CN113589671B - Conical surface holographic display method for enlarging vertical field angle - Google Patents

Conical surface holographic display method for enlarging vertical field angle Download PDF

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CN113589671B
CN113589671B CN202110751080.4A CN202110751080A CN113589671B CN 113589671 B CN113589671 B CN 113589671B CN 202110751080 A CN202110751080 A CN 202110751080A CN 113589671 B CN113589671 B CN 113589671B
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conical surface
cone
hologram
diffraction model
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CN113589671A (en
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王君
周振兴
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Sichuan University
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    • G—PHYSICS
    • G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
    • G03H—HOLOGRAPHIC PROCESSES OR APPARATUS
    • G03H1/00—Holographic processes or apparatus using light, infrared or ultraviolet waves for obtaining holograms or for obtaining an image from them; Details peculiar thereto
    • G03H1/22—Processes or apparatus for obtaining an optical image from holograms
    • G—PHYSICS
    • G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
    • G03H—HOLOGRAPHIC PROCESSES OR APPARATUS
    • G03H1/00—Holographic processes or apparatus using light, infrared or ultraviolet waves for obtaining holograms or for obtaining an image from them; Details peculiar thereto
    • G03H1/0005—Adaptation of holography to specific applications
    • G—PHYSICS
    • G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
    • G03H—HOLOGRAPHIC PROCESSES OR APPARATUS
    • G03H1/00—Holographic processes or apparatus using light, infrared or ultraviolet waves for obtaining holograms or for obtaining an image from them; Details peculiar thereto
    • G03H1/04—Processes or apparatus for producing holograms
    • G03H1/08—Synthesising holograms, i.e. holograms synthesized from objects or objects from holograms
    • G03H1/0866—Digital holographic imaging, i.e. synthesizing holobjects from holograms
    • G—PHYSICS
    • G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
    • G03H—HOLOGRAPHIC PROCESSES OR APPARATUS
    • G03H1/00—Holographic processes or apparatus using light, infrared or ultraviolet waves for obtaining holograms or for obtaining an image from them; Details peculiar thereto
    • G03H1/0005—Adaptation of holography to specific applications
    • G03H2001/0088—Adaptation of holography to specific applications for video-holography, i.e. integrating hologram acquisition, transmission and display
    • G—PHYSICS
    • G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
    • G03H—HOLOGRAPHIC PROCESSES OR APPARATUS
    • G03H1/00—Holographic processes or apparatus using light, infrared or ultraviolet waves for obtaining holograms or for obtaining an image from them; Details peculiar thereto
    • G03H1/04—Processes or apparatus for producing holograms
    • G03H1/08—Synthesising holograms, i.e. holograms synthesized from objects or objects from holograms
    • G03H1/0866—Digital holographic imaging, i.e. synthesizing holobjects from holograms
    • G03H2001/0883—Reconstruction aspect, e.g. numerical focusing

Abstract

The invention provides a conical surface holographic display method for expanding a vertical field angle. The method firstly provides a conical surface diffraction model and a rapid calculation method thereof, then provides a generation and reconstruction method of a conical surface hologram on the basis of the conical surface diffraction model, and the provided conical surface holographic display method can enlarge the vertical field angle of cylindrical surface holographic display. Since the field angle expansion research and technical methods of the holographic display are limited to only the horizontal direction, a field angle expansion method in the vertical direction has never been researched and proposed. Therefore, the method solves the problem of limited field angle in the vertical direction of holographic display for the first time, and has better creativity and novelty; in addition, the method can be applied to desktop holographic three-dimensional display and has great application potential.

Description

Conical surface holographic display method for enlarging vertical field angle
Technical Field
The invention relates to a holographic display technology, in particular to a generation method of a cone calculation hologram.
Background
Holographic display has been receiving great attention as an ideal true three-dimensional display technology. Cylindrical computed holography has become a recent research hotspot because of its 360 ° field angle. The vertical field angle of the cylindrical computer holographic display is the same as that of the flat surface, and this problem has never been studied and solved. Therefore, in order to further enlarge the vertical viewing angle of cylindrical holographic display, it is necessary to propose a new diffraction model to realize large viewing angle display of holograms.
Disclosure of Invention
The invention provides a conical surface holographic display method for expanding a vertical field angle, aiming at the problem that the vertical field angle of cylindrical surface holographic display is still limited. The method comprises two parts of cone hologram generation and cone hologram reconstruction based on a cone diffraction model.
The generation process of the cone hologram based on the cone diffraction model is specifically described as follows:
step one, the object plane of the cone diffraction model is a cylindrical surface, the radius and height of the cylindrical surface are (R, H), the complex amplitude distribution of the cylindrical surface is represented as U (theta 0, z0), the holographic recording surface is a part of a conical surface, the upper radius and height of the holographic recording surface are (R1, R0, H), the complex amplitude distribution of the cylindrical surface is represented as H (theta 1, z1), and since the radius Rz of the conical surface changes along with the change of the height z1, the radius of the conical surface can be represented as: rz-z 1 (r1-r0)/H + (r1+ r0)/2, and when the inclination angle of the tapered surface is α, tan α -r (r0-r1)/H is satisfied.
Step two, based on the cone diffraction model, the calculation method for generating the cone hologram is expressed as follows: h (θ 1, z1) ([ 1/(i × λ) ]]*∫∫U(θ0,z0)*exp[i*k*d(θ0,z0,z1)]D (theta 0, z0, z1) K (theta 0) d theta 0 dz0, wherein the distance d (theta 0, z0, z1) between the object plane and any two points on the holographic surface is sqrt [ R0, z0, z1 ] 2 +Rz 2 -2R*Rz*cos(θ0-θ1)+(z0-z1) 2 ]Where i is an imaginary unit, λ is a wavelength, K is a wave number, K (θ 0) is a tilt factor of the diffraction model, and sqrt () is an open square operation.
Step three, since the tilt factor of the diffraction model can be approximated to 1, the point spread function of the cone diffraction model can be simplified to h (θ, z0, z1) [1/(i × λ) ]]Exp { i × k × dh (θ, z0, z1) }/dh (θ, z0, z1), where dh (θ, z0, z1) ═ sqrt [ R 2 +Rz 2 -2R*Rz*cosθ+(z0-z1) 2 ]Since Rz is a constant for a specific height z1 in the generation of the cone hologram, the cone hologram generation is denoted as H (θ 1, z1) ═ jj (θ 0, z0) × H (θ 1- θ 0, z0, z1) × θ 0 × dz 0.
The process of reconstructing the conical hologram is the inverse process of the process of generating the conical hologram, and is specifically described as follows: u (θ 0, z0) ═ H (θ 1, z1) × H (θ 0- θ 1, z0, z1) × d θ 1 × dz 1.
The conical surface diffraction model is further developed into a shielding removing method of the conical surface diffraction model; the method adopts an optical path difference limiting function to realize simple and quick de-occlusion effect, and the optical path difference limiting function is expressed as holo (R) ═ sqrt (R) 2 -Rz 2 ) Then the point spread function of the cone diffraction model deblocking method is expressed as hoc (θ, z0, z1) ═ h (θ, z0, z1) × holo (r).
The conical surface diffraction model further adopts fast Fourier transform FFT to realize the fast calculation of the conical surface diffraction model; the method is specifically described as follows: first, the conical-hologram generating formula is rewritten into a form of convolution, that is, into a form of convolution
Figure GDA0003727483820000021
Figure GDA0003727483820000022
Wherein
Figure GDA0003727483820000023
The method is characterized in that the method is a one-dimensional convolution operation in an angle direction, and then a one-dimensional FFT is adopted in the generation process of the conical surface hologram to realize rapid calculation: h (θ 1, z1) ═ IFFT { FFT [ U (θ 0, z0)]*FFT[h(θ,z0,z1)]Dz0, where the IFFT is inverse fast Fourier transformAnd (5) carrying out a Rie transform.
The method has the beneficial effects that: compared with the traditional cylindrical holographic display, the conical holographic display method enlarges the field angle in the vertical direction; in addition, the fast calculation method of the conical surface diffraction model of the invention adopts fast Fourier transform, so that the fast generation of the conical surface hologram can be realized.
Drawings
FIG. 1 is a schematic diagram of a cone diffraction model of the present invention.
FIG. 2 shows the simulation results of Young's two-point interference patterns of the present invention, where 2(a) -2(d) are interference patterns with cone tilt angles of 0 °, 2 °, 4 °, and 8 °, respectively.
FIG. 3 is a schematic diagram of the vertical field angle expansion of the conical holographic display of the present invention.
FIG. 4 is a graph of the results of a conical hologram (a) and different projection angles (b) - (d) of the present invention.
Detailed Description
An exemplary embodiment of a method for extended vertical field of view holographic display according to the present invention will be described in detail below, and the method will be described in further detail. It is to be noted that the following examples are given for the purpose of illustration only and are not to be construed as limiting the scope of the present invention, and that the skilled person will be able to make insubstantial modifications and adaptations of the method based on the teachings of the method described above and still fall within the scope of the invention.
The invention provides a conical surface holographic display method for expanding a vertical field angle.
As shown in fig. 1, the cone diffraction model is specifically described as follows:
step one, the object plane of the cone diffraction model is a cylindrical surface, the radius and height of the cylindrical surface are (R, H), the complex amplitude distribution of the cylindrical surface is represented as U (theta 0, z0), the holographic recording surface is a part of a conical surface, the upper radius and height of the holographic recording surface are (R1, R0, H), the complex amplitude distribution of the cylindrical surface is represented as H (theta 1, z1), and since the radius Rz of the conical surface changes along with the change of the height z1, the radius of the conical surface can be represented as: rz-z 1 (r1-r0)/H + (r1+ r0)/2, and when the inclination angle of the tapered surface is α, tan α -r (r0-r1)/H is satisfied.
Step two, based on the cone diffraction model, the calculation method for generating the cone hologram is expressed as follows: h (θ 1, z1) ([ 1/(i × λ) ]]*∫∫U(θ0,z0)*exp[i*k*d(θ0,z0,z1)]D (theta 0, z0, z1) K (theta 0) d theta 0 dz0, wherein the distance d (theta 0, z0, z1) between the object plane and any two points on the holographic surface is sqrt [ R0, z0, z1 ] 2 +Rz 2 -2R*Rz*cos(θ0-θ1)+(z0-z1) 2 ]Where i is an imaginary unit, λ is a wavelength, K is a wave number, K (θ 0) is a tilt factor of the diffraction model, and sqrt () is an open square operation.
Step three, since the tilt factor of the diffraction model can be approximated to 1, the point spread function of the cone diffraction model can be simplified to h (θ, z0, z1) [1/(i × λ) ]]Exp { i × k × dh (θ, z0, z1) }/dh (θ, z0, z1), where dh (θ, z0, z1) ═ sqrt [ R 2 +Rz 2 -2R*Rz*cosθ+(z0-z1) 2 ]Since Rz is a constant for a specific height z1 in the generation of the cone hologram, the cone hologram generation is denoted as H (θ 1, z1) — ═ ju (θ 0, z0) · H (θ 1- θ 0, z0, z1) · d θ 0 × dz 0.
The process of reconstructing the conical hologram is the inverse process of the process of generating the conical hologram, and is specifically described as follows: u (θ 0, z0) ═ H (θ 1, z1) × H (θ 0- θ 1, z0, z1) × d θ 1 × dz 1.
The conical surface diffraction model is further developed into a shielding removing method of the conical surface diffraction model; the method adopts an optical path difference limiting function to realize simple and quick de-occlusion effect, and the optical path difference limiting function is expressed as holo (R) ═ sqrt (R) 2 -Rz 2 ) Then the point spread function of the cone diffraction model deblocking method is expressed as hoc (θ, z0, z1) ═ h (θ, z0, z1) × holo (r).
The conical surface diffraction model further adopts fast Fourier transform FFT to realize the fast calculation of the conical surface diffraction model; the method is specifically described as follows: first, the conical-hologram generating formula is rewritten into a form of convolution, that is, into a form of convolution
Figure GDA0003727483820000031
Figure GDA0003727483820000032
Wherein
Figure GDA0003727483820000033
The method is characterized in that the method is a one-dimensional convolution operation in an angle direction, and then a one-dimensional FFT is adopted in the generation process of the conical surface hologram to realize rapid calculation: h (θ 1, z1) ═ IFFT { FFT [ U (θ 0, z0)]*FFT[h(θ,z0,z1)]Dz0, where the IFFT is an inverse fast Fourier transform.
In the present example, the object plane resolution is 512 × 1024, and the wavelength λ, the outer diameter R, and the height H are 250um, 50mm, and 50mm, respectively. Two points of the Young's two-point diffraction experiment are located at (-pi/32, 0) and (pi/32, 0), respectively. FIG. 2 shows the results of two-point Young's experiments, where 2(a) -2(d) are hologram interference patterns with cone tilt angles of 0 °, 2 °, 4 °, and 8 °. FIG. 3 is a schematic diagram of the vertical field angle expansion of a cone holographic display. FIG. 4 is a graph of cone holography and different angle reconstruction results. The result shows that the method can enlarge the vertical-direction field angle of cylindrical holographic display and can realize rapid calculation generation.

Claims (2)

1. The conical surface holographic display method for expanding the vertical field angle is characterized in that: based on the cylindrical diffraction theory of two concentric cylindrical surfaces, an inner cylindrical surface serving as a holographic recording surface is inclined to form a partial conical surface, so that the vertical field angle of holographic display is enlarged; the generation process of the cone hologram based on the cone diffraction model is specifically described as follows: step one, the object plane of the cone diffraction model is a cylindrical surface, the radius and height of the cylindrical surface are (R, H), the complex amplitude distribution of the cylindrical surface is represented as U (theta 0, z0), the holographic recording surface is a part of a conical surface, the upper radius and height of the holographic recording surface are (R1, R0, H), the complex amplitude distribution of the cylindrical surface is represented as H (theta 1, z1), and since the radius Rz of the conical surface changes along with the change of the height z1, the radius of the conical surface can be represented as: rz-z 1 (r1-r0)/H + (r1+ r0)/2, and when the inclination angle of the tapered surface is α, tan α -r (r0-r1)/H is satisfied; step two, based on the cone diffraction model, the calculation method for generating the cone hologram is expressed as follows: h (θ 1, z1) ([ 1/(i × λ) ]]*∫∫U(θ0,z0)*exp[i*k*d(θ0,z0,z1)]D (theta 0, z0, z1) K (theta 0) d theta 0 dz0, wherein the distance d (theta 0, z0, z1) between the object plane and any two points on the holographic surface is sqrt [ R0, z0, z1 ] 2 +Rz 2 -2R*Rz*cos(θ0-θ1)+(z0-z1) 2 ]Wherein i is an imaginary unit, λ is a wavelength, K is a wave number, K (θ 0) is a tilt factor of the diffraction model, and sqrt () is an open square operation; step three, since the tilt factor of the diffraction model can be approximated to 1, the point spread function of the cone diffraction model can be simplified to h (θ, z0, z1) [1/(i × λ) ]]Exp { i × k × dh (θ, z0, z1) }/dh (θ, z0, z1), where dh (θ, z0, z1) ═ sqrt [ R 2 +Rz 2 -2R*Rz*cosθ+(z0-z1) 2 ]Since Rz is a constant for a specific height z1 in the generation process of the cone hologram, the cone hologram generation is expressed as H (θ 1, z1) ═ U (θ 0, z0) × (θ 1- θ 0, z0, z1) × H (θ 1- θ 0, z0, z1) × d θ 0 × dz0, and the fast calculation of the cone diffraction model is further implemented by using fast fourier transform FFT, which is specifically described as follows: first, the conical-hologram generating formula is rewritten into a form of convolution, that is, into a form of convolution
Figure FDA0003727483810000011
Figure FDA0003727483810000012
Wherein
Figure FDA0003727483810000013
The method is characterized in that the method is a one-dimensional convolution operation in an angle direction, and then a one-dimensional FFT is adopted in the generation process of the conical surface hologram to realize rapid calculation: h (θ 1, z1) ═ IFFT { FFT [ U (θ 0, z0)]*FFT[h(θ,z0,z1)]Dz0, where the IFFT is an inverse fast Fourier transform; the process of reconstructing the conical hologram is the inverse process of the process of generating the conical hologram, and is specifically described as follows: u (θ 0, z0) ═ H (θ 1, z1) × H (θ 0- θ 1, z0, z1) × d θ 1 × dz 1.
2. The holographic display method of conical surface for enlarging vertical field angle of claim 1, wherein the conical surface diffraction model is further developed into a de-occlusion method of conical surface diffraction model; the method adopts an optical path difference limiting function to realize simplicity and rapidnessThe optical path difference limiting function is expressed as holo (R) ═ (R ≦ sqrt (R) 2 -Rz 2 ) Then the point spread function of the cone diffraction model deblocking method is expressed as hoc (θ, z0, z1) ═ h (θ, z0, z1) × holo (r).
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