Chapter 1 Introduction
1.5 Organization of the dissertation
The thesis is organized as follows. In Chapter 2, we describe the general
formations for imaging system and non-imaging system including the pupil function, optical transfer function for specific shaped pupil, illumination formation, color-difference function and Ètendue function for an asymmetric pupil. In Chapter 3, we investigate optical transfer functions for the specific-shaped apertures generated by illumination with a rectangular light pipe. In Chapter 4, a programmable apodizer for imaging quality enhancement in incoherent imaging system is developed and evaluated. In Chapter 5, we introduce a new approach for extending the depth of focus and improving the image quality with structured light in a conventional imaging system, such as photography, projector and microscopy, with the defocused, spherical and coma aberrations. In Chapter 6, we develop an analytical method of illuminance formation and color-difference for mixed-color LEDs in a rectangular light pipe. In Chapter 7, a dual-f-number illumination system and its application to DMD™
projection display is investigated. Finally we draw our conclusions and future works in Chapter 8.
Chapter 2
General formalism for imaging and non-imaging optical systems
2.1 Optical systems and their general structures: light pipes, camera, projector and microscopy
The analysis of an optical system requires either analytical or numerical computation to determine the image quality on the image plane, which is taken by light rays as they come from an object and pass through the optical system. The general optical system with an imaging subsystem and a non-imaging subsystem is as schematically diagrammed in Fig. 2-1. The non-imaging system provides a desired illumination distribution and high optical collection efficiency in an incoherent imaging system. The function of the condenser is to project the light source directly into the aperture stop of the imaging lenses so that the lens aperture has the same brightness as the light source. The function of the imaging lens system is to image a bright and uniform illuminated object on the image plane. The platform of our investigations for the conventionally well-known optical systems will contain light pipe, photography, projector and microscopy as shown in Fig. 2-2, Fig. 2-3, Fig. 2-4 and Fig 2-5, respectively.
In this chapter we will introduce the optical tools for image performance evaluation in the general optical system. The subjects contain the pupil function and the optical transfer functions for an incoherent imaging system, and the illumination formation, color-difference function and Ètendue function for a non-imaging system.
In additional discussions of various aspect of the subjects, the reader may want to consult any of the following references: about Fourier optics [1] and wave aberration [2] of imaging optics, about non-imaging optics [26], about radiometry and photometry [27] and about color science [28].
Figure 2-1 General schematic diagram of the optical system with an imaging subsystem and a non-imaging subsystem
Figure 2-2 (a) The photograph of the light pipe with circle and rectangular shapes [29] and (b) the illustration of an illumination system with a light source and a light pipe for a uniform light output [30].
Illumination system provides a desired light distribution that may have little or no relationship to light distribution from the light source. Figure (b) shows an illumination system with a filament in a reflector which can provides a uniform light distribution at the end of a square light pipe.
(a) (b) Light pipe
Figure 2-3 (a) The photograph of a DSLR camera with flash light [31], (b) cutaway of an Olympus E-30 DSLR camera [32], (c) cross-section view and introduction of a DSLR camera [32] and (d) the schematic diagram of a photographic system with a camera, an imaging system, and a flash light, a non-imaging system.
(a)
(b)
(c)
Figure 2-4 (a) The photograph of a projector- Optoma EP755 model [33], (b) cross-section view and introduction of a DLP projector, and (c) the schematic diagram of a projector system with a projection lens, an imaging system, and a light pipe illumination, a non-imaging system.
(a)
(c)
(b)
Figure 2-5 (a) The photograph of an optical microscope [34], (b) cross-section view and introduction of a microscopic illumination system [35], (c) cross-section view and introduction of the illumination light path and image-forming light path in microscope [35]. The schematic diagrams of a microscopic system with an imaging lens, an imaging system, and an illumination, a non-imaging system are for (d) a transparent sample and (e) a reflective sample.
(a)
(b) (c)
2.2 Pupil function formalism
When an imaging system is diffraction limited, the point spread function (PSF) centering on the imaging point is the Fraunhofer diffraction pattern of the exit pupil.
When there is a wavefront error in an imaging system, the exit pupil is illuminated by a perfect spherical wave, but that a phase-shifting plate exists in the aperture stop, thus deforming the wavefront passing the pupil. If the phase error at the point (x, y) in the exit pupil is represented by kW(x, y), where k=2π/λ and λ is the wavelength of the light. W(x, y) is the aberration function of the effective path-length error accumulated by that ray as it passes from the Gaussian reference sphere to the actual wavefront, the latter wavefront also being defined to intercept the optical axis in the exit pupil as illustrated in Fig. 2-6. Then the complex amplitude transmittance f(x, y) of the imaginary phase-shifting phase is given by
f
( )
x,y =T′( )
x,y exp[
ikW( )
x,y]
(2-1)The complex function f (x, y) is the generalized pupil function [1]. T′
( )
x,yrepresents the amplitude transmittance over the normalized pupil coordinate.
Figure 2-6 Illustration for defining the aberration function. [1]
The pupil function of an optical system with a circular symmetrical aperture is further given by [7]
( ) ( ) ( )
where the term of the summation is the wave aberrations with the coefficients ωαβ in the polynomial. For example, ω20 is the wave aberration of the defocus coefficient;
ω40 denotes the coefficient for spherical aberration, and ω31 denotes the coefficient for coma aberration. (x ,y ) are the normalized Cartesian coordinates, and k = 2π/λ, where λ is the wavelength of the light. Function T ′( )x,y in Eq. (2-1) represents the amplitude transmittance over the normalized pupil coordinate that is scaled and normalized to make the outer periphery the unit circle, x2 +y2 ≤ 1.
2.3 Optical transfer function
The optical transfer function (OTF) represents the ratio of image contrast to specimen contrast when plotted as a function of spatial frequency, taking into account the phase shift between positions occupied by the actual and ideal image [36]. The OTF describes either the spatial or angular variation as a function of either spatial or angular frequency. When the image is projected onto a flat plane, such as photographic film or a solid state detector, spatial frequency is the preferred domain, but when the image is referred to the lens alone, angular frequency is preferred. In general terms, the optical transfer function can be described as:
( f
x, f
y) MTF ( f
x, f
y) PTF ( f
x, f
y)
OTF = ⋅
(2-3)where
( f
x, f
y) OTF ( f
x, f
y)
MTF =
(2-4)( f
x, f
y) e
i (fx,fy)PTF =
−2⋅π⋅λ (2-5) and (fx, fy) are spatial frequency in the x-plane and y-plane, respectively. The OTF accounts for aberration. Its magnitude is known as the Modulation Transfer Function (MTF) and its phase is known as the Phase Transfer Function (PTF). Fig 2-7 shows the typical form the MTF and PTF curves of an imaging system can take and also show the most usual way in which these two parameters are plotted.Figure 2-7 Typical plot of MTF and PTF versus spatial frequency. [36]
The OTF is derived from the autocorrelation of the pupil function by using the Hopkins canonical coordinate [37] and is given by
( ) ( ) ( )
is the f-number of the imaging lens system, λ is the wavelength, and N is the number of cycles per unit length in the image plane. The value of F is equal to the effectivefocal length divided by D, where D is the diameter of the effective aperture stop and the effective focal length is determined by the optical magnification of the imaging lens.
2.4 Illumination formation and color difference
The equation of the luminous intensity of the light source with Lambertian characteristics is given as
( )
J cos , -90o 90oJ
JΩ = θ = 0 θ ≤θ ≤ (2-7) where JΩ is the luminous intensity (lumen sr-1 ) of a small incremental area of the source in a direction at an angle θ from normal to the surface, and J0 is the luminous intensity of the incremental area in the direction of normal. Thus, we can derive the illuminance distribution radiated from the Lambertian light source (i.e. LED) on an irradiated surface.
With reference to Fig. 2-8, we assume that the illuminance position on a surface is (x, y), length R is the distance from the light source to the incremental area, length L is the distance from the light source to the surface, and angle θ is the angle from normal to the incremental area.
Figure 2-8 Illustration of an LED light source radiating into a surface.
The angle θ can be substituted by L, x, y and R, and is given by
(
x y)
RLThe illuminance distribution on the irradiated surface is a function ofcos3θ , which can be expressed as a function of J0, L, x and y according to Eqs. (2-7) and (2-8)
To identify the uniformity on an irradiated surface, we introduce the ANSI light uniformity defined by [38]
( )
where the maximum deviation Ur+ or Ur- from the average of nine measurements is specified as a percentage of the average light output (9 measurement locations l= 1, 2, 3…9) using the measurement described in Fig. 2-9, and at the four corners (i.e. l= 10, 11, 12, 13) on the surface.
Figure 2-9 Measurement locations at the center of nine equal rectangles of a 100% exit plane on irradiant surface. The four corner points 10, 11, 12 and 13 are located at 10% of the distance from the corner itself to the center of the measurement location 5 [38].
To identify the color uniformity on an irradiated plane, we introduce the ANSI color uniformity defined by [28]
( ) (
' 0')
2 21where u0' and v0'are the average chromatic values of the nine measurements, and
'
u and 1 v are the value with the maximum deviation of the 13 measurements from the 1' average chromatic values u0' and v0' using the measurement described in Fig. 2-9.
Finally we can derive the total color difference between two color stimuli, each given in the terms of L*, a*, b*, in CIE 1976 (L*a*b*) –space by [28] object-color stimuli given by the spectral radiant power of one of the CIE standard illuminants, for example, D65, reflected into the observer’s eyes by the perfectly reflecting diffuser.
2.5 Ètendue function for an asymmetric pupil
Ètendue is an optical invariant of a light beam relative to the beam divergence and cross-section area in the illumination system [26]. It allows us to estimate the collection efficiency of the optical system. In this section, we develop the Ètendue model of the elliptic-shaped illumination-pupil system.
The formal definition of Ètendue is given by
∫ ∫
angle of the beam divergence in the angular coordinate system [26,39]. θ is the half angles of the illumination cone and φ is the polar angle from 0 to 2π. In the elliptic-shaped illumination-pupil system, θ is a function of φ , so the Ètendue equation is given by∫
The relationship between the half angle of the illumination cone and f-number is given by
where D is the diameter of aperture-stop, NA is the numerical aperture, and f is the effective focal length. In an elliptic-shaped illumination-pupil system, the illumination-pupil shape is elliptic and is expressed as the function of the polar angle φ,
where r is the radius of the ellipse of the elliptic-shaped illumination-pupil;
a b -a2 2
ε = is the numerical eccentricity of the ellipse; a and b are the major and minor semi-axis of the ellipse of the elliptic-shaped illumination-pupil and also described as the radii of aperture-stops in the tangential and sagittal axes, respectively.
By putting Eqs. (2-17) and (2-18) into Eq. (2-16), the Ètendue expression of the elliptic-shaped illumination-pupil system is given by
∫
The integration about φ can be carried out by resolving Eq. (2-19) into the terms [40],
cos )
Then, the integral term in Eq. (2-20) can be deduced by Mathematica® software [41]
as
By putting Eq. (2-21) into Eq. (2-19), the Ètendue formula of the elliptic-shaped illumination-pupil system is given by
( ) ( ) (
T)
SWhere
( )
= 2 are the f-numbers of the elliptic-shaped illumination-pupil in the tangential and sagittal axes, respectively. As an illustration, the variation of the Ètendue with the tangential f-numbers and the sagittal f-numbers is calculated if we assume the cross-section area of light flux A is unit and shown in Fig. 2-2.
Figure 2-10 Variation of the Ètendue with the tangential f-number, (f/#)-T, for a variety of the sagittal f-number, (f/#)-S, i.e., from 1.0 to 4.0, in the optical system.
Furthermore, the shape equation of the aperture-stops of the illumination system and the projection lens is given by
( )
Where DS and DT are the diameters of the aperture-stops in the sagittal and tangential axes, respectively. It should be noted that because the ±12°-tilt angle of DMD™, the (f/#)S should be limited by a value 2.4. On the contrast, (f/#)T is determined by what the optical specifications are required; for examples, the trade-offs among the brightness, the resolution performance and the system volume, and so on. On the
other hand, DT is determined by the designs of (f/#)T and effective focal length in the illumination system and projection lens.
Besides the collection efficiency, f-number also affects the image aberrations on the illuminated plane, for example, the spherical aberration is inversely proportional to (f-number)3 , the coma aberration is inversely proportional to (f-number)2 and the astigmatism or field curvature is inversely proportional to (f-number) in terms of third-order aberration theory. Larger aberrations lead to the blurred projected image on the edge of the illuminated plane, for example the film and light valve. In order to keep the uniformity both on the center and edge of the illumination plane with the larger aberrations, it would be better to enlarge the overfill on the projected illumination spot to prevent the color band and shadow on the edge of illumination plane. However, when the overfill could be enlarged, the collection efficiency may be lost on the illumination plane and, hence, the overall throughout in the optical system will be reduced. Therefore, both the collection efficiency and the aberration for the specific f-number have to be considered in the initial phase of illumination system design.
2.6 Criteria on performance: aberration and modulation transfer function Aberration is the departure and errors occurring in the resulting image which is not conformed to the real image in the optical system. Aberration leads to blurring of the image produced by an image-forming system. It occurs when light from one point of an object after transmission through the system does not converge into (or does not diverge from) a single point. There are the following five types of monochromatic aberration which are called the Seidel aberrations: Spherical aberration, Coma, Astigmatism difference, Curvature of field and Distortion. And, there are the following two types of chromatic aberration: Longitudinal chromatic
aberration and Lateral chromatic aberration [18]. In a real optical system, the above seven aberrations are mixed. The effect of those wavefront aberrations on image quality is a fairly complex subject. Any deviation in the form of the wavefront away from spherical causes degradation in the quality of point images, thus also the quality of image as a whole. The hard part is to define the specifics of this general fact.
Assessing the relation between optical aberrations and image quality requires not only the knowledge of the mechanism by which the aberrations influence the determinants of optical quality, but also establishing the universal criteria applicable to any and all the aberrations. Such criteria need to define specific aberration tolerances as a reference point in fabrication, evaluation and the use of geometrical optics and diffraction optics.
The effect of aberrations on image quality is best assessed with the modulation transfer function (MTF) of the optical system. For that reason, this one quality indicator is addressed in more details. The MTF is a measure of the transfer of modulation (or contrast) from the subject to the image. In other words, it measures how faithfully the lens reproduces (or transfers) detail from the object to the image produced by the lens. Firstly, we have to define the type of test pattern being used. As shown in Fig 2-11, signal frequency of test pattern can be equated to a periodic line grating with the number of spacings per unit interval referred to as the spatial frequency. A common reference unit for spatial frequency is the number of line pairs per millimeter. As an example, a continuous series of black and white line pairs with a spatial period measuring 1 micrometer per pair would repeat 1000 times every millimeter and therefore have a corresponding spatial frequency of 1000 lines per millimeter.
Figure 2-11 The illustration of the MTF and spatial frequency in a lens system. [42]
The formal definition of MTF:
( ) ( )
( Max Max Intensity Intensity Min Min Intensity Intensity )
f , f MTF
x y+
= −
(2-24)In Fig. 2-11, for the first pattern group, the MTF is 0.9. For the second pattern set, the MTF can be calculated to be 0.2. The point at which you can no longer see any variation in the image is the point at which the MTF is zero, and that's the definition of the "limiting resolution" of the lens in an optical system. In this case, the final pattern set with an MTF of 0.1 would be classified as "just resolved" by this lens. In addition to the type of test pattern, the light used for illumination and the type of detector used for recording also influence MTF. When describing the performance of a lens using MTF, we typically use a plot of MTF against "spatial frequency" as shown in Fig 2-12 (a). This shows three plots. The blue curve is a perfect (diffraction limited) aberration-free lens operating at f-number equal to 4 (i.e. f/4). The black curve is the same lens operating at f/8. At f/8 many lenses are close to diffraction limited in the center of the optical field, so you could obtain a trace like this in
practice. Note that the "resolution" at f/4, defined as the spatial frequency at which the MTF drops so low that you can't see any modulation in the image, is around 450 line-pair/mm (lp/mm), while that of the perfect f/8 lens is exactly a half of that at 225 lp/mm. For a perfect lens, the resolution as defined in these terms (MTF~0) is linearly related to the aperture and is given by: Resolution (spatial frequency @ MTF=0) = 1800/(f-number). The red curve shows what you would expect from an f/4 lens with 1/2 wavelength of wavefront error (not an unreasonable amount to find in a typical camera lens). As you can see at f/8 a lens may actually be better (i.e. have a higher MTF) than the same lens at f/4 with 1/2 wavelength of wavefront error, at least for the region from 0 to about 175 lp/mm. This is the basis of the old rule that lenses are often best when stopped down a couple of stops from maximum aperture. In Fig 2-12 (b), the red region represents an MTF less than zero. What this means is that black areas appear as white and white areas appear as black, but in reality it can not be resolved. Resolution above the point at which the MTF first reaches zero is known as spurious resolution.
(a)
(b)
Figure 2-12 (a) A plot of MTF against "spatial frequency" for different designs of f /4 and f /8 lens, (b) the effect of defocus on MTF for an f /2.8 lens. [43]
Here is an example of an actual test pattern of USAF 1951 US Air Force resolution test chart [44] conform to MIL-STD-150A in Fig 2-13 (a), and recorded on film or detector in Fig 2-13 (b). As you can see the modulation in the finer spaced patterns with higher spatial frequencies in Group-II gets gradually more and more difficult to be resolved compared to the low spatial frequencies in Group-I. The higher the MTF of the lens, the finer spaced pattern that will be visible.
Figure 2-13 (a) An image of USAF 1951 US Air Force resolution test chart [44], (b) A record of the test chart in a film or a detector through the lens in an optical system.
The advantage of using OTF is both to analyzing and specifying the performance of imaging systems. One can compute the intensity distribution and contrast of an image if the intensity distribution of its corresponding object is known.
The OTF of an imaging system can be calculated and simulated theoretically from the basic design data, so that it is feasible to design a lens system to meet OTF or MTF
The OTF of an imaging system can be calculated and simulated theoretically from the basic design data, so that it is feasible to design a lens system to meet OTF or MTF