Fundamental structure of Fresnel diffraction: natural sampling grid and the fractional Fourier transform

Fresnel integrals corresponding to different distances can be interpreted as scaled fractional Fourier transformations observed on spherical reference surfaces. We show that by judiciously choosing sample points on these curved reference surfaces, it is possible to represent the diffracted signals in a nonredundant manner. The change in sample spacing with distance reflects the structure of Fresnel diffraction. This sampling grid also provides a simple and robust basis for accurate and efficient computation, which naturally handles the challenges of sampling chirplike kernels.     

Essential oil composition of four endemic Ferulago species growing in Turkey

The essential oils from aerial parts of Ferulago pachyloba (Fenzl) Boiss., F. platycarpa Boiss. & Bal., F. isaurica Penmen, and F. longistylis Boiss. (Apiaceae) were obtained by hydrodistillation and analyzed by GC and GC-MS. The highest oil yield (1.50%) was obtained from F. pachyloba followed by F. longistylis (0.16%), F. isaurica (0.08%) and F. platycarpa (0.07%). Fifty-three compounds were identified in the oil of F. pachyloba with (Z)-beta-ocimene (25.7%) and alpha-pinene (9.8%) as main constituents; sixty-seven in the oil of F. platycarpa with 2,3,6-trimethylbenzaldehyde (29.8%) and cis-chrysanthenyl acetate (24.2%) as main components; seventy-eight in the oil of F. isaurica with nonacosane (25.5%) and hexadecanoic acid (14.8%) as main constituents; and fifty-nine in the oil of F. longistylis with 2,3,6-trimethylbenzaldehyde (32.7%) and bornyl acetate (12.6%) as main components. Quantitative and qualitative differences in the oil compositions of these four species were observed.