hologhaphic coping and decoding of optical information. h.lenk (ddr).

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

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)

ℒ

and by taking the integral over o_o and dropping a constant phase term

by defining a function

(3)

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.

fig.1.

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 1^st 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 1^st degree) optical information to a fixed configuration of these parameters can be obtained (encoding of the 2^nd 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.

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 1^st 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.

extended source:

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

(5)

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 2^nd 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/).

storage medium:

are there still other possibilities of encoding in the 2^nd 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 2^nd 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 1^st 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 2^nd 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

by interaction of the coding structure with the diffracted distribution of the object we obtain in the coding plane

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

(7)

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).

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

(9)

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:

(10)

we obtain immediately behind the coder the distribution

(11)

for obtaining the distribution in the image plane this distribution must be convoluted, with

in the image plane the expression results

(12)

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

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:

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

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

verstimmungsfaktor

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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