Method for constructing a digital hologram and associated digital holography system

Disclosed is a method for constructing a digital hologram to be displayed by of a display system. The display system includes a light modulator producing a light beam and a convergent optical device designed to make the light beam converge towards a focal point. The scene is defined by a set of luminous elements. The construction method includes a step of determining values respectively associated with the pixels of the digital hologram by summing the light contributions respectively produced by the luminous elements with weighting, for each of the light contributions, by a correction coefficient depending on the area of the intersection of a surface between the convergent optical device and the focal point, and a pencil of light having a predetermined angular opening and transmitted through the convergent optical device from the luminous element producing the light contribution concerned. An associated holographic system is also described.

BACKGROUND OF THE INVENTION

Field of the Invention

The present invention generally relates to the technical field of digital holography.

It more particularly relates to a method for constructing a digital hologram and an associated digital holography system.

Description of the Related Art

Digital holography aims to reconstruct a three-dimensional scene for an observer by displaying a digital hologram by means of a light modulator.

The field of view obtained can however be limited and too restricted for the chosen application (the angle of emission being directly linked to the pixel density of the light modulator).

In order to enlarge the field of view (which is interesting in particular in case of Augmented Reality where it is desired to superimpose the displayed hologram on the real environment of the observer), it has already been proposed to display the digital hologram by means of a display system comprising a light modulator producing a light beam and a converging optical device designed to make the light beam converge towards a focal point. Such a system is commonly called FTOS (for “Fourier Transform Optical System”).

Hence, by placing the observer's eye between the converging optical device and the focal point (typically near the focal point), the observer's field of view is widened.

SUMMARY OF THE INVENTION

In this context, the invention proposes a method for constructing a digital hologram representing a scene and intended to be displayed by means of a display system comprising a light modulator producing a light beam and a converging optical device designed to make the light beam converge towards a focal point, the scene being defined by a set of luminous elements, characterized in that it comprises a step of determining values respectively associated with pixels of the digital hologram by summing the light contributions respectively produced by luminous elements with weighting, for each of the light contributions, by a correction coefficient depending on the area of the intersection of a surface located between the converging optical device and the focal point, and a pencil of light having a predetermined angular aperture and transmitted through the converging optical device from the luminous element producing the concerned light contribution.

The use of the correction coefficient makes it possible to take into account in advance (during the construction of the digital hologram) the fact that certain light rays produced by the light modulator are only partially received by the observer's pupil (phenomenon that is amplified by the convergence of these rays produced by the converging optical device). The reproduction of the three-dimensional scene by the display system is hence improved.

Other non-limitative and advantageous features of the product/method according to the invention, taken individually or according to all the technically possible combinations, are the following:the luminous elements are located on a plurality of points of the scene, respectively;the method comprises, for each point of said plurality, a step of calculating, as a function of the position of the concerned point, the correction coefficient weighting the light contributions produced by the luminous element located at the concerned point;the step of determining values comprises, for each pixel of the digital hologram, a step of determining the field generated, at the concerned pixel, by a luminous element located at a given point and a step of weighting the determined field by the correction coefficient calculated for the given point;the luminous elements are located in at least one plane;the step of determining values comprises a step of propagating the light field from a first plane to a second plane adjacent to the first plane, with application of a matrix mask (or compensation mask) whose elements are respectively associated with the different points of the first plane and have a value depending on the area of the intersection of said surface and a pencil of light having the predetermined angular aperture and transmitted through the converging optical device from the point associated with the concerned element;the method comprises a step of displaying the constructed digital hologram, by means of the display system;the display system comprises a shutter interposed between two lenses;the orientation of the shutter is determined as a function of the distribution of the luminous elements in angular sectors defined about the axis of the display system;the determined field is zero if the pencil of light emitted by the luminous element located at the given point with said predetermined angular aperture is entirely intercepted by the shutter;the two lenses have a same focal distance;the shutter is separated from each of the two lenses by a distance equal to the focal distance;said surface is a disk centered on the optical axis of the converging optical device;the predetermined angular aperture is the angular aperture of the light beam generated by the light modulator.

The invention also proposes a digital holography system comprising a module for constructing a digital hologram representing a scene and intended to be displayed by means of a display system comprising a light modulator producing a light beam and a converging optical device designed to make the light beam converge towards a focal point, the scene being defined by a set of luminous elements, characterized in that the construction module is designed to determine values respectively associated with pixels of the digital hologram by summing light contributions respectively produced by luminous elements with weighting, for each of the light contributions, by a correction coefficient depending on the area of the intersection of a surface located between the converging optical device and the focal point, and a pencil of light having a predetermined angular aperture and transmitted through the converging optical device from the luminous element producing the concerned light contribution.

Such a holography system can further comprise the above-mentioned display system.

Of course, the different features, alternatives and embodiments of the invention can be associated with each other according to various combinations, insofar as they are not mutually incompatible or exclusive.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

The digital holography system described hereinafter and shown inFIG.1comprises a module2for constructing a digital hologram H and a system10for displaying the digital hologram H.

The construction module2herein comprises a processor4and at least one memory6(such as a random memory or a rewritable non-volatile memory; it could however be, as an alternative, a drive disk). The construction module2is for example a computer.

As explained in the following, the memory6memorizes data representative of a scene to be represented. The memory6can also memorize variables used during the construction of the digital hologram H, as will described hereinafter.

The memory6further memorizes computer program instructions designed, when executed by the processor4, to implement the different operations described hereinafter and allowing the construction of the digital hologram H.

The display system10comprises a light source11(herein monochromatic of wavelength λ), a light modulator12producing a light beam (by modulation of the light emitted by the light source11) and a converging optical device14designed to make this light beam converge towards a focal point A, as will be described hereinafter with reference toFIG.2.

According to a first possible embodiment, the construction module2and the display system10can be gathered within a same holographic viewing device. The digital hologram H can then be transmitted from the construction module2to the display system10by means of an internal bus of this holographic viewing device.

According to a second possible embodiment, the construction module2and the display system10can be spaced apart from each other; the construction module2being for example located in a remote server with which the display system10exchanges data via at least one communication network. In this case, the digital hologram H can be transmitted (for example as coded data representing this digital hologram H) via this communication network.

FIG.2shows the main elements of the display system10.

As already indicated, the display system10comprises a light source11, a light modulator12(for example, of the “Spatial Light Modulator”, SLM, type) and a converging optical device, here a converging lens14.

An orthonormal reference system R (O, ux, uy, uz) is used in the following of the description, where O is the center of the light modulator12, the vector uzis orthogonal to the plane of the light modulator12and directed in the direction of propagation of the light beam generated by the light modulator12, and the vectors uxand uyare parallel to the long and short edges of the light modulator12, respectively.

As can be seen inFIG.2, the optical axis Oz of the display system10is the axis colinear to the vector uzpassing through the point O.

In the display system10, the converging lens14is positioned perpendicular to the optical axis in such a way that the optical axis Oz passes through the center CLof this converging lens14. In other words, the axis of the converging lens14is merged with (or identical to) the optical axis Oz.

The plane of extension of the light modulator12and the plane of extension of the converging lens14are hence parallel to each other.

Let's note d the distance separating the image plane (i.e., in the embodiment ofFIG.2, the plane of the light modulator12) and the converging lens14(i.e., here, d=OCL).

A light ray emitted perpendicular to the light modulator12, and hence incident on the converging lens14along the axis of this converging lens14, will hence be transmitted (after having passed through the converging lens14) towards the focal point A (located on the optical axis Oz). Let's note f the focal distance of the converging lens14: f=ACL.

As shown inFIG.2, the light beam generated by the light modulator12has an angular aperture ω (variable according to the type of light modulator12used).

Moreover, the pupil of the user that observes the light beam generated by the light modulator12after having passed through the converging lens14is represented by a disk δ. As explained hereinafter, the disk δ corresponding to the user's pupil is positioned between the converging lens14and the focal point A.

This disk δ (i.e. the user's pupil) is considered as being perpendicular to the optical axis Oz and centered on the optical axis Oz. It is also considered here that the disk δ is located on the optical axis Oz at the closest point from the center CLof the converging lens14that receives light rays from the whole light modulator12. In other words, as can be seen inFIG.2, the disk δ is considered as being positioned at the point of the optical axis Oz that is the closest to the converging lens14and that is touched by a ray emitted with an angle equal to the angular aperture w from a peripheral pixel of the light modulator12.

This situation of the disk δ (i.e. the user's pupil) is optimum in that it allows maximizing the user's field of view while allowing the latter to see the whole hologram formed by the light beam.

As can be seen inFIG.2, the distance between the converging lens14and the disk δ representing the user's pupil is denoted d1.

FIG.3shows a possible embodiment of the display system10according to which a so-called4F filtering technique is used to eliminate a symmetric ray generated by the light modulator12.

According to this technique, an optical unit comprising a shutter20interposed between two lenses16,18is provided downstream from the light modulator12.

The two lenses16,18have a same focal distance F and the shutter20is separated from each of the two lenses16,18by a distance equal to the focal distance F.

The light modulator12is itself positioned upstream from the first lens16, at a distance equal to the focal distance F.

The display system10hence comprises successively, along the optical axis Oz and with a spacing equal to the focal distance F between two successive elements: the light modulator12, the first lens16, the shutter20, the second lens18and an image plane I.

The shutter20stops the light rays crossing a given half-plane and let through the light rays located in the complementary half-plane in such a way that a light beam corresponding to the beam generated by the light modulator12is generated at the image plane I, but with elimination of a part of the spatial frequency components.

Reference can be made to the article “Band-limited zone plates for single-sideband holography”, Y. Takaki and Y. Tanemoto, in Appl. Opt. 48, H64-H70 (2009), for more details about this filtering technique.

In this exemplary embodiment, the above-mentioned distance d is hence equal to the distance between the image plane I and the converging lens14, as indicated inFIG.3.

The shutter20can be fixed (in which case the half-plane stopping the light rays is constant). As an alternative, a transmissive modulator can be used (as a shutter20) in order to chose for each frame the half-plane in which the rays are stopped.

Hereinafter, θ0denotes the angle formed, in the plane of the shutter20, between an edge of the shutter20and the direction Ox, in such a way that, for a point M of coordinates (x, y, z) in the reference system R and located in the plane of the shutter20, the cylindrical coordinates (ρ, θ, h) of M are such that ρ=SQRT(x2+y2), θ=a tan 2(y,x), z=h and the point M is located in the half-plane of the shutter20if and only ifθ0<θ<θ0+π (where SART is the square root function).

The value of the angle θ0is fixed when the shutter20is fixed, or variable (from one frame to the other one) when the position of the shutter is adjustable (using a transmissive modulator, as indicated hereinabove).

The construction of a digital hologram H intended to be displayed by means of the display system10from luminous elements located at different points P1, P2, P3, Pi, PN(here at N different points) of a scene to be represented will now be described, with reference toFIG.4. The luminous intensity of the luminous element located at point Piis denoted Ai.

The coordinates (xi, yi, zi) of the points Piand the luminous intensity Aiof the luminous elements located at points Piare here memorized in the memory6.

The construction of the digital hologram H is described here for one frame. This construction can be repeated for other frames when the scene is changed.

When the orientation of the shutter20is adjustable, the construction module2first determines the orientation of the shutter20(i.e. the above-mentioned angle θ0) as a function of the distribution of the luminous elements in angular sectors defined about the optical axis Oz of the display system10.

In practice, a predetermined number k of angular sectors and a function S(n) that indicates the number of points Pi in the angular sector of index n (with n between 1 and k) are used:S(n)=card{Pi|2π·(n−1)/k≤θ(Pi)<2π·n/k}, where θ(Pi) is the second cylindrical coordinate of point Pi: θ(Pi)=a tan 2(yi,xi).

By defining Xθas a random variable of probability density S(n) after normalization, the construction module2can chose a realization m of the random variable Xθ(by simulation using a pseudo-random method, for example by reduction to a uniform probability law) and hence determine the angle θ0:
θ0=(2m+1)·π/k.

The construction module2then proceeds to the construction of the digital hologram H. For that purpose, the following operations are implemented for each pixel pk,lof indices k, l of the light modulator12.

Hereinafter, (xk,l, yk,l, zk,l) will denote the coordinates of pixel pk,lin the reference system R and (ρk,l, θk,l, hk,l) the associated cylindrical coordinates thereof: ρk,l=SQRT(xk,l2+yk,l2), θ=a tan 2(yk,l,xk,l), Zk,l=hk,l.

For each point Piof the scene, the construction module2can hence determine the distance di,k,lbetween point Piand pixel pk,l, and the field ci,k,lgenerated, at pixel pk,l, by the luminous element located at point Pi:
di,k,l=SQRT((xk,l−xi)2+(yk,l−yi)2+(zk,l−zi)2)
ci,k,l=0 if θ0<θk,l<θ0+π, or else
ci,k,l=Ai·eXp(2π·j·di,k,l/λ),where exp is the exponential function, λ the wavelength of the light used and j the imaginary unit: j2=−1.

The construction module2then weights the contribution ci,k,lof the luminous element located at point Pi(contribution ci,k,lto the field, as just determined) by a correction coefficient ψ(Pi) calculated as described hereinafter with reference toFIGS.6et7to take into the fact that only a part of the user's pupil (represented by the disk δ) receives the light beam generated by the light modulator12(as explained hereinafter with reference toFIG.6).

The so-obtained weighted contribution is thus equal to:
c′i,k,l=ψ(Pi)·ci,k,l.

The construction module2can then determine the field Fk,lproduced at pixel pk,lby the set of N points Piby summing the weighted contributions c′i,k,lof the different points Pidetermined hereinabove:
Fk,l=Σ1≥i≥Nc′i,k,l.

The construction module2can hence determine the value Hk,lof the digital hologram H for pixel pk,l:=(Fk,l+A)2, where A is the complex amplitude of the reference wave.

By performing the following operations for all the pixels pk,l, the construction module2hence determines the values Hk,lrespectively associated with these pixels pk,lby summing (as explained hereinabove) the light contributions ci,k,lrespectively produced by the luminous elements located at points Piwith weighting, for each of the light contributions Ci,k,l, by a correction coefficient ψ(Pi) depending on the point Pi.

The construction of a digital hologram H intended to be displayed by means of a display system10from a distribution of luminous elements in a set of N planes each comprising an image Ii(pour 1≤i≤N) will now be described with reference toFIG.5.

These N images Iiare respectively located in planes of equation z=zi−1, the light modulator12being located in the plane of equation z=zN(with zN=0).

The content of the images Ii(defined by the light contributions, here the amplitude Ii(x,y) of the light wave, at the points of coordinates (x,y) of the concerned plane of equation z=zi−1) is herein memorized in the memory6.

In the described example, the memory6also memorizes binary masks Oirespectively defining the occultations in the planes of equation z=zi(for 0≤i≤N−1).

The construction of the digital hologram H is described here for one frame. This construction can be repeated for other frames when the scene is changed.

When the orientation of the shutter20is adjustable, the construction module2first determines the orientation of the shutter20(i.e. the above-mentioned angle θ0) as a function of the distribution of the passing points (points of value 1) of the binary occultation masks Oiin angular sectors defined around the optical axis Oz of the display system10.

As in the case of the embodiment described hereinabove with reference toFIG.4, a predetermined number k of angular sectors is used. The set of points in space associated with a pixel of value 1 in an occultation mask Oi is herein denoted S′: S′={P(x,y,z)| there exists i such that z=ziand Oi(x,y)=1}.

A function S(n) that indicates the number of points of the set S′ in the angular sector of index n (with n between 1 and k) can then also be used here:S(n)=card {PS′|2π·(n−1)/k≤θ(P)<2π·n/k}, where θ(P) is the second cylindrical coordinate of point P(x,y,z): θ(P)=a tan 2(y,x).

By defining Xeas a random variable of probability S(n) after normalization, the construction module2can choose a realization m of the random variable Xe(by simulation using a pseudo-random method, for example by reduction to a uniform probability law) and hence determine the angle θ0:
θ0=(2m+1)·π/k.

The construction module2then successively calculates the different fields C′irespectively present at the planes of equation z=zi(for i between 0 and N) by propagation from one plane to an adjacent plane as described now (and schematically shown by means of an arrow inFIG.5).

The construction module2performs for that purpose the following operations for each plane (starting with the farthest plane from the light modulator12, of equation z=z0, and moving towards the light modulator12), i.e. for i from 0 to N−1:the construction module2applies to the propagated field Cithe binary mask Oidefining the occultations in the plane z=zi, adds the contribution of the image Ii+1located in the plane z=zi, and applies a compensation mask ψidefined hereinafter, for example as follows for the concerned points (x,y,zi):
C′i(x,y,zi)=[Ii+1(x,y)+Oi(x,y)·Ci(x,y,zi)]·ψi(x,y);the construction module2propagates the so-obtained field C′ up to the adjacent plane (of equation z=zi+1) by means of a propagation operator Tzi:
Ci+1=Tzi(C′i).

The propagation operator Tziis here defined as follows:

The compensation mask ψiis a matrix mask whose elements are respectively associated with the different points of the plane of equation z=zi. This matrix mask ψiaims to compensate for the fact that only a part of the user's pupil (represented by the disk δ) receives the light beam generated by the light modulator12(as explained hereinafter with reference toFIG.6).

Each element is hence a correction coefficient ψ(x,y,zi) that depends on the concerned point, of coordinates (x,y,zi) in the plane of equation z=zi, similarly to what is mentioned hereinabove within the framework of the embodiment ofFIG.4. In other words, we have: ψi(x,y)=ψ(x,y,zi) (the calculation of the correction coefficient being described hereinafter with reference toFIGS.6and7).

It is hence obtained, after propagation within the different planes, the field CNpresent at the plane of equation z=zN(plane of the light modulator12).

The construction module2can hence determine the digital hologram H for the different points of the light modulator, as follows:H=(CN+A)2, where A is the complex amplitude of the reference wave.

The determination of the correction coefficient ψ aiming to compensate for the fact that only a part of the user's pupil (represented by the disk δ) receives the light beam generated by the light modulator12will now be described with reference toFIG.6.

As explained hereinabove, it is searched to determine the value of the correction coefficient ψ associated with a point P of space of coordinates (x,y,z) (this value being denoted ψ(P) within the framework ofFIG.4and ψ(x,y,z) within the framework ofFIG.5).

A pencil of light φ coming from the concerned point P and having an angular aperture ω corresponding to the angular aperture of the light modulator12is hence shown inFIG.6. This angular aperture ω is hence here constant (whatever the concerned pixel of the light modulator12, and hence of the digital hologram H).

The pencil of light φ is transmitted at the output of the light modulator12(with its central ray perpendicular to the plane of the light modulator12), then through the converging lens14(with its central ray consequently directed towards the focal point A).

As explained hereinabove, the disk δ (corresponding to the user's pupil) is positioned in such a way that a part at least of the pencil of light φ crosses the disk δ.

InFIG.7are shown the disk δ, the pencil of light φ and their intersection ε in the plane of the disk δ (plane of equation z=d+d1).

The area α of the intersection c of the disk δ and the pencil φ is equal to:
α=R2·(β−0.5·sin β)+r2·(σ−0.5·sin σ)
with
β=arcos[(R2+D2−r2)/(2·R·D)]
σ=arcos[(r2+D2−R2)/(2·r·D)]

where (as can be seen inFIG.7) r is the radius of the disk δ, R the radius of the intersection of the pencil of light φ and the plane of the disk δ, and D the distance between the center of the pencil of light φ and the center of the disk δ (that is to say the distance between the central ray of the pencil of light φ and the axis Oz at the plane of the disk δ), and hence:D=ρ·(f−d1)/f (considering the rectilinear character of the above-mentioned central ray), andR=[(1−[d−z]/f)·d1+d−z]·tan ω,with ρ the distance between point P and axis Oz (or the first polar coordinate of point P): ρ=SQRT(x2+y2) and z the third cartesian coordinate of point P.

In practice, for example, for the radius r of the disk δ a mean value generally met for the radius of the human eye's pupil is used, i.e. for example a value between 4 mm and 8 mm, here 6 mm.

Moreover, let's remind that f, d, d1and ω are characteristics of the display system10presented hereinabove with reference toFIG.2.

The construction module2can hence determine (by means of the above formulas) the area α of the intersection ε as a function of the coordinates (x,y,z) of the concerned point P.

The construction module2can determine on this basis the value of the correction coefficient ψ associated with the point P of coordinates (x,y,z):
ψ(P)=π·r2/α(x,y,z).

This correction coefficient ω (P) is inversely proportional to the proportion of the disk δ (corresponding to the user's pupil) receiving the pencil of light φ in such a way that, by weighting a light contribution received from the point P by this correction coefficient ψ (P) as proposed hereinabove, the fact that only this proportion of the user's pupil (represented by the disk δ) receives this pencil of light φ is compensated for.

Of course, various other changes can be made to the invention within the framework of the appended claims.