Simplified Aberration Analysis Method of Holographic Waveguide Combiner - MDPI
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hv
photonics
Article
Simplified Aberration Analysis Method of
Holographic Waveguide Combiner
Wei-Chia Su 1 , Shao-Kui Zhou 1,2 , Bor-Shyh Lin 2 and Wen-Kai Lin 1,2, *
1 Graduate Institute of Photonics, National Changhua University of Education, Changhua 50007, Taiwan;
wcsu@cc.ncue.edu.tw (W.-C.S.); tommy848484.cop06g@nctu.edu.tw (S.-K.Z.)
2 College of Photonics, National Chiao Tung University, Hsinchu 30010, Taiwan; borshyhlin@nctu.edu.tw
* Correspondence: alan0734.cop04g@nctu.edu.tw
Received: 5 August 2020; Accepted: 9 September 2020; Published: 10 September 2020
Abstract: Generally, the diffractive waveguide combiner and computer-generated hologram (CGH)
technique have the potential to achieve compact head-mounted display (HMD) with a natural 3D
display function. However, the diffractive waveguide combiner will degrade the image quality
because of aberration. In order to resolve this issue, the complex analysis based on the ray-tracing
method is necessary. Since the major aberration of the waveguide combiners is only astigmatism
and anamorphic distortion, only these two aberrations were discussed in this paper. Furthermore,
two common waveguide structures were discussed here. In total, four formulas were summarized
to analyze aberration and anamorphic distortion in these two structures. Finally, the simplified
formulas were verified with the commercial ray-tracing software Zemax. The calculated results of
the proposed method match the simulation of Zemax software well. Therefore, the aberration of an
arbitrary similar diffractive waveguide can be analyzed by the proposed method. This will make the
designing process simpler and faster.
Keywords: head-mounted display; computer-generated hologram; aberration analyzation;
holographic waveguide
1. Introduction
Display technology has been developing for decades. In recent years, the HMD devices have
received growing attention [1,2]. The HMDs are suitable to achieve the augmented reality (AR) function
due to their portability. Furthermore, the HMDs can provide autostereoscopic images without viewing
zone limit. However, the traditional stereoscopic imaging with two 2-D images is possibly causing the
vergence–accommodation conflicts [3]. In order to resolve this issue, providing 3D images with depth
information is necessary.
The holography technique is an ideal method to project natural 3D images [4]. However, the
traditional holography technique records information optically and is difficult to achieve dynamic
display function. In contrast, the CGH technique computes the holograms numerically, and makes the
recording process become simpler [5–7]. Furthermore, the dynamic display function can be achieved by
displaying the holograms on a spatial light modulator (SLM). Then, the HMDs with 3D information can
be achieved utilizing the CGH techniques [8,9]. Since the general SLM can only modulate amplitude
or phase distribution, the phase information or the amplitude information of the hologram must be
eliminated. However, the elimination will degrade the image quality. In order to enhance the image
quality, the iterative algorithm [10,11] and the complex-amplitude modulation method [12–14] were
proposed successively.
Excluding the image quality issue, the weight is also an important issue in similar devices. In order
to resolve this issue, the devices with waveguide combiner were also proposed [2]. In this method, the
Photonics 2020, 7, 71; doi:10.3390/photonics7030071 www.mdpi.com/journal/photonics
Photonics 2020, 7, 71 2 of 12
light is coupled into the waveguide and propagated inside the waveguide via total internal reflection
(TIR) on the waveguide surface. The in-coupler can be a diffraction element or a geometric structure
such as prism or wedge etc. Finally, the lights will be guided to the human eye by a diffraction
element. The diffraction element of in-coupler or out-coupler could be Raman–Nath grating [15,16]
or volume holographic element (VHOE) with Bragg grating [17–19]. The advantage of the former is
that achieving a wide viewing angle is easier. However, enhancing diffraction efficiency is difficult.
The VHOEs can achieve higher diffraction efficiency, but the rigorous angular selectivity will confine
the viewing angle. No matter the type, the diffraction waveguides cause the aberration and blur the
image. In order to correct the aberration, the aberration has to be analyzed in the waveguides with
symmetric [20,21] and asymmetric structures [22]. Concerning the latter, a geometric structure was
utilized as the in-coupler to reduce the power loss, and a VHOE was employed as the out-coupler. Both
the geometric structure and the diffraction element change the propagation angle of the light along a
single direction. It caused serious astigmatism and anamorphic distortion. On the contrary, the former
utilized two symmetric VHOEs as the in-coupler and the out-coupler. Since the angle changes caused
by two VHOEs are compensated to each other, the astigmatism becomes smaller, and the anamorphic
distortion is almost ignorable.
In this paper, we propose a simplified method to analyze astigmatism and anamorphic distortion of
the symmetric and asymmetric structure. When the specification of the waveguides and the diffraction
angle of the normal incident ray on the HOE are known, astigmatism and anamorphic distortion can be
calculated easily. According to the literature, the waveguides caused aberration in only one direction.
The aberration for the different field perpendicular to this direction is almost constant. Considering
the viewing angle of the current SLM devices, the proposed method was simplified based on the
paraxial approximation. In the following sections, the symmetric and asymmetric structures similar
to [21,22] were utilized to verify the proposed method. If the diffractive efficiency is not considered, the
diffraction behavior of Raman–Nath grating and VHOE is the same. Therefore, the diffraction grating
formula was employed to replace H. Kogelnik’s coupled wave theory. It makes the proposed formulas
simpler. Furthermore, the formulas are also available for Raman–Nath grating—not only for VHOE.
2. Materials and Methods
The schematic diagrams of the holographic waveguide element were shown in Figure 1. The
SLM provided object information, and the holographic waveguide combiners guide the information to
the human eye, then the observer can obtain virtual images. The symmetric structure as shown as
Figure 1a utilized two HOEs to couple optical information in and out the waveguide. The asymmetric
structure as shown as Figure 1b utilized a wedge with a polished surface as an in-coupling surface
and a HOE as an out-coupling element. The HOEs in these two structures are linear gratings in
which the grating vectors are parallel to the x-axis. Since the diopter of the HOEs and the wedge in
x-direction and y-direction is different, the waveguide element will cause astigmatism. Therefore,
the human eye obtains astigmatism virtual images when the SLM provides aberration-free objects.
On the other hand, the astigmatism objects will be obtained when the light of aberration-free images
incident the out-coupling HOE inversely. Then, the device can provide images without aberration
when the astigmatism objects are produced by the SLM.
When the light of a virtual image was coupled into the waveguide at the Out-coupling HOE,
the field curvature curve of the object is shown in Figure 2. The distance from the out-coupling HOE
to the virtual image L is 250 mm. The diffraction angle for the normal incident light is 50 degrees.
The tilt angle of the wedge-shape in Figure 1b is 17.7 degrees. The field curvature curves as shown in
Figure 2 were simulated via the commercial ray-tracing software Zemax. The horizontal and vertical
axis marks the viewing angle and the distance respect to the last surface of combiners, separately. The
red curve shows the position of the imaging points in the y-z plane (Tangential plane), and the black
one shows that in the x-z plane (Sagittal plane). The distance of black lines became shorter because of
1a utilized two HOEs to couple optical information in and out the waveguide. The asymmetric structure
as shown as Figure 1b utilized a wedge with a polished surface as an in-coupling surface and a HOE as
an out-coupling element. The HOEs in these two structures are linear gratings in which the grating
vectors are parallel to the x-axis. Since the diopter of the HOEs and the wedge in x-direction and y-
direction is different,
Photonics 2020, 7, 71 the waveguide element will cause astigmatism. Therefore, the human eye obtains 3 of 12
astigmatism virtual images when the SLM provides aberration-free objects. On the other hand, the
Photonics 2020, 7,
astigmatism x FOR PEER
objects REVIEW
will be obtained when the light of aberration-free images incident the out-coupling 3 of 11
the holographic waveguides. The red lines in the two structures were located
HOE inversely. Then, the device can provide images without aberration when the astigmatism objects at the same distance
When
because
are thethe
produced light
the of
holographic
by a virtual
SLM. imagechanged
waveguide was coupled into thedistance
the effective waveguidefor xatfan
theonly.
Out-coupling HOE, the
field curvature
Photonics 2020,curve
7, x FORofPEER
the REVIEW
object is shown in Figure 2. The distance from the out-coupling HOE
3 of 11 to
the virtual image L is 250 mm. The diffraction angle for the normal incident light is 50 degrees. The
tilt angle of When
the the light of a virtual
wedge-shape image was
in Figure coupled
1b is into the waveguide
17.7 degrees. The field at the Out-coupling
curvature curves HOE, the in
as shown
field curvature curve of the object is shown in Figure 2. The distance from the out-coupling HOE to
Figure 2 were simulated via the commercial ray-tracing software Zemax. The horizontal and vertical
the virtual image L is 250 mm. The diffraction angle for the normal incident light is 50 degrees. The
axis marks the viewing angle and the distance respect to the last surface of combiners, separately.
tilt angle of the wedge-shape in Figure 1b is 17.7 degrees. The field curvature curves as shown in
The red curve
Figure shows
2 were the position
simulated via the of the imaging
commercial points software
ray-tracing in the y-z planeThe
Zemax. (Tangential
horizontal plane), and the
and vertical
black axis
one marks
showsthe that in the x-z plane (Sagittal plane). The distance of black lines became
viewing angle and the distance respect to the last surface of combiners, separately. shorter
becauseTheofredthecurve
holographic
shows thewaveguides.
position of theThe red lines
imaging in in
points thethe
two
y-zstructures were located
plane (Tangential plane), at
andthe
thesame
distance
blackbecause the holographic
one shows that in the x-zwaveguide changed
plane (Sagittal theThe
plane). effective
distancedistance
of blackfor x fan
lines only.shorter
became
(a) (b)
because of the holographic waveguides. The red lines in the two structures were located at the same
distance
Figure
Figure because
1. The the holographic
The schematic
schematic diagrams waveguide
diagrams of changedwaveguide
of the holographic the effective
waveguide distance
element: (a)for
(a) x fan only.
symmetric
symmetric waveguide
waveguide
400
structure;
structure; (b) asymmetric waveguide structure.
(b) 400
350 x-z plane 350 x-z plane
400 y-z plane 400 y-z plane
300 x-z plane 300 x-z plane
350 350
(mm)
Distance (mm)
250 y-z plane 250 y-z plane
300 (mm) 300
Distance (mm)
200 250 200
250
Distance
150 200 150
200
Distance
100 150 150
100
50 100 100
50
50 50
0 0
-3 0 -2 -1 0 1 2 3 0-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3
x-field (degree) x-field (degree)
x-field (degree) x-field (degree)
(a) (b)
(a) (b)
FigureFigure
2. The fieldfield
2. The curvature plot
curvature plotwhich
whichexported
exported by
by commercialray-tracing
ray-tracing software shows
Figure 2. The field curvature plot which exported by commercial
commercial ray-tracing software
softwareshows
shows
astigmatism caused
astigmatism by
caused the
by waveguide
the waveguideelements: (a)
elements: (a)symmetric
symmetric waveguide
waveguide structure;
structure; (b) (b) asymmetric
asymmetric
astigmatism caused by the waveguide elements: (a) symmetric waveguide structure; (b) asymmetric
waveguide structure.
waveguide structure.
waveguide structure.
The The holographic
The holographic
holographic waveguides
waveguides
waveguides also
also
also cause
cause
cause anamorphic
anamorphic
anamorphic distortion
distortion asasshown
distortion shown in in
as shownFigure
in 3,Figure
Figure which were
3, which were
3, which
simulated
simulated via via Zemax.
Zemax. In In this
this gridgrid distortiondiagram,
distortion diagram, thethe ideal
ideal object
object without
without aberration is a is
aberration 4 by
a 4 4by 4
were simulated via Zemax. In this grid distortion diagram, the ideal object without aberration is a
grid as the solid grid. The crosses marked the real position of the grid intersection. In the asymmetric
grid
4 byas the solid
4 grid grid.
as the Thegrid.
solid crosses
Themarked
crossesthe real position
marked the realofposition
the grid of intersection. In the asymmetric
the grid intersection. In the
structure, the object with serious anamorphic distortion was enlarged in the x-direction as shown in
structure,
asymmetric the object
structure,with serious
thethe
object anamorphic
with in
serious distortion
anamorphic was enlarged
distortion in the x-direction
was enlarged in the as shown
x-directionin
Figure 3b. However, distortion the symmetric structure is not serious because the symmetric
Figure
as shown3b. However,
linearingrating
the distortion
Figure compensated
3b. However,it.the in the symmetric
In distortion
this section, in the structure
the simplified is not
symmetricformulas serious
structure because
is proposed
are not seriousthe symmetric
to because
predict the
linear grating
symmetric compensated
linear grating it. In
compensated
astigmatism and anamorphic distortion.this section,
it. In thisthe simplified
section, the formulas
simplified are proposed
formulas are to predict
proposed to
astigmatism and anamorphic distortion.
predict astigmatism and anamorphic distortion.
(a) (b)
Figure 3. The grid distortion diagram which exported by commercial ray-tracing software shows the
(a) caused by the waveguide elements: (a) symmetric waveguide
anamorphic distortion (b) structure; (b)
asymmetric waveguide structure.
Figure
Figure 3.
3. The
The grid
grid distortion
distortion diagram
diagram which
which exported
exported by
by commercial
commercial ray-tracing
ray-tracing software
software shows
shows the
the
anamorphic
anamorphic distortion
distortion caused
caused by
by the
the waveguide
waveguide elements:
elements: (a)
(a) symmetric
symmetric waveguide
waveguide structure;
structure; (b)
(b)
2.1. Astigmatism Analysis
asymmetric waveguide structure.
asymmetric
2.1. Astigmatism Analysis
Photonics 2020, 7, 71 4 of 12
Photonics 2020, 7, x FOR PEER REVIEW 4 of 11
2.1. Astigmatism Analysis
Figure 4 shows the schematic when the incident light of an arbitrary image point pass-through
Figure 4 shows the schematic when the incident light of an arbitrary image point pass-through the
the waveguides. The green line is the chief ray where the aperture stop is the human eye. The gray
waveguides. The green line is the chief ray where the aperture stop is the human eye. The gray line is
line is an arbitrary off-axis ray, which deviates from the chief ray by a small angle . Although
an arbitrary off-axis ray, which deviates from the chief ray by a small angle ∆φi . Although both the
both the off-axis rays in x-fan and y-fan were considered in this section, only the ray in x-fan was
off-axis rays in x-fan and y-fan were considered in this section, only the ray in x-fan was drawn in this
drawn in this figure. Here, we define , and as the deviation angle of the off-axis rays in
, angle
figure. Here, we define ∆φi,x and ∆φi,y as the deviation of the off-axis rays in x-fan and y-fan;
x-fan and y-fan; L is the distance from virtual images to the out-coupling
L is the distance from virtual images to the out-coupling HOE; Leye is the eye-relief; HOE; t isisthe
the thickness
eye-relief;
t is the thickness of the waveguide;
of the waveguide; θi,x and θi,y are the incident , and ,
angleareof the chief ray at the out-coupling HOEthe
the incident angle of the chief ray at out-
in the
coupling HOE in the x-direction and y-direction;
x-direction and y-direction; θd,x and θd,y are the diffraction , and angle are the diffraction angle
, in the x-direction and y-direction, in the x-
direction and
separately; θob,xy-direction,
and θob,y areseparately;
the angle of the , and , are
output light inthe
air angle
in the of the output
x-direction light
and in air in the
y-direction, andx-
direction and y-direction, and
θt is the tilt angel of the wedge-shape in Figure 4b. Notice that the angle θob respects the normal line the
is the tilt angel of the wedge-shape in Figure 4b. Notice that of
angle respects the normal line of the last surface of the waveguides. The symbols
the last surface of the waveguides. The symbols sx and s y are defined as the separation of the chief ray and are
defined
and as the separation
the off-axis rays at the of the
last chief ray
surface. Theand the off-axis
symbols dx and rays at the
d y are thelast surface.
distance Thethe
from symbols
last surfaceand to
are the
the focal pointdistance
of x-fanfrom
and the lastInsurface
y-fan. this case,to the focal pointvaried
the aberration of x-fanwithandthey-fan. In this
incident anglecase, the
in the
aberrationnot
y-direction varied with the Therefore,
significantly. incident angle in the
only the y-direction
image points onnot the significantly. Therefore,
x-axis (y = 0) were only the
considered in
image points
this section. on the x-axis (y = 0) were considered in this section.
(a) (b)
Figure4.4.The
Figure Theschematic
schematicdiagrams
diagrams ofof
thethe
rayray track
track when
when thethe information
information light
light propagated
propagated inversely:
inversely: (a)
(a) symmetric
symmetric waveguide
waveguide structure;
structure; (b) asymmetric
(b) asymmetric waveguide
waveguide structure.
structure.
2.1.1.
2.1.1.Symmetric
SymmetricStructure
Structure
InInthe
thesymmetric
symmetricstructure,
structure,the
thedistance
distancefrom
fromthethelast
lastsurface
surfacetotothe
theobject
objectininthe
thex-z
x-zplane
plane(Sagittal
(Sagittal
plane) and y-z plane (Tangential plane) of the points on the x-axis can be described
plane) and y-z plane (Tangential plane) of the points on the x-axis can be described as as
= −sx cos θob,x ,
d x = ∆
∆φob,x ,
−s (1)
d y = =∆φ y cos θob,x ,
∆ob,y ,
Then,the
Then, theastigmatism
astigmatismcan
canbe
bedefined
definedasas
∆ = − (1)
∆d = dx − d y (2)
The relationship between and can be described as
The relationship between θi and θd can be described as
, ,
, = − = , , −
θ
( i,x ) sin(θi,x )
−1 λ − sin
θ −1 sin θ
= sin = sin − (2)
d,x d,x,0
nΛ n n
,# (3)
=
"
sin ( θ )
θd,y, = sin−1
i,y
n
where is the period in the x-direction, is the wavelength, n is the refractive index of the waveguide
and , , is the diffraction angle of the normal incident ray. Accordingly, when the incident angle is
changed , the variation of diffraction angle isPhotonics 2020, 7, 71 5 of 12
where Λ is the period in the x-direction, λ is the wavelength, n is the refractive index of the waveguide
and θd,x,0 is the diffraction angle of the normal incident ray. Accordingly, when the incident angle is
changed ∆φi , the variation of diffraction angle is
cos(θi,x )
∆φd,x = − ncos(θ ) ∆φi,x
d,x (4)
∆φd,y = ∆φi,y /n
Then, the separation s in x-direction can be described as
−L
!
1 ( N + 1 ) t 1 t ∆φob,x 1
sx = ∆φi,x − ∆φd,x + cos θob,x (5)
cos(θi,x ) cos(θi,x ) cos θd,x
cos θd,x cos θ0 n cos θ0
ob,x ob,x
where N is the TIR times and θ0ob,x is the angle of output light θob,x in the waveguide considered Snell’s
law. The first term is the separation parallel to the x-axis at the out-coupling HOE. The second term is
the separation between two HOEs, and the third term is the separation from the in-coupling HOE to
the last surface. The final term behind the square bracket is utilized to obtain the component which is
perpendicular to the propagating direction. Since the angle of the incident ray and the output ray are
equal, Formula (5) becomes
−L (N + 1)tcos(θi,x ) t
sx = 2 + + cos(θi,x )∆φi,x (6)
cos (θi,x )
ncos θd,x
3 ncos θi,x
2 0
where θ0i,x is the angle of incident ray considering Snell’s law. Similarly, the separation in the y-direction
can be described as
−L∆φi,y −(N + 1)t∆φi,y t∆φi,y
sy = − + (7)
cos(θi,x )
ncos θd,x ncos θ0i,x
Finally, when Formulas (6) and (7) are inducted into Formula (2), the aberration becomes
( N + 1 ) tcos3 (θ ) tcos2 (θ ) ( N + 1 ) tcos ( θ ) tcos ( θ )
i,x i,x i,x i,x
∆d = L −
− − L − − (8)
n cos3 θd,x ncos2 θ0i,x n cos θd,x ncos θ0i,x
Considering that the incident angle θi,x of the CGH systems is very small, Formula (8) can be
simplified as
(N + 1)t 1
∆d = 1 − (9)
n cos θd,x cos θd,x
2
Here, the diffraction angle θd,x should be calculated according to Formula (3) for the arbitrary
incident angle θi,x .
2.1.2. Asymmetric Structure
In the asymmetric structure with the wedge-shaped waveguide, the diffraction angle θd,x can be
computed following Formula (3), and the output angle θob can be computed according to Snell’s law.
The angle of the output ray θob can be described as
h i
θob,x = sin nsin θt − θd,x
−1
h i (10)
θob,y = sin−1 nsin θd,y
Photonics 2020, 7, 71 6 of 12
Furthermore, when the incident angle θi is changed ∆φi , the change of the output ray is
n cos(θt −θd,x ) cos(θi,x ) cos(θt −θd,x )
∆φob,x = − cos(θ ) ∆φd,x = cos(θ ) cos(θ ) ∆φi,x
ob,x d,x ob,x (11)
∆φob,y = ∆φi,y
According to the experience in the former paragraph, the separation in at the last surface can be
described as
cos(θi,x )∆φi,x cos(θd,x ) cos(θob,x )
−L∆φi,x
s x = + D
cos2 (θi,x ) n cos2 (θd,x ) cos(θt −θd,x )
(12)
s y = −L ∆φi,y + D ∆φ
i,y
cos(θi,x ) n
where D is the distance from the out-coupling HOE to the wedge. For the x fan, the terms in the square
brackets are the separation parallel to the x-axis. The final term behind the square brackets is utilized to
obtain the component which is perpendicular to the propagating direction. Considering Formula (11)
and (12), the distance from the last surface to the object can be described as
L cos2 (θd,x ) D cos (θob,x )
2
d = − cos θ − θ
x
cos (θi,x n cos (θt −θd,x ) ob,x ob,x,0
)
3 2
(13)
n cos θob,x − θob,x,0
L
−D
dy =
cos(θi,x )
where θob,x,0 is the angle of the output chief ray when the x-field is 0. Additionally, the astigmatism
with paraxial approximation is
cos2 θd,x cos2 θob,x D cos2 θob,x
∆d = L − 1 − − 1 (14)
cos2 θ − θ n cos2 θ − θ
t d,x t d,x
Here, the output angle θob,x should be calculated according to Formulas (3) and (10) for arbitrary
incident angle θi,x .
2.2. Anamorphic Distortion
Assuming the chief ray of an image point nearby the reference point probes the Out-coupling
HOE and the incident angle of the ray is
θi,x = θi,x + ∆θi,x
(
(15)
θi,y = ∆θi,y
where ∆θi,x and ∆θi,y are equal. The chief rays of this point and the reference point will converge at
the aperture stop. Since the position of the object can be predicted, the separation of these two rays at
the object plane can also be analyzed. Then, the anamorphic distortion can be predicted via the ratio of
the components of the separation in the x-direction and y-direction.
2.2.1. Symmetric Structure
In the symmetric structure, the component in the x-direction considers paraxial approximation is
( N + 1 ) t 1 ∆θi,x ( N + 1 ) t t
∆X = Leye ∆θi,x + t + + t + L − − ∆θi,x (16)
cos θd,x cos2 θd,x n ncos θd,x nPhotonics 2020, 7, 71 7 of 12
The first term is the separation at the first surface. The second term is the total separation in
the waveguide. The third term is the product of the deviation angle ∆θi,x and the distance from the
waveguide to the object d y . Furthermore, the former formula can be simplified as
t (N + 1)t 1
t
∆X = Leye + + L + − 1∆θi,x = Leye + + L − ∆d ∆θi,x (17)
n ncos θd,x cos2 θd,x n
where ∆d is the result of Formula (9). Similarly, the separation in the y-direction can be described as
(N + 1)t ∆θi,y (N + 1)t t
t
∆Y = Leye ∆θi,y + t + + t + L − − ∆θi,y = Leye + + L ∆θi,y (18)
cos θd,x n n cos θd,x n n
Then we define the effective distance from the human eye to virtual images as
t
κ ≡ Leye + +L (19)
n
and the aspect ratio ξ of the anamorphic object can be defined as
∆X κ − ∆d
ξ≡ = (20)
∆Y κ
When the image distance L is far enough, the aspect ratio must be equal to one.
2.2.2. Asymmetric Structure
In the asymmetric structure, the separation in the x-direction can be described as
t D cos θd,x cos(θo,x ) D
∆X = ∆θi,x Leye + + + L− ∆θob,x (21)
n n cos2 θd,x cos θt − θd,x n
where ∆θob,x can be calculated as
cos θt − θd,x
∆θob,x = ∆θi,x (22)
cos θd,x cos θob,x
The first term is the separation at the wedge, and that term behind the square brackets is employed
to obtain the component which is perpendicular to the propagating direction. The second term is the
product of the deviation angle of output rays ∆θob,x and distance of object d y . Similarly, the separation
in the y-direction can be described as
t
D∆θi,y D
t
∆Y = Leye + ∆θi,y + + L− ∆θi,y = Leye + + L ∆θi,y (23)
n n n n
Finally, the aspect ratio can be described as
∆θi,x ∆θ
t D
Leye + + ∆θob,x + L − Dn ∆θob,x
∆X n n cos2 (θd,x ) i,x
ξ≡ = (24)
∆Y κ
When the image distance L is far enough, the aspect ratio can be approximated as
∆θob,x
ξ= (25)
∆θi,xPhotonics 2020, 7, 71 8 of 12
3. Results
In order to verify the proposed method, the calculated results of the formulas were compared to
the simulation results of Zemax in this section. Since the simulation in Zemax software was verified
so that results match the optical experiments in [21,22], the results of the proposed method should
match the results of Zemax. In this section, the thicknesses of the waveguide t in these two structures
were both 8 mm, and the refractive indexes were both 1.5195. The periods of three HOEs were equal
to each other. The diffraction angle of the normal incident ray was 50 degrees. The TIR times n in
the symmetric structure were 2. The tilt angle of the wedge-shape in the asymmetric structure was
17.7 degrees, and the distance from out-coupling HOE to the wedge D was about 45 mm. The field of
view (FOV) of the virtual images was ±2.4 degrees considering that the maximum diffraction angle
of the SLM with 6.4 um pixels was 2.38 degrees at 532 nm. When the image distance L was 250 mm,
Photonics 2020, 7, x FOR PEER REVIEW 8 of 11
the astigmatism of the objects is shown in Figure 5. The horizontal axis is the viewing angle in the
x-direction. Generally,
shifted in Zemax. Thethe image distance
maximum L was
deviation much
was 2.04higher than the
mm when thickness
x-field was and the eye-relief.
0°. However, the
Therefore, the viewing angle in the x-direction was approximated
percentage error was smaller than 1.09% and was ignorable. as the incident angle θi,x . The red
plots in Figure 5a,b were calculated using Formulas (9) and (14), separately.
-27 -170
Zemax Zemax
-30 formula (9) -175 Formula (14)
-33 -180
Δd (mm)
Δd (mm)
-36 -185
-39 -190
-42 -195
-45 -200
-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3
x-field (degree) x-field (degree)
(a) (b)
Figure5.
Figure 5. The
The astigmatism
astigmatism with
with different
different x-field
x-field was
was predicted
predictedvia
via Zemax
Zemax software
softwareand
andthe
thesimplified
simplified
formulas: (a) symmetric waveguide structure; (b) asymmetric waveguide structure.
formulas: (a) symmetric waveguide structure; (b) asymmetric waveguide structure.
In Figure
When the5a,image
the deviations
distance Lbetween the results the
was a variation, of the proposed method
astigmatism and Zemax
of the central field were almost
is shown in
equal ◦
to −2.4as field.
Figureto6.0,Inand the maximum
the symmetric deviation
structure, was 0.17related
astigmatism mm corresponding
to L was a constant, shownIn inFigure
Figure 5b,
6a.
the
On results of the the
the contrary, twoastigmatism
methods were separated
increased byimage
as the aboutdistance
2 mm because
L in thethe Gaussianstructure.
asymmetric image planeThe
slightly shiftedalsoin Zemax. ◦
phenomenon matches The maximum
the results deviation
in [22]. was
In Figure 6b,2.04
the mm when x-field
deviations was 0 . distances
with different However, L
the percentage
between error
different was smaller
methods were than 1.09%
all about and was
2 mm. ignorable.
According to these results, the simplified Formulas
When
(9) and (14)the
for image distance
astigmatism L waswere
analysis a variation,
verified.the astigmatism of the central field is shown in
Figure 6. In the symmetric structure, astigmatism related to L was a constant, as shown in Figure 6a.
On the contrary, the astigmatism increased as the image distance L in the asymmetric structure.
The phenomenon also matches the results in [22]. In Figure 6b, 0 the deviations with different distances
-33 Zemax Zemax
L between different methods were all about 2 mm. According to these results, the simplified Formulas (9)
Formula (9) -300 Formula (14)
and (14)-34for astigmatism analysis were verified.
-600
Δd (mm)
Δd (mm)
-35 -900
-36 -1200
-1500
-37
-1800
0 500 1000 1500 2000 0 500 1000 1500 2000
L (mm) L (mm)
(a) (b)
Figure 6. The astigmatism of the central field with different image distance was predicted via Zemax
software and the simplified formulas: (a) symmetric waveguide structure; (b) asymmetric waveguide
structure.Figure 6. In the symmetric structure, astigmatism related to L was a constant, as shown in Figure 6a.
On the contrary, the astigmatism increased as the image distance L in the asymmetric structure. The
phenomenon also matches the results in [22]. In Figure 6b, the deviations with different distances L
between different methods were all about 2 mm. According to these results, the simplified Formulas
(9) and 2020,
Photonics (14) 7,
for71astigmatism analysis were verified. 9 of 12
0 Zemax
-33 Zemax
Formula (9) -300 Formula (14)
-34
-600
Δd (mm)
Δd (mm)
-35 -900
-36 -1200
-1500
-37
-1800
0 500 1000 1500 2000 0 500 1000 1500 2000
L (mm) L (mm)
(a) (b)
Figure 6. The
Theastigmatism
astigmatism of the central
of the fieldfield
central withwith
different imageimage
different distance was predicted
distance via Zemax
was predicted via
software
Zemax and the and
software simplified formulas:formulas:
the simplified (a) symmetric waveguidewaveguide
(a) symmetric structure; (b) asymmetric
structure; waveguide
(b) asymmetric
structure. structure.
waveguide
The aspect raio raio of
ofthe
theanamorphic
anamorphicobjects
objectsin in these
these twotwo structures
structures are are shown
shown in Figure
in Figure 7. The7.
The horizontal axis is the viewing angle and the vertical axis is the aspect ratio
horizontal axis is the viewing angle and the vertical axis is the aspect ratio . The values simulated ξ. The values
simulated
by Zemax wereby Zemax werebased
calculated calculated
on thebased on the grid
grid distortion distortion
diagram. diagram.
The solid curvesThe solid curves
in Figure 7a,b werein
Figure 7a,bbywere
calculated calculated
Formulas by Formulas
(20) and (20) and
(24), separately. (24), two
In these separately.
structures,In the
these two structures,
anamorphic the
distortion
anamorphic distortion
became significant, became significant,
corresponding corresponding
to the reduction to theand
in the x-field, reduction in the x-field,
the deviation increased and
as the
the
deviation
reduction increased
in distance as L.
theInreduction
Figure 7a,in distance L. In Figure
the anamorphic 7a, thewas
distortion anamorphic distortion
not significant was not
because the
Photonics 2020,
significant 7, x FORthe
because PEER REVIEW linear grating compensated the aberration. When the distance L
symmetric 9 of
was11
symmetric linear grating compensated the aberration. When the distance L was 250 mm, the aspect
250
ratiomm, thecentral
of the aspect ratio
field of
wasthe1.13.
central
Thefield wasratio
aspect 1.13.grew
The aspect
closer ratio grewthe
to 1 with closer to 1 with
farther the farther
distance L. The
was highly significant as shown in Figure 7b. The aspect ratio grew closer to Formula (25) with the
distance
maximum L. percentage
The maximum percentage
error errorwas
in this figure in this figure
0.17%. Inwas
the 0.17%. In thestructure,
asymmetric asymmetric thestructure,
anamorphicthe
farther distance L. In this case, the aspect ratio of the central field in Formula (25) was 2.25.
anamorphic was highly significant as shown in Figure 7b. The aspect ratio grew closer to Formula (25)
Additionally, the ratio of the central field when L was 250 mm, 500 mm and infinity was 2.01, 2.12
with the farther distance L. In this case, the aspect ratio of the central field in Formula (25) was 2.25.
and 2.25, separately. The maximum percentage error in this figure was 0.32%.
Additionally, the ratio of the central field when L was 250 mm, 500 mm and infinity was 2.01, 2.12 and
2.25, separately. The maximum percentage error in this figure was 0.32%.
L=∝ Zemax
1.25 2.8 L=∝ Zemax
L=∝ Formula (20)
L=∝ Formula (24)
L=500 Zemax
L=500 Zemax
1.20 L=500 Formula (20) 2.6
L=500 Formula (24)
L=250 Zemax
L=250 Zemax
1.15 L=250 Formula (20) 2.4 L=250 Formula (24)
ξ
ξ
1.10 2.2
1.05 2.0
1.00 1.8
-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3
x-field (degree) x-field (degree)
(a) (b)
Figure 7.
Figure 7. The
Theaspect ratio
aspect of the
ratio anamorphic
of the object
anamorphic with with
object different x-fieldx-field
different was predicted via Zemax
was predicted via
software
Zemax and the and
software simplified formulas:formulas:
the simplified (a) symmetric waveguidewaveguide
(a) symmetric structure; (b) asymmetric
structure; waveguide
(b) asymmetric
structure. structure.
waveguide
4.
4. Discussion
Discussion
In
In this
thisstudy, thethe
study, equations of astigmatism
equations and anamorphic
of astigmatism distortiondistortion
and anamorphic of the diffractive
of the waveguide
diffractive
combiner
waveguide arecombiner
presented for
are the CGH technique.
presented for the Astigmatism of symmetric
CGH technique. and asymmetric
Astigmatism structures
of symmetric and
asymmetric structures can be calculated via Formula (8) and (13). The formulas were simplified as
Formulas (9) and (14) based on paraxial approximation. Furthermore, the anamorphic distortion of
these two structures can be calculated based on Formulas (20) and (24). Considering the commercial
SLM with a 6.4 um pixel size, the aberration was analyzed for ±2.4° FOV in this paper.
Dismissing the limitation of SLMs, we compared the proposed method to the simulation ofPhotonics 2020, 7, 71 10 of 12
can be calculated via Formula (8) and (13). The formulas were simplified as Formulas (9) and (14)
based on paraxial approximation. Furthermore, the anamorphic distortion of these two structures can
be calculated based on Formulas (20) and (24). Considering the commercial SLM with a 6.4 um pixel
size, the aberration was analyzed for ±2.4◦ FOV in this paper.
Dismissing the limitation of SLMs, we compared the proposed method to the simulation of Zemax
with a larger field. In the simulation configuration employed in this paper, the maximum FOV of the
symmetric structure was from −10◦ to 9.4◦ , and in the asymmetric structure was from −5.2◦ to 9.4◦ .
The upper limit was limited by the TIR condition, and the lower limit was limited by the dimension
of the in-coupling surface. Smaller than the lower limit, the chief ray exceeded the boundary of the
in-coupling surface. The calculation results and the simulation results in this range when L was
250 mm are shown in Figure 8. In Figure 8a,b, the results of Formulas (8) and (13) match the results of
Zemax. The maximum deviation between Formulas (8) and (9) was 6.18 mm, and the percentage error
was 5.67% when the x-field was −10◦ . The maximum deviation between Formulas (13) and (14) was
6.92 mm, and the percentage error was 5.36% when the x-field was 9.4◦ . In Figure 8c,d, the maximum
percentage error of the anamorphic analyzation based on Formula (20) was 2.53% when the x-field
Photonics 2020, 7, x FOR PEER REVIEW
was
10 of 11
◦
−10 , and the maximum percentage error of Formula (24) was 2.23% when x-field was 9.4 . ◦
0 Zemax -120 Zemax
Formula (8) Formula (13)
-20
Formula (9) -140 Formula (14)
-40
-160
Δd (mm)
Δd (mm)
-60
-180
-80
-200
-100
-120 -220
-10 -5 0 5 10 -6 -4 -2 0 2 4 6 8 10
x-field (degree) x-field (degree)
(a) (b)
1.5 Zemax
3.5 Zemax
Formula (20)
Formula (24)
1.4
3.0
1.3
2.5
ξ
ξ
1.2
2.0
1.1
1.5
1.0
-6 -4 -2 0 2 4 6 8 10
-10 -5 0 5 10
x-field (degree)
x-field (degree)
(c) (d)
Figure8.8. The aberration
Figure aberration analyzation
analyzation for
for large
largex-field:
x-field:(a)(a)astigmatism
astigmatismof of symmetric
symmetric structure;
structure; (b)
(b) astigmatism
astigmatism ofofasymmetric
asymmetricstructure;
structure;(c)
(c) anamorphic
anamorphic ofof symmetric structure;
structure; (d)
(d)anamorphic
anamorphicofof
asymmetric
asymmetricstructure.
structure.
Considering the above information, we believe the proposed method is available for most
5. Conclusions
diffractive waveguide HMDs, which is based on the current CGH technique. In fact, even
In this paper, the aberration of two types of the holographic waveguide combiner are discussed.
whenemploying the SLM with the smallest pixel size (3.74 um for current commercial SLMs), the
The major aberration caused by the holographic combiners is astigmatism and anamorphic distortion.
maximum FOV is about only ±4◦ . In the condition, the maximum deviation in Figure 8a was smaller
In order to provide aberration-free images to the observer, producing an object with reverse
than 0.54 mm, and the maximum deviation in Figure 8b was smaller than 2.04 mm. Furthermore, the
aberration is necessary. For each field, astigmatism is predicted via tracing the crossing point of the
percentage error of the anamorphic analyzation based on Formulas (20) and (24) was not larger than
x-fan and the y-fan separately, and the anamorphic distortion is predicted via tracing the position at
0.64%.
the object plane of the chief rays of the nearby points. In the proposed method, the astigmatism and
the anamorphic distortion can be calculated easily when the specification of the waveguides and the
diffraction angle of the HOE for the normal incident ray , , are known. Formulas (9) and (14) were
utilized to predict the astigmatism of the object in the symmetric waveguide and the asymmetric
waveguide, separately. Formulas (20) and (24) were utilized to predict the anamorphic distortion of
the objects in the symmetric waveguide and asymmetric waveguide, separately. The results of thePhotonics 2020, 7, 71 11 of 12
5. Conclusions
In this paper, the aberration of two types of the holographic waveguide combiner are discussed.
The major aberration caused by the holographic combiners is astigmatism and anamorphic distortion.
In order to provide aberration-free images to the observer, producing an object with reverse aberration
is necessary. For each field, astigmatism is predicted via tracing the crossing point of the x-fan and the
y-fan separately, and the anamorphic distortion is predicted via tracing the position at the object plane
of the chief rays of the nearby points. In the proposed method, the astigmatism and the anamorphic
distortion can be calculated easily when the specification of the waveguides and the diffraction angle
of the HOE for the normal incident ray θd,x,0 are known. Formulas (9) and (14) were utilized to
predict the astigmatism of the object in the symmetric waveguide and the asymmetric waveguide,
separately. Formulas (20) and (24) were utilized to predict the anamorphic distortion of the objects in
the symmetric waveguide and asymmetric waveguide, separately. The results of the proposed method
are very close to the simulation of ray-tracing software Zemax, where the simulation of Zemax has
been proved for aberration correcting. Therefore, the aberration of an arbitrary device with a similar
holographic waveguide element can be corrected via the proposed method. It will make the designing
process simpler and faster.
Author Contributions: Writing—original draft preparation, W.-K.L.; writing—review and editing, W.-C.S. and
B.-S.L.; methodology, W.-K.L.; software, S.-K.Z.; funding acquisition, W.-C.S. supervision, W.-C.S. All authors
have read and agreed to the published version of the manuscript.
Funding: This work is supported by the Ministry of Science and Technology of Taiwan under contract MOST
108-2221-E-018 -018 -MY3.
Conflicts of Interest: The authors declare no conflict of interest.
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