In most digital imaging applications, high-resolution imaging or videos are usually desired for later processing and analysis. The desire for high-resolution stems from two principal application areas: improvement of pictorial information for human interpretation, and helping representation for automatic machine preception. While the image sensors limit the spatial resolution of the image, the image details are also limited by the optical system, due to diffraction, and aberration<sup>1</sup>. Monocentric lens are an attractive option for gigapixel camera because the symmetrical design focuses light identically coming from any direction. Marks and Brady proposed a monocentric lens design imaging 40 gigapixels with an f-number of 2.5 and resolving 2 arcsec over a 120 degrees field of view<sup>2</sup>. Recently, Cossairt, Miau, and Nayer proposed a proof-of-concept gigapixel computational camera consisting of a large ball lens shared by several small planar sensors coupled with a deblurring step<sup>3</sup>. The design consists of a ball element resulting in a lens that is both inexpensive to produce and easy to align. Because the resolution of spherical lens is fundamentally limited by geometric aberrations, the imaging characteristics of the ball lens is expressed by the geometrical aberrations, in which the general equations for the primary aberration of the ball lens are given. The effect of shifting the stop position on the aberrations of a ball lens is discussed. The variation of the axial chromatic aberration with the Abbe V-number when the refraction index takes different values is analyzed. The variation of the third-order spherical aberration ,the fifth-order spherical aberration and the spherical aberration obtained directly from ray tracing with the f-number is discussed. The other imaging evaluation merits, such as the spot diagram, the modulation transfer function(MTF) and the encircled energy are also described. Most of the analysis of the ball lens is carried out using OSLO optics software from Lambda Research Corporation<sup>4</sup>.
The fundamental problem of the modulation transfer function(MTF) from the viewpoint of the lens designer is to find relation between the MTF and the geometrical aberrations. Let it be required to develop the spherical aberration into a polynomial expansion. The incoherent point spread function(PSF) of the optical imaging system is derived from the diffraction integral in the presence of aberrations. The optical transfer function(OTF) is the Fourier transform of the PSF, and the modulus of the OTF is the MTF. The relation between the spherical aberration and the MTF is denoted by numerical integration method. The normalized MTF is numerically calculated for various amounts of spherical aberration. A comparison is made between the MTF of the corrected spherical aberration using the optimum design for the minimum root mean square(RMS) wavefront aberration and those for the minimum peak-to-valley(P-V) wave front aberration.
The relationship between the Strehl ratio and the standard deviation of the wave front aberration of a Gaussian beam is
derived from the diffraction integral in the presence of aberrations. Let it be required to develop the spherical aberration
of a Gaussian beam into an aberration polynomial. For the maximum value of the Strehl ratio, a set of linear equations is
obtained. The optimum configuration of the balanced spherical aberration is obtained from the solution of this set of
linear equations. The coefficients of the spherical aberration of a Gaussian beam for the optimum design are illustrated in
terms of tables. A comparison is made between the Strehl ratio of the corrected spherical aberration of the Gaussian
beam using the optimum design of the uniform beam for the minimum RMS wave front aberration and those for the
minimum P-V wave front aberration. The Strehl ratio of the configuration using the optimum design of the uniform
beam changes slightly. It turns out that the spherical aberration of the Gaussian beam can be balanced with the optimum
configuration of the uniform beam. Finally, the correction of the spherical aberration of the Gaussian beam is illustrated
with an example.