Pupil phase apodization for achromatic imaging of extra-solar planets

Direct imaging of extra-solar planets in the visible and infrared region has generated great interest among scientists and the general public as well. However, this is a challenging problem. Diffculties of detecting a planet (faint source) are caused, mostly, by two factors: sidelobes caused by starlight diffraction from the edge of the pupil and the randomly scattered starlight caused by the phase errors from the imperfections in the optical system. While the latter diffculty can be corrected by high density active deformable mirrors with advanced phase sensing and control technology, the optimized strategy for suppressing the diffraction sidelobes is still an open question. In this thesis, I present a new approach to the sidelobe reduction problem: pupil phase apodization. It is based on a discovery that an anti-symmetric spatial phase modulation pattern imposed over a pupil or a relay plane causes diffracted starlight suppression sufficient for imaging of extra-solar planets. Numerical simulations with specific square pupil (side D) phase functions, such as ... demonstrate annulling in at least one quadrant of the diffraction plane to the contrast level of better than 10^12 with an inner working angle down to 3.5L/D (with a = 3 and e = 10^3). Furthermore, our computer experiments show that phase apodization remains effective throughout a broad spectrum (60% of the central wavelength) covering the entire visible light range. In addition to the specific phase functions that can yield deep sidelobe reduction on one quadrant, we also found that a modified Gerchberg-Saxton algorithm can help to find small sized (101 x 101 element) discrete phase functions if regional sidelobe reduction is desired. Our simulation shows that a 101x101 segmented but gapless active mirror can also generate a dark region with Inner Working Distance about 2.8L/D in one quadrant. Phase-only modulation has the additional appeal of potential implementation via active segmented or deformable mirrors, thereby combining compensation of random phase aberrations and diffraction halo removal in a single optical element

EPIC spectral observations of variability in earth’s global reflectance

NASA's Earth Polychromatic Imaging Camera (EPIC) onboard NOAA's Deep Space Climate Observatory (DSCOVR) satellite observes the entire sunlit Earth every 65 to 110 min from the Sun-Earth Lagrangian L1 point. This paper presents initial EPIC shortwave spectral observations of the sunlit Earth reflectance and analyses of its diurnal and seasonal variations. The results show that the reflectance depends mostly on (1) the ratio between land and ocean areas exposed to the Sun and (2) cloud spatial and temporal distributions over the sunlit side of Earth. In particular, the paper shows that (a) diurnal variations of the Earth's reflectance are determined mostly by periodic changes in the land-ocean fraction of its the sunlit side; (b) the daily reflectance displays clear seasonal variations that are significant even without including the contributions from snow and ice in the polar regions (which can enhance daily mean reflectances by up to 2 to 6% in winter and up to 1 to 4% in summer); (c) the seasonal variations of the sunlit Earth reflectance are mostly determined by the latitudinal distribution of oceanic clouds

Ciarlet–Raviart mixed finite element approximation for an optimal control problem governed by the first bi-harmonic equation

AbstractThe Ciarlet–Raviart mixed finite element approximation is constructed to solve the constrained optimal control problem governed by the first bi-harmonic equation. The optimality conditions consisting of the state and the co-state equations is derived. Also, the a priori error estimates are analyzed. In the analysis of the a priori error estimates, the improved convergent rate of the higher order than existed results is proved. Some numerical experiments are performed to confirm the theoretical analysis for the a priori error estimate