JPS6246870B2 - - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION <Industrial Field of Application> The present invention relates to a computationally synthesized wavefront recording method, and particularly to an improvement of a wavefront recording method for kinoforms.

<Prior Art> In addition to holography, which is a purely optical method, computer-generated holograms and kinoforms, which use digital processing using an electronic computer, are known as methods for recording and reproducing wavefronts containing information about objects.

These devices can record and reproduce wavefronts from imaginary objects that do not exist, so they have applications such as obtaining a three-dimensional representation of a finished product from design data and creating an inspection prototype for an aspherical reflector. can be developed.

While computer-generated holograms basically have the same properties as optical holograms, in kinoforms, only the phase component of the wavefront is recorded, and only the wavefront that forms the image of the object is generated during reproduction, making it possible to utilize the reproduced light. The efficiency is extremely high and can theoretically reach 100%.

This high light utilization efficiency means that
These kinoforms, which are very advantageous in practical terms, are generally produced according to the following principle.

To simplify the explanation, we will consider a one-dimensional object, but the following explanation can be applied to two-dimensional and three-dimensional objects as well.

As shown in FIG. 4, the object plane 1 to be recorded is
It is expressed as a set of N sample points a _k arranged at equal intervals Δa, and each arbitrary sample point a _k has a complex amplitude transmittance T(a _k ) expressed by the following equation (1). It is assumed that a point aperture exists.

T( _ak )=t( _ak )e ⁱ 〓 ^(ak) ...(1) However, t( _ak ) is the amplitude component, and in the case of complete transparency, t( _ak )=1, and it is completely transparent. If it is opaque, t( _ak )=0, but generally any intermediate value between them is taken, and 0≦t( _ak )≦1.

Further, α( _ak ) is a phase component appropriately assigned to the point aperture at the sample point _ak .

Distance Zo from object plane 1 to kinoform plane 2
is sufficiently long compared to the size of the object, and when coherent light of wavelength λ is irradiated, the complex amplitude _W ( x _n ) is the following formula (2)
is given by

In the above, c is a constant, but N is assumed to be an even number for simplicity. Further, it is assumed that the N sampling points (x _n ) for sampling the wavefront on the kinoform surface 2 are arranged at intervals Δx that satisfy the sampling theorem.

Here, if the amplitude component of W(x _n ) is A(x _n ) and the phase component is φ(x _n ), then the W(x _n ) becomes W(x _n )=A(x _n )e It can be simply expressed as ⁱ 〓 ^(xm) …(3).

Therefore, in the case of the kinoform, at all sample points, the complex amplitude W'(x _n ), W'(x _n )=e ⁱ obtained by assuming A(x _n )=1...(4) 〓 ^(xm) …(5) is recorded. This corresponds to rejecting amplitude information.

The wavefront represented by the above equation (5) is generally recorded by the following method.

First, the magnitude of the phase φ(x _n ) is converted into a light intensity proportional to it, and the phase pattern is displayed as a light intensity pattern on a CRT display or the like. At this time, the range of intensity levels of the display, that is, the dynamic range, is assigned to phases from 0 to 2π.

Next, take a picture of what is displayed on the display with a camera,
The light intensity pattern is reduced and recorded on photographic film. The reduction ratio at that time is determined depending on the wavelength of the reproduced wave and the size of the image.

This film is developed to obtain a density pattern, and further bleached to obtain a relief pattern, the entire process being controlled so that the depth of the relief is proportional to the phase size.
The relief pattern thus obtained is a kinoform.

If the created kinoform is irradiated with coherent light such as a laser beam with uniform intensity, a wavefront having only the complex amplitude of equation (5) can be obtained. In this case, if the assumption in equation (4) stated above actually holds, then
The reproduced wavefront faithfully reproduces the image of the original object, but otherwise faithful reproduction of the original object would be impossible due to errors caused by discarding amplitude information.

Therefore, when creating such a kinoform, the most important issue is to make the distribution of the amplitude component on the kinoform surface as uniform as possible, that is, to make sure that assumption (4) actually holds. And in order to do so again, the above mentioned
It becomes necessary to optimally select the phase component α( _ak ) in equation (1).

Regarding the determination of the phase component α( _ak ), the simplest method conventionally used is to divide random phases selected from uniform random numbers in the interval (0, 2π).

Physically, this method can be considered as a model for diffuse illumination of an object, but it produces a variation of about 20% in the wavefront intensity distribution on the kinoform surface, and the quality of the reconstructed image deteriorates considerably. I don't get it.

Another conventional method proposed to improve this problem is the method of creating an object-dependent phase distribution as disclosed in US Pat. No. 3,619,022.

This method is shown in a flow chart in FIG _. 5. First, in step A, a randomly selected phase α( a _k ) to obtain a sequence of T( _ak ), which is the complex amplitude distribution of the object.

Next, in step B, the complex amplitude distribution of the diffracted wavefront on the kinoform surface is calculated using the above series of T( _ak ), and assuming that the amplitude component is "1", (5)
Obtain the sequence of W′(x _n ) given by Eq.

In step C, the complex amplitude distribution of the image to be reproduced on the original object plane is calculated using W'(x _n ).

The complex amplitude of the image is obtained by inverse transformation when calculating the transformation of the wavefront from the object plane to the kinoform plane.

If the assumption (4) actually holds true, the result of the above calculation is the complex amplitude T( _ak ) of the original object.
However, in general, it is a different series of complex amplitudes T'( _ak ). This T′(a
_k ) is expressed by the following equation (6).

However, C' is a constant, t'( _ak ) is the amplitude component of T'( _ak ), and α'( _ak ) is also the phase component.

In step D, the above two amplitude components t(a
_k ) and t'(a _k ).

As a result, if the average amplitude error Er is tolerably small, the latest φ obtained in step B
(x _n ) is adopted as the phase sequence to be recorded on the kinoform surface. However, if the error Er is unacceptably large, the process proceeds to the next step E.

In this step E, t'( _ak ) is replaced by t( _ak ). The complex amplitude component obtained as a result, T″( _ak )=t( _ak )e ^ia ′ ^(ak) ...(7), is regarded as the new complex amplitude of the object, and then steps B and C are newly calculated. ,D,E are repeated.

In contrast to these conventional methods, there are other conventional methods for making the complex amplitude on the kinoform surface uniform, such as the Parity Sequence Method (PSM) and the Synthetic Coefficient Method.
There is a method called (SCM for short).

<Problems to be Solved by the Invention> The conventional method described with reference to FIG. 5 provides better results than the method of simply adding random phases.

However, as the number of repetitions increases, the average amplitude error asymptotically approaches a constant value, and this asymptotic value does not become sufficiently small.

Therefore, even with this method, it is generally impossible to faithfully reproduce the original object.

On the other hand, the PSM and SCM described above analytically obtain the phase distribution that depends on the object, and do not use an iterative algorithm like the method shown in Figure 5 above. In principle, it is difficult to increase the efficiency of light use.

Also, even when viewed individually, each has its own problems.
In PSM, the object region and dummy region are not spatially separated, so it is not suitable for making the reconstructed image visible to humans.On the other hand, in SCM, although the image can be faithfully reconstructed on the optical axis, , there is a drawback that the error increases as the distance from the optical axis increases.

As mentioned above, among the conventional kinoform recording methods, even the method shown in Fig. 5, which seems to be the best from the viewpoint of light utilization efficiency, satisfies the amplitude distribution of the wavefront on the kinoform surface. It cannot be done uniformly to the extent that
Although they are said to be superior in principle, this may be one of the major reasons why these kinoforms have not become widespread.

The present invention was made precisely to solve this problem, and it further improves the iterative algorithm type method described as the conventional method, and utilizes the advantage of the conventional method, namely, the inherent ability to achieve high light utilization efficiency. The aim was to eliminate the shortcomings without compromising the strong points.

<Means for Solving the Problems> In order to achieve the above object, the present invention provides the following configuration.

A calculation synthesis type wavefront recording method in which a complex amplitude distribution of a wavefront propagating from an object and reaching an arbitrary surface is obtained by a discrete calculation means, and only the phase component thereof is recorded on a recording medium; a sample of the object to be recorded. A set of signal sample points that are required to faithfully reproduce an image point having an amplitude component of a prespecified value, and a set of dummy sample points for which the amplitude component of the image point is not specified in advance. The amplitude component of the point aperture at the signal sampling point is set to a specified value, while the amplitude component of the point aperture at the dummy sampling point is set to an arbitrary initial value, and those amplitude components are step A of creating a complex amplitude distribution of the object by adding a randomly selected phase component to; Step B: calculating the amplitude distribution and replacing the amplitude component with a constant value; calculating the complex amplitude distribution of the image reproduced at the original object position using the complex amplitude distribution obtained in step B; Step C: Based on the comparison between the value of the amplitude component of the image at the signal sampling point and the prespecified value, the complex amplitude distribution obtained in step B is used as the complex amplitude distribution to be recorded on the recording medium. A step D in which it is determined whether or not to adopt the complex amplitude distribution obtained in step B, and if yes, the complex amplitude distribution obtained in step B is adopted as the complex amplitude distribution to be recorded on the recording medium; If the result is negative, the value of the amplitude component of the image at the signal sampling point is replaced with a prespecified value, and the complex amplitude distribution of the resulting image is changed to the new complex amplitude of the object. A calculation synthesis type wavefront recording method comprising: a step E in which the distribution is regarded as a distribution and the process returns to the above step B;

<Operation> In the present invention, as seen in the above summary structure,
First, a set of sample points of the object to be recorded is a set of signal sample points that are required to faithfully reproduce an image point having an amplitude component of a pre-specified value, and a set of signal sample points whose amplitude component of the image point is specified in advance. It consists of a set of dummy sample points that are not

The signal sample point information is essentially the sample point information itself of the object to be recorded, whereas the dummy sample point information has the function of adjusting the wavefront on the recording surface to be uniform.

The dummy sample points can be provided at positions that do not overlap with the signal sample points, and therefore, if necessary, no problem will arise even when the reproduced image is made to appeal to the human eye.

Further, the value of the amplitude component of the reconstructed image at the dummy sample point is determined as a result of executing the iterative algorithm.

In the present invention, as described above, by introducing dummy sample points into the object space to be recorded on the kinoform and using this area, the amplitude component of the wavefront on the kinoform surface is calculated by iterative calculation based on each of the above steps. It is possible to make the distribution uniform.

<Example> FIGS. 3 to 5 explain examples of the present invention. First, regarding the kinoform recording of the one-dimensional object 1, the basic aspects of the present invention will be explained in accordance with FIGS. 1 and 2. An example will be explained.

As shown in FIG. 1, the one-dimensional object 1 to be recorded is divided into an area including N signal sample points and an area including M dummy sample points.

As shown in the figure, the signal sample point area and the dummy sample point area are preferably provided at positions that do not overlap with each other, and in this case, they are two completely independent and adjacent areas. .

FIG. 2 shows a flow diagram of a basic embodiment of the method of the present invention.
It is shown in a chart and consists of the following steps A to E.

[Step A] Set a specified value S( _ak ) to the amplitude component of the point aperture at an arbitrary signal sample point a _k , and set the initial value of the amplitude component of the point aperture of the dummy sample point to a value as small as possible, preferably is set to “0”. In such a case, the amplitude component t at all sample points of the object
( _ak ) is expressed by the following equation (8).

The complex amplitude distribution T(ak ₎ of the object to be recorded is then obtained by adding a randomly selected phase component α( _ak ) to these amplitude components.

[Step B] Using the above series of T( _ak ), calculate the complex amplitude distribution of the diffracted wavefront on the kinoform surface.

The complex amplitude at the sample point x _n on the kinoform surface is given by the following equation (9).

However, c is a constant, and A(x _n ) is W(x
_n ) is the amplitude component, and φ(x _n ) is the phase component.

Here, if the amplitude component A(x _n ) can be assumed to be "1", the sequence given by the following equation (10) can be obtained.

W′(x _n )=e ⁱ 〓 ^(xm) (10) [Step C] Using the above W′(x _n ) series, calculate the complex amplitude distribution of the image to be reproduced.

This is obtained by inverse transformation when converting the wavefront from the object plane to the kinoform plane, and is given by the following equation (11).

However, c' is a constant, t'( _ak ) is the amplitude component of T'( _ak ), and α'( _ak ) is also the phase component.

[Step D] Next, compare t'( _ak ) of the image at the signal sample point with a prespecified value S( _ak ),
For example, calculate the average value Er of both errors.

As a result, if the error Er is within a predetermined tolerance, the latest φ(x _n ) sequence obtained in step B is adopted as the phase sequence to be recorded on the kinoform surface.

[Step E] If the average error Er is outside the allowable range, the amplitude component t'(a
_k ) with a prespecified value S( _ak ).

As a result, the obtained complex amplitude is expressed by the following equation (12).

After this, the series of complex amplitudes T'' ( _ak ) is regarded as a new complex amplitude distribution of the object, and the process returns to step B described above.

The above is a basic embodiment of the present invention, but the present invention is not limited thereto. For example, the initial value of the amplitude component at the dummy sample point may be determined in an arbitrary design.

However, the value of the amplitude component of the image obtained after execution of the iterative algorithm according to the invention is influenced by the initial value, and therefore, in order to sufficiently reduce the light energy reaching the dummy region during the regeneration stage of the kinoform, it is necessary to As this initial value, as seen in the above embodiment, it is desirable to select a sufficiently small value from "0".

Furthermore, as a standard for comparing the value of the amplitude component of an image at a signal sampling point with a predetermined value, in addition to the concept of average error as in the above embodiment, the maximum error may also be used. good.

Particularly in applications where binary information is stored without error using kinoforms, comparison from the perspective of minimizing the maximum error may be more effective.

Furthermore, the layout of the dummy area can be arbitrarily determined.

For example, FIG. 3 shows several examples of the arrangement relationship between the signal sample point collection area 5 and the dummy sample point collection area 4 regarding one-dimensional and two-dimensional objects.

Among these, the ones that are generally advantageous are A,
As shown in B, C, D, and E, a set of signal sample points is included in one continuous area 5, and no dummy sample points are included in this area 5, and the set of dummy sample points is It is arranged in a separate and independent area 4. Rather, this is almost an essential condition when the reproduced image is to be recognized visually by humans.

In particular, as shown in FIG. It will be extremely advantageous.

In addition to the configuration shown in Fig. 3D, the method shown in Fig. 3E, in which a small rectangular area is provided in the center and used as a dummy sample point gathering area 4 together with the frame area, creates a phase mismatch during the kinoform creation process. Therefore, it is effective in cases where there is a risk that a high-intensity spot may occur in the center of the reproduced image, and is particularly suitable for storing binary information.

However, if no particular problem arises, as shown in FIG. It may also be arranged in sections.

According to experiments in which the method of the present invention was applied to two-dimensional objects as described above, both the average amplitude error and the maximum amplitude error were reduced by more than an order of magnitude compared to the case using the conventional iterative method. As a result, we succeeded in deriving a phase distribution to obtain a reconstructed image that matches the original object to an acceptable degree.

<Effects of the Invention> According to the present invention, it is possible to record the original object with high fidelity without impairing the light utilization efficiency, which is an essential advantage of the kinoform.

That is, in the present invention, dummy sample points are introduced in order to improve the fidelity of the reproduced image, but these dummy sample points do not need to be included in the signal sample point area from the essential point of view, so they are not necessary. By confining these within separate and independent areas, it is possible to prevent any inconvenience from occurring even when the reproduced image is viewed by human eyes.

In addition, there is essentially no limit to the initial value set for the amplitude component of the dummy sample point, and therefore,
Since it can be set to a sufficiently small value from "0", the proportion of reproduced light that reaches the dummy sample point can be made sufficiently small, which is the essential advantage of the above-mentioned kinoform. This allows the high efficiency to be maintained.

[Brief explanation of the drawing]

Fig. 1 is an explanatory diagram of an example of the geometrical relationship between a signal sample point set area and a dummy sample point set area for explaining the principle of the present invention, and Fig. 2 is a diagram showing the principle or basic implementation of the method of the present invention. An example iterative algorithm illustration,
Figure 3 is an explanatory diagram of an example of the arrangement relationship between the signal sample point collection area and the dummy sample point collection area for one-dimensional and two-dimensional objects, and Figure 4 is an illustration of the geometric relationship between the object plane and the kinoform plane. FIG. 5 is an explanatory diagram of the principle of a conventional iterative algorithm. In the figure, 1 is the object plane, 2 is the kinoform plane, 3 is the coherent light, 4 is the dummy sample point gathering area, and 5
is the signal sample point set area.

Claims (1)

[Scope of Claims] 1. A calculation-synthesis type wavefront recording method in which a complex amplitude distribution of a wavefront propagating from an object and reaching an arbitrary surface is obtained using a discrete calculation means, and only the phase component thereof is recorded on a recording medium. ; A set of sample points of an object to be recorded is a set of signal sample points that are required to faithfully reproduce an image point having an amplitude component of a prespecified value, and a set of signal sample points whose amplitude component of the image point is specified in advance. a set of dummy sample points that are not set; and a specified value is set for the amplitude component of the point aperture at the signal sample point, while an arbitrary initial value is set for the amplitude component of the point aperture at the dummy sample point. step A of creating a complex amplitude distribution of said object by setting and adding a randomly selected phase component to those amplitude components; step B, which calculates the complex amplitude distribution of the wavefront that has reached the surface and replaces its amplitude component with a constant value; and an image that is reproduced at the original position of the object using the complex amplitude distribution obtained in step B. a step C of calculating the complex amplitude distribution of the image; based on the comparison between the value of the amplitude component of the image at the signal sampling point and the prespecified value, the complex amplitude distribution obtained in the step B is recorded on the recording medium; A step D in which it is determined whether or not to adopt the complex amplitude distribution to be recorded, and if yes, the complex amplitude distribution obtained in step B is adopted as the complex amplitude distribution to be recorded on the recording medium; If the result of the judgment in step D is negative, the value of the amplitude component of the image at the signal sampling point is replaced with a value specified in advance, and as a result, the complex amplitude distribution of the image obtained is A calculation synthesis type wavefront recording method comprising: a step E in which the object is regarded as a new complex amplitude distribution and the step B is returned to;