hologhaphic coping and decoding of optical information. h.lenk (ddr).
in the first part some general considerations on the problem of holographic encoding will be given, the second part contains theoretical and experimental work on the special subject of encoding the signal wave. holography as a procedure of storing optical information by interferential superposition of wave fields offers some certain possibilities in the encoding of optical information. as encoding we consider in a very general sense the transformation of the optical distribution which exists at the locus of a two-dimensional or three-dimensional оobject; into another distribution by means of an unequivocal and reversible attaching process.
in optics this attaching process can be based simply on the wave equation and the propagation laws of light. let us consider an object with the complex object function
in the ξ, η -plane being illuminated with a plane monochromatic wave. the object function can be split into an undiffracted
part and signal-bearing one: o=oo+ob. the resulting optical distribution i.e. the diffraction figure shall be observed in the x,y - plane a distance a, behind the ξ, η-plane (fig.l).
by using kirchgoff diffraction theory for not too great diffraction angles, the simplified diffraction integral can be written in the form:
with d denoting the distance between a certain point in the object and a point in the x,y-plane and k=2π/λ. expanding d into a power series yields
and we obtain from eq. (l)
and by taking the integral over oo and dropping a constant phase term
by defining a function
eq. (2) can be explained in the form of a convolution and gives the expression:
with * denoting the convolution. equation (4) will be the star-
ting point for some later considerations.
it can be seen that the object, distribution is carried over into an amplitude and phase distribution which is given by the convolution of the original distribution with a function characterizing the influence of the diffraction process.
before the invention of holography it was necessary to connect this encoding step by diffraction immediately with a decoding one by means of a lens, i.e. the conventional process of optical imaging.
a successful encoding by diffraction is only possible, if the distribution of the main parameters of the wave-field (amplitude, phase) can be stored, called off and transformed into an image distribution being equivalent to the original object distribution. this problem is solved by holography in a well known way. the holographic process can therefore be considered as an encoding process (encoding process of the 1st degree).
the main components of the holographic process are the object wave as the carrier of the object information encoded by
difraction, the reference wave as a defined relation information and to certain sense the storage medium.
we take up the question in which way and to which extent by appropriate influencing of these components, once more a well defined and reversible attaching of the code(coding of the 1st degree) optical information to a fixed configuration of these parameters can be obtained (encoding of the 2nd degree).
reference wave respects reference source: considering first a point reference source (enclosed the case of the infinitely far point) there exists the possibility of changing the position or the spectral range of the source in order to obtain attaching the stored information to a certain configuration. restricting ourselves for simplicity to a plane problem (the hologram extends only in the x-direction) the relative phase of the reference at a point of the hologram is given by ( is the distance between reference source and actual hologram point). the course of the phase function upon the hologram is a monotonic function depending on . (the amplitude dependence wlllbe omitted). if the reconstruction takes place with the same source in the same position & or its conjugate) the same phase function (its respective conjugate) is incident on the hologram add the reconstruction of the virtual and the real image term results. condition for exact reconstruction is the exact duplicating of the former reference function on the hologram.
changing the position or spectral range of the source means changing its position vector by some amount Δ or its wavelength by Δλ. bacause of the phase dependence
both cases result in an alteration of the course of the phase
function upon the hologram. this new function is still a monoto-
nic one. the difference between the former phase function and current one depends on the amount and the direction of the position change vector. the function respects to the change in the wavelength. from the theory of holographic imaging it is well known such changes lead to changes of the position and size of the images and to distortions. but a rapid dropout of information does not occur. therefore we have still the case of encoding the 1st degree with an additional introduction of mistakes into the holographic process, the character and size of which is determined by the change of the reference function for taking and reconstruction the hologram.
the situation is remarkable changed, if we use an extended source especially one with a statistical phase distribution. such a source can be realized e.g. by introducing a ground glass as a secondary source into the reference path. stroke /1/ showed, that for the case of fourier-holography with extended sources sharp images can be obtained when using the same source for taking and
reconstructing the hologram (source compensation effect.) as kiemle /2/ pointed out, this principle is valid also for other types of holograms.
in the scope of our hitherto existing considerations this fact is easily understood. the reference function of the hologram originating from an extended source can be calculated by using eq. (4) with the source function q inserted instead of the object function
ℒref h(x y)~q*f(x,y). (5)
as eq. (5) shows, ℒ will be no monotonic function but a varying or statistic one, the variations of which are determined by the source function q, source extension and distance between the source and hologram. if the same source in the same position is used for reconstruction, the reference function is exactly duplicated and the stored information can be retrieved. changing the position of extended source between taking and reconstructing the hologram causes a rapid drop out of the information. reference encoding by moving a diffusor in the reference path was reported e.g. by la macchia /3/. a coarse estimate of the position sensitivity can be given in the following way. if dr denotes the diameter of the source and a is the distance between the source and hologram, a far-field consideration gives for the diffraction spot, i.e.the area of equal phase approximately:
considering the source as statistic, this amount will be the mean sise of the area of constant phase of the reference function. taking the source aperture ratio as 1:10, the resulting sensitivity is about 6μ. therefore it is sufficiently to use statistical elements of the source, the mean size of which is of the same order of the magnitude.
examination of eq.(5) shows, that the phase distribution of the reference function upon the hologram depends on the wavelength too. it must be therefore possible, to obtain encoding of the holographic information not only by attaching a specific holog-
ram to a certain position of the extended statistic reference and moving the reference between sequential exposures, but also by using an extended statistical reference in fixed position and changing the wavelength slightly between the exposures. thus the use of an extended source offers the possibility of a coding process of the 2nd degree by attaching the stored information to a certain position or certain wavelength of the reference source. the use of the same source for the reconstruction can be avoided by combining a hologram of the extended source taken against a point reference with a hologram of the object taken under equivalent geometrical conditions with the extended source as a reference (see for instance lanzl, et.al./4/).
are there still other possibilities of encoding in the 2nd degree? the next main parameter of the holographic process is the storage medium. if its depth is great compared with the wavelength, we obtain the well known effect of selecting either the angle of the light incidence or its wavelength /5,6.7,8,9/. the basic effect is the bragg-condition which must be fulfilled in the reconstruction process. the holographic image is now displayed by the interaction of multiple diffracted and reflected partial waves, so that proper phase matching of these waves is a very critical condition. that means that the slight deviation in the course of the reference function which results from changing either the position of the reference and or the object or the wavelength between construction and reconstruction of the hologram is sufficient to code out the stored information, thus re-
sults the possibility of attaching the optical information to a certain geometrical configuration or to a certain wavelength also if a point reference is used. a coding effect of the 2nd degree is obtained, if the storage medium has spatial properties. signal wave:
next we shall deal more extensively with a coding process working with a coding structure in the path of the signal wave /10/.in the systematic description given here this process is considered an an operation on the third main component of holography, the signal wave. this is a coding process which takes place inside of the coding process of the 1st degree. as will be seen, also in this way on attaching of information to a certain position of coding structure can be obtained or because of the wavelength dependence of the coding process the attaching of the information can also be obtained to a certain wavelength if the position of the coding structure is fixed. in both cases the reference is point-like and the rcoonstruction takes place by using the conjugate reference wave for illuminating the hologram. thus in this way results on encoding process of the 2nd degreeand and cases dealed with can be summarized in the following scheme:
in principle a combination of these processes could be thinkable but this question is beyond the scope of this paper. encoding of the signal wave:
the basis of the following considerations will be eq.(4)
the process of the reconstruction of the hologram with a coding structure in the signal path is shown in fig.3. the object is
again situated in the (ξ,η)-plane, the coder, the coding function (τ(xc,yc) of which we assume to be complex for the present, is arranged in the (xc,yc)-plane a distance a1 away. the hologram is located in the (x, y)-phase a distance away from the coder.
by interaction of the coding structure with the diffracted distribution of the object we obtain in the coding plane
ooτ(xc,yc)+os*foc τ(xc,yc) (6)
where f denotes the expression
now the resulting distribution must be calculated. by using the
f-expression for this case
for diffraction figure in the hologram plane we obtain
in this expression the space dependence is omitted, for simplicity. in the hologram plane the reference wave αr is superposed. in order to obtain 1:1 imaging, as the reference wave a plane ware impinging in the x,z-plane is closed, that is
αr=ar exp[ik sinθx]. (6)
let us further suppose, that the resulting amplitude transmittance of the hologram is proportional to the stored intensity ta ~ j. for reconstruction the hologram is illuminated with the conjugate of the reference wave, for practical use the hologram together with the coder is rotated by an angle of 180° and illuminated with the reference again (fig. 4).
of all of the waves which originate by the reconstruction only those containing the real image will be considered.
we obtain by raconstruction amongst others
the first main term has the direction of the axis, the other parts of the reconstructed field are separated with regard to their direction. now the diffracted distribution of the first wave must be calculated in the coding plane (now being the decoding plane) by means of the same scheme already used in the construction process. it is obtained:
the multipliсation indicates the interaction with the coder. because of
we obtain immediately behind the coder the distribution
for obtaining the distribution in the image plane this distribution must be convoluted, with
in the image plane the expression results
that means, the real image wave multiplied with the square of the
amount of the complex coding function and a convolution. starting from this expression special cases can be discussed. phase coding structure:
the coding function has the form
because of results from. eq.(12)
the real image wave is obtained in undisturbed from apart from multlplicative factor.
the following model experiments will demonstrate this effect. first we used as
object a simple slit, the coding structure was a crossed phase grating (grating spacing about 50 μ ). fig.5 shows the reconstruction without decoder and fig.6 obtains the formation of the several, diffraction orders by an interaction with the coder several slit images. with the coder inserted into the image wave only one image results (fig.7). for some other experiments a coder with a statistical but rather coarse phase structure was used. fig.7 shows a picture of it in transmitted laser light. object was a plane
line-drawed figure (fig.8). fig.9 again shows the reconstructed image without using the coder for dacodlag. evidently the image
is severely degraded. after insertion and carefully adjustment an almost undisturbed image is obtained (fig.10). the effect is still more evident in the next example. fig.11 shows the object as a text
object. as fig.12 demonstrates (reconstruction without decoding) the information is completely destroyed. after insertion of the coder in a proper position the decoded image appears (fig.13). amplitude "coding" structure:
starting again with eq.12 it can be seen that for amplitude coding structures τ(xc,yc) = ta(xc,yc) (ta is the amplitude transmission) the square of the coder function coupled with a convolution enters the expression. therefore in general no aliaplification of the equation is possible, as well as no coding in the sense formulated in the beginning of this paper is possible. the "coding process" is not reversible, because information lost by absorbtion cannot be restored. the absorbing "coding structure" acts like a real mask or a frequency filter and drop-out of information with respect to decrease in resolution occurs.
fig.14 shows the model of a binary amplitude structure which was insetted into the object wave(object in fig.15) for the reconstruction and reconstruction of the hologram. the reconstructed picture in fig.16 shows the image degrading effect of the structure.
an experiment with a special "amplitude coder" (grating with
50μ spacing) was carried out for demonstrating the effect of altering the "coder" (in this case the grating constant is g) between exposure and reconstruction. as object is need a slit. a theoretical analysis, based on the superposition of the two different periodic coding factions
indicates that a splitting of the slit images occurs. the amount
of this splitting in proportional to the difference ()
i.e. to the difference frequency. fig.17 shows the splitting of
the alit images in dependence on the amount of the alteration of the grating spacing. the numbers indicate the ratio .
the last fig.18 gives a sight of the apparatus used for the experiments. the left part is mirrors mounting for the division of the incident light and for the generation of the reference wave. the object is mounted in the middle part. between object and hologram (to the right) the coder is inserted. coder and hologram mounted are fixed on a base-plate and can be rota-
ted together around a vertical axis lying in the hologram plane. after rotation around 180˚ degree the reference acts as a reconstruction wave, while illumination is interrupted. the reconstructed real image term is displayed through the coder, now acting as a decoder and can be observed in the image plane. hologram and coder mounted are adjustable in several degrees of freedom with regard to translation and rotation.
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