# Cross-Medium Photoacoustic Communications: Challenges, and State of the Art
## 4 Properties of Nonlinear Photoacoustic
### Remote Underwater Acoustic Source  
In order to generate a remote underwater laser acoustic source, the Naval Research Laboratory (NRL) investigated the physics of intense underwater laser propagation and acoustic generation, including linear group velocity dispersion (GVD), nonlinear refractive index effects such as nonlinear self-focusing (NSF), filamentation, self-phase modulation; scattering, absorption, and laser-induced breakdown. An ultrashort laser pulse can propagate relatively long distances underwater (up to distances on the order of the attenuation length, approximately 10 m in seawater) at a moderate intensity using the NSF and GVD mechanisms. Then the laser pulse quickly converges to an intense focus within a few centimeters at a predetermined remote location and generates a plasma and an acoustic shock wave at this location [69]. Due to GVD the laser optical pulse longitudinally compresses and increases the intensity of the negatively chirped optical pulse, triggering the self-focusing effect for transverse compression. The propagation distance \(L_{GVD}\) needed to produce maximum longitudinal pulse compression for a negatively chirped pulse in a medium with linear GVD is approximately [69],  
\[L_{GVD}\approx\,\frac{T(0)}{\beta_{2}\partial\omega} \tag{18}\]  
Here, \(T(0)\) is the initial pulse duration, \(\partial\omega\) is the frequency bandwidth and \(\beta_{2}\) is the GVD parameter. Thus, the longitudinal compression range can be controlled by controlling the initial pulse length and/or the laser bandwidth. Longitudinal pulse compression increases the intensity of the negatively chirped optical pulse and transverse compression of the pulse generally occurs when the intensity of the pulse exceeds a threshold represented by,  
\[P_{critical}=\frac{\lambda^{2}}{2\pi n_{0}n_{2}} \tag{19}\]  
where \(n_{0}\) is the linear index of refraction and \(n_{2}\) is the nonlinear index of refraction. \(P_{critical}\) is of the order of 1 MW for visible wavelengths in water. A characteristic distance for the transverse compression by NSF is approximately [70],  
\[L_{NSF}=\frac{z_{R}}{\sqrt{\frac{P(z)}{P_{NSF}}-1}} \tag{20}\]Here, \(z_{R}\) is the Raleigh range. For optimal pulse compression, initial beam power and beam radius should be selected such that \(L_{NSF}=L_{GVD}\) and the transverse and longitudinal compression occur simultaneously at a chosen distance. Thus, the intensity of the pulse is high enough to cause optical breakdown at a chosen distance and generate the acoustic signal. The laser wavelength is preferably chosen to have low attenuation in water, as attenuation is highly dependent on the wavelength. An attenuation length \((L_{atten})\) can characterize attenuation of light in water, with the beam intensity decreasing with propagation distance \(z\) according to \(I(z)=I(0)\exp(-z/L_{atten})\). The underwater propagation path length should be selected to be less than \(L_{atten}\) for applications where maximum energy is required at the acoustic source. However, the total underwater propagation path can be a few times greater than the attenuation length for lower energy applications. The maximum transmission (and least absorption) occurs in pure water in the wavelength range 300-500 nanometers, with a maximum attenuation length of around 50 m in this region. However, the attenuation length in seawater is typically 5 to 10 m because of impurity concentrations. The global average attenuation length is approximately 4 m but can reach 10 m or more in relatively clear ocean water [70]. NRL has investigated underwater optical filaments, including the determination of optimal generation parameters, maximum propagation length, and filament plasma lifetime to create the underwater acoustic source. Filamentation is observed in water when laser power is above the critical power \((P/P_{critical}>1)\) with focused above a critical \(f\)-number.