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Practical Holography XV

Stephen A. Benton

3. ABSTRACT TITLE: Nonparaxial Talbot effect.

AUTOR LISTING:

Valeriy S. Feshchenko, Benderskogo vosstanja str. No 3, flat 22, Benderi 278100, Moldova, feshchenko@mail.ru

Alexander N. Malov, Baikalskaja str., No 196, flat 35, Irkutsk-31, 664031, Russia, malov@physdep.isu.ru

Olesya A. Rogodjnikova, Karla Libknekhta str. No 205/3, flat 17, Tiraspol, 3300, Moldova

Yury N. Vigovsky, Novoslobodskaja str. No 31/1, Moscow, 103055, Russia, vigovsky@dol.ru

5. PRESENTATION: Poster Presentation

6. ABSTRACT TEXT:

One of important applied directions in diffraction physics is optical data processing, that in general kind is fulfillment of operations with multidimensional signals. Airborne reconnaissance data, x-ray photographs, radar images and etc. is multidimensional signals.

In present article the phenomenon of a self-imaging of the images of periodic object (Talbot effect) in nonparaxial area of a diffraction is experimentally investigated. Transformations of periodic fields of an optical range for optical processing of the two-dimensional images are offered and investigated. Is shown, that a choice of various geometry of the optical scheme for synthesis of the images of periodic object in the nonzero diffraction orders, permits to execute transformation the two-dimensional image in one-dimensional and back - one-dimensional to two-dimensional image.

Method of formation of images of the periodic objects in nonzero diffraction orders is offered and investigated, which can are used the basis for systems of images multiplication.

7. KEY WORDS: Array , nondestructive testing, optical signal processing, Talbot effect.

8. BRIFF BIOGRAPHY: Valeriy S. Feshchenko received the MSc degree in physics from Moldavian State University in 1991 and the DSc in radiophysics from the Irkutsk State University, Russia, 1998. He work as senior scientist at the laboratory of Coherent Optic and Holography in Dniester State University, 25 October str. 128, Tiraspol, Moldavia. His research interest include holography, optical data processing, nondestructive testing and laser light interaction with biotissue.

Nonparaxial Talbot effect.

Valeriy S. Feshchenko1, Aleksander N. Malov2, Olesya A. Rogozhnikova1,

Yury N. Vigovsky3

1Dniestr State University, 25 October str, 128, Tiraspol, Moldavia, feshchenko@mail.ru.

2Irkutsk State University, Gagarin str, 20, Irkutsk, Russia, malov@physdep.isu.ru

3MeDia Co, Novoslobodskaja str. No 31/1, Moscow, Russia, vigovsky@dol.ru.

Abstract

The phenomenon of a self-imaging of the images of periodic object (Talbot effect) in nonparaxial area of a diffraction is experimentally investigated. Method of formation of images of the periodic objects in nonzero diffraction orders is offered and investigated, which can are used the basis for systems of images multiplication.

Keywords: optical image processing, periodical objects, Talbot effect.

  1. Introduction
  2. One of important applied directions in diffraction physics is optical processing of information files, that in general way assumes fulfillment of operations with multidimensional signals which are functions of several variables. As multidimensional signals it is possible to consider airborne reconnaissance data, x-ray photographs, radar images and etc.

    The given article shows, that the structure of the arising self-images is connected with the presence of local cross nonuniformities of a field within the limits of one space period of structure, and with unlocalized influence on field of all aperture of a subject or optical system. The self-imaging phenomenon of periodic object images (Talbot effect) in essentially nonparaxial area of diffraction is experimentally investigated. Periodic fields transformations of an optical range for optical processing of the two-dimensional images are offered and investigated too. Is shown, that a choice of various of geometry of optical schemes for synthesis of periodic subject images in nonzero diffraction orders permits to execute transformation of the two-dimensional image into one-dimensional and on the contrary.

    The method of formation of periodic objects images in nonzero diffraction orders is offered and investigated, which can form the basis for systems multiplication of the images.

  3. Main theory.
    1. Diffraction of light on periodic object.

    Let the transmission of the individual image will be te (x, y), then the transmission of a rectangular object in the way of a matrix from MxN of individual elements fig.1 can be presented such as follows:

    (1)

    where M and N - integers, dx - period of a matrix along axis x, dy - period of a matrix along axis y, Ä - operation of convolution, d - Dirac delta-function.

    It is known, that the peak distribution appropriate to this transmission will be observed on so-called Talbot distance [1]:

    , , k=1, 2,..., (2)

    where l and m - mutually simple numbers, l - wavelength of radiation, d-period of a object.

    The distribution of amplitude in Talbot’s plane will be:

    (3)

    Thus the periodic object may be to presented as a superposition of a diffraction grating and mosaic image made from individual elements.

    Fig.1. Scheme of distribution of electromagnetic radiation behind periodic object. 1-incident electromagnetic wave, 2-periodic object, 3-paraxial Talbot’s planes, 4 - diffracted beams, 5- nonparaxial Talbot’s planes, 6,7 - various cells of periodic object.

    The property of diffraction gratings is known to multiply the image of periodic object in both parties from optical axis (fig.1). In this case in Talbot plane, in nonzero diffraction orders, following distribution the amplitudes is formed:

    , (4)

    for diffraction orders multiplied along axis X, and

    , (5)

    for diffraction orders multiplied along axis Y.

    Because of filtering property d - function in nonzero diffraction orders there will be the subtraction of those elements of object, which are not perpendicular axes, along which there is the multiplication. Two-dimensional object will become one-dimensional object.

    In fig.1 this process is submitted in the way of the scheme. In paraxial areas, where all diffracting light beams are crossed, classical Talbot effect is observed, i.e. the Talbot planes, designated in fig.1 with the point 3, placed along axis on distances satisfying to a condition (2). There is space area, where the diffraction orders with different signs are not crossed. To this area ratio (4) and (5) correspond. In these space areas the self-image occurs owing to coincidence phases of wave fronts from identical elements of different (points 6 and 7 in fig.1) individual images. As outside of axis there is a overlapping only nonzero orders of one sign, and the image will be formed only by the nonzero diffraction orders.

    We shall find out now, on which distance between them the self-image planes in nonzero diffraction orders will place.

    It is known, that the distance between self-image planes in axial Talbot effect is defined by that change of a wave front phase, at passing by them of this distance in free space, should be divisible 2p . Distance between self-image planes in nonzero diffraction orders will depend on this factor too.

    We shall consider, that a phase of the image in nonzero diffraction orders from images phase in zero order differs. If we present amplitude of an optical field at periodic object [2] near to axis of a incident radiation beam in the following way:

    (6)

    where k=2p /l , dp=d (sinq -sinq 0) - complete difference of a course between light ray, come in a remote point of supervision from appropriate points of two next individual images, where

    distribution of amplitudes from individual image. The phase of this image in this case will be j = (2N-1) kdp.

    For that of a object on edge of a beam the condition (6) can be written in way:

    (7)

    The phase of image in this case will be j 1 = Nkdp. At N® ¥ , j /j 2.

    Thus, if the phase of image in zero order will satisfy to a condition:

    j =2p n, where n=1,3,5... N. (8)

    The image in zero order will have positive contrast, and in the field of geometrical shadow and nonzero diffraction orders - negative contrast.

    In case if the phase of image in zero order will satisfy to a condition::

    j =2p n, where n=2,4,6... 2N. (9)

    And in zero diffraction orders and in nonzero diffraction orders there will be the positive contrast.

    Thus, distance between planes of self-image in nonzero diffraction orders along axis z, will be in two times more, than distance between paraxial self-image planes.

    In planes of observing not coincident with the plane of periodic object, spectrum of field of object is principle finite. This occurs in consequence of divergences in the space of high harmonics of limited bunches, as well as in consequence of the wave nature of light, thanks to influence of quickly damped waves. As a result images located in nonparaxial area will be subjected to distortion. Known [66], that in the spectrum of limited subject in paraxial area when moving from image centre to edge in Talbot's planes total quantity of overlapped harmonics does not decrease. Number of symmetric overlapped harmonics decreases only, thanks to that the quality of images is deteriorated. When diffraction of radiation into nonparaxial area is occuring, the image is primordially formed by the orders of diffraction of one sign and the threshold frequency which according to [4] equal to

    (10)

    where p- real number, N - quantity of elementary cells in object, d - the period of a object, is achieved on edge of a paraxial zone. And further there is only the decrease of general number of space harmonics in image. Therefore the best images is located on edge of a paraxial zone. The deterioration of images quality at the expense of separation of nonzero harmonics can be described, as well as in paraxial case [4] by Linfoot's quality factor:

    (11)

    where ~ - mark of Fourier transformation of function, f - spatial frequency, s - coherent transmission function of a periodic object as of optical system, l - wavelength, xn - coordinate of the elementary image, t(x) - function of transmission of the elementary image. The counting of coordinate xn it is necessary to begin from edge of a paraxial zone but not from center of the image, as in paraxial case, and as t(x) is necessary to select an one-dimensional component of transmission function of the elementary image.

    Thus, the shaping the self-imaging on the mechanism of Talbot’s effect corresponds to a modified principle E. Abbe about double diffraction in the time of formation of the image [5]. According to him, when imaging possible to select diffraction on elementary components of object and diffraction on totality of components located periodically on the field of subject. Hereupon delay of phase of various components of angular spectrum which required for imaging occurs because of what a radiation that is diffracted on identical elements of various elementary images make a various way in the free space. Since both types of a diffraction occur in one plane, it is possible to form the images in nonzero diffraction orders of radiation on periodic subject, and not just along optical axis.

  4. Experimental research of light diffraction process on periodic object.

3.1. Scheme of transformation of a two-dimensional image of object in one-dimensional.

The scheme (see fig.2) consist of helium-neon laser, a lens, periodic object and screen. The object is demonstrated on fig. 3, was produced by a method hardening bleaching on photographic film "Mikrat-izopan"(Slavich, Russia) on technique, described in [6]. The periodic object was purely phase. The example of subtraction is submitted on fig. 4.

As it is visible on photo, in nonzero diffraction orders the elements, which are not perpendicular axes, along which there is the multiplication will be lacking.

Fig.5 explains, which just the elements were suppressed in these images. The spatial filtering occur at the expense of three-dimensional geometry of the scheme, instead of two-dimensional geometry as in classical schemes of matched spatial filtering. The special holographic matched Vanderlugt filter [5] is not required in this case. Therefore and the accuracy of such processing is not limited, for example, properties of the holographic filter and its noise.

Using same optical scheme the change dependence of the phase of image which formed in nonzero diffraction orders from change of phase of image in paraxial area was investigated.

Fig.2. Optical scheme for observation of Talbot’s effect. Z – Talbot’s distance. Fig.3. Periodic object.

 

Fig.4a. Multiplication along axis X.

Fig.4b. Multiplication along axis Y.

Fig.5a. The elementary image of a of a periodic object with fig.3. The parts of the elementary image are shaded which were subtract at multiplication along axis X.

Fig.5b. The elementary image of a of a periodic object with fig.3. The parts of the elementary image are shaded which were subtract at multiplication along axis Y.

On photos presented on fig.6a Talbot's plane the phase of image in which satisfy to a condition (8) is demonstrated. It is visible, that dark points in paraxial image (noted on fig.6a by a square) is corresponded bright points in image generated in the nonzero orders (on fig.6a is noted by a rectangular). In other words the image in nonzero diffraction orders is inverted concerning image in paraxial area. It means, that the phase of image in nonzero diffraction orders is shifted relatively of the phase of image in paraxial area in odd number p .

And on fig.6b. the image in other Talbot's plane is demonstrated and the phase of image in this plane is satisfied to a condition (9). It is visible, as in nonzero diffraction orders (is noted on fig.6b by a rectangular) and in paraxial area the central points of the images remain dark. It means, that the phase of image in nonzero diffraction orders is shifted in relation to phase of image in paraxial area in even number p .

Fig.6a. The image of a periodic object in Talbot’s plane. The image phase in nonzero diffraction orders is displaced on p .

Fig.6b The image of a periodic object in Talbot’s plane. The image phase in nonzero diffraction orders is displaced on 2p .

Thus, there is the opportunity in same plane to obtain the multiplied positive and negative images

If the matrix consists of individual elements having a way shown on fig.7. multiplcation occurs in a direction perpendicular toward side of triangle In this case we also are using scheme (fig.2) but then it is possible to obtain only three groups of the multiplied images and not four. Quantity of axes along which images are multiplied equally to quantity of axes of symmetry, as is demonstrated on fig.8a.

Fig.7. Periodical object, which three axes of symmetry has.

On fig.8b the paraxial image of a object in Talbot's plane is demonstrated. According to theoretical accounts [7] in this case we must observe amplitude distribution in the form of dark triangles and bright places between them. But in fact we see in the image only contours of the elementary objects. This fact is possible to explain by that a object is transparent (phase) and in place where the phase of a wave varies the poorly essential contribution to image add the light which pass through periodic object without a diffraction. It is in other words possible to say that the small variations of an intensity are overlaped by high coherent noise.

Fig.8a.

Fig.8b.

3.2. Synthesis of the images in nonzero diffraction orders.

In previous paragraph the mechanism of formation of the Talbot images in nonzero diffraction orders was considered. Taking into account this mechanism it is possible to offer set of interesting practical applications for effects which arise on the diffraction of light in area where nonzero the orders of a various sign are not overlaped.

So, for example, it is possible to synthesize the image of periodic object in nonzero diffraction orders. If the image with distribution of amplitudes as in (4) is superimposed on another image with distribution of amplitudes as in (5) in Talbot's plane we shall obtain:

(12)

That coincides expression (3). That is there is the inverse transformation of one-dimensional object in two-dimensional object.

For confirmation this we made following experiment. In optical scheme represented on fig.9 a periodic object (fig.10a) are located. Since the periodic object had two axes of symmetry therefore in optical scheme also is used two beams of coherent radiation.

On photo presented on fig.10b. the result of this synthesis is demonstrated As we see in fig.10b. the image in nonzero diffraction orders is similar to image to zero order.

But there is and difference so for example in images generated in nonzero orders of diffraction there is not powerful coherent background, therefore they will be more qualitative. Such scheme permits to obtain in one plane four identical images that can be used in the systems of processing information where is required small multiplicity of multiplication.

Fig.9. Optical scheme for synthesis of the images of periodical object in nonzero diffraction orders.

 

 

Fig.10a. Periodical object

Fig.10b. Synthesis of periodical object images in nonzero diffraction orders.

-

Fig.11. Optical schemes for contouring effect obtaining of phase object. Z –Talbot’s distant.

Fig.12. Example of phase object contouring.

Other effect - contouring of phase objects arise because in the field of geometrical shadow are interacting nonzero diffraction orders containing the information on thin structure of object. Hence, if the phase subject has smooth change on D j area and sharp change on edges, in nonzero diffraction orders the information only about borders of object will be saved.

For confirmation of this effect we made the following experiments. On fig.11. the optical scheme with three laser beams is demonstrated. The quantity of laser beams in optical scheme for contouring of phase objects in nonzero diffraction orders depends on symmetry of periodic object. If object has N of axes of symmetry hence for observation of contouring effect in nonzero diffraction orders it is necessary N of laser beams. In our particular case the object had three axes of symmetry (matrix of triangles) therefore and is used in scheme three beams of coherent radiation.

We see that in nonzero diffraction orders in center of area where are overlaped three beams the image of contours of individual elements of the phase object is formed.

Conclusions.

On the basis of materials presented in this article it is possible to make the following conclusions:

  1. Since the periodic object is not only image but also diffraction grating there is the opportunity to transform image of subjects from two-dimensional in one-dimensional and on the contrary in nonzero the diffraction orders. That permits to use this effect for multiplication of the images.
  2. Because of various speed of separation of the diffraction orders there is the opportunity to make in nonzero diffraction orders contouring of the elementary images of periodic objects using the schemes offered in this article.
  3. The image which construct in nonzero diffraction orders provided that the subject has rather thin structure of the image and many of nonzero harmonics (about ten) has phase which twice times less than phase of the image formed in paraxial areasThat permits to obtain in one plane as negative images, and positive images.
  4. Fresnel approach for description of an electromagnetic field, diffracting on periodic object correctly only up to the distance z=Ld/l , when all nonzero diffraction orders will separate among themselves in space.

Reference

  1. Patorski K. The self-imaging phenomenon and its application./ Progress in optics, 1989, v.XXVII, p.3-108.
  2. Born M., Wolf E. Principles of optics - Oxford-London-Edinburgh-New York-Paris-Frankfurt: Pergamon Press, 1964, 884 p.
  3.  

  4. Andreev U.S., Sadovnik L.Sh., Tarnovethkiy V.V. The final sizes influence of periodic objects to intensity of a Fresnel images. /Opt. and Spect.(URSS), 1987, v.63, №3, pp.620-627.
  5.  

  6. Smirnov A.P. The influence the finite sizes of objects on intensity of Fresnel images. /Opt. and Spect.(URSS), 1978, v.44, №.2, pp.359-365.
  7. Goodman J.W. Introduction to Fourier optics. San-Francisco:McGraw-Hill, 1968.- 364 p.
  8. Malov A.N., Feshchenko V.S. Hydrotipical Methods of Producing of Relief Holograms. /In: II International Conference on Microelectronics and Computer Science ICMCS'97. - Kishinev: Tehnica Publishing House, 1997, v.2, p. 217-218.
  9. Porfiryev L.F. The principles of theory of the signal transformation of in optic-electronic systems. Leningrad: Mashinostroenie.-1989, 420p.
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