Integer, fractional and fractal Talbot effects Export

@article{citeulike:2098743, abstract = {Self-images of a grating with period <i>a</i>, illuminated by light of wavelength \&b.lambda;, are produced at distances <i>z</i> that are rational multiples <i>p/q</i> of the Talbot distance <i>z</i>_T = <i>a</i>²/\&b.lambda;; each unit cell of a Talbot image consists of <i>q</i> superposed images of the grating. The phases of these individual images depend on the Gauss sums studied in number theory and are given explicitly in closed form; this simplifies calculations of the Talbot images. In \&lsquo;transverse\&rsquo; planes, perpendicular to the incident light, and with \&b.zeta; = <i>z</i>/<i>z</i>_T irrational, the intensity in the Talbot images is a fractal whose graph has dimension <a name="ILM0001">\&nbsp;</a>. In \&lsquo;longitudinal\&rsquo; planes, parallel to the incident light, and almost all oblique planes, the intensity is a fractal whose graph has dimension <a name="ILM0002">\&nbsp;</a>. In certain special diagonal planes, the fractal dimension is <a name="ILM0003">\&nbsp;</a>. Talbot images are sharp only in the paraxial approximation \&b.lambda;/<i>a</i> \&rarr; O and when the number <i>N</i> of illuminated slits tends to infinity. The universal form of the post-paraxial smoothing of the edge of the slit images is determined. An exact calculation gives the spatially averaged non-paraxial blurring within Talbot planes and defocusing between Talbot planes. Similar calculations are given for the blurring and defocusing produced by finite <i>N</i>. Experiments with a Ronchi grating confirm the existence of the longitudinal fractal, and the transverse Talbot fractal at the golden distance \&b.zeta; = (3 - 5^1/2)/2, within the expected resolutions.}, author = {Berry, M. V. and Klein, S.}, citeulike-article-id = {2098743}, citeulike-linkout-0 = {http://dx.doi.org/10.1080/09500349608232876}, doi = {10.1080/09500349608232876}, journal = {Journal of Modern Optics}, keywords = {talbot}, number = {10}, pages = {2139--2164}, posted-at = {2007-12-28 20:52:26}, priority = {5}, publisher = {Taylor \&amp; Francis}, title = {Integer, fractional and fractal Talbot effects}, url = {http://dx.doi.org/10.1080/09500349608232876}, volume = {43}, year = {1996} }

TY - JOUR ID - citeulike:2098743 L3 - citeulike-article-id:2098743 N2 - Self-images of a grating with period <i>a</i>, illuminated by light of wavelength &b.lambda;, are produced at distances <i>z</i> that are rational multiples <i>p/q</i> of the Talbot distance <i>z</i>_T = <i>a</i>²/&b.lambda;; each unit cell of a Talbot image consists of <i>q</i> superposed images of the grating. The phases of these individual images depend on the Gauss sums studied in number theory and are given explicitly in closed form; this simplifies calculations of the Talbot images. In &lsquo;transverse&rsquo; planes, perpendicular to the incident light, and with &b.zeta; = <i>z</i>/<i>z</i>_T irrational, the intensity in the Talbot images is a fractal whose graph has dimension <a name="ILM0001">&nbsp;</a>. In &lsquo;longitudinal&rsquo; planes, parallel to the incident light, and almost all oblique planes, the intensity is a fractal whose graph has dimension <a name="ILM0002">&nbsp;</a>. In certain special diagonal planes, the fractal dimension is <a name="ILM0003">&nbsp;</a>. Talbot images are sharp only in the paraxial approximation &b.lambda;/<i>a</i> &rarr; O and when the number <i>N</i> of illuminated slits tends to infinity. The universal form of the post-paraxial smoothing of the edge of the slit images is determined. An exact calculation gives the spatially averaged non-paraxial blurring within Talbot planes and defocusing between Talbot planes. Similar calculations are given for the blurring and defocusing produced by finite <i>N</i>. Experiments with a Ronchi grating confirm the existence of the longitudinal fractal, and the transverse Talbot fractal at the golden distance &b.zeta; = (3 - 5^1/2)/2, within the expected resolutions. IS - 10 JF - Journal of Modern Optics EP - 2164 TI - Integer, fractional and fractal Talbot effects PB - Taylor &amp; Francis VL - 43 SP - 2139 KW - talbot AU - Berry, MV AU - Klein, S PY - 1996/// UR - http://dx.doi.org/10.1080/09500349608232876 ER -