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2020,  2 (1):   70 - 78   Published Date:2020-2-20

DOI: 10.1016/j.vrih.2019.10.005
1 Introduction2 Analyzing the factors behind ghost images3 Example analysis4 Optimal design5 Conclusion

Abstract

Background
This paper introduces a polarized catadioptric virtual reality optical system. With a focus on the issue of serious ghost image in the system, root causes are analyzed based on design principles and optical structure.
Methods
The distribution of stray light is simulated using Lighttools, and three major ghost paths are selected using the area of the diffuse spot,
Sd
and the energy ratio of the stray light, K as evaluation means. A method to restrain the ghost image through optimization of the structure of the optical system by controlling the focal power of the ghost image path is proposed. Results/Conclusions The results show that the
Sd
for the ghost image path increases by 40% and K decreases by 40% after optimization. Ghost image is effectively suppressed, which provides the theoretical basis and technical support for ghost suppression in a virtual reality optical system.

Content

1 Introduction
In the field of virtual reality (VR), an immersive and interactive technology, rapid developments have been seen in Smart Wear[1]. An optical system is a key equipment in virtual reality technology that simulates visual information. VR optical systems are divided into four types: single aspheric, single Fresnel, multi-stacks and polarized catadioptric[2]. The polarized catadioptric VR optical system adopts a folded optical structure[3], which has a large field of view (FOV) as a characteristic, as well as small size and light weight. It can meet the requirements of compact, lightweight, wireless[4], and has generated extensive research.
At present, the main impediment to the application of this system comes from complex Ghost images[5]. The convergence point of abnormal transmission lights near the image plane in the optical system is called a ghost image[6]. It will affect the actual imaging effect and degrade the users' experience. Most methods aiming at suppressing ghost images consist of restricting the spread of ghost images in the human eyes, including adding diaphragms or extinction paints[7,8], or to improve the processing technology, such as coating anti-reflective films on optical surfaces or selecting specific polarization film layers and matching characteristics between them[9]. These methods can reduce the influence of the ghost image to a certain extent, but a residual ghost image persists in the system due to limitations in processing accuracy.
2 Analyzing the factors behind ghost images
A polarized catadioptric VR optical system usually consists of two lenses with distinct refractive indices. The Fresnel loss is caused by the transmission of light between these optical media with distinct refractive indices, and these residual reflected lights will be produced when the lights pass through the medium[10]. Ghosts will be formed when these lights reach the image plane. According to Fresnel's formula, the relationship between the surface residual reflectance R and the refractive index n of the optical materials is as follows:
R = ( n - 1 ) 2 ( n + 1 ) 2
Equation (1) shows the residual reflectivity of common optical materials, which is between 3% and 8%. However, the optical surface produced is usually not ideal, so in practice the residual reflectivity is higher. Considering that the actual product is coated with anti-reflective film on the optical surface, the residual reflective energy of the surface is reduced to less than 1% of the incident energy. The residual reflectivity of each optical surface is set to 2%, considering the actual visible anti-reflective film and roughness, and retaining a certain margin of tolerance. At this stage, the energy of the light decreases to
10 - 2
times the pre-reflection energy every time the light is reflected, and the energy of the light can be neglected after multiple reflections. Therefore, only the ghost image from direct transmission and the ghost image formed by one reflection are considered in this study, as shown as the B and C beams in Figure 1.
The system contains many types of polarization elements. The polarization state of the light in the system changes imperfectly because of the inherent polarization error of these elements[11,12], or the mismatch between the spectral range and the display. Ghosts can also appear if these lights are transmitted through the system.
The arrangement of polarizers in the system[13] is shown in Figure 2. A quarter-wave plate (QWP) is used to convert linearly polarized light into circularly polarized light or to convert circularly polarized light into linearly polarized light. A reflective polarizer (RP) can reflect linearly polarized light in one direction and transmit linearly polarized light in another direction that is orthogonal to the polarized direction of reflected light. These two devices, together with the partial reflector (PartRef), make up the catadioptric optical path. The polarizer (POL) and QWP 1 near display are used to convert the light emitted into a circularly polarized light to meet the system's requirements for the polarization of the incident light.
When light penetrates the surface of an optical element at a non-vertical angle, the reflection and transmission characteristics depend on the polarization phenomenon[14]. If the polarization vector of the light is in the plane containing the incident and reflected beams, we call it a p-polarized light; if it is perpendicular to the plane, we call it an s-polarized light. Any input polarization state can be expressed as the vector sum of s and p components. When light travels in a wave plate, two refracted beams are produced, which will no longer be described as s and p polarization. One of the refracted rays follows the refraction law, so it is called an ordinary light or o-light; the other refracted ray is different, because it does not obey the refraction law and is called extraordinary light or e-light. When a beam of light with wavelength
λ
, angle of incidence
θ 1
and azimuth angle of incidence plane
φ
is incident on a wave plate with thickness d, the change in phase difference for the emitted light can be calculated as follows[15]:
δ = 2 π λ n B O ¯ + B E ¯ - n o A O ¯ = 2 π λ ( n d S x + A B ¯ s i n   θ 1 - n o 2 d n o 2 - s i n 2   θ 1 )
where
n o
and
n e
are the two principal refractive indices for the wave plate for o and e lights, and n is the refractive index for the e-light, which can be derived from the generalized refraction law:
s i n   θ 1 = n o s i n   θ o = n s i n   θ e
n = n o n e ( n e 2 c o s 2   θ + n o 2 s i n 2   θ ) - 1 / 2
c o s   θ = s i n   θ e c o s   φ
where
θ o
is the refractive angle for the o light,
θ e
is the angle between the wave normal of e-light and the wave plate normal, and
θ
is the angle between the wave normal of the e-light and the optical axis.
S x
is the x-direction component of the Poynting vector
S e
of the e-light,
S x = c o s   α c o s   θ e + s i n   θ e s i n   α c o s   φ ( 1 + t a n 2   θ e s i n 2   φ ) - 1 / 2
According to Equation (2), for a beam of light with an azimuth of 0° and a wavelength of 632nm, the phase difference curve can be plotted (Figure 3); for a beam of complex color light incident vertically to 632nm QWP, the phase difference curve can be plotted (Figure 4).
Obviously, the phase delay of the wave plate is related to the wavelength, incidence angle and azimuth of the incident light, so the polarization distribution of the divergent beam is not uniform after passing through the wave plate, and the polarizers will also introduce a complex ghost image.
3 Example analysis
The ghost image analysis[16,17] of the optical system in Figure 5 is carried out using Lighttools, an illumination analysis software. This system consists of four optical surfaces, L1, L2, L3 and L4, with corresponding polarization film layers attached to different surfaces (not shown in the figure). The role of the ideal lens is to simulate the human eyes in the image.
The transmittance of each optical surface is 98%. The residual reflectance is 2%, and the absorptivity is not reported. The reflectivity of some reflective films is 50%. The optical path difference of QWP is shifted to 0.23
φ
. The polarized reflectance of RP is 98%, and the transmission of polarized transmission axis is 98%. A Monte Carlo method[18] is used to analyze the actual propagation of light in the system. The results show that there is a lot of stray light in the system. Figure 6 is the result of ray tracing at 0° FOV in Lighttools. A large number of scattered light points are distributed around the actual image points.
There are two main approaches to evaluate ghosts: the area of ghost diffuse spot
S d
and the energy ratio of the ghost path stray light K. The diffuse spot is the intensity distribution of the diffraction image formed by a point light source passing through an optical system on different cross sections before and after the image plane. The diffuse spots produced by ghosts on the image surface can reflect the sensitivity of the system to the ghosts. The smaller the diffuse spot is, the more concentrated its energy is, and the brighter the ghost image is relative to the image plane, the clearer the ghost image is to the human eye, pointer to a higher impact for the ghost image; conversely, the larger the diffuse spot is, the more divergent the energy is, and the lower the influence of the ghost image on the system is.
In ideal point source imaging, the intensity of the light is symmetrical before and after the image plane and varies with FOV. In actual optical system imaging, aberration and other defects can easily break this symmetry. Hence, to make computations more convenient, the scattered speckle is regarded as an ellipse in the process of simulation, that is, the formula of the area is
S d = π a b
where a and b are the radii of the two axes of the speckle respectively.
The stray light energy ratio K refers to the ratio of stray light energy and effective light energy on the imaging plane. It is used to characterize the impact of the stray light on the optical system. The smaller the value, the smaller the impact of stray light on the system. For a specific optical path, the formula is
Κ = E g h o s t / E e f f e c t
where
E g h o s t
refers to the stray light energy for the path and
E e f f e c t
refers to the effective light energy for the system.
Combining the two methods above, the three ghost paths that will form ghost images and seriously affect the imaging quality can be extracted from all the stray light paths of the system. The specific light paths are shown in Figure 7.
Figure 7a is a direct transmitted ghost image (Path 1). Because of defects in the related polarization film in the system, the light does not propagate according to the ideal polarization state. Some of the light energy directly transmits through the system to form a ghost image. Because there is no reflection, this segment of stray light has the most significant impact, and the K value can reach 5% and even higher.
Figure 7b and 7c show the L2 surface reflection ghost (Path 2) and the L3 surface reflection ghost (Path 3). The light reflected through the L4 surface fails to reach the second reflective surface L1. Because of the residual reflection at the L2 outer surface and the L3 inner surface, the light is reflected by the two surfaces. Although the light energy of these two parts decreases by a reflection, the polarization state of the light is the same as that of the main light. Stray light can completely emit the system, therefore K can still exceed 2%.
4 Optimal design
The focal power is usually denoted by the letter
  ϕ
. When
ϕ > 0
, light converges; when
  ϕ < 0
, light diverges; finally, when
ϕ = 0
, there is no deflection effect on light, and the direction for the transmission of light will not be changed. For the single refraction spherical focal power,
ϕ = ( n ' - n ) / r = n ' / f ' = - n / f
where
n '
is the refractive index of the image, n is the refractive index of the object, r is the radius of the sphere,
f '
is the focal length of the image and f is the focal length of the object.
For a group of light consisting of a birefringent sphere, if the focal powers of the two sides are
ϕ 1   a n d   ϕ 2
, and the distance is d, then the total focal power can be calculated using the following formula:
ϕ t o t a l = ϕ 1 + ϕ 2 - d ϕ 1 ϕ 2
Therefore, as long as the focal power of the ghost path is less than 0 in the design process, the ghost path diffuse spot can be increased, the contrast of the ghost image can be reduced, and the ghost image can be effectively suppressed.
The refractive indices of Lens 1 and 2 in the system are denoted by
n 1
,
n 2
, while
r 1
r 2
r 3
r 4
f 1
f 1 '
f 2
f 2 '
f 3
f 3 '
f 4
f 4 '
represent the curvature radius, object focus and image focus for L1, L2, L3 and L4. Finally, the intervals of the four surfaces are denoted by
d 1
,
d 2
and
d 3
. The following then holds:
f 1 = - r 1 n 1 - 1 ,    f 1 ' = n 1 r 1 n 1 - 1
f 2 = n 1 r 2 n 1 - 1 ,    f 2 ' = - r 2 n 1 - 1
f 3 = - r 3 n 2 - 1 ,    f 3 ' = n 2 r 3 n 2 - 1
f 4 = n 2 r 4 n 2 - 1 ,    f 4 ' = - r 4 n 2 - 1
If the direction from Display to Pupil is specified to be positive, then in the positive direction, the focal powers of the four surfaces are respectively
ϕ 1 = n 1 f ' ,   ϕ 2 = 1 f 2 ' ,   ϕ 3 = n 2 f 3 ' ,   ϕ 4 = 1 f 4 '
In the opposite direction, the focal powers of L2 and L3 are respectively
ϕ 2 ' = - ϕ 2 ,   ϕ 3 ' = - ϕ 3
Therefore, the impact of the L2 and L3 light intensity can be counteracted in the two opposite processes of incident light from L2 to L3 and from L3 to L2.
Regarding the reflection process, the reflective focal powers of the four surfaces are respectively
ϕ 1 r = - 2 n 1 n 1 - 2 ϕ 1 ,   ϕ 2 r = - 2 n 1 - 2 ϕ 2 ,   ϕ 3 r = - 2 n 2 n 2 - 2 ϕ 3 ,   ϕ 4 r = 2 n 2 n 2 - 2 ϕ 4
Then, with regard to the corresponding path of the system:
ϕ e f f e c t :   ϕ 1 ϕ 2 ϕ 3 ϕ 4 r ϕ 3 ' ϕ 2 ' ϕ 1 r ϕ 2 ϕ 3 ϕ 4
ϕ 1 ϕ 4 r ϕ 1 r ϕ 2 ϕ 3 ϕ 4
ϕ P a t h   1 :   ϕ 1 ϕ 2 ϕ 3 ϕ 4
ϕ P a t h   2 :   ϕ 1 ϕ 2 ϕ 3 ϕ 4 r ϕ 3 ' ϕ 2 r ϕ 3 ϕ 4
ϕ 1 ϕ 2 ϕ 4 r ϕ 2 r ϕ 3 ϕ 4
ϕ P a t h   3 :   ϕ 1 ϕ 2 ϕ 3 ϕ 4 r ϕ 3 r ϕ 4
Because the calculation of optical focusing is complex, the symbol
" "
is used instead of the superposition of the focal power produced by the optical surface. We can then get the following requirements for the focal power of the corresponding path:
ϕ e f f e c t > 0 ϕ P a t h   1 < 0 ϕ P a t h   2 < 0 ϕ P a t h   3 < 0
Based on Equation (22), the optical system can be optimized in Zemax. Because the ghost effect of Path 1 is the most significant, the weight of ghost optimization is increased commensurately in the process of optimization. The optimized optical structure is shown in Figure 8 and Figure 9 displays the
S d
curve of the speckle area before and after optimization. As is apparent there, the area of the scattered speckles in each FOV of Path 1 increases as per Figure 9a; the scattered speckles in 0° FOV (i.e., central FOV) of Path 2 and Path 3 increase significantly, and the 35° FOV (i.e., edge FOV) is shown in Figure 9b and 9c. As is apparent in Figure 10, the aggregate area of the system increases by approximately 40%.
The stray energy ratio K of the optimized structure decreases as well. Figure 11 is a sketch of ray tracing in Lighttools (taking 0° FOV as an example). Each light color in the diagram represents an optical path. The system clearly contains many stray light paths. Setting up the number of light rays emitted by the light source (the energy of each ray being the same), the number of light rays and the light energy can be measured by a detector. Table 1 shows data regarding the light for the effective light path and three ghost light paths entering human eyes before and after optimization. As can be seen here, the K of 0° FOV decreases by approximately 40%.
Comparison of ray data of 0° FOV before and after optimization
Path Before K After K
Ray number E/Lumen Ray number E/Lumen
Effect 13106 60.2876 11576 53.2496
Path 1 237 1.0902 1.81% 164 0.7544 1.42%
Path 2 164 0.7544 1.25% 95 0.437 0.82%
Path 3 309 1.4214 2.36% 107 0.4922 0.92%
Total 5.42% 3.16%
5 Conclusion
In this study, the factors influencing ghost images in a polarized catadioptric VR optical structure were analysed in detail, and three ghost image generation paths were simulated by Lighttools and Zemax. A method was proposed to lower the focal power of the corresponding ghost image path to less than 0 by controlling parameters consisting of the lens shape, curvature and radius in the system. In the optimized system, the ghost path diffuse spot enlarges by 40% and K decreases by 40%. This method improves the contrast for effective light, enhances the optical performance of the system, and can effectively suppress ghost images in the system. Unlike the established process optimization schemes, in this study, we analysed and suppressed ghost images in the system based on the source of these ghost images. This method can be widely used in the design and processing of VR products. It establishes a key reference for improving the imaging quality in compact VR equipment.

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