Chapter 5 Extending the depth of field in conventional imaging system with
5.8 Image performance evaluation
To highlight the capability of our approach, we took a resolution pattern to explore the image performance in an incoherent imaging system. This pattern is 1951 USAF resolution test chart [44] conforms to MIL-STD-150A standard with resolution 600 (dpi) × 600 (dpi).
(a) (b)
Case 1: Defocus
Referring to Fig. 5-16, in column (a), one could see the images for the clear aperture, while in column (b) and (c), the images for the specifically shaped apertures with scale ratios K=0.05 and K=0.3 are shown, respectively. The images were generated by the multiplication of OTF in the Fourier domain using the convolution technique. Furthermore, the images with defocus coefficients of ω20= 5λ/π, ω20= 10λ/π, ω20= 15λ/π, and ω20= 20λ/π, are shown with lines (1)-(4) respectively. Comparing with the images for the specifically shaped apertures, the images of typical aperture show a clearer loss in contrast at high spatial frequencies with larger ω20. Especially for ω20≥10λ/π, there is a significant enhancement of the image resolution at high spatial frequency by the use of a specifically shaped aperture with K=0.3.
Figure 5-16 The computer-simulated images of resolution patterns for (a), a clear aperture, and (b), a specifically shaped aperture with the scale ratio K=0.05, and (c), a specifically shaped aperture with the scale ratio K=0.3, obtained with different defocus coefficients: (1)ω20= 5λ/π, (2) ω20= 10λ/π, (3) ω20= 15λ/π and (4) ω20= 20λ/π.
(a) (b)
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Case 2: Spherical aberration
Referring to Fig. 5-17, in column (a), one could see the images for the clear aperture, while in column (b) and (c), the images for the specifically shaped apertures with scale ratios K=0.05 and K=0.3 are shown, respectively. Furthermore, the
images with spherical-aberration coefficients of ω4 0 = 5λ/π, ω4 0 = 10λ/π, ω4 0 = 15λ/π, and ω4 0 = 20λ/π, are shown with lines (1)-(4),
respectively. Spherical aberration could make the image of a bright point source surrounded by a halo of light. The effect of spherical aberration on an extended image is to soften the contrast of the image and to blur its details with symmetrical distribution. Comparing with the images for the specifically shaped apertures, the images of the clear aperture show a clearer loss in contrast and a seriously blurred flare at all spatial frequency with larger ω40 even tough the three-bar charts for all spatial frequencies is resolved for the clear aperture. Especially for ω40≥10λ/π, there is a significant enhancement of the imaging contrast by the use of a specifically shaped aperture with K=0.3 and 0.05.
Figure 5-17 The computer-simulated images of resolution patterns for (a), a clear aperture, and (b), a specifically shaped aperture with the scale ratio K=0.05, and (c), a specifically shaped aperture with the scale ratio K=0.3, obtained with different spherical aberration coefficients: (1)ω40= 5λ/π, (2) ω40= 10λ/π, (3) ω40= 15λ/π and (4) ω40= 20λ/π.
(a) (b)
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Case 3: Coma aberration
In order to obviously show the effect of coma aberration by the use of a suitable test chart, we utilized a concentric-circles pattern with resolution 96 (dpi) × 96 (dpi) [60]. Referring to Fig. 5-18, in column (a), one could see the images for the clear aperture, while in column (b) and (c), the images for the specifically shaped apertures with scale ratios K=0.05 and K=0.3 are shown, respectively. Furthermore,
the images with coma-aberration coefficients of ω31= 5λ/π, ω31 = 10λ/π, ω31 = 15λ/π, and ω31 = 20λ/π, are shown with lines (1)-(4)
respectively. Coma aberration could make the image of a point source spread out into a comet-shaped flare with the non-symmetrical distribution. Comparing with the images for the specifically shaped apertures, the images of the clear aperture show a seriously blurred flare at all spatial frequency with larger ω31 along the vertical direction. Especially for ω31≥ 10λ/π, there is a significant enhancement of the imaging resolution by the use of a specifically shaped aperture with K=0.3 and 0.05.
Figure 5-18 The computer-simulated images of resolution patterns for (a), a clear aperture, and (b), a specifically shaped aperture with the scale ratio K=0.05, and (c), a specifically shaped aperture with the scale ratio K=0.3, obtained with different coma aberration coefficients: (1) ω31= 5λ/π, (2) ω31= 10λ/π, (3) ω31= 15λ/π and (4) ω31= 20λ/π.
(a) (b)
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Case 4: Combined aberration (defocus, Spherical aberration, and Coma)
Referring to Fig. 5-19, in column (a), one could see the images for the clear aperture, while in column (b) and (c), the images for the specifically shaped apertures with scale ratios K=0.05 and K=0.3 are shown, respectively. Furthermore, the images with the aberration coefficients of -ω20 = ω4 0 = ω31 = 5λ/π, 10λ/π, and 20λ/π including defocus, spherical and come aberrations are shown with lines (1)-(3), respectively. Comparing with the images for the specifically shaped apertures, the images of typical aperture show a significant loss in contrast at all spatial frequency.
Especially for the aberration coefficients ≥5λ/π, there is a significant enhancement of the image resolution at all spatial frequency by the use of a specifically shaped aperture with K=0.05 and K=0.3.
Figure 5-19 The computer-simulated images of resolution patterns for (a), a clear aperture, and (b), a specifically shaped aperture with the scale ratio K=0.05, and (c), a specifically shaped aperture with the scale ratio K=0.3, obtained with different defocus coefficients ω20 , different spherical aberration coefficients ω41 , and different coma aberration coefficients ω31 , when -ω20= ω40 = ω31 = (1) 5λ/π, (2) 10λ/π, and (3) 20λ/π.
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Case 5: the influence of fill factor
We take a resolution pattern to simulate the imaging performance for this specific shaped aperture with the scale ratio a/D =0.3 and different fill factors ranged from 100% to 50% in a defocus system with the defocused coefficient ω20= 10λ/π as shown in Fig. 5-20. Comparing with the images in the red frames shown in the figures, the image of the specific shaped aperture with the fill factor 60% shows a higher resolution at the middle spatial frequencies. Hence, it is evident that the image quality at specific spatial frequency will be improved once the specific shaped aperture is implemented.
Figure 5-20 The computer-simulated images of resolution patterns for a specific shaped aperture with the scale ratio a/D =0.3 and different fill factors (a) 100%, (b) 90%, (c) 80%, (d) 70%, (e) 60%
and (f) 50%, in a defocus system with the defocused coefficient ω20= 10λ/π.
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(c) (d)
(e) (f)
5.9 Summary and remarks
We have provided a new approach for improving the image quality for the incoherent imaging systems, including photography, projector and microscopy, with a specific illuminator modulator. The semi-analytical results using the optical transfer function (OTF) indicated that the depth of focus can be extended with specifically shaped illumination, which is generated by a digital micromirror device and LED array, on the aperture stop in the imaging system with a specific defocus coefficient, and the specific coefficients for spherical aberration and coma aberration.
In summary, (1) The limiting resolution of a defocused imaging system with a specific defocus coefficient can be improved by its corresponding binary shaped pupil.
It has been shown that a shaped pupil with a scale ratio K equal to 0.05 is more helpful for extending the depth of focus at low spatial frequency, while a shaped pupil with a scale ratio K equal to 0.3 is more useful for extending the depth of focus at high spatial frequency. (2) In an imaging system with the coefficient for spherical aberration or coma aberration, respectively, the OTF for the shaped pupil with a scale ratio K of less than 0.3 is significantly larger than that for the clear aperture at all spatial frequencies. Especially for ω40≥10λ/π and ω31≥10λ/π, there is a significant enhancement of the imaging contrast by the use of a specifically shaped aperture with K=0.3 and 0.05 according to the computer-simulated image. (3) the OTFs of a specifically shaped aperture with K=0.05 and K= 0.3 are greater than the OTFs of a clear aperture especially when the aberration coefficients ≥5λ/π. It is evident that the image quality will be enhanced as the specifically shaped aperture is used, especially for the imaging system with large aberration coefficients including defocus, spherical and come aberrations. (4) The image quality could not be significantly influenced as the fill factor (i.e. aperture ratio) of the spatially shaped pupil varied from 100% to 80%.
In the point of view of optical design, the pupil plane at the aperture stop in the imaging subsystem should be designed to be a conjugate at the other pupil plane on the digital micromirror device for the non-imaging subsystem. The pupil matching between the imaging unit and the non-imaging one could significantly affect its practical implementation and hence the matching sets one limitation on performance.
Overall, the pupil matching will be determined by the precise alignment between the digital micromirror device and the lens elements in the whole optical system.
Overall, the proposed approach of shaped pupil from an illumination modulator is a dynamically programmable method to achieve aberration compensation for imaging applications, such as photography, projector and microscopy. This method provides a connection between non-imaging and imaging systems for enhancing the image quality. Regarding the pupil aberration is considered in the illumination system, this influence may be incorporated with the corresponding spherical aberration and even off-axis coma with different levels of coefficients.
Hence, practically, the pupil aberration could be included partially and further explored once the spherical aberration and coma are included. It is worth noting that this proposed model can rapidly and field-sequentially generate a specifically shaped pupil with 10 gray levels within the very short image processing time of 0.22 ms in the case of K =0.05. Different spatial frequencies represent different imaging information from an object in photography or a sample in microscopy or a light valve in a projector. High spatial frequencies represent sharp spatial changes in the image, such as edges, and generally correspond to local information and fine detail, while the portion of low spatial frequencies represent global information about the shape, such as general proportion and orientation. On the other hand, this programmable structured light can be substituted for the conventional flash light and illumination light in the conventional incoherent imaging system, in order to extend the depth of
focus (EDOF), and for dynamically providing improved imaging quality for many varied scenes combined with the technology of high dynamic range (HDR) imaging.
For instance, by the use of this dynamically programmable structured light with EDOF and HDR technologies, we can capture an sense-of-depth-enhanced image by combining a sub-image with global information (i.e., low spatial frequencies) within first half period of exposure time and another sub-image with local information (i.e., high spatial frequencies) within the other half period of exposure time, which have different shaped pupils with specific scale ratios K.
Chapter 6
Illuminance formation and color-difference of mixed-color LEDs in a rectangular light pipe: an analytical approach
6.1 Introduction
In this section we developed an analytical method of illuminance formation and color-difference for mixing colored LEDs in a rectangular light pipe using the methods of non-imaging optics [26], photometry [27], and colorimetry [28]. The different illuminance formations and color-differences studied in this chapter are the types in which ANSI light uniformity [38] and ANSI color uniformity [38] vary with the different geometric structures of light pipes. For the purpose of comparison, this chapter also includes the formation of illuminance distribution for a single LED source using the same analytical method.
In order to obtain a convenient form of the illuminance formation, we employed the hollow straight light pipe with a perfect reflectivity of 100% for the calculations in this chapter. In reality, the finite absorption loss is inevitable. With more times of reflection, the flux of the ray will decrease, which can affect the illuminance distribution. This influence has been explored by Cheng and Chern [19]. For a common coating with a reflectivity of 90%, the difference of the uniformity deviation from that of 100% reflectivity is less than 1% at the length scale L/A equal and greater than unity, where L is the length of the light pipe and A is a constant which is a geometric parameter for the scale unit of the light pipe’s input face. To have a best
approximation for the analytical evaluation, we make another assumption that the LED is a point light source with Lambertian characteristics. In most applications this means that the dimension of LED source should be much less than the cross-sectional dimension of the light pipe. The considerations of a hollow straight light pipe with a perfect reflectivity and a point light source may limit this study, but it will provide an analytical base for the connection between non-imaging and coloring in order to obtain a “uniform” source for illumination applications.
The remainder of this section is organized as follows. In Section 2 we revisit on the chromatic issue and LED RGB color mixing. In Section 3, we describe the light pipe illumination system. In Section 4, we derive illumination formation and function of color difference. In Section 5 we analyze the non-imaging performance for a rectangular light pipe system with mixed-color LEDs. Finally we draw our summary in Section 6.
6.2 Chromatic issue and LED RGB color mixing
Today’s light emitting diode (LED) technology is widely applied in vehicles, architecture, signal lighting, backlighting and in projection microdisplays [20]. Most of these applications require the shaping of a uniform beam illuminance profile, managing color quality and saving power consumption while maintaining high luminous efficiency in the illumination systems. There are two kinds of approaches to generate white light with LEDs. One is the phosphor-converted white LED which provides a compact integrated package but has a relatively lower luminous efficiency.
The other is the mixed-color LED which provides more light throughput compared to a single phosphor-converted white LED with the same operating power. In practice, however, there are several technical challenges to creating a mixed-color LED, such as white light homogenizing with the acceptably lowest spatial variation, color mixing
and color balancing with acceptably lowest chromatic variation.
Rectangular light pipes are commonly optical devices that manage light properties in illumination systems, especially where extremely uniform illuminations with specific illuminance distributions are required [13]. Typical applications are the illuminations in the projection display [22], lithography [15], endoscopes [16] and in optical waveguides [17]. A rectangular light pipe is made with parallel reflective sides with a square or rectangular cross section. The light source can be located on one end of the light pipe and the other end is then the uniformly illuminated plane [18]. The shape of the light pipe can modify the original characteristics of the spatial distribution of the light source but not the angular distribution. The uniform illumination on the exit end of the light pipe is determined by the ratio of the length to the cross-sectional dimension of the light pipe [19].
In the literature, the mention of the properties of light pipes is mostly concentrated on transmittance, flux analysis and irradiance formation for the single light source. Although Derlofske and Hough developed a flux confinement diagram model to discuss the flux propagation of square light pipes and angular distribution [21], and Cheng and Chern developed a semi-analytical method to investigate the formation of irradiance distribution, [19] to the best of our knowledge few articles have investigated the formation of illuminance and the color-differences in mixed-color light sources. An earlier investigation of the use of mixed-color LEDs in a light pipe was done by Zhao et al., and it comprised two parts: a computer simulation using a commercial ray tracing software package with Monte Carlo algorithm; and an experimental study verifying the results obtained from the simulation [30].
6.3 Configuration of the light pipe illumination system
A schematic sketch of an optical system using a rectangular light pipe is shown in Fig.
6-1. The optical system consists of an LED light source with Lambertian angular distribution and a rectangular light pipe, where a and b are the width and the height of the cross section of the light pipe, respectively, L is the length of the light pipe, c is the size of the square light source and n is the optical index of the light pipe’s material.
The LED light source is located at the entrance of the light pipe, and the light source has an angular distribution extending from +90° to -90°. The exit is uniformly illuminated by the multiple reflections throughout the light pipe.
Figure 6-1 Schematic diagram and dimension of the light pipe and the single LED light source.
The light pipe is made with parallel reflective sides with a rectangular cross section. The virtual image of the light at the entrance of the light pipe has the checkerboard-array-shaped light distribution which results from the multiple reflections of the light source through the pipe as shown in Fig. 6-2. The spaces between each adjacent light spot are: a on the vertical axis and b on the horizontal axis, which are equal to the width and height of the cross section of the light pipe, respectively. And, the size of each light spot is equal to that of the light source. The radiant intensity of each light spot is denoted as J’i,j as shown in Fig. 6-2 (b). The subscripts m and n stand for the reflective times on the horizontal axis and vertical
axis, respectively, and the plus and minus values of i and j denote the opposite directions of the light source. For example, J’1,2 means that the luminous flux of the light spot is reflected one time on the horizontal axis and two times on the vertical axis and its location is shown in Fig. 6-2 (b).
(a) (b)
Figure 6-2 (a) Principal of the operation of a light pipe. (b) Virtual image at the entrance of the light pipe. The light pipe is made with parallel reflective sides with a rectangular cross section. The multiple reflections of the light source through the pipe can produce a spatial checkerboard-array-shaped light distribution.
The schematic sketch of one specific light pipe optical system with mixed-color LEDs is illustrated in Fig. 6-3 (a). The optical system consists of four colored LED light sources, including one red-colored LED, two green-colored LEDs and one blue-colored LED, and a rectangular light pipe, where a and b are the width and the height of the cross section of the light pipe, respectively, where L is the length of the light pipe, c is the side of the square light source, and n is the optical index of the light pipe material. The light sources are located at the entrance of the light pipe as shown in Fig. 6-3 (b), where the red LED is located at the coordinate (-P, Q), the first green
LED is located at (P, Q), the second green LED is located at (-P, -Q) and the blue LED is located at (+P, -Q), respectively, assuming that the coordinate of the center for the entrance of the light pipe is (0, 0).
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(b)
Figure 6-3 (a) Schematic diagram and dimension of the light pipe. (b) Locations of red, green and blue LED light sources, i.e. mixed-color LEDs, on the entrance of light pipe.
6.4 Optical computation for illumination formation and function of color difference
The illuminance distribution on the end plane of the light pipe is referring to Eq. (2-9) ands given by
( ) (
2 2 2)
22 2 0
3
y x L J L L
J cos y)
H(x, = θ θ = × + + (6-1)
Then, we determine the individual contribution of illuminance by the virtual light spots on the entrance plane of the light pipe. The illuminance H0,0is radiated without
any reflection through the light pipe about x from –a/2 to +a/2 and about y from –b/2 and +b/2 as shown in Fig. 6-4.
Figure 6-4 Illustration of a Lambertian light source radiating into the exit plane of the light pipe for the different virtual light spot on the entrance plane of the light pipe
and as given by
Where the values a and b are the width and the height of the cross section of the light pipe, respectively. Also, we can derive the illuminances H’1,0 and H’2,0 that are radiated with one time reflection and a two times reflection , respectively, through the light pipe along the +x axis direction, as shown in Fig. 5 . They are given by
(
L xL y)
, 2a x 3a2 ,- 2b y 2b. derive the practical illuminance distribution, which is radiated from the individual virtual light spot on the entrance plane of the light pipe, on the exit plane of the lightpipe about x from –a/2 to +a/2 and about y from –b/2 and +b/2 using the mirror mapping method according to Eq. (6) and Eq. (7) as given by
( )
Then, we can extend the expression of the illuminance for each virtual light spot on the exit plane of the light pipe with one time reflection and a two times reflection through the light pipe along the ±x axes and the ±y axes directions about x from –a/2 to +a/2 and about y from –b/2 and +b/2, respectively as follows,
( ) ( )
Finally we can summarize the expression of the total illuminance for each virtual light
Finally we can summarize the expression of the total illuminance for each virtual light