We consider mode coupling in multimode optical fibers using either two Bragg gratings or a Bragg grating and a long-period grating. We show that the magnitude of the band edge curvature can be controlled leading to a flat, quartic band-edge or to two band edges at distinct, nonequivalent k-values, allowing precise control of slow light propagation.

We develop a way to enhance the amplitudes of the nonpropagating evanescent orders of resonant dielectric gratings. We use this blazing to design gratings with spectra tailored to generate steerable sub-Rayleigh field concentrations on a surface. We investigate the enhancement and customization of evanescent fields necessary to create a virtual and passive scanning probe with no moving parts. Spot size can be decreased 1 order of magnitude below the free-space Rayleigh limit. © 2010 Optical Society of America OCIS codes: 050.6624, 050.1960.The wavelength-dependent response of spatially periodic optical structures (gratings) has been exploited for many years, especially in spectroscopy. Gratings can be blazed to favor the reflection or transmission of light into a desired propagating order with high resolving power [1]. The realization that gratings can also excite useful subwavelength evanescent fields, necessary for achieving resolution beyond the Rayleigh limit, has generated a number of proposals [2-4]. Here we develop an approach to enhance ("blaze") and tailor desirable evanescent fields. The Fraunhofer equation sin θ m ¼ sin θ 0 þ mλ=d governs the angles θ m at which a grating of period d with an incident plane wave of wavelength λ at an angle θ 0 generates plane-wave orders. Writing the spatial part of the field as expðik x y þ ik y yÞ, the Fraunhofer equation is cast in terms of parallel (x) and perpendicular (y) components of the incident and scattered wave vectors (k):Here k 0 ¼ 2π=λ is the incident wave number and m indices the generated orders. For sufficiently large jmj, jk xm j > k 0 , so k ym is imaginary; the sign is chosen such that these evanescent orders decay exponentially away from the grating. Thus Eq. (1) takes the form k ym ∼ ijk xm j ≈ 2iπjmj=d, independent of wavelength. This suggests the possibility of sub-Rayleigh resolution, if we can excite these large jmj orders. By the field continuity conditions, subwavelength field features on the grating surface require orders with subwavelength variation. As these orders are evanescent, this fine detail is not visible in the far field. In the absence of losses or scatterers, the energy in evanescent fields remains trapped on the surface, and so their amplitude is not limited by energy conservation. Thus while blazing propagating orders is well known, blazing evanescent orders should be particularly effective and would provide strong fields at a scale below the Rayleigh limit, an essential feature of superresolution [3,4]. Below, we present a way to blaze evanescent orders and propose a method to exploit this to provide sub-Rayleigh limit resolution.Our approach requires multiple high-Q internal grating resonances. These resonances, most simply described by guided-mode resonances in a homogeneous dielectric waveguide [5], are due to Fabry-Perot reflection between the grating top and bottom surfaces. Their high Q is due to total internal reflection (TIR), so they couple very weakly to the propagating orders through which energy would otherwise l...

A method is developed to enhance the amplitudes of the non-propagating evanescent orders of resonant dielectric gratings. The origin of these resonances is analyzed in detail. The method relies on interactions between stacked gratings with different periods, and so a formalism is developed to model such stacks mathematically. In addition, a theoretical approach is developed to design gratings that enhance or blaze desired orders. These orders, controlled independently by incident fields from different angles, interfere and are optimized to produce steerable sub-Rayleigh field concentrations on a surface. These spots may function as a virtual scanning probe for non-invasive sub-Rayleigh microscopy. Optimization is conducted using a Monte Carlo Markov chain, and spots are generated which are both 1 order of magnitude narrower than the free space Rayleigh limit and robust to noise in the incident fields.