KR101308011B1 - fast holographic fringe pattern generation - Google Patents

Abstract

를 사용한다.The present invention relates to a digital hologram generation method, and more particularly to a method for rapid digital hologram generation to improve the quality of the reconstructed image through accurate phase correction by the same amount of calculation.
The simplified method for generating a digital hologram of the present invention is an algorithm for generating a digital hologram at high speed and generating it at a high speed.

Lt; / RTI >

Description

Fast holographic fringe pattern generation

The present invention relates to a digital hologram generation method, and more particularly to a method for rapid digital hologram generation to improve the quality of the reconstructed image through accurate phase correction by the same amount of calculation.

As is well known, holograms are created using the principles of holography and refer to a fringe pattern (Fringe Pattern) that reproduces a three-dimensional image or a medium on which such interference fringe is recorded. The principle of holography is to split a coherent beam, for example a beam from a laser, into two by a beam splitter so that one beam directly illuminates the recording medium and the other beam is directed to the object we want to see. will be. In this case, the light directly shining on the recording medium is called a reference light, and the light shining on the object is called an object light. Since the object light is light reflected from each surface of the object, the phase difference (distance from the object surface to the recording medium) appears differently depending on the object surface. At this time, the unmodified reference light interferes with the object light, and the recording medium in which the interference fringe is stored is called a hologram.

Next, in order to reconstruct the image stored in the hologram, the light beam may be irradiated to the recording medium again. In this case, the reference light having the same wavelength and phase as the recording time should be irradiated.

Meanwhile, with the development of computer technology, it is possible to artificially generate a hologram by a numerical method, and thus a computer generated hologram (CGH) is optically restored. For example, a method has been developed for directly recording Fresnel holograms on a Charge Coupled Device (CCD), which is now an intermediate process that allows full digital recording and processing of holograms without any photographic recording. . Such digital sampling and numerical hologram reconstruction are called digital holography.

Various types of algorithms have been developed to generate digital holograms at high speed by calculating the digital holograms in real time. A simplified method called diffraction-specific fringe computation was proposed by Lucente. This method is based on partitioning the hologram plane into a number of segments, in which the holographic fringe pattern corresponding to a single object point called "hogel" is associated with the weight of the underlying fringe pattern. Computed by nesting This approach results in a computational speed of 70 times faster than typical non-simplified methods.

A similar approach is called a coherent holographic stereogram (CS). This method significantly reduces the computational complexity, using a segmented local spatial frequency for each segment based on the partition of the hologram. In addition, a single stage Fast Fourier Transform (FFT) is used to obtain the hologram pattern. The Phase-Added Stereogram (PAS) is the original version of the CS, and since it was proposed in 1993, a modified and improved version of the PAS has been steadily being proposed. Compensated Phase-Added Stereograms (CPAS), Accurate Compensated Phase-Added Stereograms (APAS) and Accurate Compensated Phase-Added Stereograms (ACPAS) are typical versions of PAS. CPAS uses a phase correction method, and APAS uses a large IFFT size per segment to reduce errors due to segmented spatial frequency domain. A later version, ACPAS, improves the quality of reconstruction by performing both of these refinements, namely the phase correction method and the large FFT size per segment.

CPAS, on the other hand,

를 사용한다. 위의 수학식에서 관련 파라미터에 대해서는 후술한다.

Lt; / RTI > Related parameters in the above equation will be described later.

와

항목을 사용하는 것에 의해 정밀한 빔 스티어링이 악화되어 복원 영상의 질이 저하되는 문제점이 있었다.But according to these CPAS

Wow

Precise beam steering is deteriorated by using the item, and the quality of the restored image is deteriorated.

An object of the present invention is to provide a simplified method for rapid digital hologram generation that can improve the quality of a reconstructed image through accurate phase correction by the same amount of computation.

A simplified method for generating a rapid digital hologram of the present invention for achieving the above object is an algorithm for generating a digital hologram at high speed to generate at high speed.

를 사용한다.

Lt; / RTI >

According to the simplified method for rapid digital hologram generation of the present invention, the quality of the reconstructed image can be improved by correcting the phase by the same amount of computation.

1 is a diagram for explaining a CS calculation principle.
2 shows the spectrum of a segment and the transformed fringe pattern per segment.
FIG. 3 is a diagram for explaining the propagation of light from one point to the hologram plane, (a) for spherical waves, and (b) for simplified plane waves.
4 is a view showing a generated holographic fringe pattern, (a) is a fringe pattern by RS, (b) a fringe pattern by a simplified method.
5 is a diagram for explaining the principle of dividing the digital hologram plane into rectangular segments.

Hereinafter, with reference to the accompanying drawings will be described in detail a preferred embodiment of the simplified method for rapid digital hologram generation of the present invention.

1 shows the geometry of a CS operation. In a first step the hologram plane is partitioned into suitable rectangular segments. The fringe pattern for each segment is simplified by the superposition of a two-dimensional complex sinusoid, where each of these sine curves simplifies the distribution of each object point into a hologram pattern per segment. Accordingly, the spectrum associated with each segment can be easily restored by placing each complex sine curve at the corresponding frequency position.

The spatial frequency (cycles per unit length) associated with the dispersion of points in a point cloud is determined by moving them to the allowed discrete frequency value closest to them without changing its complex amplitude. Is quantized.

This deformation becomes an additional source of distortion upon restoration. The three methods mentioned above have already been proposed to reduce errors.

Returning to FIG. 1 again, in the second step, the segment spectrum consisting of the complex amplitudes of the two-dimensional complex sine curve is converted by the IFFT to change the spectrum into an associated fringe pattern. This process is repeated for each segment to complete the operation.

Any hologram function per segment representing spatial coordinates in the plane can be written as a superposition of harmonic functions, as shown in FIG. Each harmonic function is the inverse Fourier transform of a single impulse function corresponding to a single object point with complex amplitude. Because of the linearity of the Fourier transform, the IFFT of the segment spectrum, consisting of the variance from all three-dimensional object points, produces a hologram of that segment.

CPAS

Fringe patterns created using PAS with FFT cannot accurately adjust the diffracted light in the desired direction because of the frequency error as mapped to a predetermined discrete value. The first improvement, CPAS, compensates for the errors caused by this mapping. As described above, precise direction adjustment can be obtained by adding some phase correction. Phase correction can be calculated from the difference between spatial frequency values in the continuous and discrete domains. The reconstructed image of CPAS has the same amount of computation but improves over PAS.

ACPAS

Two improvements, namely phase correction and reduction due to precise segmentation in the spatial frequency domain of error, can be integrated in one step, which is called ACPAS because it has the advantages of both CPAS and APAS. The quality of the reconstructed image by the image is almost similar to that of the image reconstructed from the reference hologram.

Improvement according to the present invention

The reference hologram can be obtained from the Rayleigh-Sommerfeld (RS) integration because this approach causes accurate and complex light disturbances on the plane at some distance from the object of interest.

The plural function form of the RS integral is shown in Equation 1 below.

In Equation 1 above, N is the number of object points. A = α _p exp ( φ _p ) is a complex constant, where α _p and φ _p are the amplitude and phase of the p- th object point of the point cloud, respectively. The wave number k is 2π / λ, where λ is the free space wavelength of the coherent light.

사이의 거리 r _p 는

이다. RS 적분은 물체 점으로부터의 전파 광을 구면파로 취급하고, 홀로그램 평면상에서 원하는 복소 진복 분포는 정확하게 결정될 수 있다. 전파하는 구면파는 물체 점에서 분절화 및 방향화된 복수의 평면파로 취급될 수 있다. 관련 도면이 도 3에 도시된다. pth object point and point on hologram

The distance between r _p is

to be. The RS integration treats the propagation light from the object point as a spherical wave, and the desired complex amplitude distribution on the hologram plane can be accurately determined. Propagating spherical waves can be treated as a plurality of plane waves segmented and directed at an object point. The related figure is shown in FIG. 3.

평면에서 PAS의 복수 함수 형태는 아래의 수학식 2와 같이 표현될 수 있다.If the width of the plane wave in Fig. 3B is sufficiently small, this approach can be equivalent to that of the spherical wave. This simplification method uses local spatial frequency to determine the two-dimensional sine harmonic function corresponding to the object point and associated distance phase.

The plural function form of the PAS in the plane may be expressed as Equation 2 below.

와

방향에서의 국소 공간주파수이다. 위의 수학식 2에서 두 번째 항목은 거리 위상이고, 세 번째 항목은 국소 공간주파수 u _p 와 v _p 를 갖는 싸인 고주파 함수이다.Where u _p and v _p correspond to the pth object point, respectively

Wow

Local spatial frequency in the direction. In Equation 2 above, the second item is a distance phase, and the third item is a signed high frequency function having local spatial frequencies u _p and v _p .

Local spatial frequency

의 국소 공간주파수는 물체 점으로부터 홀로그램 평면상의 점

까지의 거리에 따라 변화한다. 국소 공간주파수를 결정하기 위해 수학식 1의 위상 항목은 상수 항목과 두 개의 1차 편미분 항목을 갖는 테일러 급수(Taylor series)를 사용하여 아래의 수학식 3과 같이 정해질 수 있다.Function in Equation 1

The local spatial frequency of is the point on the hologram plane from the object point.

It changes depending on the distance to. In order to determine the local spatial frequency, the phase item of Equation 1 may be determined as shown in Equation 3 below using a Taylor series having a constant item and two first-order partial derivatives.

의 국부 공간주파수 쌍은 아래의 수학식 4 및 수학식 5와 같이 정해진다.function

The local spatial frequency pair of is determined by Equation 4 and Equation 5 below.

와

는 한 점으로부터 홀로그램 평면의 점

에 대한 조사 각도이다. 그러나 이 경우에는 간략화의 정확도가 테일러 급수 확장의 세 가지 항목만을 사용하기 때문에 물체 조사 각도가 증가함에 따라 감소될 수 있다.In Equations 4 and 5 above

Wow

Is a point on the hologram plane from one point

The survey angle for. In this case, however, the accuracy of the simplification can be reduced as the object irradiation angle increases because only three items of Taylor series expansion are used.

와 간섭파

의 간섭을 사용하여 정해질 수 있다. 상기한 두 평면파의 간섭 패턴의 복소 진폭 트랜스미턴스는 아래의 수학식 6과 같이 정해질 수 있다.For high accuracy, the local spatial frequency is two monochromatic plane waves, i.e. object waves with spatial frequencies u _p and v _p in the z = 0 plane.

And interference wave

Can be determined using The complex amplitude transmittance of the interference patterns of the two plane waves may be determined as shown in Equation 6 below.

The last item of Equation 6 may be determined as Equation 7 below.

From Equation 7, the local spatial frequency is defined as Equations 8 and 9 below.

와

는 각각

와

축에서 조사 물체파 각도이고,

와

는 각각

와

축에서 조사 참조파이다. 따라서 홀로그램 평면상에서 물체 점으로부터 수용 가능한 국부 공간주파수 쌍은 수학식 8 및 9에 의해 결정될 수 있다.In Equations 8 and 9

Wow

Respectively

Wow

Irradiation object wave angle in the axis,

Wow

Respectively

Wow

Irradiation reference wave on the axis. Thus, the acceptable local spatial frequency pairs from the object point on the hologram plane can be determined by equations (8) and (9).

The fringe pattern generated using the RS integration and simplification method is shown in FIG.

-Partitioned hologram plane

는 ΔS*ΔS의 장방형 세그먼트들로 구획될 수 있다.

는 (n,m)번 째의 구획된 홀로그램을 나타내고,

는 (n,m)번째 세그먼트의 중심을 나타낸다. 도 5는 관련 도면이다. 이러한 구획에 따르면,

는

에 의해 다시 쓰여질 수 있는데, 여기에서

이고

이다. 위의 수학식 2의 구획된 형태는 아래의 수학식 10과 같이 나타낼 수 있다.In the simplification method, the hologram fringe pattern is appropriate because the hologram fringe pattern is determined by the sine function and the relative distance phase. Hologram plane of size N * M (N, M: natural number greater than 1)

Can be partitioned into rectangular segments of ΔS * ΔS .

Represents the ( n , m ) partitioned hologram,

Denotes the center of the ( n , m ) th segment. 5 is a related diagram. According to these compartments,

The

Can be rewritten by

ego

to be. The partitioned form of Equation 2 above may be expressed as Equation 10 below.

Where r _nmpc is the distance between the pth object point and the ( n , m ) th segment, and u _npc and v _mpc are the local spatial frequencies defined at the center of the ( n , m ) th segment. In Equation 10, the plurality of functions of each segment may be composed of a distance phase, a position phase of the segment, and a sine function corresponding to each object point. A sinusoidal function having a Fourier transform and an associated phase of a segment is expressed by Equation 11 below.

Where u ' = u _npc and v' = v _mpc . This means that the spectrum of the segment corresponding to each object point is made up of the complex amplitude of the two-dimensional sine function and the inverse Fourier transform of the spectrum of the segment produces a harmonic function (hologram fringe pattern).

Thus, the exact hologram fringe pattern in the simplified scheme can be generated by the inverse Fourier transform of the complex amplitude of the two-dimensional complex sine function in the continuous domain.

과 I _nm (u',v')의 복소 함수인

와

는 아래의 수학식 12와 13으로 표현될 수 있다.Discrete spatial frequencies and discrete spatial domains should be considered for digital systems. In the discrete area, Equations 10 and 11

And the complex function of I _nm ( u ' , v' )

Wow

May be represented by Equations 12 and 13 below.

이고

이다. 공간주파수

와

는

와

이고, 여기에서 Δu _npc 와 Δv _mpc 는 미지이다. 따라서, 디지털화 에러는 공간주파수 영역의 디지털화에 의해 야기될 수도 있는데,

에 의해 표현될 수 있다. 이러한 에러는 디지털화된 공간주파수 영역상에서 더 작은 디지털화 단계에 의해 감소될 수 있다.From here,

ego

to be. Spatial frequency

Wow

The

Wow

Where Δu _npc and Δv _mpc are unknown. Therefore, the digitization error may be caused by the digitization of the spatial frequency domain,

Lt; / RTI > This error can be reduced by smaller digitization steps on the digitized spatial frequency domain.

The present invention is not limited to the above-described embodiments, and various modifications can be made without departing from the scope of the technical idea of the present invention.

Claims (2)

An algorithm for generating digital holograms,
Partitioning the digital hologram plane having a (M * N) size (M, N: natural number greater than 1) for an object composed of a plurality of object points into rectangular segments;
Generating the rectangular partitioned segment into a hologram fringe pattern using the algorithm and an inverse Fourier transform,
The algorithm is based on the distance between an object point at a particular point and a segment at a particular point, a local spatial frequency defined at the center of the (n, m) th segment (n: natural number less than N, m: natural number less than M), And a complex function comprising a distance function, a position phase of the segment and a sine function corresponding to each object point.
The method of claim 1, wherein the inverse Fourier transform generates the hologram fringe pattern using the following equation.

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