Nonlinear Optics

Abstract

The field of nonlinear optics ranges from fundamental studies of the interaction of intense laser light with matter to applications such as frequency conversion, non-resonant amplification, all-optical switching. The treatment developed in this chapter restricts the attention to those topics that are of interest for applications, and limits the notational complications to a minimum. Second-harmonic generation is described in some detail in Sect. 7.2, because it is considered a case study for illustrating the mathematical approach and the experimental methods that are generally utilized in nonlinear optics. Section 7.3 completes the description of second-order effects with a concise treatment of parametric amplifiers and oscillators and of sum-frequency generation. Section 7.4 is devoted to a review of the main phenomena arising from the optical Kerr effect. The subject of Sect. 7.5 is stimulated Raman scattering, with particular emphasis on applications based on optical-fiber devices. Stimulated Brillouin scattering is described in Sect. 7.6.

Problems

7.1

The frequency of a Nd-YAG laser output ($\lambda = 1064$ nm) is to be doubled in a lithium niobate crystal by type-I birefringence phase matching. Knowing that, for $LiNbO_3$, $n_o(\lambda = 1064~\mathrm{nm}) =2.2347$, $n_e(\lambda = 532~\mathrm{nm}) =2.2316$, and $n_o(\lambda = 532~\mathrm{nm}) =2.3301$, calculate the phase-matching angle between the optical axis and the direction of the incoming beam.

7.2

Calculate the second-harmonic conversion efficiency for type-I harmonic generation in a perfectly phase-matched 2-cm long KTP crystal with an incident beam at $\lambda = 1064$ nm having an intensity of $3~\mathrm{MW}/\mathrm{cm}^2$. (For KTP, $n_{\omega } = n_{2\omega } \approx 1.8$, and $\chi ^{(2)} = 13$ pm/V).

7.3

The generation of the second-harmonic of 1064-nm light is performed by using a periodically poled lithium niobate crystal ($n_e=2.1550$ at $\lambda =1064$ nm and $n_e=2.2316$ at $\lambda =532$ nm). Calculate the period of periodical poling required for phase matching of second-harmonic generation.

7.4

A parametric amplifier uses a 2-cm long KTP crystal ($n_1 \approx n_2 \approx 1.77$, $\chi ^{(2)}=10$ pm/V) to amplify light of wavelength 580 nm in a collinear wave configuration. The pump wavelength is $\lambda = 400$ nm and its intensity is $4~\mathrm{MW}/\mathrm{cm}^2$. Determine the gain coefficient and the overall gain.

7.5

A Q-switched Nd-YAG laser produces an output pulse containing 10 mJ of energy with a pulse duration of 10 ns. The pulse generates an optical Kerr effect in a 2-cm thick carbon disulfide ($n_2=3.2 \times 10^{-14}~\mathrm{cm}^2/\mathrm{W}$) cell. Assuming that $w_o=100~{\upmu }$m, calculate the focal length due to the self-focusing effect.

7.6

A 1540-nm 2-ns laser pulse impinges on a 1-mm thick GaAs plate. Assuming a spot size of 0.2 mm and a two-photon absorption coefficient $\beta _a=10.2$ cm/GW, calculate the transmittance of the plate for the two cases in which the pulse energy is 1 mJ and $100~{\upmu }$J.

7.7

A laser pulse at $\lambda _o=1064$ nm propagating into an optical fiber excites a cascade of stimulated Raman scattering processes. Assuming that the Raman frequency shift is 13 THz, calculate the wavelength of the second Raman Stokes line, $\lambda _{s2}$.