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Multifrequency cw solid-state laser
1 May 1995
OPTICS COMMUNICATIONS Optics Communications 116 (1995) 383-388
Multifrequency cw solid-state laser V.L. Kalashnikov, International
V.P. Kalosha, V.P. Mikhailov, I.G. Poloyko
Laser Center, Belarus State University, 7 Kurchatov St., Minsk 220 064, Belarus
Received 19 October 1994
Abstract Modulators for cw solid-state lasers additional cavity and antiresonant loop main and secondary (Stokes or double) numerical simulation of the generation
based on stimulated Raman scattering and second harmonic generation effects in the are proposed. They are shown to be an effective tool for ultra-short pulse generation at frequencies simultaneously. The static properties of these modulators are analysed. The dynamics of the Ti : sapphire laser with proposed modulators enables one to determine
laser parameter ranges where effective mode-locking takes place.
The generation of powerful ultra-short pulses tunable in a wide spectral range is a very actual problem for different scientific and technical applications. A great progress in this field has been made since the new wide-band solid-state active media became widely used providing pulse durations close to the theoretical minimum [ 1 ] . There are various recent techniques for the mode-locking of cw solid-state lasers capable to produce subpico- and femtosecond pulses using nonresonant nonlinearities, such as additive-pulse mode-locking with nonlinear [2-X] and linear [9,10] resonator, Kerr-lens mode-locking [ 1 l-171, the use of the nonlinear optical effect [ l&20] and phase selfmodulation in a birefringence medium [ 21-231 or in the antiresonant loop [ 24-261. The wavelength tunability of these methods, however, is restricted by active medium gain band width. To extend the tunable range a frequency conversion is usually applied. Traditional systems of such type consist of two independent units providing mode-locking and frequency conversion. This complicates the system and increases the power losses due to the additional optical elements introduced into the cavity. The optical parametric oscillators (OPOs) [27] or lasers with intracavity second har0030-4018/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIOO30-4018(95)00058-5
manic generation (SHG) [28] for ultra-short modelocked pulses experience also the problem of dispersion distortion of the pulse passing the nonlinear medium, which reduces the conversion efficiency. If we succeed to combine in one optical element the mode-locking and frequency conversion properties we may overcome this probleni and realize an effective tool for ultra-short pulse generation tunable far beyond the gain band width. It is known that SHG may be used in the modulator containing the nonlinear crystal, the phase element and the dichroic mirror. The direct SHG action leads to a nonlinear attenuation of the field because the more intensive fluctuation spikes convert into the second harmonic more efficiently. So, the dichroic mirror and the phase element are used to provide the back frequency conversion (from the second harmonic to the fundamental). This ensures the decrease of the nonlinear losses at the modulator with the increase of the incident field intensity, i.e. the amplitude fluctuation discrimination. As the back frequency conversion from Stokes to the fundamental has an extremely low efficiency, using the scheme analogous to that of the SHG-based modulator for SRS in the single cavity is impossible.
384
V.L. Kalashnikov
et al. /Optics
Here we propose, for the first time to our knowledge, SHG- and SRS-based modulators for the mode-locking of cw solid-state lasers. Under certain phase relations between main and additional cavity fields or between counterpropagating beams inside the antiresonant loop, an interferometrical subtraction from the main cavity field of its replica nonlinear attenuated due to SHG or SRS ensures a saturable absorber-like action. A long time fundamental and Stokes ultra-short pulse formation, respectively, enables one to use a short nonlinear crystals with a practically negligible dispersion walkoff for the main and secondary frequency pulses and the small profile distortion, although the one-pass conversion efficiency is small. The simplest scheme of the proposed mode-locking technique is the laser with an additional cavity. A nonlinear crystal placed in the additional cavity provides a second harmonic generation, a stimulated Raman scattering or a frequency summing [ 281 depending on the crystal orientation. An additional cavity may be treated as a nonlinear Fabry-Perot interferometer with its characteristic time equal to the main cavity round-trip time. The nonlinear reflection coefficient of the interferometer is [ 5,271 R
=R
Ill
LfR+exp(i$) LR+exp(i4)
(1)
’
where R is the reflectivity of the coupling mirror between the main and additional cavity, C$is the phase detuning between the cavities, L is the nonlinear loss in the additional cavity. For the SHG and SRS the losses are L = sech ( $Z$ , Z exp( - gZZ) E’$ + EZ exp( -gZZ)
’
(2)
respectively, where y and g are nonlinear coupling coefficient, y is proportional to the third order susceptibility components, g is proportional to the Raman scattering coefficient, I is the nonlinear crystal length, 1 -R,,R E=Ein (l_R2)1/2 is the fundamental field before the nonlinear element in the additional cavity, Ei, is the incident additional cavity field and Es is the Stokes field before the nonlinear element in the additional cavity, Z= EZ + Ez.
Communications
116 (1995) 383-388
Eq. (2) is valid under phase and group synchronism conditions. It has been assumed also that in the SRS case the ultra-short pulse duration is greater than the nonlinear medium transverse relaxation time determining the spontaneous scattering band width. These approximations are true for example up to a 5 ps generation pulse duration for a KTP-crystal length I= 3 cm, nondimensional coefficients $I (a,,T,,) ‘I2 = 10 and gl/( u,,T,,) = 20, where ~32 is the Ti : sapphire emission cross section and T,, = 5 ns is the cavity round-trip time. The nonlinear reflection coefficient cannot be expressed from Eqs. ( l)-( 3) as an explicit function of the interferometer parameters and incident field intensity, because the fundamental intensity in Fq. (3) depends on R,,. This may provide a wide range of nonlinear Fabry-Perot interferometer properties [ 271; one of them is the discrimination property which may be characterized by the derivative of the nonlinear reflectivity over the incident field intensity for a small signal. The dependence of 3R I, 113U on R and C$in the case of SHG in the additional cavity is presented at Fig. la, where U = CT~~T,,E~?,is the normalized field intensity. In the case of SRS this characteristic behavior is analogous. It is seen that the highest positive values of the derivative are located near + = r and R = 0.6-0.95, where the modulator discriminates the intensity fluctuations most efficiently. With these parameters one should anticipate a mode-locking occurrence. Now we proceed further with a numerical simulation of the generation dynamics of the cw solid-state laser with a SRS-based modulator. It may verify weather our conclusions based on the modulator’s static properties analysis were right or not. The numerical simulations were based on the fluctuation theory [ 10,331. The generation is initiated by a spontaneous emission of the active medium. Then the field is affected by the four-level active medium amplification with a maximal gain a;, and the absorption and emission cross sections u3, and LT~~,respectively pumped by a continuous photon flux Zr, a spectral filtering with a group time delay tf, the modulator action and linear losses. Numerical simulation enables one to determine the ranges of the laser parameters, where the efficient mode-locking takes place. We name these ranges the mode-locking ranges. In Figs. 2 and 3, for example, the mode-locking ranges are plotted in the R and Up = a14TcavZpcoordinates, where U, is the nor-
V.L. Kalashnikov et al. /Optics
Communications I16 (1995) 383-388
malized pump intensity, for SHG and SRS, respectively. Any point inside the range gives a pump intensity level UP and a reflection R that provide an efficient mode-locking satisfying the following criteria: no less than 95% of the generation energy at the cavity period is concentrated in the main peak, and this peak remains stable for a long enough time period (200 T,,=20 ps in our case) [33]. Under a continuous pump the overall action of all pulse shaping factors becomes balanced and a high-stable ultra-short pulse train is produced. When the laser turns into the modelocking regime the ultra-short pulse train at the fundamental frequency is accompanied by the train at the Stokes frequency and/or the second harmonic with a lower intensity. If the operation point lies below the lower mode-locking range boundary, a free oscillation regime takes place. The rising above the upper range boundary causes the unstable multipulse generation. As one can see, the mode-locking range is found near the values of R and 4 predicted from the static properties of the modulator analysis. The lowest pump intensities and the widest mode-locking range extension, which are of practical interest, lie near the maximum of the modulator’s reflection derivative. A key parameter enabling to control the mode-locking dynamics in the SRS case is the Stokes field fraction circulating inside the additional cavity. It determines the fundamental-to-Stokes conversion efficiency. This
a
b
“;II.-:-.v Fig. 1. The “discrimination strength” of the SHG-based modulators (a) in the additional cavity and (b) in the antiresonant loop as function of R and 4. The normalized coefficient of nonlinearity is $/(oJca”)“2= 10.
u,
I 2
O.l-
n
1
:B Q
rr,
0.1
385
t
A 2/
CLOl1
O.OlO.OOlr/:--’
IV
w
0.001 0
I 0.4
0.8
I R
Fig. 2. Effective mode-locking ranges of the Ti: sapphire laser with SHG-based modulators (1) in the additional cavity and (2) in the antiresonant loop. The simulation parameters are 4= Z-, tf= 1 ps, c&= 1.5, T,,,= 1 ns, $l(~jJ~~~)“~= 10.
0
3
,
0.4
I
I
0.8
g
Fig. 3. Effective mode-locking ranges of the Ti : sapphire laser with SRS-based modulators ( 1) in the additional cavity and (2,3) in the antiresonant loop; gZlu3zT,, = 30 and the other parameters are as in Fig. 2. The Stokes field fraction extended from the system is ( 1) 50%, (2) lOO%, (3) 10%.
386
V.L. Kalashnikov et al. /Optics Communications I16 (1995) 383-388
mittive for the Stokes field and has a transmission of (30-70) % for the fundamental, while for the output mirror the inverse situation holds, i.e. (30-70) % and 90% for Stokes and the fundamental, respectively. Such an optics (multilayer dielectric mirrors) is available now and for the frequency shift order of 10’ cm- ’ between the fundamental and Stokes fields which is typical for SRS it is quite functional. The nonlinear reflection of the modulator is Rn1=R4L2+2R2(1-R2)Lcos M3
4
Fig. 4. Schematic diagram of the SHG- or SRS-based modulator in the antiresonant loop. A: active medium, BF: birefringence filter, MI, MT,M3, Ma: mirrors, BS: beam splitter, NE: nonlinear element providing SHG or SRS, PE: phase element providing a nonreciprocal phase shift, IS: isolator.
(4)
where the nonlinear losses L are the same as in Eq. (2). In case of SRS the field before the SRS-crystal is E = RE,,, where Ei, and Es are the fundamental and Stokes amplitude before the ARL, respectively. After the ARL the Stokes field is E$OUt)
=E,
G
I
g+E’exp(-gll)
efficiency becomes small when a large amount of the Stokes field is extended from the cavity. If a large amount of the Stokes field is locked inside the system the SRS-conversion efficiency may become large enough to make the nonlinear losses for the fundamental wave too high to support the generation. So, there is an optimal loss level for the Stokes field in the additional cavity. In our case this was determined to be near 50%, which provides the maximum size of the modelocking range. It is well known that the additional cavity suffers from one substantial shortcoming, namely, the need for interferometric precise control of the main and additional cavity lengths. Using the antiresonant loop enables one to avoid the above disadvantage. The laser consists of a unidirectional ring, coupled to the ARL (Fig. 4). The last contains a nonlinear crystal NE and the element providing the nonreciprocal phase shift 4 between counterpropagating beams inside the ARL. The counterpropagating beams inside the ARL travel along equal optical paths and there is no need in the electronic systems for a mismatch control. The desired phase mismatch 4 between the beams is easily controlled by the nonreciprocal phase element PE (Fig. 4) based on Faraday rotators. It is supposed that SRS or SHG is effective only for the one of two beams in the ARL, and the coupling plate is transparent for both the fundamental, Stokes or second harmonic fields. For example, the coupling plate may be almost fully trans-
++(l-R2)2,
(5)
As it seen from Fig. lb the discrimination “strength’ ’ of the modulator with SHG in the ARL is slightly lower than for the additional cavity modulator. In case of SRS in the ARL the discrimination “strength” behavior is similar but depends strongly on the intensity of the Stokes component locked inside the system. The rise of Es leads at first to the rise of the reflectivity derivative maximum and shifts this maximum toward smaller reflectivities R. Then a sharp fall of R,,, and dR$/aU occurs, because of the very strong SRS. Range 2 in Fig. 2 is a mode-locking range of the Ti : sapphire laser with a SHG-based modulator in the ARL. The range extension over U, is wider than the one for the additional cavity more than an order of magnitude due to the substantial lowering of the modelocking threshold. This is connected with the difference in the stationary transmissions of the two modulators. For the modulator with the ARL the transmission varies from 68% at the right side of the mode-locking range to 92% at the left, the minimal pump intensity U, corresponding to the aRz, /aU maximum. The minimal ultra-short pulse duration, both Stokes and fundamental, determined by the spectral filter characteristic time tf is observed inside the mode-locking range and lies near (5-lO)t,, which makes 5-10 ps. This is not too far from the shortest duration in the lasers with intracavity frequency conversion. For example, OPO in [ 341 produces 400 fs signal pulses,
V.L.. Kalashnikav et al. /Optics
which after extracavity dispersion compensation reduce to 29 fs. It should be noted, also, that the oscillator was synchronously pumped by 80 fs pulses from the Ti: sapphire source, while our scheme uses cw pumping. Limitations on minimal pulse duration in SRS-lasers are imposed by two factors: ( 1) the dispersion coefficient of the nonlinear medium and (2) its transverse relaxation time T2determining the inertional response of the SRS. For commonly used solid-state media this limit is near 1 ps. Shorter pulse durations are available in optical fibers (see, for example, [ 351, where 80 fs Stokes pulses have been generated). But the use of the fibers as an intracavity nonlinear element is impossible because of the high intracavity powers, which may destroy the fiber. Correspondingly, only low output powers are produced in traditional SRSconverters with the fibers. The mode-locking of the laser with an SRS-based modulator has some peculiarities (Fig. 3, ranges 2 and 3). When the complete Stokes field is extended from the system, at each transit the mode-locking is analogous to the one provided by the SHG-based modulator (range 2). However, the mode-locking threshold is somewhat higher in this case, because for an effective SRS with a low initial Stokes field higher pump intensities are needed. The increase of the fraction of the Stokes component extended from the system leads to the rise of the lower boundary of the mode-locking range, because of the effective SRS (range 2, Fig. 3). Locking the Stokes component lowers the modelocking threshold substantially and shifts the whole range toward smaller R (range 3). The upper boundary, however, is lowered too, because at high pump intensities the SRS becomes effective enough to break down the generation. It should be noted that locking the Stokes component inside the system causes the pulse temporal shape distortion, due to a mismatch of the effective cavity periods for the fundamental and Stokes components. This effect results in a pulse duration growth of up to 10-20 times minimal duration determined by the band limiting spectral filter characteristic time tp However, the Stokes train intensity also grows. In conclusion, we have demonstrated the possibility of the efficient mode-locking of cw wide-band solidstate lasers using the effect of stimulated Raman scatand second harmonic generation. The tering mode-locking features of such systems have been investigated, which enables one to determine the opti-
Communications 116 (1995) 383-388
387
mal parameters for efficient mode-locking. The possibility of generating stable ultra-short pulse trains at the fundamental and secondary (double and Stokes) frequencies was shown. The proposed systems enable one to extend the spectral range of mode-locked solid-state lasers and to achieve a discrete-continuous tuning of the generation wavelength up to 0.3-2 pm. The work was partially supported by the BuelorusSian Foundation for Fundamental Researches, grant #MP12.
References [l] W.H. Knox, Opt. Photon. News 5 ( 1992) 10. [2] K.J. Blow and D. Wood, J. Opt. Sot. Am. B 5 (1988) 629. [3] P.N. Kean, X. Zhn, D.W. Crust, R.S. Grant, N. Langford and W. Sibbett, Opt. I&t. 14 (1989) 39. [4] J. Mark, L.Y. Liu, K.L. Hall, H.A. Haus and E.P. Ippen, Opt. Lett. 14 (1989) 48. [5] H.A. Haus, J.G. Fujimoto and E.P. Ippen, J. Opt. Sot. Am. B 8 (1991) 2068. [6] R.S. Grant, P.N. Kean, D. Burns and W. Sibbett, Opt. Lett. 16 (1991) 384. [ 71 U. Keller and T.H. Chiu, IEEE J. QuantumElectron. 28 ( 1992) 1710. [S] V.L. Kalashnikov, V.P. Kalosha, V.P. Mikhailov and MI. Demchuk, Kvant. Electron. (Moscow) 19 (1992) 1159. [9] P.M.W. French, D.U. Noske, N.H. Rizvi, J.A.R. Williams and J.R. Taylor, Opt. Commnn. 83 (1991) 185. [lo] MI. Demchuk, V.L. Kalashnikov, V.P. Kalosha and V.P. Mikhailov, Opt. Commun. 96 ( 199 1) 249. [ 111 G. Gab&a, D. Huang, J. Jacobson, M. Ramaswamy, E.P. Ippen and J.G. Fujimoto, Opt. Lett. 16 (1991) 1. [ 121 DE. Spence, P.N. Kean and W. Sibbett, Opt. Lett. 16 (1991) 42. [ 131 N. Sarukma, Y. Ishida and H. Nakano, Opt. Lett. 16 (1991) 153. [ 141 G.P.A. Malcolmand AI. Ferguson, Opt. Lett. 16 (1991) 1967. [ 151 P. LiKamWa, B.H.T. Chai and A. Miller, Opt. Lett. 17 (1992) 1438. [ 161 J.M. Evans, D.E. Spence, W. Sibbett, B.H.T. Chai and A. Miller, Opt. L&t. 17 (1992) 1447. [ 171 Y. Pang, V. Yanovsky, F. Wise and B.I. Minkov, Opt. L&t. 18 (1993) 1168. [ 181 L. Dahlstrom, Opt. Commun. 4 (1971) 214. [ 191 G.B. Alt’shu1er.E.G. Dulneva, V.B. Karasev and S.F. Sharlay, Zh. Tech. Fiz. 49 (1979) 143. [ 20 ] V.L. Kalashulkov, V.P. Kalosha, V.P. Mikhallov, LG. Poloyko andM.l. Demchuk, J. Opt. Sot. Am. B 10 (1991) 1443. [21] M. Hofer, M.H. Ober, F. Haberl and M.G. Fermann, IEEE J. Quantum Electron. 28 (1992) 720. [22] C.J. Chen, P.K.A. Wai andC.R. Menyuk, Opt. I&t. 17 (1992) 417.
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et al. /Optics
[23] V.L. Kalashnikov, V.P. Kalosha and V.P. Mikhailov, Kvant. Electron. (Moscow) 21 (1994) 101. [24] T.F. Carmthers and I.N. Dulling, Opt. Lett. 15 (1990) 804. [25] A.G. Bulushev, E.M. Dianov and O.G. Okhotnikov, Opt. Lett. 16 (1991) 88. [26] V.L. Kalashnikov,V.P. Kalosha,V.P. Mikhailov, LG. Poloyko and M.I. Demchuk, Kvant. Electron. (Moscow) 21 (1994) 37. [27] I.R. Shen, Principles of nonlinear optics (Nauka, Moskow, 1989). [28] A.M. Belskii, I.M. Gulis, V.P. Mikhailov, K.A. Saecbnikov and E.A. Tsvirko, Kvant. Electron. (Moscow) 19 (1992) 769. 1291 K.A. Stankov, Appl. Phys. B 45 (1988) 191.
Communications
116 (1995) 383-388
[30] K.A. Stankov and J. Jethwa, Opt. Commun. 66 (1988) 41. [31] J.R.M. Barr and D.W. Huges, Appl. Phys. B 49 (1989) 423. [32] S.A. Akhmanov, V.A. Vyslonkh and A.S. Cbirkin, Optics of femtosecond laser pulses (Nauka, Moskow, 1988). [33] V.L. Kalasbnikov, V.P. Kalosha, V.P. Mikhallov and MI. Demchuk, Kvant. Electron. (Moscow) 19 (1992) 479. [34] W. Sibbett, M. Ebrahimzadeh, D.T. Reid and J.M. Dudley, in: Abstract VIlIth Int. Symp. on Ultrafast processes in spectroscopy, Vilnius 1993, Ch. 11.1. [35] B. Zysset, P. Beaud, W. Hodel and H.P. Weber, Ultrafast phenomena, Vol. V, eds. G.R. Flemming and A.E. Siegman (Springer, Berlin 1986) p. 54.
OPTICS COMMUNICATIONS Optics Communications 116 (1995) 383-388
Multifrequency cw solid-state laser V.L. Kalashnikov, International
V.P. Kalosha, V.P. Mikhailov, I.G. Poloyko
Laser Center, Belarus State University, 7 Kurchatov St., Minsk 220 064, Belarus
Received 19 October 1994
Abstract Modulators for cw solid-state lasers additional cavity and antiresonant loop main and secondary (Stokes or double) numerical simulation of the generation
based on stimulated Raman scattering and second harmonic generation effects in the are proposed. They are shown to be an effective tool for ultra-short pulse generation at frequencies simultaneously. The static properties of these modulators are analysed. The dynamics of the Ti : sapphire laser with proposed modulators enables one to determine
laser parameter ranges where effective mode-locking takes place.
The generation of powerful ultra-short pulses tunable in a wide spectral range is a very actual problem for different scientific and technical applications. A great progress in this field has been made since the new wide-band solid-state active media became widely used providing pulse durations close to the theoretical minimum [ 1 ] . There are various recent techniques for the mode-locking of cw solid-state lasers capable to produce subpico- and femtosecond pulses using nonresonant nonlinearities, such as additive-pulse mode-locking with nonlinear [2-X] and linear [9,10] resonator, Kerr-lens mode-locking [ 1 l-171, the use of the nonlinear optical effect [ l&20] and phase selfmodulation in a birefringence medium [ 21-231 or in the antiresonant loop [ 24-261. The wavelength tunability of these methods, however, is restricted by active medium gain band width. To extend the tunable range a frequency conversion is usually applied. Traditional systems of such type consist of two independent units providing mode-locking and frequency conversion. This complicates the system and increases the power losses due to the additional optical elements introduced into the cavity. The optical parametric oscillators (OPOs) [27] or lasers with intracavity second har0030-4018/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIOO30-4018(95)00058-5
manic generation (SHG) [28] for ultra-short modelocked pulses experience also the problem of dispersion distortion of the pulse passing the nonlinear medium, which reduces the conversion efficiency. If we succeed to combine in one optical element the mode-locking and frequency conversion properties we may overcome this probleni and realize an effective tool for ultra-short pulse generation tunable far beyond the gain band width. It is known that SHG may be used in the modulator containing the nonlinear crystal, the phase element and the dichroic mirror. The direct SHG action leads to a nonlinear attenuation of the field because the more intensive fluctuation spikes convert into the second harmonic more efficiently. So, the dichroic mirror and the phase element are used to provide the back frequency conversion (from the second harmonic to the fundamental). This ensures the decrease of the nonlinear losses at the modulator with the increase of the incident field intensity, i.e. the amplitude fluctuation discrimination. As the back frequency conversion from Stokes to the fundamental has an extremely low efficiency, using the scheme analogous to that of the SHG-based modulator for SRS in the single cavity is impossible.
384
V.L. Kalashnikov
et al. /Optics
Here we propose, for the first time to our knowledge, SHG- and SRS-based modulators for the mode-locking of cw solid-state lasers. Under certain phase relations between main and additional cavity fields or between counterpropagating beams inside the antiresonant loop, an interferometrical subtraction from the main cavity field of its replica nonlinear attenuated due to SHG or SRS ensures a saturable absorber-like action. A long time fundamental and Stokes ultra-short pulse formation, respectively, enables one to use a short nonlinear crystals with a practically negligible dispersion walkoff for the main and secondary frequency pulses and the small profile distortion, although the one-pass conversion efficiency is small. The simplest scheme of the proposed mode-locking technique is the laser with an additional cavity. A nonlinear crystal placed in the additional cavity provides a second harmonic generation, a stimulated Raman scattering or a frequency summing [ 281 depending on the crystal orientation. An additional cavity may be treated as a nonlinear Fabry-Perot interferometer with its characteristic time equal to the main cavity round-trip time. The nonlinear reflection coefficient of the interferometer is [ 5,271 R
=R
Ill
LfR+exp(i$) LR+exp(i4)
(1)
’
where R is the reflectivity of the coupling mirror between the main and additional cavity, C$is the phase detuning between the cavities, L is the nonlinear loss in the additional cavity. For the SHG and SRS the losses are L = sech ( $Z$ , Z exp( - gZZ) E’$ + EZ exp( -gZZ)
’
(2)
respectively, where y and g are nonlinear coupling coefficient, y is proportional to the third order susceptibility components, g is proportional to the Raman scattering coefficient, I is the nonlinear crystal length, 1 -R,,R E=Ein (l_R2)1/2 is the fundamental field before the nonlinear element in the additional cavity, Ei, is the incident additional cavity field and Es is the Stokes field before the nonlinear element in the additional cavity, Z= EZ + Ez.
Communications
116 (1995) 383-388
Eq. (2) is valid under phase and group synchronism conditions. It has been assumed also that in the SRS case the ultra-short pulse duration is greater than the nonlinear medium transverse relaxation time determining the spontaneous scattering band width. These approximations are true for example up to a 5 ps generation pulse duration for a KTP-crystal length I= 3 cm, nondimensional coefficients $I (a,,T,,) ‘I2 = 10 and gl/( u,,T,,) = 20, where ~32 is the Ti : sapphire emission cross section and T,, = 5 ns is the cavity round-trip time. The nonlinear reflection coefficient cannot be expressed from Eqs. ( l)-( 3) as an explicit function of the interferometer parameters and incident field intensity, because the fundamental intensity in Fq. (3) depends on R,,. This may provide a wide range of nonlinear Fabry-Perot interferometer properties [ 271; one of them is the discrimination property which may be characterized by the derivative of the nonlinear reflectivity over the incident field intensity for a small signal. The dependence of 3R I, 113U on R and C$in the case of SHG in the additional cavity is presented at Fig. la, where U = CT~~T,,E~?,is the normalized field intensity. In the case of SRS this characteristic behavior is analogous. It is seen that the highest positive values of the derivative are located near + = r and R = 0.6-0.95, where the modulator discriminates the intensity fluctuations most efficiently. With these parameters one should anticipate a mode-locking occurrence. Now we proceed further with a numerical simulation of the generation dynamics of the cw solid-state laser with a SRS-based modulator. It may verify weather our conclusions based on the modulator’s static properties analysis were right or not. The numerical simulations were based on the fluctuation theory [ 10,331. The generation is initiated by a spontaneous emission of the active medium. Then the field is affected by the four-level active medium amplification with a maximal gain a;, and the absorption and emission cross sections u3, and LT~~,respectively pumped by a continuous photon flux Zr, a spectral filtering with a group time delay tf, the modulator action and linear losses. Numerical simulation enables one to determine the ranges of the laser parameters, where the efficient mode-locking takes place. We name these ranges the mode-locking ranges. In Figs. 2 and 3, for example, the mode-locking ranges are plotted in the R and Up = a14TcavZpcoordinates, where U, is the nor-
V.L. Kalashnikov et al. /Optics
Communications I16 (1995) 383-388
malized pump intensity, for SHG and SRS, respectively. Any point inside the range gives a pump intensity level UP and a reflection R that provide an efficient mode-locking satisfying the following criteria: no less than 95% of the generation energy at the cavity period is concentrated in the main peak, and this peak remains stable for a long enough time period (200 T,,=20 ps in our case) [33]. Under a continuous pump the overall action of all pulse shaping factors becomes balanced and a high-stable ultra-short pulse train is produced. When the laser turns into the modelocking regime the ultra-short pulse train at the fundamental frequency is accompanied by the train at the Stokes frequency and/or the second harmonic with a lower intensity. If the operation point lies below the lower mode-locking range boundary, a free oscillation regime takes place. The rising above the upper range boundary causes the unstable multipulse generation. As one can see, the mode-locking range is found near the values of R and 4 predicted from the static properties of the modulator analysis. The lowest pump intensities and the widest mode-locking range extension, which are of practical interest, lie near the maximum of the modulator’s reflection derivative. A key parameter enabling to control the mode-locking dynamics in the SRS case is the Stokes field fraction circulating inside the additional cavity. It determines the fundamental-to-Stokes conversion efficiency. This
a
b
“;II.-:-.v Fig. 1. The “discrimination strength” of the SHG-based modulators (a) in the additional cavity and (b) in the antiresonant loop as function of R and 4. The normalized coefficient of nonlinearity is $/(oJca”)“2= 10.
u,
I 2
O.l-
n
1
:B Q
rr,
0.1
385
t
A 2/
CLOl1
O.OlO.OOlr/:--’
IV
w
0.001 0
I 0.4
0.8
I R
Fig. 2. Effective mode-locking ranges of the Ti: sapphire laser with SHG-based modulators (1) in the additional cavity and (2) in the antiresonant loop. The simulation parameters are 4= Z-, tf= 1 ps, c&= 1.5, T,,,= 1 ns, $l(~jJ~~~)“~= 10.
0
3
,
0.4
I
I
0.8
g
Fig. 3. Effective mode-locking ranges of the Ti : sapphire laser with SRS-based modulators ( 1) in the additional cavity and (2,3) in the antiresonant loop; gZlu3zT,, = 30 and the other parameters are as in Fig. 2. The Stokes field fraction extended from the system is ( 1) 50%, (2) lOO%, (3) 10%.
386
V.L. Kalashnikov et al. /Optics Communications I16 (1995) 383-388
mittive for the Stokes field and has a transmission of (30-70) % for the fundamental, while for the output mirror the inverse situation holds, i.e. (30-70) % and 90% for Stokes and the fundamental, respectively. Such an optics (multilayer dielectric mirrors) is available now and for the frequency shift order of 10’ cm- ’ between the fundamental and Stokes fields which is typical for SRS it is quite functional. The nonlinear reflection of the modulator is Rn1=R4L2+2R2(1-R2)Lcos M3
4
Fig. 4. Schematic diagram of the SHG- or SRS-based modulator in the antiresonant loop. A: active medium, BF: birefringence filter, MI, MT,M3, Ma: mirrors, BS: beam splitter, NE: nonlinear element providing SHG or SRS, PE: phase element providing a nonreciprocal phase shift, IS: isolator.
(4)
where the nonlinear losses L are the same as in Eq. (2). In case of SRS the field before the SRS-crystal is E = RE,,, where Ei, and Es are the fundamental and Stokes amplitude before the ARL, respectively. After the ARL the Stokes field is E$OUt)
=E,
G
I
g+E’exp(-gll)
efficiency becomes small when a large amount of the Stokes field is extended from the cavity. If a large amount of the Stokes field is locked inside the system the SRS-conversion efficiency may become large enough to make the nonlinear losses for the fundamental wave too high to support the generation. So, there is an optimal loss level for the Stokes field in the additional cavity. In our case this was determined to be near 50%, which provides the maximum size of the modelocking range. It is well known that the additional cavity suffers from one substantial shortcoming, namely, the need for interferometric precise control of the main and additional cavity lengths. Using the antiresonant loop enables one to avoid the above disadvantage. The laser consists of a unidirectional ring, coupled to the ARL (Fig. 4). The last contains a nonlinear crystal NE and the element providing the nonreciprocal phase shift 4 between counterpropagating beams inside the ARL. The counterpropagating beams inside the ARL travel along equal optical paths and there is no need in the electronic systems for a mismatch control. The desired phase mismatch 4 between the beams is easily controlled by the nonreciprocal phase element PE (Fig. 4) based on Faraday rotators. It is supposed that SRS or SHG is effective only for the one of two beams in the ARL, and the coupling plate is transparent for both the fundamental, Stokes or second harmonic fields. For example, the coupling plate may be almost fully trans-
++(l-R2)2,
(5)
As it seen from Fig. lb the discrimination “strength’ ’ of the modulator with SHG in the ARL is slightly lower than for the additional cavity modulator. In case of SRS in the ARL the discrimination “strength” behavior is similar but depends strongly on the intensity of the Stokes component locked inside the system. The rise of Es leads at first to the rise of the reflectivity derivative maximum and shifts this maximum toward smaller reflectivities R. Then a sharp fall of R,,, and dR$/aU occurs, because of the very strong SRS. Range 2 in Fig. 2 is a mode-locking range of the Ti : sapphire laser with a SHG-based modulator in the ARL. The range extension over U, is wider than the one for the additional cavity more than an order of magnitude due to the substantial lowering of the modelocking threshold. This is connected with the difference in the stationary transmissions of the two modulators. For the modulator with the ARL the transmission varies from 68% at the right side of the mode-locking range to 92% at the left, the minimal pump intensity U, corresponding to the aRz, /aU maximum. The minimal ultra-short pulse duration, both Stokes and fundamental, determined by the spectral filter characteristic time tf is observed inside the mode-locking range and lies near (5-lO)t,, which makes 5-10 ps. This is not too far from the shortest duration in the lasers with intracavity frequency conversion. For example, OPO in [ 341 produces 400 fs signal pulses,
V.L.. Kalashnikav et al. /Optics
which after extracavity dispersion compensation reduce to 29 fs. It should be noted, also, that the oscillator was synchronously pumped by 80 fs pulses from the Ti: sapphire source, while our scheme uses cw pumping. Limitations on minimal pulse duration in SRS-lasers are imposed by two factors: ( 1) the dispersion coefficient of the nonlinear medium and (2) its transverse relaxation time T2determining the inertional response of the SRS. For commonly used solid-state media this limit is near 1 ps. Shorter pulse durations are available in optical fibers (see, for example, [ 351, where 80 fs Stokes pulses have been generated). But the use of the fibers as an intracavity nonlinear element is impossible because of the high intracavity powers, which may destroy the fiber. Correspondingly, only low output powers are produced in traditional SRSconverters with the fibers. The mode-locking of the laser with an SRS-based modulator has some peculiarities (Fig. 3, ranges 2 and 3). When the complete Stokes field is extended from the system, at each transit the mode-locking is analogous to the one provided by the SHG-based modulator (range 2). However, the mode-locking threshold is somewhat higher in this case, because for an effective SRS with a low initial Stokes field higher pump intensities are needed. The increase of the fraction of the Stokes component extended from the system leads to the rise of the lower boundary of the mode-locking range, because of the effective SRS (range 2, Fig. 3). Locking the Stokes component lowers the modelocking threshold substantially and shifts the whole range toward smaller R (range 3). The upper boundary, however, is lowered too, because at high pump intensities the SRS becomes effective enough to break down the generation. It should be noted that locking the Stokes component inside the system causes the pulse temporal shape distortion, due to a mismatch of the effective cavity periods for the fundamental and Stokes components. This effect results in a pulse duration growth of up to 10-20 times minimal duration determined by the band limiting spectral filter characteristic time tp However, the Stokes train intensity also grows. In conclusion, we have demonstrated the possibility of the efficient mode-locking of cw wide-band solidstate lasers using the effect of stimulated Raman scatand second harmonic generation. The tering mode-locking features of such systems have been investigated, which enables one to determine the opti-
Communications 116 (1995) 383-388
387
mal parameters for efficient mode-locking. The possibility of generating stable ultra-short pulse trains at the fundamental and secondary (double and Stokes) frequencies was shown. The proposed systems enable one to extend the spectral range of mode-locked solid-state lasers and to achieve a discrete-continuous tuning of the generation wavelength up to 0.3-2 pm. The work was partially supported by the BuelorusSian Foundation for Fundamental Researches, grant #MP12.
References [l] W.H. Knox, Opt. Photon. News 5 ( 1992) 10. [2] K.J. Blow and D. Wood, J. Opt. Sot. Am. B 5 (1988) 629. [3] P.N. Kean, X. Zhn, D.W. Crust, R.S. Grant, N. Langford and W. Sibbett, Opt. I&t. 14 (1989) 39. [4] J. Mark, L.Y. Liu, K.L. Hall, H.A. Haus and E.P. Ippen, Opt. Lett. 14 (1989) 48. [5] H.A. Haus, J.G. Fujimoto and E.P. Ippen, J. Opt. Sot. Am. B 8 (1991) 2068. [6] R.S. Grant, P.N. Kean, D. Burns and W. Sibbett, Opt. Lett. 16 (1991) 384. [ 71 U. Keller and T.H. Chiu, IEEE J. QuantumElectron. 28 ( 1992) 1710. [S] V.L. Kalashnikov, V.P. Kalosha, V.P. Mikhailov and MI. Demchuk, Kvant. Electron. (Moscow) 19 (1992) 1159. [9] P.M.W. French, D.U. Noske, N.H. Rizvi, J.A.R. Williams and J.R. Taylor, Opt. Commnn. 83 (1991) 185. [lo] MI. Demchuk, V.L. Kalashnikov, V.P. Kalosha and V.P. Mikhailov, Opt. Commun. 96 ( 199 1) 249. [ 111 G. Gab&a, D. Huang, J. Jacobson, M. Ramaswamy, E.P. Ippen and J.G. Fujimoto, Opt. Lett. 16 (1991) 1. [ 121 DE. Spence, P.N. Kean and W. Sibbett, Opt. Lett. 16 (1991) 42. [ 131 N. Sarukma, Y. Ishida and H. Nakano, Opt. Lett. 16 (1991) 153. [ 141 G.P.A. Malcolmand AI. Ferguson, Opt. Lett. 16 (1991) 1967. [ 151 P. LiKamWa, B.H.T. Chai and A. Miller, Opt. Lett. 17 (1992) 1438. [ 161 J.M. Evans, D.E. Spence, W. Sibbett, B.H.T. Chai and A. Miller, Opt. L&t. 17 (1992) 1447. [ 171 Y. Pang, V. Yanovsky, F. Wise and B.I. Minkov, Opt. L&t. 18 (1993) 1168. [ 181 L. Dahlstrom, Opt. Commun. 4 (1971) 214. [ 191 G.B. Alt’shu1er.E.G. Dulneva, V.B. Karasev and S.F. Sharlay, Zh. Tech. Fiz. 49 (1979) 143. [ 20 ] V.L. Kalashulkov, V.P. Kalosha, V.P. Mikhallov, LG. Poloyko andM.l. Demchuk, J. Opt. Sot. Am. B 10 (1991) 1443. [21] M. Hofer, M.H. Ober, F. Haberl and M.G. Fermann, IEEE J. Quantum Electron. 28 (1992) 720. [22] C.J. Chen, P.K.A. Wai andC.R. Menyuk, Opt. I&t. 17 (1992) 417.
388
V.L. Kalashnikov
et al. /Optics
[23] V.L. Kalashnikov, V.P. Kalosha and V.P. Mikhailov, Kvant. Electron. (Moscow) 21 (1994) 101. [24] T.F. Carmthers and I.N. Dulling, Opt. Lett. 15 (1990) 804. [25] A.G. Bulushev, E.M. Dianov and O.G. Okhotnikov, Opt. Lett. 16 (1991) 88. [26] V.L. Kalashnikov,V.P. Kalosha,V.P. Mikhailov, LG. Poloyko and M.I. Demchuk, Kvant. Electron. (Moscow) 21 (1994) 37. [27] I.R. Shen, Principles of nonlinear optics (Nauka, Moskow, 1989). [28] A.M. Belskii, I.M. Gulis, V.P. Mikhailov, K.A. Saecbnikov and E.A. Tsvirko, Kvant. Electron. (Moscow) 19 (1992) 769. 1291 K.A. Stankov, Appl. Phys. B 45 (1988) 191.
Communications
116 (1995) 383-388
[30] K.A. Stankov and J. Jethwa, Opt. Commun. 66 (1988) 41. [31] J.R.M. Barr and D.W. Huges, Appl. Phys. B 49 (1989) 423. [32] S.A. Akhmanov, V.A. Vyslonkh and A.S. Cbirkin, Optics of femtosecond laser pulses (Nauka, Moskow, 1988). [33] V.L. Kalasbnikov, V.P. Kalosha, V.P. Mikhallov and MI. Demchuk, Kvant. Electron. (Moscow) 19 (1992) 479. [34] W. Sibbett, M. Ebrahimzadeh, D.T. Reid and J.M. Dudley, in: Abstract VIlIth Int. Symp. on Ultrafast processes in spectroscopy, Vilnius 1993, Ch. 11.1. [35] B. Zysset, P. Beaud, W. Hodel and H.P. Weber, Ultrafast phenomena, Vol. V, eds. G.R. Flemming and A.E. Siegman (Springer, Berlin 1986) p. 54.