Basic Formulation of Ray Tracing
Referring to Fig. 5-2-1, the entrance angle of ray from a source, which was denoted by a grid of field points, essentially determines the distance traveled by the ray in the wedge plate.
This also implies that the entrance angle of ray can be converted to the exit position of ray on the screen as to be shown below. Therefore, by controlling the entrance angle of ray, one can guide the ray to a desired position on the screen and hence, enhance display quality. It would be worthwhile to emphasize that even without imaging optics; the grid of field points can still form the image over the exit port of the wedge plate, i.e., the upper screen, provided the ray is well limited to some specified emitted angle.
Figure 5-2-1 Schematic diagram of a wedge plate display where D is the length of screen diagonal.
Basically, when one ray propagates inside the wedge plate, if the reflective angle does not reach to the critical angle, the ray will encounter a total internal reflection (TIR) by which the interface between the plate and air would act as a mirror, and the corresponding reflected angle will decrease gradually until it reaches the critical angle such that the ray exits away from the plate. To deduce the relation between entrance angle and emergent position, Travis and Zhong assumed that ray enters into wedge plate with an incident angleθiwould exit the plate at a position X and the exit angleθois the same with the critical angle of wedge
plateθC , and hence, the relation follows
L
X C
i
θ
cosθ
sin = ⋅ , where L is the side length of
wedge plate, as shown in Fig. 5-2-2(a) [5-5]. However, the exit angle does not have to equal the critical angle: the ray would exit the wedge plate even when the reflective angle is smaller than the critical angle because of finite vertex angle of wedge plate. In other word, the formula in prior work [5-5] works for specific case and has to be reconsidered.
Figure 5-2-2 Schematic diagram of ray propagation: (a) a virtually-folded wedge plate and (b) a virtually-folded wedge plate with non-zero entrance height.
Let us assume that one ray propagated inside the wedge plate and finally exited away from the upper side of the wedge plate, then the times of reflections could be denoted as 2m, and the exit angle follows θo =90o −θi−2m⋅θV , where θV is the vertex angle of wedge plate. When the exit angleθ is slightly smaller than the critical angleo θC , the corresponding
incident ray will have the same number of reflections, and ] 2
approximately, where ceiling is a function that gives the nearest integer value that lager than the exact value (e.g., ceiling[π]=4). With these considerations, we have which specifies the required entrance angleθifor some specific X. Referring to Fig.
5-2-2(b), if ray enters the wedge plate at a height h, Eq.(5-2-1) can be rewritten as
where b is the height of the incident ray at entrance and the wedge thickness of entrance port is b+h. Equations 5-2-1 and 5-2-2 are the basic formulas of ray tracing in wedge plate.
When a skew ray is incident along the non-meridional direction of wedge plate, the corresponding (effective) vertex angle θV' would be small than the original θV, as seen in Fig.
5-2-1, and follows θV'=tan−1(tanθV cosΦ) whereΦis the angle between the meridional and non-meridional planes. Therefore, to have the ray-tracing formula of non-meridional ray, one could simply replaceθ of Eq.(5-2-2) byV θ . Equations (5-2-1) and (5-2-2) and V' their extensions provide the complete formulas of ray tracing for wedge plate and hence, a base for the analysis of the formation of dark zone and display quality without additional imaging optics.
Aberration Analysis
Eq. (5-2-2) describes basic behavior of ray propagating in wedge plate. For further illustration, when ray propagated in the meridional plane, Eq. (5-2-2) can be rewritten as
( )
And when a skew ray is incident along the non-meridional direction of the wedge plate, the ray-tracing formula should be
( )
whereΦis the angle between the meridional and non-meridional planes.
Next we explain the wedge plate aberration form. Referring to Fig. 4-2-3(a), if an object with height h and distance d emits one ray into the wedge plate at aperture height ρ, the ray angle after entering wedge is
(
*)
]Substituting this equation into Eq.(5-2-3) produces the actual exit location of this ray in the wedge screen, based on system parameters. However, some exceptions must be noted in practical situations. Figure 5-2-3(b) shows that when the entrance angleθ is smaller than vertex angle θ (even v θ <0), the ray will touch the bottom surface of the wedge plate first.
This situation assumes that the ray is emitted from a virtual source location A which is separated from the top of entrance port by distance b’, and the entrance angle should be adapted to2θv−θ . By trigonometry,
With this assumption, Eq. (5-2-3) is suitable for all practical rays emitted outside the wedge plate.
Figure 5-2-3 (a) Schematic diagram of a ray emitted from a finite size object; (b) Schematic diagram when ray incidents to bottom of wedge
The ideal location for a ray in the screen should be the emitting position (height) multiplied by the corresponding magnification. For example, if the system magnification is m, the object height is h, and the ray is emitted from field point fp, then the ideal exit location
in the wedge screen should be
h m
perfect =(1− fp)⋅ ⋅
X (5-2-7)
For an optical system, the aberration is the deviation between the ideal and real locations.
The following formula can identify the aberration Dx of the wedge plate:
( ) (
fp)
m hEquation (5-2-8) can determine the aberration relative to different ray locations on entrance port for some particular wedge systems. Consider an illustration with object height h=1, object distance d=10, aperture height Τ=5 and the refractive index of wedge plate n=1.51872 (BK7) for a 50-inch wedge plate display, L= 62.25 and w=43.578 inches. In this study, the aberration curve in field points h=0.0, 0.7, and 1.0 were sketched by Mathematica.
Figure 5-2-4 Analytical result of total aberration plots of the wedge-plate display
Contrary to the generally accepted concept of aberration theory, the results in Fig. 5-2-4 show that the ray intercept curve exhibits non-differential characteristics, i.e., kink-like features, as the signature of aberration behavior for wedge-plate type displays. The onset of this kink-like signature is linked to the appearance of the imaging dark zone. The dark zone was
excited because the finite vertex angle which is inevitable in wedge-plate displays [5-4] [5-10]
. The following contains a short summary of the physical mechanism, i.e., the dark zone for completeness. The dark zone is caused by different output ray reflections. The distribution of emerging ray positions “jumps” where the emerging angle is equal to the critical angle of the wedge plate, and the reflection number increases after each “jump.”
The dark zone affects image quality and the brightness uniformity of the wedge-plate display.
The analysis and reduction of dark zone will be discussed in the next section.
Numerical Verification
Ray tracing can be performed by using TracPro. We build a dielectric-filled wedge plate, laying on the z-axis, with a 50 inch diagonal screen length, 5 inch wedge thickness, length of 62.25 inch, width of 43.578inch, and refractive index n=1.51872(BK7). The bottom surface produces 100% reflection so that rays can only leave the wedge from the upper side. Other surfaces produce 100% transmittance to prevent ghost image formation by Fresnel loss, while still retaining total internal reflection. Figure 5-2-3(a) shows an object with height h=1 placed in front of the wedge plate at distance d=10. Figure 5-2-5 shows the tracing
from ρ =−1 to ρ =1 (with normalization) for the field point h=0.0 at the meridional plane.
Figure 5-2-5 Simulation result of total aberration plots of the wedge-plate display
Dark zones obviously exist, as indicated by the dashed circle in Fig. 5-3-3. Equation (5-2-8)
shows that the aberration can be calculated from practical position to the ideal location for each ray, and Fig. 5-2-5 shows that the ray tracing was also extended to another two field points h=0.7, and 1.0. The simulation results show precise conformity to the analytical results. Comparing the top bend and the non-continuous kink-like feature caused by the dark zone in three field points with Fig. 5-2-4 shows that Eq. (5-2-8) is highly accurate.
Third-Order Aberration Coefficients
Without loss of generality, a Taylor series expansion can be made for the aberration along the meridional plane. The leading terms of the third-order aberration are listed below:
2 ....
where a0, a1, a2….are the corresponding coefficient terms listed in increasing order.
According to the method in last section, the curves of a1 to a9 terms where field points are 0, 0.7, and 1.0, are sketched as shown in Figs. 5-2-6(a)~5-2-6(i).
Figure 5-2-6 (a)-(i) Aberration plots of the first three-order terms of the wedge-plate display.
These graphs reveal that each term in Eq. (5-2-9) as a non-continuous reverse curve located at the sameρvalue as the top bend in Fig. 5-2-5. The a4ρ2 and a8ρ3 terms correspond to the traditional spherical aberration; the a7ρ2h term corresponds to coma; the a9ρh2 term corresponds to astigmatism; a4ρ2 and a8ρ3 terms correspond to distortion; the
h
a5ρ term belongs to second order aberration, and is difficult to refer to as regular aberration. Table 5-2-1 lists a few leading aberration coefficient items.
Table 5-2-1 Aberration coefficients
Figure 5-2-7 outlines the sum of the first three-order approximation of aberration for a 50-inch wedge plate, where the wedge parameters are the same as those in simulation. In this numerical example, the third-order aberration approximation corresponds well with the total aberration. This provides a useful guideline in designing the display. Referring to Fig. 5-2-7, two angular solutions with zero aberration corresponding to Case 1 (Fig. 5-2-3(a))
and Case 2 (Fig. 5-2-3(b)) would be found for each field point. The advantage of aberration analysis is that it can determine the best emitting angle for each field point of object.
Figure 5-2-7 Sum of the first three-order aberration.
Wedge Parameter Analysis
Adjusting wedge plate parameters can minimize aberration and optimize image quality. To this end, the aberration of the 50-inch wedge is deconstructed as a function of the wedge vertex angleθ and refractive index n for the same incident position v ρ=0, as shown in Figs.
5-2-8(a) and 5-2-9(a). The field point fp=0.7 was chosen in this representative case. The results indicate that it can be difficult to efficiently reduce aberration during the discussing range. This study assumes rays incident through entrance pupil center (ρ=0). As long as the field point fp is selected, the traced ray angle is also determined. This angle may have large discrepancy with the ideal incident angle, and cause a lot of aberration. It is hard to eliminate aberration using only the system parameters (refractive index n, wedge vertex angleθ , etc.). In another words, determining a good initial condition is a perquisite to v wedge optimization.
Figure 5-2-8 Aberration plots with variable n: (a) when ρ=0; (b) when ρ=0.18; (c) whenρ=0.65.
According to Fig. 5-2-4, when a ray emits from field point fp =0.7, less aberration will occur when rays pass through ρ=0.18 and ρ=0.65. The analysis for the refractive index n is reconfirmed with these two incident angles, as shown in Figs. 5-2-8(b) and 5-2-9(c). These results show that the aberration would equal zero if the refractive index n was about 1.535.
While n varies from 1.5 to 1.55, the aberration fluctuates smoothly in Fig. 5-2-8(b) and rapidly in Fig. 5-2-8(c). This shows that the two zero-aberration initial designs would have different refractive index tolerance, and the analysis in this study can help select the better one. The relation between aberration and the wedge vertex angle θ when rays emit from v fp=0.7 and pass through ρ=0.18 and ρ=0.65 can also be plotted by the same method.
Figures 5-2-9(b) and 5-2-9(c) show the results.
Figure 5-2-9 Aberration plots with variableθv: (a) when ρ=0; (b) when ρ=0.18; (c) whenρ=0.65.