Laser Physics, Vol., No.,, pp. 46 466. Original Text Copyright by Astro, Ltd. Copyright by MAIK Nauka /Interperiodica (Russia). SOLID STATE LASERS AND NONLINEAR OPTICS Neodymium Laser Q-Switched with a Cr 4+ : YAG Crystal: Control over Polarization State by Exterior Weak Resonant Radiation A. V. Kir yanov*, V. Aboites*, and N. N. Il ichev** ** Centro de Investigaciones en Optica A.C., Apartado Postal 948, Leon, GTO, 37 Mexico e-mail: kiryanov@foton.cio.mx ** General Physics Institute, Russian Academy of Sciences, ul. Vavilova 38, Moscow, 794 Russia Received July, 999 Abstract Polarization bistability is investigated theoretically in a Nd : YAG laser passively Q-switched with a Cr 4+ : YAG crystal under the weak resonant exterior radiation control.. INTRODUCTION Experimental observation [] and theoretical simulation [] of strong dependence of state of polarization of a neodymium laser passively Q-switched with a Cr 4+ : YAG crystal (see Fig. a) on relative orientations in the laser cavity of a passive switch (PS) and partial polarizer (PP) have been previously reported. It has beehown that polarizatiotate of the laser is governed by relative orientations of the Cr 4+ : YAG PS and intracavity PP as well as the energy density of a giant pulse (GP) in the switch. Complicate behavior of polarization state is due to interplay between the PP action (which causes linear anisotropy of the cavity) and that of the Cr 4+ : YAG crystal (which causes nonlinear anisotropy of the cavity). It has been also shown that along with a GP formation in the laser cavity some change (rotation at certain angle) in the electric vector of radiation is observed. This paper reports the results of novel theoretical simulations demonstrating the possibility to control over polarizatiotate of a neodymium laser Q-switched with a Cr 4+ : YAG crystal by exterior weak resonant signal.. MODEL In accordance with the existing concept, Cr 4+ phototropic centers in YAG are considered as groups of linearly absorbing dipoles oriented along the principal crystallographic axes of YAG [], [], and [] [3, 4] (see Fig. b). As it is known [4 7], propagation of high-power resonant radiation in a crystal with different orientations of dipole moments is accompanied by appearance of self-induced anisotropy of resonant absorption, when differences in the crystal transmission coefficient are observed at the absorptioaturation stage. This effect has beehown [4, 7, 8] to appear when probing a Cr 4+ : YAG crystal with powerful radiation (λ =.6 μm) in extratracavity experimental configuration Also this effect is observed whetudying the output characteristics of a neodymium laser pas- 7 8 4 5 6 3 9 θ β ψ X β Z E CW Y E GP [] [] Cr 4+ Cr 4+ ϕ θ Z, [] X Cr 4+ Fig.. Schematic layout of laser and diagram of relative orientation of Cr 4+ centers in YAG crystal. 46
46 KIR YANOV et al. sively Q-switched with a Cr 4+ : YAG crystal and when intracavity multiple interaction of high-power laser radiation with PS takes place [, 9, ]. Note that if an additional partly polarizing element (PP) is inserted in the cavity, the mentioned interplay between the PP linear anisotropy and Cr 4+ : YAG PS nonlinear absorption anisotropy results not only GP output energy angular dependence, but also in drastic changes in polarization state of the laser. The basic idea of this work is to analyze the possibility of changing the initial relationship between the ground-state populations of Cr 4+ centers groups in Cr 4+ : YAG PS by weak controlling cw signal, whose wavelength falls into the Cr 4+ : YAG resonant absorption band (.8. μm) [] and whose polarization plane coincides with orientation of one of the Cr 4+ centers group (see Fig. ). In the absence of cw radiation, there are the same numbers of Cr 4+ centers in the ground state for both ([] and []) groups and no initial linear anisotropy in the Cr 4+ : YAG PS. But once the cw signal is switched on, some part of Cr 4+ centers oriented in the direction [] is bleached (being transferred from the ground state to the excited one with a lifetime τ s = 3 μs []). The result is the removal of the ground state population equilibrium between the groups [] and [] and, hence, slight linear anisotropy in the Cr 4+ : YAG PS. Dynamical interplay between this initial linear absorption anisotropy in PS, the linear anisotropy in PP, and the self-induced nonlinear anisotropy in PS leads, as analysis shows, to rather different scenarios in the polarizatiotate of the laser. 3. OPTICAL SCHEME The laser is formed by two [rear () and output ()] mirrors, an active element (3) (isotropic neodymiumcontaining material), Cr 4+ : YAG PS (4), PP (5) (glass plate oriented at the angle β with respect to the axis orthogonal to the cavity axis), and aperture (6). Also assume that the optical axis of the cavity, Z, coincides with one of three groups of Cr 4+ centers, []. Two other groups are, hence, in the plane of Fig. b and oriented by the angles θ [] and θ + π/ [] with respect to the axis X. The axes X and Y correspond to the orientations of minimum (α X ) and maximum (α Y ) losses of the glass plate (5). Linear anisotropy of the cavity is determined by the specific orientation (the angle β) of the glass plate. Define the polarization ellipse azimuth of the GP as E GP, assuming its orientation to be given by the angle ϕ. Suppose also that the controlling linearly polarized weak signal is delivered from a cw laser or laser diode (7), wavelength of which falls into the Cr 4+ : YAG PS resonant absorption band (its polarization azimuth is shown in Fig. by the vector E CW ). The cw radiation is supposed to propagate collinearly with that of GP in the cavity of the main neodymium laser (i.e., along the axis Z). Assume that the polarization plane of the cw source is fixed and parallel to one of the orientations of Cr 4+ centers in PS (direction [] in Fig. b). Chopper (8) may be inserted between the laser (7) and the Cr 4+ : YAG crystal, playing the role of an intensity modulator. Polarizatiotate of the main neodymium laser can be measured with an extracavity total polarizer, for instance, with a Glan prism. So, let us introduce this element as (9), taking into account the possibility of its virtual rotation in the plane of Fig. b (with the angle ψ fixing its maximum transmission). The neodymium laser output characteristics can be analyzed experimentally with a pair of photodiodes (, ), allowing one to determine the GP output and polarization ellipse azimuth as well. Photodetector () measures the controlling beam power delivered from the laser (7). 4. THEORY Theoretical analysis is performed under the assumption that the state of polarization always represents an eigenstate corresponding to the lowest intracavity losses. Similarly to [9], the analysis exploits the polarizabilitiy calculation for the Cr 4+ : YAG PS and PP. One can obtain for the GP polarization ellipse azimuth ϕ(t): ϕ() t = -- arctan ( ) sin( θ) ----------------------------------------------------------------------------, α cos( θ) Y α ------------------------------------------------- X ( l s σ s n ) ( ( s () t n ) s () t ) () where (t) and (t) are the ground state populations of the Cr 4+ centers groups [] and [], respectively; σ s is the resonant absorption cross section at the wavelength λ =.6 μm; l s is the length of Cr 4+ : YAG crystal; (α Y α X ) is the PP linear losses difference; t is time. Note that in the case under study we do not take into account the ellipticity degree of radiation. ( ) We need to find the population difference (t) (t), which arises during the intracavity radiation build-up, as well as to determine the GP intensity, I(t), for the cases of presence (absence) of exterior cw signal. For this aim, let us use the system of equations [, ], modified in accordance with our particular problem: df a F -------- a ( ) = ----- σ dt t a N a l a σ s l s ( cos ( θ ϕ)-- R (.) + sin ( θ ϕ) ) ln -- r α X cos ϕ α Y sin ϕ, LASER PHYSICS Vol. No.
NEODYMIUM LASER Q-SWITCHED 463 ( ) d = ------------, (3.) + B where B is the saturation parameter, B = P/P sat is the power density of the laser (7) and P sat = / σ s * τ s is the saturating power density. One can estimate for a Cr 4+ : YAG crystal: P sat 4 W/cm, since σ s *, the resonant absorption cross section of a Cr 4+ : YAG at the wavedn --------- a = γσ, (.) dt a N a F a c ( ----------- σ, (.3) dt s n ) s F a ckcos = ( θ ϕ) + ------------------- dn ------------ s ( ----------- σ (.4) dt s n ) s F a cksin + B n ( ) s = ( θ ϕ) + ---------------------------, where (see Fig. a) F a is the photon density inside the cavity at the position of active medium (AM), N a is the population inversion in AM, σ a is the lasing cross section of AM, l a is the AM length, K = S a /S s is the ratio of cross sections of the laser beam at the positions of AM and Cr 4+ : YAG PS, is the initial nondisturbed ground-state population of Cr 4+ centers oriented by [] and [], B is the parameter representing the influence of exterior controlling cw radiation delivered to the Cr 4+ : YAG PS, r is the reflection coefficient of the output mirror (), γ is the factor accounting for reduction of population inversion in AM, t R is the round-trip time for the cavity of the length L (t R = L/c, where c is the velocity of light). In the system of equations () interaction of intracavity radiation with the two orthogonal groups of Cr 4+ centers is directly accounted. Such an approach is shown [9, 3] to be productive for the analysis of media possessing the self-induced absorption anisotropy. Crucially important in the problem under study are the initial conditions, since, from one side, cw radiation delivered by the laser (7) determines the initial conditions for and ; from the other side, it is neces- ( ) sary to determine the corresponding initial condition for the angle ϕ, characterizing the initial linear anisotropy of the cavity caused by the action of PP (5) and the effect of additional illumination of Cr 4+ : YAG PS by the laser (7). It is easy to demonstrate that the initial conditions ( ) for the ground state populations and are written as: ( ) t = t = t = t, (3.) τ t τ s ( ) length of the cw laser (7), is virtually the same as σ s (λ =.6 μm) []. Then, from equation (.), minimizing all the linear losses of the cavity, one can find ϕ* = ϕ t = t (4) = -- sin( θ) arctan -------------------------------------------------------------------. ( α cos( θ) Y α X )( + B) ----------------------------------------- Bl s σ s The last expression corresponds to the general equation governing the temporal dependence ϕ(t) [see formula ()]. Since the value can be easy calculated from the data of initial, T in, and final, T fin, transmission coefficients of the Cr 4+ : YAG crystal ( = (ln(/t in ) ln(/t fin ))/σ s l s ), the corresponding value for the initial inversion population is N a t = t ------------ σ sl s n s ( + Bcos ( θ ϕ* )) = ------------------------------------------------------------------- σ a l a + B + ln -- r + α X cos ϕ* + α Y sin ϕ* (5) (let us note that T fin reflects the presence of the excited state and/or nonsaturating residual absorption in Cr 4+ : YAG crystal). 5. RESULTS Consider the results of the laser modeling (Figs. 4), with the basic parameters being listed in the table. It is seen from Fig. that for the cases of switched off (Figs. a d) and switched on (Figs. e h) signals from the controlling laser (7) one obtains different regimes for the main neodymium laser. Although the envelope of output radiation as well as inversion population dynamics in AM (see Figs. a, d, e, h) are virtually the same, in the first case Cr 4+ centers of the [] orientation are preferably bleached, whereas in the second one, Cr 4+ centers of the [] orientation are preferably bleached (compare curves and in Figs. b, Table AM: Nd 3+ : YAG PS: Cr 4+ : YAG Cavity σ a = 6.5 9 cm σ s = 5.6 8 cm r = 85% l a = 5. cm l s =. cm L = 3 cm hν =.85 9 J T in = 8%, T fin = 98% K =. γ =.6 τ s = 3. 6 s Spot size =.75 cm LASER PHYSICS Vol. No.
464 KIR YANOV et al. Population inversion (AM), 7...8.6.4....8.6.4. (e) Doped centers number (PS), 7 4. 3.5 3..5..5..5.5 5 Polarization azimuth, deg. 4 3 (c) 4. 3.5 3..5..5..5.5 3 4 5 (f) (g) 5 (d) 5 (h) Intensity, mw/cm 4 3 4 3 3 4 5 Cavity round-trip numer 3 4 5 Fig.. Results of numerical calculations by using equations () for Nd : YAG laser passively Q-switched with Cr 4+ : YAG PS (a d) without and (e h) with additional illumination by resonant cw radiation [θ = 44, β =, (a d) B = and (e h).5]. Curves and correspond to populations of Cr 4+ centers oriented in [] and [] directions, respectively. f). Meanwhile, polarization azimuth behavior (see Figs. c, g) is also strictly determined by the presence or absence of additional illumination of the Cr 4+ : YAG crystal. One caee abrupt (45 ) flips of resultant polarization azimuth, while their initial values are very close to each other. Hence, polarization azimuth of the laser makes 9 flips, depending on whether Cr 4+ : YAG PS is illuminated by the exterior cw radiation or not. LASER PHYSICS Vol. No.
NEODYMIUM LASER Q-SWITCHED 465 Relative energies in orthogonal polarizations Polarization azimuth in pulse maximum, deg..8.6.4. 6 4 4 6 8 6 4 ln B 3.6.59.58.57.56 5 5 5 3 Fig. 3. Dependences of output energy () and its relative weights in orthogonal polarizations [() ψ = +44 and (3) ψ = 46 ] on logarithm B (θ = 44, β = ). Dependences of polarization azimuth () in GP maximum and () at the initial stage of GP formation on logarithm B (θ = 44, β = ). Output energy, mj Initial polarization azimuth, deg Polarization azimuth, deg 5 4 4 3 5 4 3 3 4 5 3 3 4 6 8 PP orientation, deg Fig. 4. Dependences of polarization azimuth in GP maximum on orientations of PP in the cavity: () θ = 5, () 5, (3) 3, and 44 (4) (B = ) and () B =, ().5, and (3).5 (θ = 44 ). Note that the corresponding powers of the controlling signal are evaluated to be rather moderate, <~5 mw (considerably less than high-power GP intensity and the saturating power P sat ). One can also find dependences of output parameters GP energy [4] and polarization azimuth orientation of the main neodymium laser against changes in the parameter B [i.e., versus power of the controlling laser (7)]. It is seen from Fig. 3 that a very narrow dip should be observed in GP output energy (Fig. 3a, curve ) within the value of B corresponding to the flip of polarizatiotate (Fig. 3b, curve ). It is also characteristic that linearity of polarization for both branches of polarization bistability is very high (compare curves, 3 in Fig. 3a). Figure 4 shows the dependences of polarization azimuth of the main neodymium laser on orientation of the glass plate (the angle β). Figure 5a shows a set of such curves for different orientations of the Cr 4+ : YAG PS, when cw signal is switched off. Fig. 5b shows the influence of additional illumination of the Cr 4+ : YAG crystal. It is seen that increase in its power results ihift of the bistability point of the device. 6. CONCLUSIONS We proposed the device based on the dependence of polarizatiotate of neodymium laser passively Q-switched with Cr 4+ : YAG crystal upon relative orientations of the switch and intracavity PP. We theoretically demonstrated that one of the main parameters (polarization) of the neodymium laser can be controlled by the relatively weak signal delivered from a cw laser or laser diode, the wavelength of which falls into the Cr 4+ : YAG resonant absorption band. We found the parameters ranges for both the powerful neodymium and weak controlling lasers when polarization bistable regime of the device is observed. ACKNOWLEDGMENTS This work is supported partly by the Russian Foundation for Basic Research (project no. 98--7676). LASER PHYSICS Vol. No.
466 KIR YANOV et al. REFERENCES. Il ichev, N.N., Kir yanov, A.V., Gulyamova, E.S., and Pashinin, P.P., 997, Quantum Electron., 7, 98.. Il ichev, N.N., Kir yanov, A.V., Gulyamova, E.S., and Pashinin, P.P., 998, Quantum Electron., 8, 8. 3. Eilers, H., Hoffman, K.R., Dennis, W.M., et al., 99, Appl. Phys. Lett., 6, 958. 4. Il ichev, N.N., Kir yanov, A.V., Pashinin, P.P., and Shpuga, S.M., 994, JETP, 78, 768. 5. Il ichev, N.N., Kir yanov, A.V., Malyutin, A.A., Pashinin, P.P., 993, Laser Phys., 3, 8. 6. Dykman, M.I. and Tarasov, G.G., 977, Sov. Phys. JETP, 45, 8. 7. Il ichev, N.N., Kir yanov, A.V., Pashinin, P.P., and Shpuga, S.M., 994, Quantum Electron., 4, 77. 8. Damzen, M.J., Camacho-Lopez, S., and Green, R.P.M., 996, Phys. Rev. Lett., 76, 894. 9. Il ichev, N.N., Kir yanov, A.V., Gulyamova, E.S., and Pashinin, P.P., 998, Laser Phys., 8, 8.. Il ichev, N.N., Kir yanov, A.V., and Pashinin, P.P., 998, Quantum Electron., 8, 47.. Eilers, H., Dennis, W.M., Yen, W.M., et al., 993, IEEE J. Quantum Electron., 9, 58.. Zhang, X., Zhao, S., Wang, Q., et al., 997, IEEE J. Quantum Electron., 33, 86. 3. Brignon, A., 996, J. Opt. Soc. Am. B, 3, 54. 4. Koechner, W., 99, Solid State Laser Engineering, 3rd ed. (Berlin: Springer-Verlag), Ch. 8. LASER PHYSICS Vol. No.