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1. #Bragg condition for diffraction of a wave of wave vector k series#
2. #Bragg condition for diffraction of a wave of wave vector k free#

Young, “Criterion for Bragg and Raman–Nath diffraction regimes”, Appl. Phariseau, “On the diffraction of light by progressive supersonic waves”, Proc. Carvalho, “Compression, self-bending, and collapse of Gaussian beams in photorefractive crystals”, Opt. Chimenti, “Focal shift of convergent ultrasonic beams reflected from a liquid–solid interface”, J. Tamir, “Non-specular reflection of convergent beams from liquid–solid interface”, Trait. Tamir, “Spatial modifications of Gaussian beams diffracted by reflection gratings”, J. Hoshino, “Studies on dynamical diffraction phenomena of neutrons using properties of wave fan”, J. Shull, “Perfect crystals and imperfect neutrons”, J. Shull, “Observation of Pendellösung fringe structure in neutron diffraction”, Phys. Henceforth the member ∂Ψ1/∂μcan be neglected.Ģ2.Ĝ. 58–71.Ģ1.ěecause the space harmonics Ψ1and Ψ2propagate in the direction normal to the ∇μvector, it is to be expected that the field evolution along the direction given by ∇μ will be slower than the evolution along the direction of propagation (Fig. Sobolev, Partial Differential Equations of Mathematical Physics (Dover, New York, 1984), pp. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. For example, if the following typical values are considered-x= 0.1 m, y= 10−2m, λ0= 500 nm, ∊r′=2.25, d= 0.1 m-the term β0dtan δsin ν≫ 1 has a value of approximately 3 × 103.ġ9. The condition β0dtan δsin ν≫ 1 does not restrict the applicability of the model to the majority of devices. 29–31 below.ġ7.ěecause of the Pendellösung effect, the assumption that the reconstructed field Ψ1has a Gaussian amplitude distribution is strictly true (as is shown in Section 3) only at the beginning of the diffraction process.ġ8. For a detailed discussion of the transition between the Raman–Nath and the Bragg regimes, see Refs. It is assumed that the medium is sufficiently thick, with a high value of the dielectric permittivity modulation, to ensure Bragg behavior of the grating even in the paraxial zone. Cooke, Volume Holography and Volume Gratings (Academic, Oxford, 1981).ġ6. where d is the interplanar spacing, the angle between the wave vector of the incident plane wave, k o, and the lattice planes, its wavelength and n is an integer, the order of the reflection. Leonard, “Dichromated gelatin for the fabrication of holographic optical elements”, Appl. Braggs law provides the condition for a plane wave to be diffracted by a family of lattice planes: 2 d sin n.

#Bragg condition for diffraction of a wave of wave vector k series#

11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. Lin, “Design of achromatic grating couplers for backplane optical interconnections”, in Diffraction Optics, Vol. Moreover, if laser diode sources are used, wavelength shifts of several nanometers can arise as a result of the thermal and bias fluctuations.ġ3. Special care should be taken when sources with a broader spectral band are implemented or when a wavelength multiplexed system is to be designed. Kostuk, “Diffractive optic design for board-level free-space optical interconnects”, in Diffractive Optics, Vol. Calvo, “Wave-front conversion between a Gaussian beam with a cylindrical phase function and a plane wave in a monomode on-axis transmission holographic coupler”, J. Kong, “Diffraction of optical beams with arbitrary profiles by a periodically modulated layer”, J. Tamir, “Bragg diffraction of Gaussian beams by periodically modulated media”, J.

Tamir, “Full-wave multiple-scattering analysis of diffraction by superimposed gratings”, J. Lee, “Multiple-scattering theory of wave diffraction by superimposed volume gratings”, J. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction”, J.

#Bragg condition for diffraction of a wave of wave vector k free#

In the context of nearly free electrons, diffraction causes the E Vs k relationship to distort in the. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction”, J. 'We have noted that diffraction occurs when one end of a wave vector touches a Bragg plane. Next, we derive the angular direction of the diffracted beam wave vector for Bragg-mismatched readout using the mathematical formalism of the CWT. Gaylord, “Rigorous 3-D coupled-wave diffraction analysis of multiple superimposed gratings in anisotropic media”, Appl. That is, to obtain the direction of the diffracted wave in Bragg-mismatched readout, one should draw a vector in the z-direction from the tip of the + K vector to the surface of the Ewald sphere. Schenfeld, “An optical interconnection network with 3-D layout and distributed control”, in Optical Interconnections and Networks, H. Goodman, “Fan-in and fan-out with optical interconnections”, Opt.