International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.7, pp. 208-209

Section 1.7.3.3.2.4. Resonant SHG

B. Boulangera* and J. Zyssb

aInstitut Néel CNRS Université Joseph Fourier, 25 rue des Martyrs, BP 166, 38042 Grenoble Cedex 9, France, and bLaboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France
Correspondence e-mail:  benoit.boulanger@grenoble.cnrs.fr

1.7.3.3.2.4. Resonant SHG

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When the single-pass conversion efficiency SHG is too low, with c.w. lasers for example, it is possible to put the nonlinear crystal in a Fabry–Perot cavity external to the pump laser or directly inside the pump laser cavity, as shown in Figs. 1.7.3.6[link](b) and (c). The second solution, described later, is generally used because the available internal pump intensity is much larger.

We first recall some basic and simplified results of laser cavity theory without a nonlinear medium. We consider a laser in which one mirror is 100% reflecting and the second has a transmission T at the laser pulsation ω. The power within the cavity, Pin(ω), is evaluated at the steady state by setting the round-trip saturated gain of the laser equal to the sum of all the losses. The output laser cavity, Pout(ω), is given by (Siegman, 1986[link])[P_{\rm out}(\omega)=TP_{\rm in}(\omega)]with[P_{\rm in}(\omega)={2g_oL'-(\gamma+T)\over 2S(T+\gamma)}.\eqno(1.7.3.62)][L'] is the laser medium length, [g_o=\sigma N_o] is the small-signal gain coefficient per unit length of laser medium, σ is the stimulated-emission cross section, No is the population inversion without oscillation, S is a saturation parameter characteristic of the nonlinearity of the laser transition, and [\gamma=\gamma_L=2\alpha _LL'+\beta] is the loss coefficient where αL is the laser material absorption coefficient per unit length and β is another loss coefficient including absorption in the mirrors and scattering in both the laser medium and mirrors. For given go, S, αL, β and [L'], the output power reaches a maximum value for an optimal transmission coefficient Topt defined by [[\partial P_{\rm out}(\omega)/\partial T]_{T_{\rm opt}}=0], which gives[T_{\rm opt}=(2g_oL'\gamma)^{1/2}-\gamma.\eqno(1.7.3.63)]The maximum output power is then given by[P_{\rm out}^{\rm max}(\omega)=(1/2S)[(2g_oL')^{1/2}-\gamma^{1/2}]^2.\eqno(1.7.3.64)]

In an intracavity SHG device, the two cavity mirrors are 100% reflecting at ω but one mirror is perfectly transmitting at 2ω. The presence of the nonlinear medium inside the cavity then leads to losses at ω equal to the round-trip-generated second harmonic (SH) power: half of the SH produced flows in the forward direction and half in the backward direction. Hence the highly transmitting mirror at 2ω is equivalent to a nonlinear transmission coefficient at ω which is equal to twice the single-pass SHG conversion efficiency ηSHG.

The fundamental power inside the cavity Pin(ω) is given at the steady state by setting, for a round trip, the saturated gain equal to the sum of the linear and nonlinear losses. Pin(ω) is then given by (1.7.3.62)[link], where T and γ are (Geusic et al., 1968[link]; Smith, 1970[link])[T=2\eta_{\rm SHG}=[P_{\rm out}(2\omega)/P_{\rm in}(\omega)]\eqno(1.7.3.65)]and[\gamma=\gamma_L+\gamma_{NL}.\eqno(1.7.3.66)]ηSHG is the single-pass conversion efficiency. γL and γNL are the loss coefficients at ω of the laser medium and of the nonlinear crystal, respectively. L is the nonlinear medium length. The two faces of the nonlinear crystal are assumed to be antireflection-coated at ω.

In the undepleted pump approximation, the backward and forward power generated outside the nonlinear crystal at 2ω is[P_{\rm out}(2\omega)=2KP_{\rm in}^2(\omega)\eqno(1.7.3.67)]with[K=B(L^2/w_o^2)\sin c^2(\Delta kL/2), ]where[B={32\pi\over\varepsilon_oc}{2N-1\over N}{d_{\rm eff}^2\over\lambda_\omega^2}{T^{2\omega}_3T_1^\omega T_2^\omega\over n^{2\omega}_3n_1^\omega n_2^\omega}\quad({\rm W}^{-1}).]

The intracavity SHG conversion efficiency is usually defined as the ratio of the SH output power to the maximum output power that would be obtained from the laser without the nonlinear crystal by optimal linear output coupling.

Maximizing (1.7.3.67)[link] with respect to K according to (1.7.3.62)[link], (1.7.3.65)[link] and (1.7.3.66)[link] gives (Perkins & Fahlen, 1987[link])[K_{\rm opt}=(\gamma_L+\gamma_{NL})S\eqno(1.7.3.68)]and[P_{\rm out}^{\rm max}(2\omega)=(1/2S)[(2g_oL')^{1/2}-(\gamma_L+\gamma_{NL})^{1/2}]^2.\eqno(1.7.3.69)](1.7.3.69)[link] shows that for the case where [\gamma_{\rm NL}\ll\gamma_L] ([\gamma\simeq\gamma_L]), the maximum SH power is identically equal to the maximum fundamental power, (1.7.3.64)[link], available from the same laser for the same value of loss, which, according to the previous definition of the intracavity efficiency, corresponds to an SHG conversion efficiency of 100%. [P_{\rm out}^{\rm max}(2\omega)] strongly decreases as the losses ([\gamma_L + \gamma_{\rm NL}]) increase . Thus an efficient intracavity device requires the reduction of all losses at ω and 2ω to an absolute minimum.

(1.7.3.68)[link] indicates that Kopt is independent of the operating power level of the laser, in contrast to the optimum transmitting mirror where Topt, given by (1.7.3.63)[link], depends on the laser gain. Kopt depends only on the total losses and saturation parameter. For given losses, the knowledge of Kopt allows us to define the optimal parameters of the nonlinear crystal, in particular the figure of merit, [d_{\rm eff}^2/n_3^{2\omega}n_1^\omega n_2^\omega] and the ratio (L/wo)2, in which the walk-off effect and the damage threshold must also be taken into account.

Some examples: a power of 1.1 W at 0.532 µm was generated in a TEMoo c.w. SHG intracavity device using a 3.4 mm Ba2NaNb5O15 crystal within a 1.064 µm Nd:YAG laser cavity (Geusic et al., 1968[link]). A power of 9.0 W has been generated at 0.532 µm using a more complicated geometry based on an Nd:YAG intracavity-lens folded-arm cavity configuration using KTP (Perkins & Fahlen, 1987[link]). High-average-power SHG has also been demonstrated with output powers greater than 100 W at 0.532 µm in a KTP crystal inside the cavity of a diode side-pumped Nd:YAG laser (LeGarrec et al., 1996[link]).

For type-II phase matching, a rotated quarter waveplate is useful in order to reinstate the initial polarization of the fundamental waves after a round trip through the nonlinear crystal, the retardation plate and the mirror (Perkins & Driscoll, 1987[link]).

If the nonlinear crystal surface on the laser medium side has a 100% reflecting coating at 2ω and if the other surface is 100% transmitting at 2ω, it is possible to extract the full SH power in one direction (Smith, 1970[link]). Furthermore, such geometry allows us to avoid losses of the backward SH beam in the laser medium and in other optical components behind.

External-cavity SHG also leads to good results. The resonated wave may be the fundamental or the harmonic one. The corresponding theoretical background is detailed in Ashkin et al. (1966[link]). For example, a bow-tie configuration allowed the generation of 6.5 W of TEMoo c.w. 0.532 µm radiation in a 6 mm LiB3O5 (LBO) crystal; the Nd:YAG laser was an 18 W c.w. laser with an injection-locked single frequency (Yang et al., 1991[link]).

References

Ashkin, A., Boyd, G. D. & Dziedzic, J. M. (1966). Resonant optical second harmonic generation and mixing. IEEE J. Quantum Electron. QE2, 109–124.
Geusic, J. E., Levinstein, H. J., Singh, S., Smith, R. G. & Van Uitert, L. G. (1968). Continuous 0.532-m solid state source using Ba2NaNbO15. Appl. Phys. Lett. 12(9), 306–308.
LeGarrec, B., Razé, G., Thro, P. Y. & Gillert, M. (1996). High-average-power diode-array-pumped frequency-doubled YAG laser. Opt. Lett. 21, 1990–1992.
Perkins, P. E. & Driscoll, T. A. (1987). Efficient intracavity doubling in flash-lamp-pumped Nd:YLF. J. Opt. Soc. Am. B, 4(8), 1281–1285.
Perkins, P. E. & Fahlen, T. S. (1987). 20-W average-power KTP intracavity-doubled Nd:YAG laser. J. Opt. Soc. Am. B, 4(7), 1066–1071.
Siegman, A. E. (1986). Lasers. Mill Valley, California: University Science Books.
Smith, R. G. (1970). Theory of intracavity optical second-harmonic generation. IEEE J. Quantum Electron. 6(4), 215–223.
Yang, S. T., Pohalski, C. C., Gustafson, E. K., Byer, R. L., Feigelson, R. S., Raymakers, R. J. & Route, R. K. (1991). 6.5-W, 532-nm radiation by CW resonant external-cavity second-harmonic generation of an 18-W Nd:YAG laser in LiB3O5. Optics Lett. 16(19), 1493–1495.








































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