tailieunhanh - Handbook of algorithms for physical design automation part 73

Handbook of Algorithms for Physical Design Automation part 73 provides a detailed overview of VLSI physical design automation, emphasizing state-of-the-art techniques, trends and improvements that have emerged during the previous decade. After a brief introduction to the modern physical design problem, basic algorithmic techniques, and partitioning, the book discusses significant advances in floorplanning representations and describes recent formulations of the floorplanning problem. The text also addresses issues of placement, net layout and optimization, routing multiple signal nets, manufacturability, physical synthesis, special nets, and designing for specialized technologies. It includes a personal perspective from Ralph Otten as he looks back on. | 702 Handbook of Algorithms for Physical Design Automation It can be shown that when the diffraction pattern of Equation is placed at exactly the focal distance in front of the lens the field at the focal plane a distance f behind the lens allows this phase factor to cancel the phase factor in the Fraunhofer diffraction formula and the field in the image plane becomes 1 E p q a i EJ œ œ J J M x y e-i 2n k xP yq dx dy œ œ This form will be recognized as a mathematical representation that corresponds to the 2D Fourier transform 30 of the mask pattern E p q a FT M x y To actually form the image of the mask at position x1 y1 the lens aperture and behavior represented by a pupil function designated as P a b are multiplied with the diffraction pattern at the focal point. This image in the focal plane is in turn transformed by a second lens at a distance f E xi yi a FT P a b E p q FT P a b FT M x y where P represents the pupil function encompassing the wavefront transforming behavior of the lens. This is illustrated in Figure . Pupil functions can be simple mathematical structures such as pr m 1 a2 b2 d 1 P rl ocka P a b 2 w 0 a2 b2 r 0 p r representing the physical cutoff of the circular lens housing or radius r shown in both Cartesian a b and polar coordinates p 0 . However additional phase behavior of the lens can also be included in the pupil function. Lens aberrations can be represented by an orthonormal set of polynomials called Zernike polynomials each representing a specific aberration 29 . The Zernike polynomials are generally represented in polar coordinates following the form Zj p 0 amnRm p Ym 0 Table below shows a few of the Zernike polynomials and the corresponding aberration. More detail on these functions can be found in Ref. 29 . . . -. ._ . FIGURE Simplified representation of the optical system of an imaging tool. At the pupil plane the amplitude of the field represents a two-dimensional Fourier transform .