Critical angle light scattering from bubbles: an asymptotic series approximation

The critical scattering angle at 82.8 degrees from an air bubble in water locates the transition from partial to total re-flection from elementary geometrical optics. The irradiance scattered into a narrow angular region near the critical scattering is a monotonically increasing function of bubble radius a provided a >> lambda, and the weak contributions from rays reflected internally from the far side of the bubble are neglected. The asymptotic series for critical angle scattering derived here leads to a simple approximation for the irradiance. It also describes the breakdown of elementary geometrical optics for reflection at the critical angle from a curved surface. The leading correction to the scattering amplitude relative to the perfect reflection amplitude is found to be O(beta(-(1/4))), where beta = 2pia/lambda is the size parameter and lambda is the wavelength of light in water. The series is confirmed by comparison (as a function of beta) with smoothed Mie computations. The leading correction is significant for beta as large as 20,000, and it is larger when the light is polarized with the E field parallel to the scattering plane rather than perpendicular to it. The dependence on beta(-(1/4)) is also shown from an average of the reflection coefficient over a Fresnel zone. Applications to optical bubble sizing are noted, and the nature of approximations in previous physical-optics models of critical angle scattering is clarified.