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The general results presented above for Hamilton's principle can be applied to optics using the Lagrangian defined in Fermat's principle.The Euler-Lagrange equations with parameter σ =x 3 and N=2 applied to Fermat's principle result in ˙ = with k = 1, 2 and where L is the optical Lagrangian and ˙ = /.
Hamilton's principle states that the true evolution q(t) of a system described by N generalized coordinates q = (q 1, q 2, ..., q N) between two specified states q 1 = q(t 1) and q 2 = q(t 2) at two specified times t 1 and t 2 is a stationary point (a point where the variation is zero) of the action functional [] = ((), ˙ (),) where (, ˙,) is the Lagrangian function for the system.
A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depends upon the material of the medium. One corresponding concept in mechanics is the principle of least/stationary action. Many important problems involve functions of several variables.
There is the technical difficulty of achieving a large illumination area without destroying the imaging optics. One approach is the so-called spatiotemporal focusing [ 4 ] [ 5 ] in which the pulsed beam is spatially dispersed by a diffraction grating forming a 'rainbow' beam that is subsequently focused by an objective lens. [ 5 ]
The resolution of a digital imaging device is not only limited by the optics, but also by the number of pixels, more in particular by their separation distance. As explained by the Nyquist–Shannon sampling theorem , to match the optical resolution of the given example, the pixels of each color channel should be separated by 1 micrometer, half ...
Hamilton's optico-mechanical analogy is a conceptual parallel between trajectories in classical mechanics and wavefronts in optics, introduced by William Rowan Hamilton around 1831. [1] It may be viewed as linking Huygens' principle of optics with Maupertuis' principle of mechanics.
Principles of Optics, colloquially known as Born and Wolf, is an optics textbook written by Max Born and Emil Wolf that was initially published in 1959 by Pergamon Press. [1] After going through six editions with Pergamon Press, the book was transferred to Cambridge University Press who issued an expanded seventh edition in 1999. [ 2 ]
In optics, an image-forming optical system is a system capable of being used for imaging. The diameter of the aperture of the main objective is a common criterion for comparison among optical systems, such as large telescopes. The two traditional optical systems are mirror-systems and lens-systems .