Thick-grating non-divergent focussing device: 2-D analysis

The two-dimensional analysis of a recently reported thick grating Bragg diffraction focussing device is carried out using Solymar's coupled-wave approach. It is shown that the device is capable of operating as a non-divergent lens with beam contraction ratios of better than 100:1, and conversion efficiencies of nearly 100%. It is also shown that it can operate as a novel kind of Fourier spectrum analyser, the focussed diffracted intensity being proportional (as a function of Bragg condition violation) to the modulus squared of the Fourier transform of the incident finite beam. The focussing properties are studied as functions of incident beam profile, width and position, grating strength and Bragg condition violation. Poynting vector optics is used successfully to predict (in conjunction with the dispersion surfaces of x-ray dynamical theory) the off-Bragg behaviour in the focal plane. It is likely that the device could be used as avariable ratio beam contractor in integrated optics, where the grating strength could be controlled (interdigital electrode system) electrooptically.     

An analysis of optical waveguide tapers

The characteristics of slab waveguide tapers with finite-thickness cladding are investigated by approximating the tapered section as a series of uniform five-layer waveguides with successive changes in the core and cladding thickness. The behaviour of multimode coupling and transmission characteristics for typical tapers are studied in detail by computer simulation, employing parameter values appropriate for practical systems. Strong coupling and efficient conversion between core and cladding modes are demonstrated. In addition, it is shown that for tapers of moderate slopes, the nearest-neighbour coupling approximation is quite adequate for the calculation of the individual mode amplitudes. For steeper tapers, however, additional neighbouring modes have to be taken into account, and the modal energy tends to spread further towards the higher-order modes. It is also demonstrated that relatively large step-size can be used in the iterative numerical calculations. This, in conjunction with the choice of an appropriate coupling scheme (e.g., the nearest-neighbour approximation for mild tapers), makes it possible to reduce the cost of computer simulation. The relevance of the present analysis to fiber tapers is also discussed. Supported in part by the Department of Communications of Canada     

Propagation and deflection of guided modes in planar waveguides via grating rotation

Propagation and deflection of guided modes in a polymer slab waveguide is studied. Rotation of the grating structure with respect to the plane of incidence deflects the waveguide modes in the plane of the sample so that the propagation direction is no longer perpendicular to the grating grooves. The deflection angles are measured as a function of the grating rotation angle and the experimental results are compared with theoretically expected values. The difference in polarization behaviour between waveguide and plasmon surface polariton grating rotation coupling is shown.     

Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals

The use of surface-impedance and surface-admittance concepts for analyzing reflection and refraction at an isotropic dielectric interface (first developed about 1938) is extended to include an interface between uniaxial birefringent dielectrics. Total internal reflection and the polarizing (Brewster) angle at an anisotropic interface are shown to be naturally explainable in terms of surface impedance (for TM polarization) and surface admittance (for TE polarization). The allowable modes in an integrated optical uniaxial asymmetric dielectric slab waveguide are also shown to be directly obtainable using the surface impedance/admittance approach. Numerical examples are presented.     

Measurements of the velocity gradient using a laser Doppler phenomenon

This paper reports on a method for measuring the velocity gradient using a laser Doppler phenomenon. The velocity gradient is determined from the velocity difference between two different points of the probing object and is actually obtained from the optical heterodyning of two differently Doppler-shifted scattered light fields from two points of the object having different velocities. The properties of output beat signals are theoretically investigated from their spectral broadening including the effect of the detecting aperture. The preliminary experiment was performed by using a rotating glass disk whose velocity is different from its center to the outside. The experimental results shows the usefulness of the present method for measurements of the velocity gradient.     

Optics of Floquet-Bloch waves in dielectric gratings

The behavior of light in dielectric gratings is discussed in terms of the optical Floquet-Bloch waves (or modes). The emphasis is on the development of a good physical understanding of the nature of these waves, using the wavevector diagram to summarize their spatial dispersion and spectra. It is shown that Floquet-Bloch theory offers some advantages conceptually over the commonly used coupled-wave theory, because the rays of the Floquet-Bloch waves (given by their group velocities) play the same role in a periodic medium as do those of plane waves in isotropic or graded-index media. The effect on power conservation of truncating the Floquet expansions for the Floquet-Bloch waves is considered in detail. Using the greater intuitive power of Floquet-Bloch theory, it is shown (in contrast to recent claims to the contrary) how rigorous coupled-wave theory can be applied to symmetrical reflection gratings, and secondly how the light in these gratings can be viewed in terms of the multiple-beam interference of Floquet-Bloch waves, leading to behavior reminiscent of a low-finesse Fabry-Perot cavity.     

Bragg diffraction of gaussian beams by thick gratings: Numerical evaluations by plane-wave decomposition

The validity of 2-D coupled-wave theory is investigated for the case of Bragg diffraction of a Gaussian beam by a thick unslanted phase grating in transmission mode. Fourier plane wave decomposition theory, as presented in a companion article, is used to test 2-D coupled-wave theory under circumstances when its validity is in question, namely when the incident distribution is very narrow or fast-varying, or when the parameterv _o, due to Kogelnik (1969), which is proportional to the product of coupling rate and grating thickness, is very large. Numerical evaluations of the field patterns at output from the grating, obtained using plane-wave decomposition, are used to explore the effect of the gradual violation of the conditions (from the previous article) under which 2-D coupled-wave theory is valid.     

Faraday rotation in a single-mode fiber with controlled birefringence

Faraday rotation within a polarization non-conserving single-mode optical fiber is used for ac magnetic field measurement. For that purpose the intrinsic fiber birefringence on the path to and from the Faraday rotation section is controlled. As control devices externally applied stress birefringence and phase retarder plates were used. With respect to magnetic field measurements on high electric potentials all control devices are placed at fiber input and output only.     

Mode dispersion characteristics of graded-index optical waveguides

A method to compute the propagating modes of graded index waveguides with an arbitrary refractive index distribution is proposed, based on the variational method applied to the integral wave equation. The method is applied to two refractive index distributions: a Gaussian index distribution and a complementary error-function index distribution. The results are compared with the predictions of perturbation analysis and are experimentally verified by measuring the mode spectrum of a Ti-diffused LiNbO₃ waveguide.     

Low-frequency TM modes guided in a dielectric slab deposited on a conducting substrate

A dielectric slab deposited on a conducting substrate can guide different types of electromagnetic modes. We present the study of the lowest-order TM mode, which is a distortion of surface polaritons at the air/substrate and layer/substrate junctions.     

Mode coupling in graded-index optical fibres due to microbending

The effects of microbending on mode coupling in graded-index optical fibres are investigated through a ray-optics approach and the standard Fokker-Planck theory. This model can be applied to any kind of index profile and statistics of the perturbation. In particular, for a parabolic index fibre and for a perturbation with microscopic correlation, a closed form expression of the optical power steady-state distribution is derived.     

Bragg diffraction of finite beams by thick gratings: Two rival theories

Two rival techniques used in the analysis of Bragg diffraction of two-dimensional light beams by thick unslanted phase gratings, namely Fourier plane wave decomposition and 2-D coupled-wave theory, are compared. The advantages and regions of applicability of each are discussed, and the conditions found under which they yield identical results. In particular, the results of plane-wave decomposition are used to provide quantitative conditions for validity of 2-D coupled-wave theory. These conditions are not easily derived using any other technique, and set quantitative limits to how narrow or fast-varying the incident distribution may be before 2-D coupled-wave theory fails. It is also shown that 2-D coupled-wave theory is inadequate when the parameterv ₀, due to Kogelnik (1969), which is proportional to the product of coupling rate and grating thickness, is very large.     

Beam propagation through parabolic-index waveguides with distorted optical axis

An integral expression is derived for the fields in parabolic-index waveguides with distorted optical axis. It is based on well-known ray matrix techniques and generalized Huygens integrals. The distortions give rise to beam oscillations which in turn are identified as the excitation mechanism for radiation modes. The model gives some better understanding of the transition regions between waveguides. On leave from the Department of Electronics onthe Weizmann Institute of Science, Rehovot, Israel     

Formation of holographic diffraction gratings in photoresist

The exposure-development process for the fabrication of holographic diffraction gratings for integrated optics in the positive photoresist Shipley AZ-1350 is theoretically and experimentally investigated. An analysis of the light intensity distribution is carried out, taking into account the reflectivity at the photoresist-substrate interface and the attenuation in the photoresist for two holographic exposure arrangements. The influence of the exposure energy, the development time for a fixed concentration of the developer and the initial photoresist thickness are described. Various grating profiles are calculated for photoresist films, coated on matched and reflective substrates. Scanning electron micrographs demonstrate very good agreement between the actual and calculated gratings profiles.     

Improved inverse WKB procedure to reconstruct refractive index profiles of dielectric planar waveguides

A new method to reconstruct smooth refractive-index profiles from measured effective indices is presented. The profile form near the surface and its approach towards the bulk value are explicitely taken into account. Well behaved profiles such as Gaussian or exponential can be reconstructed within experimental accuracy from three modes only, and even two modes may provide useful information. The method is tested for numerical stability.     

Reduced depth polarizer for integrated optics

Taking advantage of the different cut-off thickness of the two fundamental modes in an optical waveguide, a polarizing structure has been realized. It consists of a narrow stripe (∼100 μm) of reduced diffusion depth in an ion-exchanged waveguide. The TE₀-mode passes the polarizer with negligible losses, whereas the TM₀-mode is forced to couple out into the substrate. The extinction ratio is better than −16 dB.     

Bisection algorithm to calculate hybrid modes of birefringent planar graded index waveguides

Guided modes of a planar dielectric waveguide which encounter a nondiagonal permittivity tensor are neither TE nor TM, but hybrid. They are described by a pair of coupled second-order differential equations for the transversal electric and magnetic field components. We construct a real-valued function which plays the role of the transversal electric or magnetic field in the uncoupled Sturm-Liouville differential equation for TE or TM modes. The number of zeroes, or nodes, of this function labels the modes. The nodes increase with the prospective propagation constant. This fact is proven by constructing suitable self-adjoint operators and referring to the minimax principle. The nodal properties allow to formulate an efficient bisection algorithm for effective indices and field distributions of guided hybrid modes.     

Formation of blazed holographic gratings

Various methods for the recording of blazed holographic diffraction gratings are presented and the regions in the wavelength-groove density plane for which they can be used are shown. The gratings are made in the positive photoresist Shipley AZ-1350. The diffraction efficiency for s and p polarization was measured as a function of wavelength and angle of incidence. An absolute efficiency of 80% was achieved.     

Measurements of velocity using a lenticular grating

A new method is reported for measuring the velocity using a lenticular grating. The principle of the method is theoretically described on the basis of transmission-grating velocimetry. The theoretical studies are performed to estimate the deflection and collection characteristics of the light passing through the lenticular grating. The method is used to measure the velocity of a rotating random pattern. The experimental results show the usefulness of the method for measurements of the velocity.