Talbot Planes Research Articles

Positioning method based on digital moiré

An alternative object positioning technique based on digital moiré is presented. The moiré pattern is obtained by the digital subtraction of different self-images under predetermined conditions including a grating rotation. The principle of the method relies on the fringe visibility matching in the moiré process while moving from one self-image plane to another. The motion produces both loose of fringe contrast and rotation of the moiré fringes. The theoretical model of this matching is provided. It applies the concept of measuring a global maximum contrast to each image, which allows to obtain both the best contrast and the adequate rotation. This allows a precise identification of each Talbot plane associated to an object to be positioned.

Fractional Talbot field and of finite gratings: compact analytical formulation.

We present a compact analytical formulation for the fractional Talbot effect at the paraxial domain of a finite grating. Our results show that laterally shifted distorted images of the grating basic cell form the Fresnel field at a fractional Talbot plane of the grating. Our formulas give the positions of those images and show that they are given by the convolution of the nondistorted cells (modulated by a quadratic phase factor) with the Fourier transform of the finite-grating pupil.

Motionless multiresolution interferometry based on Talbot effect

We describe the principle of a motionless interferometer based on the Talbot effect. The test object is scanned by a convergent spherical beam with variable curvatures and lateral shifts. The wave- fronts are generated by a programmable off-axis Fresnel zone plate, implemented on a high resolution and high speed binary phase spatial light modulator. The aberrated field is then analyzed by passage through a fixed binary phase grating, detection by a sensor located in a fractional Talbot plane, and numerical phase retrieval from the Fourier analysis of the reconstructed intensity distribution. We describe the principle of the object multiresolution analysis. The system operation is then illustrated experimentally in a simple case exhibiting subwavelength resolution ca- pability. Finally, we give some orders of magnitude for the dynamic range that can be expected with such a system, as well as possible extension toward very high resolution measurements. © 1999 Society of Photo-Optical Instrumentation Engineers. (S0091-3286(99)01212-X)

Wavefront sensor using the Talbot effect

A method for measuring optical wavefronts is proposed by properly sampling the light amplitude distribution with a two-dimensional periodic mask. At the Talbot planes, where a reference wavefront (spherical or plane) would originate replicas of the aperture, distortions of the resulting selfimages take place. These intensity patterns are detected and processed for obtaining the required wavefront information. The approach is illustrated with some experimental results.

Defect enhancement using Fresnel imaging and a TN-LCD

In this paper, we discuss defect enhancement in periodic images by Fresnel imaging. A periodic image is written in a liquid crystal display (matrix LCD) and read out by a coherent beam, a plane wave. The LCD is of twisted-nematic type (TN). The polarization of the read-out wave is rotated as a function of the gray value of the image. Because the input image is periodic, self-imaging (also called Fresnel imaging or Talbot imaging) occurs at certain distances (Talbot planes) behind the LCD. A mirror is put in one of these planes and the wave is redirected back to the LCD. The wave reads out the LCD again, and the rotation of polarization is performed again for all pixels, but in opposite direction; consequently, the input polarization state is reached again. If the input image has a defect, then this pixel is not present in the redirected wave, because self-imaging only occurs for periodic parts of an image. This pixel leaves the LCD with a different polarization state and can be detected.

Apparent superresolution in near-field optical imaging of periodic gratings

A superresolution phenomenon in images of diffraction gratings obtained with a near-field optical microscope probe positioned at distances of the order of 1>mu;m from the surface was recently reported [Appl. Opt. 33, 876 (1994)]. This experiment was interpreted in terms of evanescent waves observed in the far field. We show that the effect observed in that experiment is a very well known far-field optical phenomenon called the Talbot effect. The existence of self-imaging Talbot planes for gratings with periods of the order of a few micrometers appears promising for many practical applications.

Optical diffraction inspection of periodic structures using neural networks

An optical diffraction method is described for inspecting peri- odic structures such as combs or semiconductor leads. Coherent light passing between the prongs of the structure self-interfere at the frac- tional Talbot plane to provide a simple method of inspection. Computer simulation and laboratory experiments show the viability of this ap- proach. The theory assumes infinite structures. In practice, large end effect signals arise due to the finiteness of the periodic structure. A neu- ral network is demonstrated that learns to distinguish end effect signals from prong damage signals. The variability of the measuring process in a production environment makes neural networks an appropriate approach for this task. © 1998 Society of Photo-Optical Instrumentation Engineers. (S0091-3286(98)02303-4)

Practical limitations of Talbot imaging with microlens arrays

The Talbot effect which also appears for the foci of microlens arrays can, in principle, be used for array illuminators. In particular, the multiplication of the foci in the fractional Talbot planes seems to be promising. Certainly, in practice there are three effects which limit the application of the fractional Talbot effect for microlens arrays. Theoretical, numerical and experimental results are given to demonstrate these limitations.

Photoablation and lens damage from fractional Talbot images of Dammann gratings

UV photoablation of materials is recorded for both the near and far fields after transmission through a Dammann grating. The fused silica Fourier lens used for far-field imaging was damaged by a near-field intensity pattern with the same periodicity as the Dammann grating. The lens was located inadvertently at one eighth of the Talbot distance Z (T) behind the Dammann grating. Patterns recorded in copper film at the even-fractional Talbot planes compare qualitatively with calculated intensities. On the basis of these findings, a near-field intensity pattern was used to ablate vias in copper and polyimide films. The pattern at a distance of Z(T)/8 was used for via ablation because it is the pattern with the most fluence per spot.

Fractional Talbot effect for periodic microlens arrays

It is well known that the foci of a periodic array of microlenses are reproduced and multiplied in the fractional Talbot planes. But the intensity between the multiplied foci varies in general in one period, es- pecially if the light in the spacing between the lenses is not absorbed but transmitted. Additionally, in the case of microlenses with a high numeri- cal aperture the parabolic approximation or Fresnel diffraction, which is normally used to explain the Talbot effect, is no longer valid. A theoret- ical explanation and numerical simulations are given together with ex- perimental results. © 1997 Society of Photo-Optical Instrumentation Engineers. (S0091-3286(97)02904-8)

Integer, fractional and fractal Talbot effects

Abstract Self-images of a grating with period a, illuminated by light of wavelength λ, are produced at distances z that are rational multiples p/q of the Talbot distance z T = a 2/λ; each unit cell of a Talbot image consists of q 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 ‘transverse’ planes, perpendicular to the incident light, and with ζ = z/z T irrational, the intensity in the Talbot images is a fractal whose graph has dimension . In ‘longitudinal’ planes, parallel to the incident light, and almost all oblique planes, the intensity is a fractal whose graph has dimension . In certain special diagonal planes, the fractal dimension is . Talbot images are sharp only in the paraxial approximation λ/a → O and when the number N 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 N. Experiments with a Ronchi grating confirm the existence of the longitudinal fractal, and the transverse Talbot fractal at the golden distance ζ = (3 − 51/2)/2, within the expected resolutions.

Split Talbot array illuminators

We describe periodic binary patterns, with a basic cell formed by two or more bright intervals, that can be generated at fractional Talbot planes of multilevel phase gratings. These structures are proposed as alternate forms of Talbot array illuminators. A simple example is experimentally verified.

Fractional Talbot effect: analysis in phase space

The Fresnel diffraction of periodic objects at rational fractions of the Talbot distance is described in terms of the Wigner distribution function (WDF). The analysis provides a heuristic model for understanding the formation of the diffraction patterns as well as for evaluating the complex amplitude at any fractional Talbot plane. Furthermore, certain symmetry properties of the Fresnel-diffracted wave field can be derived directly from the WDF. Additionally, a discussion is given on how periodic signals and information about the phase are encoded in the WDF.

Influence of the finite aperture and observation of higher longitudinal-order fringes in Fizeau interferometry

The interference pattern of a Fizeau interferometer, ideally a (nearly) self-imaging field, is examined under true measurement conditions. The quality of the higher longitudinal-order fringes is investigated by finite-aperture diffractive calculations. A specific symmetry property of the volume interference pattern, namely the invariance of the intensity structure under a rotation of the interferential wedge about its apex, is studied by nondiffractive simulations that take into account the instrument geometry and finite incident wave. The formation of the higher longitudinal-order fringes is verified experimentally. The possibility of transferring these fringes close to the device by a suitable non-normal incidence is demonstrated. The existence of various fractional Talbot planes is also confirmed.

Fractional Talbot four-level phase-only holograms using ferroelectric liquid-crystal spatial light modulators

We propose a new technique for generating a four-level reconfigurable phase-only hologram, using two binary phase {0,pi} ferroelectric liquid-crystal spatial light modulators. The principle is based on the self-imaging properties of periodic structures operating in a Fresnel field double-diffraction system. Fractional Talbot planes are shown to be of particular interest. The problem of hologram bandwidth reduction that results from the holographic pattern replication is addressed. A trade-off between diffraction efficiency and hologram coding capacity is considered.

Talbot interferometer with circular gratings for the measurement of temperature in axisymmetric gaseous flames

A detailed study for measuring the temperature distribution in axisymmetric flames by using a Talbot interferometer with circular gratings is presented. We increased the sensitivity of the interferometer by optimizing the pitch of the grating and the Talbot plane. We compare the experimental results with the values that were measured with a thermocouple to an accuracy of ±0.2% of full scale ±4 digits. Good agreement is seen between the temperatures measured by use of a thermocouple and those measured by use of Talbot interferometry.

Automated phase sensing and control of an external Talbot cavity laser with phase-contrast imaging

We describe a novel phase-sensing and control system, based on phase-contrast imaging, operating within a linear external cavity laser consisting of 18 GaAlAs edge-emitting gain stripes. The system is used to achieve single-spatial-mode operation and diffraction-limited output from the linear cavity, which uses diffractive coupling at a Talbot plane to achieve coherent operation.

Talbot effect and Cornu’s spiral

Cornu's spiral is used in the testing and extension of a recent explanation of the Talbot effect, which occurs in the Fresnel domain. The results confirm all parts tested. They also indicate that each band of a Talbot plane is controlled primarily by light from only a few slits of the grating, contrary to earlier assumptions. Increases in slit width are found to limit the number of Talbot planes that can be observed.

Fresnel diffraction in fractional Talbot planes: a new formulation

We consider field distributions in fractional Talbot planes behind a periodic two-dimensional complex-amplitude transparency that is illuminated by a unit-amplitude plane wave. In the paraxial approximation the field in various fractional Talbot planes is expressed as a sum of contributions from a finite number of points in the plane of the transparency, yielding compact algebraic formulas for the diffracted field. Given the desired intensity distribution in the fractional Talbot plane, we synthesize the transmission function from nonlinear equations. An experimental illustration that uses a binary phase grating is given.

Talbot plane patterns: grating images or interference effects?

It has been widely said that with the Talbot effect a grating makes images of itself unaided. However, the effect as produced by simple amplitude gratings was recently redefined in terms of interference-pattern visibility or contrast instead of self-imaging. Then, by starting with a pair of slits instead of the usual infinite grating, a new and more general description of the effect was developed. Now numerical methods and tools from physical optics are used further to characterize the influences of the grating, light parameters, and the position of the plane of observation on pattern form, fine structure, band positions, and phases. It is found that none of the patterns in the Talbot planes actually approximates grating images in terms of all of these properties. Hence the Talbot effect should be defined in terms of interference effects, not grating images.