Chapter 2 General formalism for imaging and non-imaging optical systems
2.6 Criteria on performance: aberration and modulation transfer function
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 requirements with a desired image quality. The OTF of an entire system, such as a digital camera system can be predicted by computing the product of the OTFs of the individual sub-units in the system. The digital camera system consists of a lens, a CCD, an electronic circuit and a display, as shown in Fig 2-14. The MTF of an entire system at a spatial frequency s is given by
Group-II Group-I
(a) (b)
( )
s MTF( )
k s MTF( )
k s MTF( )
k s MTF( )
k sMTFSystem = Lens 1 × CCD 2 × Electronic 3 × Display 4
(2-25) Where MTFLens(k1s) etc are the MTFs of the sub-units in the system and k1 etc are the constants that relate to the spatial frequency at that sub-unit with spatial frequency s of the entire system. The PTF of the entire system is given by
( )
s PTF( )
k s PTF( )
k s PTF( )
k s PTF( )
k s TFP System = Lens 1 + CCD 2 + Electronic 3 + Display 4
(2-26)
Figure 2-14 Schematic diagram of MTFs in an entire system
The MTF of a system by multiplying together the MTFs of the sub-units is very valuable in optical design. The designer can optimize the performance of each sub-unit individually and then match with each other for the final image quality of the entire system.
An important point of note is that an optical system, such as a photography or a microscopy consists of not only a lens system (an imaging system) but also an illumination system (a non-imaging system), but unfortunately MTF of an illumination light was generally assumed to be equal to unity in optical design. So, MTFs of a non-imaging system could be an effective factor for the final performance of an optical system. This issue will be investigated in this thesis as follows.
Chapter 3
Optical transfer functions for the specific-shaped apertures generated by illumination with a
rectangular light pipe
3.1 Introduction
In this chapter, we investigate, both analytically and numerically, the imaging properties of the optical system with specific shaped apertures produced by the Lambertian illumination with a rectangular light pipe by calculating the OTF using the Hopkins method [37]. The different shaped apertures studied here are the types in which the amplitude transmittances vary with the different geometric structures of light pipes from the center of the pupil toward its rim. In order to do an image evaluation, the purpose of this study is to calculate the OTF of the aberration-free optical system (i.e. in-focus system and a diffraction-limited system) and the defocused optical system. For the purpose of comparison, this chapter also includes the OTFs of these two optical systems with a clear aperture as carried out by Hopkins [37].
The remainder of this chapter is organized as follows. In Section 2 we revisit on light pipes and light guides. In Section 3, we describe the optical projection system with a light pipe. In Section 4, we derive the pupil functions of different geometric structures of light pipes. In Section 5 we derive and analyze the corresponding OTFs and analyze the image quality in an aberration-free optical system and in a defocused optical system, respectively. Finally we draw our summary in Section 6.
3.2 Revisit on light pipes and light guides: academic interest and current trends A light pipe is a commonly optical element that manages light properties in illumination systems, especially where extremely uniform illumination with specific illuminance distribution is required. Typical applications are in the illumination engines in the projectors, lithography, endoscopes and in optical waveguides. In those applications, the illumination system is mostly a combination of a light pipe and the corresponding imaging system with a projection lens. A light pipe is made with parallel reflective sides, either as a cylinder or 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 uniform illumination on the exit end of the light pipe is determined by the ratio between the length to the diameter and also the cross-sectional shape of the light pipe [19]. The multiple reflections of the light source through the pipe can produce a spatially checkerboard-array-shaped light distribution at the aperture stop of the illumination system which is then projected to the aperture stop of the corresponding imaging system. Hence, we need to determine the pupil function with a specific shaped aperture generated by the light pipe in the optical transfer function (OTF) for the image evaluation in the optical system. The earlier investigations of the use of shaped apertures are all in annular apodizers [3], radial Walsh filters [5], multiple mirror telescopes [45], and the diffraction by an aperture with central obstruction [46], and non-uniform amplitudes [6,7].
There are two kinds of the light pipes in term of the properties of the materials:
hollow light pipe and solid light pipe. A hollow light pipe is made with parallel mirrors where the sidewalls join together and where coatings have the reflectivity of less than 100 and angular/color dependence. A solid light pipe is made of the dielectric material with the addition of the refraction, Fresnel loss, material absorption and a total internal reflection. Additionally, such a dielectric-filled light pipe brings a
factor of an index through Snell's law into the treatment of the angular profile and requires a longer length to obtain a given uniformity [13]. In order to have a convenient form of the pupil function, we employed the hollow straight light pipe in the calculation of the OTF in this chapter. The consideration of a hollow straight light pipe may be limited, but the study will provide a semi-analytical base of the connection between the non-imaging and the imaging in order to have a “uniform”
source for imaging applications.
3.3 Configuration of the optical system
The schematic sketch of an optical projection system using a rectangular light pipe is illustrated in Fig. 3-1 showing the relationships between the pupils and fields. The system consists of the following subsystems: a light source with Lambertian angular distribution, an illumination system using a rectangular light pipe, a light valve and a projection system.
Figure 3-1 Schematic diagrams of the optical projection system with the light pipe and light-valve to illustrate the relationship between pupils and fields.
The Lambertian 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 through the light pipe. The uniform illuminance on the exit plane of the light pipe is transferred to its conjugate imaging plane on the light valve. The multiple reflections of the light source through the pipe can produce a spatial checkerboard-array-shaped light distribution that is a virtual image at the entrance of the light pipe. This virtual image is imaged onto the aperture stop of the illumination system by lens #1. Then, the image at the pupil of the illumination system is conjugately imaged onto the aperture stop of the corresponding projection lens system by lens #2 in the illumination system and lens #3 in the projection system. Finally, the image on the light valve is projected onto the screen by the projection lenses lens #3 and lens #4.
The dimension of the light pipe and the light source is shown in Fig. 3-2, 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 and c is the size of the square light source.
Figure 3-2 Schematic diagrams and Dimension of the light pipe and LED source.
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 that is the virtual image of the light at the entrance of the light pipe as shown in Fig. 3-3 (a). 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 luminous flux of each light spot is noted as Fm,n as shown in Fig. 3-3 (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 m and n denote the opposite directions of the light source.
For example, F1,2 means 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.
3-3 (b).
(a) (b)
Figure 3-3 (a) Principle operation of light pipe. (b) Image at the aperture stop in the optical system.
The light pipe is made with the parallel reflective sides with the rectangular cross section. The multiple reflections of the light source through the pipe can produce the spatially checkerboard-array-shaped light distribution, a virtual image at the entrance of the light pipe.
3.4 Optical computation for pupil function
The equation of the luminous exitance from the light source is assumed as one pulse function shown in Fig. 3-4.
-c/2
Figure 3-4 LED spatial light-intensity distribution’s chart.
and expressed as,
where c is the size of the square light source, B0 is the luminous exitance (lumen cm-2) of the uniform light source, and H(x+c/2), H(x-c/2), H(y+c/2) and H(y-c/2) are the step functions. The luminous intensity of the light source with Lambertian characteristics is given by
( )
I cos , -90o 90o IIΩ = θ = 0 θ ≤θ ≤ (3-2) where IΩ is the luminous intensity (lumen ster-1 ) of a small incremental area of the source in a direction at an angle θ from normal to the surface, and I0 is the luminous intensity of the incremental area in the direction of normal. Then, we can derive the luminous flux (lumen) radiated from the Lambertian light source into the exit end of the light pipe.
With reference to Fig. 3-5, we assume that the incremental rectangular area dA on the end plane of the light pipe is dxdy, length R is the distance from the light source to the incremental area, length L is the distance from the light source to the end plane of the light pipe, and angle θ is the angle from normal to the incremental area.
The values a and b are the width and the height of the cross section of the light pipe, respectively.
Figure 3-5 Geometry of a LED source radiating into the exit plane of the light pipe.
The angle θ can be substituted by L, x, y and R, and is given by
The luminous flux intercepted by the incremental area is the product of the luminous intensity of the light source and the solid angle, which can be expressed as the function of I0, L, x and y according to Eqs. (3-2) and (3-3) as given by
Then, we determine the individual luminous fluxes of the virtual light spots on the entrance plane of the light pipe. The luminous flux F0,0of the virtual light spot on the entrance plane of the light pipe is equivalent to the luminous flux F’0,0on the exit end of the light pipe, which is radiated without any reflection through the light pipe as shown in Fig. 3-6. We integrate Eq. (3-4) about x from –b/2 to +b/2 and about y
Also, we can derive the powers F0,1 and F0,2 that are radiated with one time reflection and a two times reflection through the light pipe, respectively, as shown in Fig. 3-6.
Figure 3-6 Illustration of a LED 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 they are given by where ρ is the reflectivity and the exponent of ρ is the reflective time. Finally we can then summarize the expression of the luminous flux for each virtual light spot on the entrance plane of the light pipe as follows,
( )
where |m| and |n| stand for the reflective times through the light pipe on the horizontal and vertical axis, respectively.
In order to have a convenient form for the numerical evaluation, we make the
reasonable assumption, 1
. In most application cases that means that
the dimension of the cross section of the light pipe should be much less than the
length of the light pipe. In this case, it is certain that the Lambertian light source is uniformly transferred on the exit plane of the light pipe [19]. The numerical aperture N.A. can be generally assumed around 0.5 as a minimum (i.e., the corresponding f-number is equal to unity) because of “Ètendue limited” in most of projection systems [39] and reducing an optical system to the commercial practice. Based on the assumptions, the term in the integration in Eq. (3-8) can be approximately derived as the following expression,
( )
Then, the integration of Eq. (3-8) can be carried out by substituting Eq. (3-9). Finally, the expression of the luminous flux for each virtual light spot on the entrance plane of the light pipe is given by
( )
nonlinearly decreased with the different reflective times m and n through the light pipe because of the characteristics of the angular profile for luminous intensity of a Lambertian light source.The pupil function of a defocused optical system with a circular aperture is referring to Eq. (2-2) and given by [6]
( ) ( ) [ ( ) ]
where ω20 is the wave aberration of the defocused coefficient, (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. (3-11) represents the amplitude distribution over the pupil.
The amplitude transmittance T(x, y) has a circular symmetrical pupil coordinate that is scaled and normalized to make the outer periphery of the unit circle, x2 +y2 ≤ 1.
The amplitude transmittance T(x, y) has a circular symmetrical pupil coordinate that is scaled and normalized to make the outer periphery of the unit circle, x2 +y2 ≤ 1.