Digital projectors are widely used in our daily life. Novel projection technologies are emerging to meet the requirements of the handheld and wearable devices. There are mainly three types of digital projection technology. The first is to use projection lenses to relay and magnify the images on the micro Spatial Light Modulator (SLM) to the projection screen based on geometric imaging rules^[1]. This type of projector is now commercially available^[2-3]. However, complex lens systems including spherical and aspheric lenses are usually required to correct the aberrations, which makes the projectors bulky^[4-5]. If the projection lenses are miniaturized, image resolution and light efficiency will be lost inevitably. Because the miniaturized lenses reduce the luminous flux and increase the negative diffraction effect on the lens's aperture^[6]. Furthermore, the Depth of Focus (DOF) of this type of projectors is also limited.

The second is to use a scanning laser beam to draw the image on a screen at a very high frequency^[7-8]. Since there is no projection lens in the projector, the total volume and cost of the system is reduced. The image distortion of scanning laser projectors is more easily to correct than conventional projectors. The disadvantage of this type of projector is that the image resolution is limited by the divergence angle and diffraction of the laser beam in free space^[8].

The third is to use holograms to project an image based on diffractive imaging rules^[9-12]. Instead of physical lenses, digital lens phase factors are coded in the holographic projector, which makes the system compact and versatile. The advantages of this type of projectors include lens-less projection, high energy efficiency, wide color gamut and potentials for 3D display. Recently, both the image size and imaging quality of the holographic projector have been improved remarkably^[13-14]. Full-HD resolution image is achieved experimentally using a 4k SLM^[15]. Miniature holographic projector with cloud computing capabilities is also demonstrated^[16]. Although a considerable amount of research has been dedicated to suppress the noise, the speckle noise is still an inherent challenge that deteriorates the image quality^[17-21]. Moreover, the throw ratio is still limited by the diffraction angle of the light from the finite size pixels of the SLM. There are two effective methods to get a short throw ratio holographic projector, one is to use zoom lenses, the other is to use a diverging spherical wave illumination system^[22-25]. In such projectors, additional lenses are usually required. To avoid using optical lens, photolithography or electron beam lithography is used to make a high resolution hologram to get a short throw ratio projection. However, the projection systems can not achieve dynamic projection^[26-27]. In addition, the complicated calculation is still the bottleneck of current holographic projectors. Although the remote computing servers can effectively reduce the size and power consumption of the holographic projector, the total energy consumption of the projection system is still high.

To address the limitations of the projectors mentioned above, a short throw ratio diffractive projection technique with infinite DOF and without holographic computation and coherence illumination is proposed and demonstrated. A compact Light-emitting Diode (LED) is employed to illuminate the projector, which makes the image free of coherent artifact noises. A Fourier Transform Lens (FTL) is used to generate the spatial frequency spectrum of the projected image in real time. The Rayleigh-Sommerfeld diffraction of the frequency spectrum in free space is used to project the image. High quality and short throw ratio diffractive projectors are demonstrated and image with huge DOF is realized.

Traditional holographic projectors usually use diverging spherical wave illumination to get big images within a short throw distance. As shown in Fig. 1(a), a laser beam is focused by an objective lens to generate a diverging spherical wave. After reflection by the beam splitter, the diverging spherical wave illuminates the hologram on the Liquid Crystal on Silicon (LCOS) device. The hologram then reflects and modulates the spherical wave and generates an image on the projection screen by Fresnel diffraction. The holograms are calculated either in real time or offline. In such a holographic projector, a spherical wave having a large divergence angle can be used to get a short throw ratio, but the image quality is limited by speckle noise, ringing artifacts, zeroth-order and high-order diffracted light. In addition, the complexity of holographic calculations remains unresolved.

In comparison, the proposed non-holographic diffractive projector uses a FTL and LED to achieve short throw ratio projection without holographic calculations. As shown in Fig. 1(b), a collimated LED light is used to illuminate the Digital Mirror Device (DMD) with the help of the Total Reflection (TR) prism. The DMD is located in front focal plane of the FTL. The image to be projected is shown on the DMD directly. The light reflecting from the image goes through the FTL and generates a frequency spectrum of the image in back focal plane of the FTL. Then, by the Rayleigh-Sommerfeld diffraction of the image's frequency spectrum, an image will be projected on the screen clearly. Different from the traditional holographic projector, hologram and holographic calculation is no longer needed, which makes the projector simple and efficient. In addition, coherent light source is not necessary, so that the image is free from speckle, ringing artifacts, and other diffraction noises.

To get a short throw ratio diffractive projection, a large diffraction angle is required. Paraxial approximations of the scalar diffraction theory are not applicable to the projector. Rayleigh-Sommerfeld diffraction theory instead of Fresnel or Fraunhofer diffraction should be used in such condition Fig. 2 shows the simplified coordinate system of the diffraction projector without counting the reflection. The image to be projected is located in the (x₀, y₀) plane, the frequency spectrum of the image is located in the (x₁, y₁) plane, the projection screen is located in the (x₂, y₂) plane with a distance z from the (x₁, y₁) plane. The front and back focal length of the FTL is F₁ and F₂ respectively. The amplitude and intensity distribution of the image to be projected are U₀(x₀, y₀) and I₀(x₀, y₀) respectively. According to the property of the FTL, the frequency spectrum of the image's amplitude distribution on the (x₁, y₁) plane U₁(x₁, y₁) is given by

1 $ {U_1}({x_1},{y_1}) = {\rm{F}}\{ {U_0}(\frac{{{x_0}}}{M},\frac{{{y_0}}}{M})\} $

where M=F₁/F₂, F{ } is the Fourier transform operator. The complex amplitude distribution of the light filed on the screen is given by the Rayleigh-Sommerfeld diffraction integral of the image's frequency spectrum, which is as follows.

2 $ U _ { 2 } ( x _ { 2 } , y _ { 2 } ) = \int\int U _ { 1 } ( x _ { 1 } , y _ { 1 } ) \frac { z \operatorname { exp } ( {\rm i} k r ) } { 2 \pi r ^ { 3 } } ( 1 - {\rm i} k r ) {\rm d} x _ { 1 } {\rm d} y _ { 1 } $

where r is the distance between the two corresponding points in the (x₁, y₁) and (x₂, y₂) planes.

3 $r = \sqrt { ( x _ { 2 } - x _ { 1 } ) ^ { 2 } + ( y _ { 2 } - y _ { 1 } ) ^ { 2 } + z ^ { 2 } }$

For short throw ratio projection, the spectrum area in (x₁, y₁) plane is much smaller than the projection image in (x₂, y₂) plane and the image distance z meets the conditions z>>x₁, and z>>y₁. So Eq. (3) can be simplified to Eq. (4) as follows.

4 $r = \sqrt { x _ { 2 } ^ { 2 } + y _ { 2 } ^ { 2 } + z ^ { 2 } }$

Using Eq. (4), the diffraction integral of Eq. (2) can be simplified to Eq. (5) as follows.

5 $ {U_2}({x_2},{y_2}) = \frac{1}{{i\lambda {\rm{z}}}} {\rm exp} ({\rm{i}}kr)\frac{{{z^2}}}{{{r^2}}}\int {\int {{U_1}} } ({x_1},{y_1}) \cdot {\rm exp} [\frac{{ - {\rm{i}}2\pi }}{{\lambda z}}({x_1}\frac{{{x_2}z}}{r} + {y_1}\frac{{{y_2}z}}{r})]{\rm{d}}{x_1}{\rm{d}}{y_1} $

If the coordinate transformation is defined as follows.

6 $ {X_2} = \frac{{{x_2}z}}{r},{Y_2} = \frac{{{y_2}z}}{r}, \xi = \frac{{{X_2}}}{{\lambda z}},\eta = \frac{{{Y_2}}}{{\lambda z}} $

Eq. (5) can be rewritten as follows.

7 $ {U_2}(\xi ,\eta ) = \frac{1}{{{\rm{i}}\lambda z}} exp ({\rm{i}}kr)\frac{{{z^2}}}{{{r^2}}}{\rm{F}}\{ {U_1}({x_1},{y_1})\} $

According to Eq. (1) and (7), the complex amplitude distribution on the screen can be rewritten as follows.

8 $ {U_2}(\xi ,\eta ) = {\rm{C}}\frac{{{z^2}}}{{{r^2}}}{U_0}( - M{x_0}, - M{y_0}){\rm{ = C}}\int {\int {{U_0}} } (\alpha ,\beta )\frac{{{z^2}}}{{{r^2}}}\delta (\alpha + M{x_0},\beta + M{y_0}){\rm{d}}\alpha {\rm{d}}\beta $

where C is a complex constant that can be neglected for the intensity information on the screen, δ(x) is Dirac function. The amplitude impulse response function of the projector is as follows.

9 $h ( \xi , \eta ) = \frac { z ^ { 2 } } { r ^ { 2 } } \delta ( \alpha + M x _ { 0 } , \beta + M y _ { 0 } )$

According to Eq. (9), the intensity impulse response function of the projector is as follows.

10 $h _ { I } ( \xi , \eta ) = | h ( \xi , \eta ) | ^ { 2 } = \frac { z ^ { 4 } } { r ^ { 4 } } \delta ( \alpha + M x _ { 0 } , \beta + M y _ { 0 } )$

According to the properties of linear optical system, the intensity distribution of the image on the screen is as follows.

11 $ {I_2}(\xi ,\eta ) = {I_0}(\xi ,\eta )*{h_I}(\xi ,\eta ){\rm{ = }}\frac{{{z^4}}}{{{r^4}}}{I_0}(\alpha ,\beta )*\delta (\alpha + M{x_0},\beta + M{y_0}) = \frac{{{z^4}}}{{{r^4}}}{I_0}( - M{x_0}, - M{y_0}) $

where I₀(ξ, η) is the ideal intensity distribution of the image to be projected on the screen, the operator * denotes the convolution. According to Eq. (6) and Eq. (11), the intensity distribution on the screen can be rewritten as follows.

12 $I _ { 2 } ( X _ { 2 } , Y _ { 2 } ) = \frac { z ^ { 4 } } { r ^ { 4 } } I ( - M \lambda z x _ { 0 } , - M \lambda z y _ { 0 } )$

For paraxial approximation, the parameter z meets the condition z≈r, Eq. (9) can be simplified as follows.

13 $I _ { 2 } ( x _ { 2 } , y _ { 2 } ) = I ( - M \lambda z x _ { 0 } , - M \lambda z y _ { 0 } )$

Eq. (12) means that in the non-paraxial approximation region, with some coordinate transformation and intensity variation, the light field′s intensity distribution on the screen is a magnified version of the image′s intensity distribution to be projected. Eq. (13) means that in paraxial approximation region, without any transformation, the light field′s intensity distribution on the screen is a magnified version of the image′s intensity distribution exactly. We can use Eq. (12) and Eq. (13) to design the diffractive projector. The magnification of the projector is Mλz. The projection image will always keep in focus in any projection distance if the approximate condition in Eq. (4) is valid.

From the perspective of geometric optics, this method extends the DOF of the projection system by essentially reducing the aperture diameter of the imaging system. As shown in Fig. 2, there will be a focal spot on the back focal plane of the FTL when the reflected light from the DMD is focused by the FTL. For short throw ratio projector, the focal spot is small enough to be considered as a point source. This system can be considered as a pinhole projector. Similar to pinhole camera, the DOF of the imaging system is infinite.

It has been proven that the proposed projector is experimentally feasible. Figure 3 is the experimental apparatus of the non-holographic diffractive projector. A blue collimated LED is used to illuminate the DMD through a TR prism. A pair of relay lens is used to relay the image on the DMD to the front focal plane of the FTL. A custom-designed FTL is used to generate the frequency spectrum of the image. A mirror is used to reflect the frequency spectrum to the screen. Through the Rayleigh-Sommerfeld diffraction of the frequency spectrum, images are projected on the screen clearly. The collimated LED light (Thorlabs SOLIS-445C) has a wavelength of 445 nm and a clear aperture of 48 mm, respectively. The resolution of the DMD is 1024×768, and the pixel pitch is 13.68 μm (TI DLP7000). The magnification of the relay lens is 0.5 (Edmund Optics). The parameter M and numerical aperture (N.A.) of the FTL are 10 and 0.5 respectively.

Fig. 4 shows the imaging results of the diffractive projector. Fig. 4(a) is the target grid image to be projected. Fig. 4(c) is the projection result with pincushion distortion and intensity variation when Fig. 4 (a) is showed on the DMD. The imaging results are coincided with the calculation of Eq. (6) and Eq. (12). To correct the distortion and the variation, inverse transformations are digitally done to the grid image according to Eq. (6) and Eq. (12), as shown in Fig. 4(b). Fig. 4(d) is the projection result when Fig. 4 (b) is shown on the DMD. As shown in Fig. 4(d), the curves due to the distortion of the large-angle diffraction return to a straight line. The intensity variation of the projection image which is similar to the vignetting effects of the lens but due to the big angle diffraction again is compensated digitally. These results show that the pincushion distortion and intensity variation are well corrected.

Fig. 5 gives the projection result of the grayscale image. Comparing with the holographic projection, no obvious speckle and other diffraction noises are observed. The contrast and the grayscale representation of the image are fine. Thanks to the characteristics of far-field diffraction, the proposed diffraction projector has near-infinite DOF. Fig. 6 shows the imaging result of the ultra large DOF projector. It projects images on three different depth screens clearly and synchronously. Fig. 6(a) is the target character image to be projected, the character A, B, and C are located in different area of the image frame. Fig. 6(b) is inverse transformed image of character A, B, and C. The projection result is shown in Fig. 6(c). The screens to capture the images are located in the plane at z=20 cm, 40 cm, and 100 cm apart from the projector, respectively. The projected character A, B, and C are all in focus and clear. If we move the screen further away, the projection image will always keep in focus, although the intensity of the image will decrease as the image leaving away from the projector. If the screen is moved close to the projector, the projection image will always keep in focus until the projection distance z is so small that Eq.(3) can't be simplified to Eq.(4). In experiments, images are blurred when the projection distance z is smaller than 3 cm. So the DOF range of the projector is z∈[3, ∝]. The infinite DOF projector can be widely used in 3D projection display and augmented reality.

All the analysis above did not consider the lens aberration which will actually deteriorate the image quality. So a high quality FTL is requried in the proposed projector. The additional optical lenses in the projector will challenge the limited space of the compact projector. Like holographic projection systems, these limitations can be addressed by using high resolution SLM or compact optical lenses.

To summarize, the concept of non-holographic diffractive projection technique using incoherent light was demonstrated. A collimated LED light combined with a FTL is used to generate the frequency spectrum of the projected image in real time. Rayleigh-Sommerfeld diffraction of the frequency spectrum is used to generate the clear image. The advantages of the projectors include: 1) No coherent light and holographic computation are required. 2) Images are free of coherent artifact noises. 3) Big image in a short throw distance. 4) Infinite DOF. The main drawback of the presented technique is that additional optical lenses are required in the projector. Optimal design of the optics and mechanical structures are required in future work.