Raman amplification of a coherent Stokes wave which was generated by FPM at a collinear interaction of all waves involved.
2. Stokes wave generationin single-mode fibers The experimental methods used in the present investigation is the same as those described in Ref. [4]. Pulsed YAG: Nd and LiF: Fi color center lasers were used for biharmonic pumping of fibers. The color center laser [7] was pumped by a YAG:Nd laser of AI= 1064 nm wavelength. The tuning wavelength of the color center laser is A2= 1090-1230 nm. These lasers were operated with a repetition rate of 10Hz. The pulses of the YAG: Nd and color center lasers have a duration of 10 and 8 ns, respectively, and their spectral bandwidth is of less than 0.1 nm. The peak power PI of the YAG:Nd laser is 1-2 kW which is almost the same as the power P2 of the color center laser. Fused silica fibers doped with 3% Ge02 in the core were used in our experiments. A tunable anti-Stokes radiation of frequency Va (= 2vI - V2, VI> V2) is generated by the biharmonic VI and V2 pumping in multi-mode fibers. The tuning bandwidth is 260 em -I [4]. The anti-Stokes generation and the tuning range are determined by the IPM conditions and the parametric amplification bandwidth, respectively [4,6]. A high efficient Stokes of frequency Vs (= 2 V2 - VI) is generated together with the anti-Stokes. The tuning range of the Stokes frequency is extended to VI - V2=V2-Vs= 1200 cm- I. The high efficient Stokes generation was obtained for the shift VI - V2 up to 610 em -I. As a result, we could tune the Stokes radiation in a wide spectral region of wavelength from 1116 to 1330 nm. From the biharmonic pumping for a 3 m long single-mode fiber, it was found that the efficiency of the Stokes generation is extraordinary high and it reaches to 40%. However, contrary to the case of the multimode fibers, the anti-Stokes was observed to disappear in the single-mode fiber. The spectral bandwidth of the Stokes radiation is about 1.5 em - I under a low-intensity pumping and it increases with the increase of the pumping power.
499
The IPM conditions do not playa principal role for the Stokes generation in the single-mode fibers. This is confirmed from the following evidences. Firstly, the biharmonic pumping with wavelengths A1= 1064 nm and A2= 1117 nm (i.e, vI- 112=440 cm :") causes a phase mismatch due to the refractive index dispersion of tll=2k2-k l - k s= 1.57 cm" ' (k;=2nn;/A;). Since the coherent length of wave interaction is given by le= Ln] Sk, Ie is estimated to be about 4 em. We obtained the efficient Stokes generation in several long single-mode fibers oflength I» Ie. Secondly, the conversion efficiency from pump to Stokes generation is not changed essentially between the singlemode and multi-mode fibers. Thirdly, an efficiency ofthe FPM is mainly determined by the nonresonant cubic susceptibility (x~i) which exceeds the resonant one (Xf[l ) in disordered solid materials such as glass by several times [3] (see Sect. 3), i.e. the parametric process (i.e. FPM) is also involved for the large frequency shift of 111-112> 1200 cm- I where Xf[l =0. However, the high efficient Stokes of frequency lis is generated at II 1- 112 < 1200 em - I which corresponds to the range of vibronic resonances of fused silica [8]. Therefore, the nonlinear process in the generation of the powerful Stokes wave should be considered as a consequence of both the FPM and Raman amplification which does not need the IPM.
3. Raman amplification ofthe coherentStokes wave 3.1. Theoretical model
The effect due to the coexistence of the FPM and Raman amplification was studied theoretically in Refs. [9-14]. It was experimentally shown that a stimulated Stokes Raman is efficiently generated in H 2 gas by the collinear biharmonic pumping where the interaction length is longer than the coherent length t: [12,13]. The following equation was obtained [11] for the power P, of the Stokes wave which is generated by the biharmonic pumping of a medium with length I and then Raman-amplified, under conditions of a large Raman amplification gI2/» 1 (g is Raman gain, 12 is pumping intensity of the frequency 112) and a phase mismatch of k» gI2 ;
500
A.Ya. Karasik, T. Tsuboi / Optics Communications // 3 (/995) 498-504
Here PI and P2 mean pumping powers of frequencies VI and V2 ( VI> V2), respectively, D. v= VI- Vb andx (3 ) is a complex:
3
X(3)(D.v ) =X' (3 )(D.v) - iX"(3) (D.v ) .
2
A nonlinear self-phase modulation was not considered in th is approach. It was assumed that the amplification process does not deplete the pumping. For a small increment of gI2 /« I, Eq, (I ) transforms to a well-known equation for FPM [II]:
1\
I~
/ I
z «
o z
o LI I d 'I I
"$ 'e :::> Li
S
z
~
I.
z «
~1
:2:
«
2;i
0::
0
0
r, ~ PI P~ IX(3) (D.v) 12/~ sin? (D.kl/2) / (D.kl/2)2 . (2) We can rewrite Eq. (I ): P; =Pso exp(l2g (D.v)/ ) ,
~ 'I II
( 3)
where (4 )
Eq. (3) means that the Stokes wave with power of Pso is generated by the biharmonic pumping within the coherent length Ie and it propagates in a medium receiving the Raman ampl ification. We define the Stokes wave of power PsOas the coherent prob e Stokes wave.
3.2. Tunable coherentprobe wave From Eq. (4) , we assum e that the Stokes probe wave is generated by the nonresonant xfJJ predominantly, because IxfJJ I » Ixf2) I in a spectral region of A. = 400-2000 nm in fused silica [15]. Hence, the probe wave must be generated at VI - V2> 1200 cm - I, i.e. in a region where the bih armonic pumping frequenc y difference does not correspond to the phonon vibra tion region of the fused silica. Th e vibration region is given by the spontaneous Raman scattering spectrum offused silica, which is shown by solid curve of Fig. 1. In order to check this assumption, we pumped a I m long single-mode fiber by biharmonic radiation with wavelengths )' 1= 532 nm (SHG of YAG: Nd laser ) and )' 2=560-620 nm (Rh6G dye laser ). Indeed, the tunable Stokes wave of frequency Vs (=2 V2 - VI ) was generated by varyi ng the frequenc y
Fig. I. Comparison of the Raman ampl ification spectrum of the coherent Stokes wave of frequen cy V,=2V2 - v, at biharmonic pumping of frequency difference v, - V2 (open circles, dashed curve ) in a I m long single-mode fiber with a spontaneo us Raman scatteri ng spectrum (solid curve) and with the Raman amplification spectrum oflow-frequency wave (V2) in the high-frequency VI pump ing field (fi lled circles) . In the latter case, the gain mcans P2(l )I P2(l =0) , i.e. ratio of the out put to input power of the V2 wave. Pump ing wavelengths are )' 1 1tv, 1064 nm and ). 2= I I V2= 1090-1 230 nm. The inset shows a pan of the spectrum of the Stokes radiation which was obta ined at biharmonic pumping with wavelengths ). , = 532 nm and )' 2= 560-570 nm .
=
=
difference VI - V2 from 1200 to 3000 cm - I. Th e tunable anti-Stokes of frequen cy Va ( = 2 VI - V2) was generated simultaneously with the Stokes wave. The intensities of both the Stokes and anti-Stokes waves are approximately equal and they are little changed during the VI- V2 tun ing from 1200 to 3000 em - I . The latter fact corresponds to a small dispersion of the xfJJ in glasses [15]. As shown theoretically in Ref. [5] and experimentally in Refs. [3,6], the nonregularities of fiber (i.e. variations of a core diameter and core-cladding refractive index difference ) give rise to an increa se of effective bandwidth of the phase mat ching. Th is gives rise to a decrease of the frequency dependence of the Ie values in the broad bandwidth , leading to an effective coher ence-length value le.err over the frequ ency range. Therefore it is understood that we have not observed the VI - V2 dependence for the Stokes power Pso and the anti-Stokes power PaO' A successive shortening of the fiber length from 1 m into 0.2 m also did not lead to an essent ial change of the Stokes and ant i-Stokes rad iat ion powers.
A.Ya. Karasik. T. Tsuboi I Optics Communications 113 (1995) 498-504
Therefore, the anti-Stokes is believed to be generated in a short fiber length of about a few em. We did not find an apparent dependence of the P so (Pao) power on the PI and P2 pumping powers, which is expected from Eq. (4). The increase of pumping powers leads to a broadening of the bandwidth of phase matching and to a shortening of the coherent length Ie [5,6]. This decrease of Ie compensates the increase of pumping powers, and therefore we can understand the non-observation of change of the coherent Stokes (anti-Stokes) power by the pumping power. We investigated a change of the power of Stokes radiation by varying the frequency difference VI- V2 from 200 to 3000 cm- I. At vI - V2> 1200 cm- I the power of the coherent probe Stokes wave was Pso= 10- 3 P., while P, took a maximum value at V\ - V2 = 450 cm -1 as shown in Fig. 1.
3.3. Raman amplification According to the theory by Smith [16], when the gain of Eq. (1) is gIII= 16, powers of the pump and Stokes radiation are equal in the case of stimulated Raman scattering (SRS) which is produced from spontaneous Raman scattering noises. In this case the power of spontaneous radiation Pso(sp) is given by Pso(sp ) = 10- 7P, from Eq. (3). Therefore we obtain Pso=104pso(sp) using P so=10-3Ps (see Sect. 3.2). This suggests that the Stokes generation process by the biharmonic pumping is much dominate when compared with the process due to the SRS. Our experimental results show that, in the case of the biharmonic pumping with powers of P I=P2 at VI - V2 < 1200 em - I, a threshold for the Stokes of us (=2V2- VI) is essentially less than the threshold for and V2 pumpings. When the SRS by each of the SRS threshold is exceeded for either of VI and pumpings, for example for VI pumping, the second V2 pumping leads to a suppression of the SRS at Vs= vl - 450 cm- I by the simultaneous creation of the intense Stokes radiation of V s (= 2 V2 - V I). The frequency shift of 450 em -1 corresponds to the maximum intensity in the Raman amplification spectrum (Fig. 1). We have not observed anti-Stokes Raman generation by the biharmonic pumping at VI - V2 < 1200 em - I in the single-mode fibers. This is explained by
"1
"2
501
an absorption of the anti-Stokes radiation [5,11] at frequency Va (=2"I-V2). We shall investigate the pumping-power dependence of the Stokes wave experimentally. Since the Raman amplification of the coherent probe Stokes wave is caused by either the VI or V2 pumping (see Eqs. (3) and (4) ), we have chosen the pumping frequency difference V 1 - V2 = 350 em-I in order to suppress the SRS by the v I pumping. In this case, the FPM-induced Stokes wave appears at 700 ern- I from the frequency Vb whose position corresponds to a minimum intensity in the Raman amplification spectrum (Fig. I). Therefore we can neglect the effect of the Raman amplification by the VI pumping. The power dependence of the Stokes radiation was measured using a 3 m long single-mode fiber. The result is shown in Figs. 2 and 3. A dependence of the Stokes power (Ps) on single-mode fiber length was also measured under the condition that the PI and P2 values are kept constant. The result is shown in Fig. 4. Saturation is obtained at relatively high P2 power (see Fig. 3). This is believed to be caused by the generation of the 2nd Stokes component of "s(2) (= 2 Vs- V2)' The scattering of P, values appearing in Figs. 2-4 is connected with the detection time of several hundred laser pulses. In the case that the Raman amplification by the VI pumping can be neglected, the power of the probe wave P so has a linear dependence on the PI power as expected from Eq. (4). The solid line of Fig. 2 shows
~5
~ c
::J
4
.Q
....
III
~3
cf' 0::
W2
~
eLl O,)f"------=---+---.l;---~--+_-_+--+_'
POWER
F\
(arb. units)
Fig. 2. Power of the coherent Stokes wave radiation of frequency Vs=2VI- VI at biharmonic pumping of a single-mode fiber of 3 m length as a function of the high-frequency VI pumping power (}'1 = I IVI = 1064 nm). The difference of the pumping frequencies is fixed at VI - VI= 350 em-I.
A.Ya. Karasik, T. Tsuboi/ Optics Communications Jl3 (1995) 498-504
502
~2
~
I
'c
I
I
::J
.D \...
~
~O -1 POWER ~ (arb. units) Fig. 3. Power of the coherent Stokes wave rad iation of frequency at biharmonic pumping of a single-mode fiber of 3 m length as a function of the low frequency V2 pumping power. The difference of the pumping frequencies is fixed at VI- v2=350 em-I . P,=O at P2=0 .
V.=2V2- VI
3
~2
I I
'c::J
.d 1 \...
.s
~ 01
0
.2
-1
FIBER
LENGTH
(ern)
Fig. 4. Power of the coherent Stokes wave radiation of frequency at biharmonic pumping of a single-mode fiber as a function of the fiber length. The difference of the pumping frequencies is fixed at VI- V2= 350 cm " ' . P,=O at 1=0. V.=2V2- VI
a linear P.-power dependence of the Stokes wave power, in agreement with Eqs. (3) and (4). In the analysis of Sect. 3.1 is neglected the self-phase modulation effect [12-14]. This is consistent with the observation that the tuning range of the Stokes wave, which corresponds to the bandwidth of Raman amplification (V2 - Vs < 610 cm - I, see Fig. 1), does not depend on the biharmonic pumping powers. The Stokes wave power is observed to increase exponentially with the low-frequenc y V2 pumping power
and fiber length, in agreement with Eq. (3) (see solid straight lines in Figs. 3 and 4, respectively). Contrary to this case of the Stokes power P., it is noted that the power Pso of the coherent probe Stokes wave depends on neither the fiber length nor the pumping PI and P2 powers (see Sect. 3.2 ). The dashed curve of Fig. 1 shows the Raman gain of the coherent Stokes wave of frequency Vs (=2 V2 - v d as a function of vI - V2' We shall describe how to obtain this Raman amplification spectrum of the coherent probe Stokes wave. At first, we measured the Raman amplification of a low-intensity wave of frequency V2 under the VI pumping. The filled circles of Fig. I show the measured Raman gain (i.e . ratio P2(l)/P2(l=O) of the output to input V2 powers) as a function of V I- V2' At the small increment of gI2 !« I in Eq. (I ), the increment is proportional to a cross-section of the scattering [ 17] and its frequency shift dependence coincides with the spontaneous Raman spectrum (solid curve of Fig. I). Measurements of Raman gain of the coherent probe Stokes wave by the biharmonic pumping are rather complicated because the Raman amplification is also caused by each of the VI and V2 pumpings and because it is difficult to know the intensity value of the probe Stokes wave. However, since the probe Stokes intensity does not depend on VI - V2, the estimation of the Raman gain is simplified. We measured the intensity of the Stokes radiation of Vs (=2V2- vd at extremely low PI and P2 pumping powers in a I m long single-mode fiber. For the Raman gain estimation, we put a low-intensity broad-band probe signal offrequency Vs [4] into the fiber and then measured its amplification under the V2 pumping without VI pumping. After the measurement of gain , we took off the broad-band probe wave, and then we put the second VI pumping beam into the fiber by choosing the power value which makes the power of r, wave equal to the amplified signal of the broad-band probe wave. After such a gain calibration, we measured the power of the Stokes radiation P, as a function of V I - V 2 under the chosen PI and P2 power values . The obtained Raman gain spectrum does not coincide with the spontaneous Raman scattering spectrum (solid curve of Fig. I) because of the high increment, but the two curves are similar to each other as a whole. This agrees with our suggestion that Ra-
A.Ya. Karasik. T. Tsuboi/ Optics Communications J J3 (1995) 498-504
man amplification is made for the coherent probe Stokes wave generated by FPM. The intensity of the coherent probe wave generated by FPM is mainly determined by the nonresonant susceptibility x~J , while its Raman amplification is due to the resonant one X~3 ) only. A nonresonant spectral pedestal, which is usually induced by the FPM, indeed does not appear in the Raman amplification spectrum under the biharmonic pumping as seen in a dashed curve of Fig. I. This nonresonant pedestal is an obstacle in the the CARS investigations when a vibronic resonance structure is investigated in disordered condensed media where Ix~J I » Ix~) I. Therefore, the obtained high intensity of the Stokes signal and the absence of a nonresonant pedestal make this nonlinear process attractive for spectroscopic applications as mentioned below. The inset of Fig. I shows a Stokes spectrum in a region of 950-1250 cm - I which was obtained by the biharmonic fiber pumping using a laser beam of Al = 532 nm and tunable beam of )' 2 =560-570 nm. Two peaks are observed at frequency shifts 1080 em -I and 1160 cm - I . The resolution of these lines is significantly high when compared with the spontaneous Raman spectrum. Therefore, the present method is useful to clarify the fine structure appearing in low-intensity vibronic resonances. Another application using the biharmonic pumping method is suggested as follows. When we used laser pulses with multi-spike temporal structure due to a partial mode-locking in the pumping [4] , we observed weak Stokes and anti-Stokes radiation with the same multi-spike structure in a tuning range of VI - V 2 = 1200-3000 cm -1. When the difference V I - V 2 was decreased from 1200 em- I , one of more intense spikes in the probe Stokes wave was enhanced considerably by the Raman amplification. As a result, a single picosecond pulse was generated. Therefore the biharmonic pumping is useful to obtain such a tunable, intense picosecond narrow pulse. However, when the biharmonic pumping power was increased to give rise to a saturation of the Stokes power (Figs. 3 and 4), it was observed that the other low-intensity spikes are also amplified and its pulse envelope consists of several picosecond subpulses.
503
4. Conclusion We have investigated nonlinear processes under biharmonic laser pumping of single-mode silica fibers. We tuned continuously biharmonic pumping frequency difference both in the region of vibronie resonances of fused silica (v I - V2 < 1200 ern- I ) and out of this region (v 1 - v2 > 1200-3000 cm :") , using a pulsed YAG :Nd laser and a tunable dye or LiF :F2' color center laser. When the pumping frequency difference is changed continuously in the range of 1200-3000 cm -I , both the low-intensity Stokes and anti-Stokes of frequencies Vs ( = 2 V2 - v\) and Va (= 2 VI - V2), respectively, are generated. The power of the Stokes Pso (antiStokes PaD) has little dependence on the fiber length and on the pumping powers. It is concluded that this process involves a generation of the coherent probe Stokes (anti-Stokes) due to the FPM at the collinear interaction of all waves in a fiber length of a few em. An absence of obvious dependences of Pso (PaD) on the pumping power and fiber length is explained by the influence of fiber imperfections and pumping intensity to the phase matching bandwidth ofFPM. The observed small frequenc y dispersion of the Stokes (anti-Stokes) intensity is in a good agreement with a small dispersion of the nonresonant cubic susceptibility, which is responsible for the generation of the coherent probe Stokes wave. When the pumping frequency difference is changed in a region of V 1- V 2< 1200 em-I , it was observed that the Stokes power at the output of long fibers of several meters increases up to three orders in magnitude, while the anti-Stokes disappears. We have measured the Stokes power by the biharmonic pumping as a function of the fiber length and pumping power. These results show that exponential amplification by the low-frequenc y V2 pumping is mainly responsible for a high conversion efficiency of the Stokes generation. We also obtained the Stokes amplification spectrum plotted against the frequency difference VI - V 2' The spectrum is similar to the spontaneous Raman scattering spectrum. The obtained experimental results and the analysis show that the Vs Stokes generation does not need the IPM conditions over the whole fiber length. We have determined the mechanism of the nonlinear process as a result of Raman amplification of the coherent
504
A.Ya. Karasik, T. Tsuboi / OpticsCommunications 113 (1995) 498-504
probe Stokes wave generated by FPM; the low-intensity probe Stokes wave (vs ) is generated in a fiber length of several em by the coherent excitation of phonons offrequency Vph= VI - V2 = V2 - Vs and by the collinear interaction of four waves, and subsequently it is Raman-amplified exponentially over the fiber length.
Acknowledgements We thank E.A. Zachidov, P.G. Zverev and T.T. Basiev for their assistances and helpful discussions. The present work was partially supported by a Grantin-Aid from the Japanese Ministry of Education and Science.
References [I) R.H. Stolen, IEEE 1. Quantum Electron. QE-11 (1975) 100. (2) E.M. Dianov, E.A. Zakhidov, A.Ya. Karasik, P.Y. Mamyshev and A.M. Prokhorov, JETP Lett. 34 (1981 ) 38; Sov. Phys. JETP 56 (1982) 21.
[3) R.H. Stolen, J.E. Bjorkholm and A. Ashkin, Appl. Phys. Lett. 24 (1974) 308. [4) T.T. Basiev, E.M. Dianov, E.A. Zakhidov, A.Ya. Karasik, S.B. Mirov and A.M. Prokhorov, JETP Lett. 37 (1983) 229. [5) R.H. Stolen and J.E. Bjorkholm, IEEE J. Quantum Electron. QE-11 (1982) 1062. [6) E.M. Dianov, A.Ya. Karasik and E.A. Zakhidov, SOy. J. Quantum Electron. 16 (1986) 382. [7] T.T. Basiev, V.V. Osiko and S.B. Mirov, IEEE J. Quantum Electron. QE-24 (1988) 1052. [8) G.E. Walrafen and P.N. Krishnan, Appl. Optics 21 (1982) 351. [9) N.1. Koroteev, Opt. Spectrosc. 29 (1970) 286. [10) N.1. Koroteev and I.L. Shumai, SOy. J. Quantum Electron. 4 (1975) 2489. [II) N.1. Licholit, V.I. Kislenko, V.L. Strishevskii and Yu.N. Yashkir, Ukrainian Fizicheskii Zh. (Kiev) 21 (1976) 1012. [12) G.V. Venkin, G.M. Krochik, L.L. Kulyuk, 0.1. Maleev and Yu.G. Khronopulo, JETP Lett. 21 (1975) 105. [13) V.S. Butylkin, G.V. Venkin, L.1. Kulyuk, OJ. Maleev, Yu.G. Khronopulo and M.F. Shalyaev, SOy. J. Quantum Electron. 7 (1977) 867. [14) G.V. Venkin, G.M. Krochik, L.1. Kulyuk, 0.1. Maleev and Yu.G. Khronopulo, SOy. Phys. JETP 43 (1975) 873. [15) A. Owyoung and N. George, Phys. Rev. B 5 (1972) 628. [16) R.G. Smith, Appl. Optics II (1972) 2489. [17) P. Lallemand, P. Simova and G. Bret, Phys. Rev. Lett. 17 (1966) 1239.
I January 1995
ELSEVIER
Optics Communications 113 (1995) 498-504
OPTICS COMMUNICATIONS
UJ5
Raman amplification of coherent Stokes wave at biharmonic laser pumping of single-mode silica fibers A.Ya. Karasik a, T. Tsuboi b • General Physics Institute, Vavilov St. 38. Moscow 11 7333. Russian Federation b Faculty ofEngineering, Kyoto Sangyo University, Kam igamo, Kyoto 603. Japan Received 3 May 1994; revision received 24 August 1994
Abstract
We have studied a nonlinear process in the generation of narrow-band tunabl e Stokes radiation of frequency v. = 2 V2 - VI under a biharmonic tunable pulsed laser pumped with VI and V2 frequencies ( VI > vz) in single-mode silica fibers . The Stokes radiation is generated at a high conversion efficiency up to 40% when the pumping frequency difference is VI - V2 < 1200 em -I which corresponds to the region of vibronic resonances of fused silica, while its power is decreased by several orders at 1200 em - I < VI - V2 < 3000 ern - I. The process is explained by Raman amplification of the coherent probe Stokes wave which is generated by the four-photon mixing.
1. Introduction
detected by varying
VI - Vz
continuously, i.e. by tun-
ing in ac cordance with the law ofFPM under the IPM
The nonlinear four-photon mixing (FPM) in multimode silica fibers, which is caused by a monochromatic laser pumping, leads to an efficient Stokes and anti-Stokes generation [1,2]. Such a coherent light scattering is due to an intermode phase matching (IPM) over a large fiber length. In this case, waves of different frequencies propagate in different waveguide modes, and the modes of more than one with different effective refractive indexes take part in the nonlinear process. The IPM conditions are achieved when the phase mismatch due to frequenc y (v) dispersion of refractive index n (v) is compensated by the intermode dispersion. By the biharmonic pumping of the multi-mode fibers, where two laser frequencies VI and Vz (VI> vz) are used, the excitation of coherent molecular vibrations with frequenc y Vph (= V I - vz) generates an efficient anti-Stokes wave with frequency Va (=2Vl- V2)' Experimentally the anti-Stokes wave is
conditions ka =Lk, -k2 for mode wave-vectors [3,4]. The tunable range is determined by the spectrum of a parametric amplification [5] . The bandwidth of the spectrum depends on the fiber imperfections and pumping power [5,6]. As a result, the continuous tuning range of anti-Stokes radiation is relatively broad and reaches to 260 ern - I [6] . In addition to the coherent ant i-Stokes Raman scattering (CARS), an intense narrow band Stokes radiation of frequenc y Vs ( = 2 Vz - VI) was found to be generated by the biharmonic pumping in not only multi-mode fibers but also single-mode fibers [4] . The high efficiency of this process at collinear interaction of waves and the broad range of frequency tuning (vz- Vs~ 1000 cm- I ) indicate an absence of the IPM conditions. The reason of this nonlinear process was discussed briefly in Ref. [4] , but the mechan ism has not been clarified . The present work was undertaken to elucidate the
0030-4018/95/$ 09.50 e 1995 Elsevier Science B.V. AlI rights reserved. SSDl 0030·4018 (94 )00508-7
A.Yo. Karasik, T. Tsuboi I Optics Communications 113 (J 995) 498-504
mechanism of the nonlinear process caused by the biharmonic pumping in silica fibers. We show that the mechanism is determined by the Raman amplification of a coherent Stokes wave which was generated by FPM at a collinear interaction of all waves involved.
2. Stokes wave generationin single-mode fibers The experimental methods used in the present investigation is the same as those described in Ref. [4]. Pulsed YAG: Nd and LiF: Fi color center lasers were used for biharmonic pumping of fibers. The color center laser [7] was pumped by a YAG:Nd laser of AI= 1064 nm wavelength. The tuning wavelength of the color center laser is A2= 1090-1230 nm. These lasers were operated with a repetition rate of 10Hz. The pulses of the YAG: Nd and color center lasers have a duration of 10 and 8 ns, respectively, and their spectral bandwidth is of less than 0.1 nm. The peak power PI of the YAG:Nd laser is 1-2 kW which is almost the same as the power P2 of the color center laser. Fused silica fibers doped with 3% Ge02 in the core were used in our experiments. A tunable anti-Stokes radiation of frequency Va (= 2vI - V2, VI> V2) is generated by the biharmonic VI and V2 pumping in multi-mode fibers. The tuning bandwidth is 260 em -I [4]. The anti-Stokes generation and the tuning range are determined by the IPM conditions and the parametric amplification bandwidth, respectively [4,6]. A high efficient Stokes of frequency Vs (= 2 V2 - VI) is generated together with the anti-Stokes. The tuning range of the Stokes frequency is extended to VI - V2=V2-Vs= 1200 cm- I. The high efficient Stokes generation was obtained for the shift VI - V2 up to 610 em -I. As a result, we could tune the Stokes radiation in a wide spectral region of wavelength from 1116 to 1330 nm. From the biharmonic pumping for a 3 m long single-mode fiber, it was found that the efficiency of the Stokes generation is extraordinary high and it reaches to 40%. However, contrary to the case of the multimode fibers, the anti-Stokes was observed to disappear in the single-mode fiber. The spectral bandwidth of the Stokes radiation is about 1.5 em - I under a low-intensity pumping and it increases with the increase of the pumping power.
499
The IPM conditions do not playa principal role for the Stokes generation in the single-mode fibers. This is confirmed from the following evidences. Firstly, the biharmonic pumping with wavelengths A1= 1064 nm and A2= 1117 nm (i.e, vI- 112=440 cm :") causes a phase mismatch due to the refractive index dispersion of tll=2k2-k l - k s= 1.57 cm" ' (k;=2nn;/A;). Since the coherent length of wave interaction is given by le= Ln] Sk, Ie is estimated to be about 4 em. We obtained the efficient Stokes generation in several long single-mode fibers oflength I» Ie. Secondly, the conversion efficiency from pump to Stokes generation is not changed essentially between the singlemode and multi-mode fibers. Thirdly, an efficiency ofthe FPM is mainly determined by the nonresonant cubic susceptibility (x~i) which exceeds the resonant one (Xf[l ) in disordered solid materials such as glass by several times [3] (see Sect. 3), i.e. the parametric process (i.e. FPM) is also involved for the large frequency shift of 111-112> 1200 cm- I where Xf[l =0. However, the high efficient Stokes of frequency lis is generated at II 1- 112 < 1200 em - I which corresponds to the range of vibronic resonances of fused silica [8]. Therefore, the nonlinear process in the generation of the powerful Stokes wave should be considered as a consequence of both the FPM and Raman amplification which does not need the IPM.
3. Raman amplification ofthe coherentStokes wave 3.1. Theoretical model
The effect due to the coexistence of the FPM and Raman amplification was studied theoretically in Refs. [9-14]. It was experimentally shown that a stimulated Stokes Raman is efficiently generated in H 2 gas by the collinear biharmonic pumping where the interaction length is longer than the coherent length t: [12,13]. The following equation was obtained [11] for the power P, of the Stokes wave which is generated by the biharmonic pumping of a medium with length I and then Raman-amplified, under conditions of a large Raman amplification gI2/» 1 (g is Raman gain, 12 is pumping intensity of the frequency 112) and a phase mismatch of k» gI2 ;
500
A.Ya. Karasik, T. Tsuboi / Optics Communications // 3 (/995) 498-504
Here PI and P2 mean pumping powers of frequencies VI and V2 ( VI> V2), respectively, D. v= VI- Vb andx (3 ) is a complex:
3
X(3)(D.v ) =X' (3 )(D.v) - iX"(3) (D.v ) .
2
A nonlinear self-phase modulation was not considered in th is approach. It was assumed that the amplification process does not deplete the pumping. For a small increment of gI2 /« I, Eq, (I ) transforms to a well-known equation for FPM [II]:
1\
I~
/ I
z «
o z
o LI I d 'I I
"$ 'e :::> Li
S
z
~
I.
z «
~1
:2:
«
2;i
0::
0
0
r, ~ PI P~ IX(3) (D.v) 12/~ sin? (D.kl/2) / (D.kl/2)2 . (2) We can rewrite Eq. (I ): P; =Pso exp(l2g (D.v)/ ) ,
~ 'I II
( 3)
where (4 )
Eq. (3) means that the Stokes wave with power of Pso is generated by the biharmonic pumping within the coherent length Ie and it propagates in a medium receiving the Raman ampl ification. We define the Stokes wave of power PsOas the coherent prob e Stokes wave.
3.2. Tunable coherentprobe wave From Eq. (4) , we assum e that the Stokes probe wave is generated by the nonresonant xfJJ predominantly, because IxfJJ I » Ixf2) I in a spectral region of A. = 400-2000 nm in fused silica [15]. Hence, the probe wave must be generated at VI - V2> 1200 cm - I, i.e. in a region where the bih armonic pumping frequenc y difference does not correspond to the phonon vibra tion region of the fused silica. Th e vibration region is given by the spontaneous Raman scattering spectrum offused silica, which is shown by solid curve of Fig. 1. In order to check this assumption, we pumped a I m long single-mode fiber by biharmonic radiation with wavelengths )' 1= 532 nm (SHG of YAG: Nd laser ) and )' 2=560-620 nm (Rh6G dye laser ). Indeed, the tunable Stokes wave of frequency Vs (=2 V2 - VI ) was generated by varyi ng the frequenc y
Fig. I. Comparison of the Raman ampl ification spectrum of the coherent Stokes wave of frequen cy V,=2V2 - v, at biharmonic pumping of frequency difference v, - V2 (open circles, dashed curve ) in a I m long single-mode fiber with a spontaneo us Raman scatteri ng spectrum (solid curve) and with the Raman amplification spectrum oflow-frequency wave (V2) in the high-frequency VI pump ing field (fi lled circles) . In the latter case, the gain mcans P2(l )I P2(l =0) , i.e. ratio of the out put to input power of the V2 wave. Pump ing wavelengths are )' 1 1tv, 1064 nm and ). 2= I I V2= 1090-1 230 nm. The inset shows a pan of the spectrum of the Stokes radiation which was obta ined at biharmonic pumping with wavelengths ). , = 532 nm and )' 2= 560-570 nm .
=
=
difference VI - V2 from 1200 to 3000 cm - I. Th e tunable anti-Stokes of frequen cy Va ( = 2 VI - V2) was generated simultaneously with the Stokes wave. The intensities of both the Stokes and anti-Stokes waves are approximately equal and they are little changed during the VI- V2 tun ing from 1200 to 3000 em - I . The latter fact corresponds to a small dispersion of the xfJJ in glasses [15]. As shown theoretically in Ref. [5] and experimentally in Refs. [3,6], the nonregularities of fiber (i.e. variations of a core diameter and core-cladding refractive index difference ) give rise to an increa se of effective bandwidth of the phase mat ching. Th is gives rise to a decrease of the frequency dependence of the Ie values in the broad bandwidth , leading to an effective coher ence-length value le.err over the frequ ency range. Therefore it is understood that we have not observed the VI - V2 dependence for the Stokes power Pso and the anti-Stokes power PaO' A successive shortening of the fiber length from 1 m into 0.2 m also did not lead to an essent ial change of the Stokes and ant i-Stokes rad iat ion powers.
A.Ya. Karasik. T. Tsuboi I Optics Communications 113 (1995) 498-504
Therefore, the anti-Stokes is believed to be generated in a short fiber length of about a few em. We did not find an apparent dependence of the P so (Pao) power on the PI and P2 pumping powers, which is expected from Eq. (4). The increase of pumping powers leads to a broadening of the bandwidth of phase matching and to a shortening of the coherent length Ie [5,6]. This decrease of Ie compensates the increase of pumping powers, and therefore we can understand the non-observation of change of the coherent Stokes (anti-Stokes) power by the pumping power. We investigated a change of the power of Stokes radiation by varying the frequency difference VI- V2 from 200 to 3000 cm- I. At vI - V2> 1200 cm- I the power of the coherent probe Stokes wave was Pso= 10- 3 P., while P, took a maximum value at V\ - V2 = 450 cm -1 as shown in Fig. 1.
3.3. Raman amplification According to the theory by Smith [16], when the gain of Eq. (1) is gIII= 16, powers of the pump and Stokes radiation are equal in the case of stimulated Raman scattering (SRS) which is produced from spontaneous Raman scattering noises. In this case the power of spontaneous radiation Pso(sp) is given by Pso(sp ) = 10- 7P, from Eq. (3). Therefore we obtain Pso=104pso(sp) using P so=10-3Ps (see Sect. 3.2). This suggests that the Stokes generation process by the biharmonic pumping is much dominate when compared with the process due to the SRS. Our experimental results show that, in the case of the biharmonic pumping with powers of P I=P2 at VI - V2 < 1200 em - I, a threshold for the Stokes of us (=2V2- VI) is essentially less than the threshold for and V2 pumpings. When the SRS by each of the SRS threshold is exceeded for either of VI and pumpings, for example for VI pumping, the second V2 pumping leads to a suppression of the SRS at Vs= vl - 450 cm- I by the simultaneous creation of the intense Stokes radiation of V s (= 2 V2 - V I). The frequency shift of 450 em -1 corresponds to the maximum intensity in the Raman amplification spectrum (Fig. 1). We have not observed anti-Stokes Raman generation by the biharmonic pumping at VI - V2 < 1200 em - I in the single-mode fibers. This is explained by
"1
"2
501
an absorption of the anti-Stokes radiation [5,11] at frequency Va (=2"I-V2). We shall investigate the pumping-power dependence of the Stokes wave experimentally. Since the Raman amplification of the coherent probe Stokes wave is caused by either the VI or V2 pumping (see Eqs. (3) and (4) ), we have chosen the pumping frequency difference V 1 - V2 = 350 em-I in order to suppress the SRS by the v I pumping. In this case, the FPM-induced Stokes wave appears at 700 ern- I from the frequency Vb whose position corresponds to a minimum intensity in the Raman amplification spectrum (Fig. I). Therefore we can neglect the effect of the Raman amplification by the VI pumping. The power dependence of the Stokes radiation was measured using a 3 m long single-mode fiber. The result is shown in Figs. 2 and 3. A dependence of the Stokes power (Ps) on single-mode fiber length was also measured under the condition that the PI and P2 values are kept constant. The result is shown in Fig. 4. Saturation is obtained at relatively high P2 power (see Fig. 3). This is believed to be caused by the generation of the 2nd Stokes component of "s(2) (= 2 Vs- V2)' The scattering of P, values appearing in Figs. 2-4 is connected with the detection time of several hundred laser pulses. In the case that the Raman amplification by the VI pumping can be neglected, the power of the probe wave P so has a linear dependence on the PI power as expected from Eq. (4). The solid line of Fig. 2 shows
~5
~ c
::J
4
.Q
....
III
~3
cf' 0::
W2
~
eLl O,)f"------=---+---.l;---~--+_-_+--+_'
POWER
F\
(arb. units)
Fig. 2. Power of the coherent Stokes wave radiation of frequency Vs=2VI- VI at biharmonic pumping of a single-mode fiber of 3 m length as a function of the high-frequency VI pumping power (}'1 = I IVI = 1064 nm). The difference of the pumping frequencies is fixed at VI - VI= 350 em-I.
A.Ya. Karasik, T. Tsuboi/ Optics Communications Jl3 (1995) 498-504
502
~2
~
I
'c
I
I
::J
.D \...
~
~O -1 POWER ~ (arb. units) Fig. 3. Power of the coherent Stokes wave rad iation of frequency at biharmonic pumping of a single-mode fiber of 3 m length as a function of the low frequency V2 pumping power. The difference of the pumping frequencies is fixed at VI- v2=350 em-I . P,=O at P2=0 .
V.=2V2- VI
3
~2
I I
'c::J
.d 1 \...
.s
~ 01
0
.2
-1
FIBER
LENGTH
(ern)
Fig. 4. Power of the coherent Stokes wave radiation of frequency at biharmonic pumping of a single-mode fiber as a function of the fiber length. The difference of the pumping frequencies is fixed at VI- V2= 350 cm " ' . P,=O at 1=0. V.=2V2- VI
a linear P.-power dependence of the Stokes wave power, in agreement with Eqs. (3) and (4). In the analysis of Sect. 3.1 is neglected the self-phase modulation effect [12-14]. This is consistent with the observation that the tuning range of the Stokes wave, which corresponds to the bandwidth of Raman amplification (V2 - Vs < 610 cm - I, see Fig. 1), does not depend on the biharmonic pumping powers. The Stokes wave power is observed to increase exponentially with the low-frequenc y V2 pumping power
and fiber length, in agreement with Eq. (3) (see solid straight lines in Figs. 3 and 4, respectively). Contrary to this case of the Stokes power P., it is noted that the power Pso of the coherent probe Stokes wave depends on neither the fiber length nor the pumping PI and P2 powers (see Sect. 3.2 ). The dashed curve of Fig. 1 shows the Raman gain of the coherent Stokes wave of frequency Vs (=2 V2 - v d as a function of vI - V2' We shall describe how to obtain this Raman amplification spectrum of the coherent probe Stokes wave. At first, we measured the Raman amplification of a low-intensity wave of frequency V2 under the VI pumping. The filled circles of Fig. I show the measured Raman gain (i.e . ratio P2(l)/P2(l=O) of the output to input V2 powers) as a function of V I- V2' At the small increment of gI2 !« I in Eq. (I ), the increment is proportional to a cross-section of the scattering [ 17] and its frequency shift dependence coincides with the spontaneous Raman spectrum (solid curve of Fig. I). Measurements of Raman gain of the coherent probe Stokes wave by the biharmonic pumping are rather complicated because the Raman amplification is also caused by each of the VI and V2 pumpings and because it is difficult to know the intensity value of the probe Stokes wave. However, since the probe Stokes intensity does not depend on VI - V2, the estimation of the Raman gain is simplified. We measured the intensity of the Stokes radiation of Vs (=2V2- vd at extremely low PI and P2 pumping powers in a I m long single-mode fiber. For the Raman gain estimation, we put a low-intensity broad-band probe signal offrequency Vs [4] into the fiber and then measured its amplification under the V2 pumping without VI pumping. After the measurement of gain , we took off the broad-band probe wave, and then we put the second VI pumping beam into the fiber by choosing the power value which makes the power of r, wave equal to the amplified signal of the broad-band probe wave. After such a gain calibration, we measured the power of the Stokes radiation P, as a function of V I - V 2 under the chosen PI and P2 power values . The obtained Raman gain spectrum does not coincide with the spontaneous Raman scattering spectrum (solid curve of Fig. I) because of the high increment, but the two curves are similar to each other as a whole. This agrees with our suggestion that Ra-
A.Ya. Karasik. T. Tsuboi/ Optics Communications J J3 (1995) 498-504
man amplification is made for the coherent probe Stokes wave generated by FPM. The intensity of the coherent probe wave generated by FPM is mainly determined by the nonresonant susceptibility x~J , while its Raman amplification is due to the resonant one X~3 ) only. A nonresonant spectral pedestal, which is usually induced by the FPM, indeed does not appear in the Raman amplification spectrum under the biharmonic pumping as seen in a dashed curve of Fig. I. This nonresonant pedestal is an obstacle in the the CARS investigations when a vibronic resonance structure is investigated in disordered condensed media where Ix~J I » Ix~) I. Therefore, the obtained high intensity of the Stokes signal and the absence of a nonresonant pedestal make this nonlinear process attractive for spectroscopic applications as mentioned below. The inset of Fig. I shows a Stokes spectrum in a region of 950-1250 cm - I which was obtained by the biharmonic fiber pumping using a laser beam of Al = 532 nm and tunable beam of )' 2 =560-570 nm. Two peaks are observed at frequency shifts 1080 em -I and 1160 cm - I . The resolution of these lines is significantly high when compared with the spontaneous Raman spectrum. Therefore, the present method is useful to clarify the fine structure appearing in low-intensity vibronic resonances. Another application using the biharmonic pumping method is suggested as follows. When we used laser pulses with multi-spike temporal structure due to a partial mode-locking in the pumping [4] , we observed weak Stokes and anti-Stokes radiation with the same multi-spike structure in a tuning range of VI - V 2 = 1200-3000 cm -1. When the difference V I - V 2 was decreased from 1200 em- I , one of more intense spikes in the probe Stokes wave was enhanced considerably by the Raman amplification. As a result, a single picosecond pulse was generated. Therefore the biharmonic pumping is useful to obtain such a tunable, intense picosecond narrow pulse. However, when the biharmonic pumping power was increased to give rise to a saturation of the Stokes power (Figs. 3 and 4), it was observed that the other low-intensity spikes are also amplified and its pulse envelope consists of several picosecond subpulses.
503
4. Conclusion We have investigated nonlinear processes under biharmonic laser pumping of single-mode silica fibers. We tuned continuously biharmonic pumping frequency difference both in the region of vibronie resonances of fused silica (v I - V2 < 1200 ern- I ) and out of this region (v 1 - v2 > 1200-3000 cm :") , using a pulsed YAG :Nd laser and a tunable dye or LiF :F2' color center laser. When the pumping frequency difference is changed continuously in the range of 1200-3000 cm -I , both the low-intensity Stokes and anti-Stokes of frequencies Vs ( = 2 V2 - v\) and Va (= 2 VI - V2), respectively, are generated. The power of the Stokes Pso (antiStokes PaD) has little dependence on the fiber length and on the pumping powers. It is concluded that this process involves a generation of the coherent probe Stokes (anti-Stokes) due to the FPM at the collinear interaction of all waves in a fiber length of a few em. An absence of obvious dependences of Pso (PaD) on the pumping power and fiber length is explained by the influence of fiber imperfections and pumping intensity to the phase matching bandwidth ofFPM. The observed small frequenc y dispersion of the Stokes (anti-Stokes) intensity is in a good agreement with a small dispersion of the nonresonant cubic susceptibility, which is responsible for the generation of the coherent probe Stokes wave. When the pumping frequency difference is changed in a region of V 1- V 2< 1200 em-I , it was observed that the Stokes power at the output of long fibers of several meters increases up to three orders in magnitude, while the anti-Stokes disappears. We have measured the Stokes power by the biharmonic pumping as a function of the fiber length and pumping power. These results show that exponential amplification by the low-frequenc y V2 pumping is mainly responsible for a high conversion efficiency of the Stokes generation. We also obtained the Stokes amplification spectrum plotted against the frequency difference VI - V 2' The spectrum is similar to the spontaneous Raman scattering spectrum. The obtained experimental results and the analysis show that the Vs Stokes generation does not need the IPM conditions over the whole fiber length. We have determined the mechanism of the nonlinear process as a result of Raman amplification of the coherent
504
A.Ya. Karasik, T. Tsuboi / OpticsCommunications 113 (1995) 498-504
probe Stokes wave generated by FPM; the low-intensity probe Stokes wave (vs ) is generated in a fiber length of several em by the coherent excitation of phonons offrequency Vph= VI - V2 = V2 - Vs and by the collinear interaction of four waves, and subsequently it is Raman-amplified exponentially over the fiber length.
Acknowledgements We thank E.A. Zachidov, P.G. Zverev and T.T. Basiev for their assistances and helpful discussions. The present work was partially supported by a Grantin-Aid from the Japanese Ministry of Education and Science.
References [I) R.H. Stolen, IEEE 1. Quantum Electron. QE-11 (1975) 100. (2) E.M. Dianov, E.A. Zakhidov, A.Ya. Karasik, P.Y. Mamyshev and A.M. Prokhorov, JETP Lett. 34 (1981 ) 38; Sov. Phys. JETP 56 (1982) 21.
[3) R.H. Stolen, J.E. Bjorkholm and A. Ashkin, Appl. Phys. Lett. 24 (1974) 308. [4) T.T. Basiev, E.M. Dianov, E.A. Zakhidov, A.Ya. Karasik, S.B. Mirov and A.M. Prokhorov, JETP Lett. 37 (1983) 229. [5) R.H. Stolen and J.E. Bjorkholm, IEEE J. Quantum Electron. QE-11 (1982) 1062. [6) E.M. Dianov, A.Ya. Karasik and E.A. Zakhidov, SOy. J. Quantum Electron. 16 (1986) 382. [7] T.T. Basiev, V.V. Osiko and S.B. Mirov, IEEE J. Quantum Electron. QE-24 (1988) 1052. [8) G.E. Walrafen and P.N. Krishnan, Appl. Optics 21 (1982) 351. [9) N.1. Koroteev, Opt. Spectrosc. 29 (1970) 286. [10) N.1. Koroteev and I.L. Shumai, SOy. J. Quantum Electron. 4 (1975) 2489. [II) N.1. Licholit, V.I. Kislenko, V.L. Strishevskii and Yu.N. Yashkir, Ukrainian Fizicheskii Zh. (Kiev) 21 (1976) 1012. [12) G.V. Venkin, G.M. Krochik, L.L. Kulyuk, 0.1. Maleev and Yu.G. Khronopulo, JETP Lett. 21 (1975) 105. [13) V.S. Butylkin, G.V. Venkin, L.1. Kulyuk, OJ. Maleev, Yu.G. Khronopulo and M.F. Shalyaev, SOy. J. Quantum Electron. 7 (1977) 867. [14) G.V. Venkin, G.M. Krochik, L.1. Kulyuk, 0.1. Maleev and Yu.G. Khronopulo, SOy. Phys. JETP 43 (1975) 873. [15) A. Owyoung and N. George, Phys. Rev. B 5 (1972) 628. [16) R.G. Smith, Appl. Optics II (1972) 2489. [17) P. Lallemand, P. Simova and G. Bret, Phys. Rev. Lett. 17 (1966) 1239.