Programmable apodizer in incoherent imaging
systems using a digital micromirror device
Chu-Ming Cheng Jyh-Long Chern
National Chiao Tung University Institute of Electro-Optical Engineering Department of Photonics
Microelectronics and Information System Research Center
1001 University Road Hsinchu, 300 Taiwan
E-mail: [email protected]
Abstract. We propose a programmable apodizer using a digital micro-mirror device and the total-internal-reflection prism subsystem for inco-herent imaging systems. It is shown that the proposed programmable apodizer can extend the depth of focus with the specific shaped aperture generated by the digital micromirror device. With a scale ratio of K 艋0.05, one can achieve almost the same level of imaging quality as provided by the conventional annular apodizer, where K represents the ratio between the integer multiple of the micromirror’s square pixel size and the diameter of the effective aperture stop. © 2010 Society of Photo-Optical
Instrumentation Engineers. 关DOI: 10.1117/1.3314306兴
Subject terms: apodizer; optical transfer function; incoherent imaging system. Paper 090331R received May 8, 2009; revised manuscript received Oct. 28, 2009; accepted for publication Dec. 18, 2009; published online Feb. 23, 2010.
1 Introduction
The current consumer applications of optical instruments and equipment demand high imaging quality, optical effi-ciency, and high resolution with the volume of the machine, nevertheless, being compact. At the same time, extending the depth of focus共EDOF兲 in an imaging system has been a long-standing issue in optical designs. Enhancing the quality of an image can be achieved and determined not only by the pupil function but also by its amplitude transmittance.1 Nonuniform amplitude transmission filters can be employed to vary the response of an optical imaging system, for instance, to increase the focal depth and to de-crease the influence of spherical aberration. Earlier EDOF investigations and experiments were carried out on annular apodizers,2,3 the radial Walsh filter,4 nonuniform-shaped apertures,1,5 and wavefront coding6,7 in imaging systems, where the nature of light is incoherent. However, none of those are programmable for the amplitude transmission at the aperture stop. From the point of view of potential ap-plications as well as from a purely academic interest per-spective, it is worthwhile to explore the possibility of how to realize a programmable apodizer for incoherent imaging systems.
In the literature, amplitude-transmitting filters for apodizing and hyperresolving can be implemented with a programmable liquid-crystal spatial light modulator operat-ing in a transmission-only mode in a coherent imagoperat-ing sys-tem with the laser light source, polarizers, and quarter-wavelength plates.8,9 In this paper, we proposed a programmable apodizer using the digital micromirror de-vice 共DMD™; Texas Instrument, Dallas, Texas兲10 and the total-internal-reflection 共TIR兲 prism subsystem in a polarization-free mode in an incoherent imaging system. We evaluated the imaging properties of the incoherent im-aging system with a specific shaped aperture generated by DMD by calculating the optical transfer function 共OTF兲 using the Hopkins method.11We also included the OTF of
the specific shaped aperture for the conventional annular apodizer, which has been demonstrated, both theoretically and experimentally, by Mino and Okano1 to show that the proposed programmable apodizer can not only extend the depth of focus but can also achieve almost the same level of imaging quality as the conventional annular apodizer in an incoherent imaging system.
The remainder of this paper is organized as follows. In Section 2, the configuration of the proposed system, which consists of a DMD and a TIR prism subsystem is illus-trated. In Section 3, we derive the pupil functions of the differently shaped apertures, which are generated by the DMD. Then, in Section 4, we calculate the OTF in such an incoherent imaging system. Furthermore, the corresponding OTF is evaluated and then to identify the imaging perfor-mance for a system of perfect imaging共aberration free兲 as well as the defocused one in Section 5. Finally, the conclu-sions are given in Section 6.
2 Configuration of Optical System
The schematic sketch of an incoherent imaging system us-ing one DMD and a charge-coupled device 共CCD兲 imager is illustrated in Fig.1. The system is formed by an image-taking lens module and a prism module. By following the paths of the axial rays as indicated by the solid lines in Fig.
0091-3286/2010/$25.00 © 2010 SPIE Object (Object plane) Lens #1 TIR prism DMD™ (Aperture stop) CCD (Image plane) On-state light Off-state light Lens #2
Fig. 1 Schematic diagram of the incoherent imaging system with
1, the rays starting from a point in the object pass through lens 1 and a prism module. The size of the axial cone of energy from the object is limited by the aperture stop on the DMD. The DMD consists of hundreds of thousands of moving micromirrors that are made to rotate to either +12 or −12 opositions depending on the binary state, i.e., on-state or off-on-state, of the underlying complementary metal oxide semiconductor synchronized dynamic random access memory cells below each micromirror.10
The DMD array size is 1024⫻768, and the pixel micro-mirrors measure ⬃13.7m2 to form a matrix having a high fill factor of⬃90%. The prism system comprises two transparent prisms, with an air gap between them. TIR at the interface between the prism and the air gap is utilized to separate the rays by their angle. The TIR prism has been applied into the DMD-based projection display in practice.12 The prism system can guide the rays onto and away from the DMD, simultaneously. The rays indicated by the dotted lines in Fig.1 from the object are imaged and focused onto the CCD by lens 2 when the configuration of the DMD is the on state. When the configuration of the DMD is the offstate, the rays indicated by the dashed lines in Fig.1are steered away in the opposite direction and the rays from the object are not imaged on the CCD. The DMD performs a spatial light modulation to rapidly generate a specific shaped aperture with either uniform or nonuniform illumination distribution at the aperture stop in an imaging system within the limited exposure time. This digital mi-cromirror device can provide a programmable apodizer with a specific binary transmission for the incoherent im-aging system. This implementation is not limited by this practical device. The TIR prism performs light separation to manage the illuminations and also make the normal vec-tors of the object, aperture stop, and image planes, respec-tively, coincide with the optical axis of the optical imaging system with the most compact volume.
3 Calculation of Pupil Functions
The pupil function of a defocused optical system with a circular symmetrical aperture is given by1
f共x,y兲 = T
⬘
共x,y兲expik20共x2+ y2兲 x2+ y2艋 1=0 x2+ y2⬎ 1, 共1兲
where20 is the wave aberration of the defocused coeffi-cient,共x,y兲 are the normalized Cartesian coordinates, and k = 2/, where is the wavelength of the light. Function T
⬘
共x,y兲 in Eq.共1兲represents the binary amplitude distribu-tion over the normalized pupil coordinate that is scaled and normalized to make the outer periphery the unit circle, x2 + y2艋1. The binary amplitude transmittance T⬘
共x,y兲 is generated by the DMD as shown in Fig.2. We can derive the amplitude transmittance of the shaped aperture T⬘
共x,y兲 in an on-state configuration as follows:T
⬘
共x,y兲 = E⬘
共x,y兲丢兺
m兺
n T共x,y兲␦冉
x −2mc D冊
␦冉
y − 2nc D冊
共2兲 0艋 兩m兩,兩n兩 艋 Int冋
D/c − 1 2册
, 共3兲where 丢 represents the convolution operation. T共x,y兲=1 −共x2+ y2兲 denotes the amplitude transmittance with a con-tinuous profile at the aperture stop which can extend the focal depth in the imaging system with a conventional an-nular apodizer.1D is the corresponding diameter of the ef-fective aperture stop. c represents the width of each square individual aperture generated by DMD in the pupil plane, which is equal to an integer multiple of the value d, with d being the width of each square pixel in the DMD. ␦关x −共2mc/D兲兴␦关y−共2nc/D兲兴 denotes the␦ function, indicat-ing the location of the individual aperture in the normalized coordinate on the aperture stop. E
⬘
共x,y兲=关H共x+c/D兲 − H共x−c/D兲兴⫻关H共y+c/D兲−H共y−c/D兲兴 is the binary am-plitude transmittance of the individual shaped aperture, which is then scaled and normalized into the pupil coordi-nate. Int关共D/c−1兲/2兴 is the interpart of 关共D/c−1兲/2兴. H共x+c/D兲, H共x−c/D兲, and H共y+c/D兲 and H共y−c/D兲 are the step functions. It is evident that the total aperture tion is formed by convolving the individual aperture func-tion with an appropriate array of the ␦ function, each lo-cated at one of the coordinate origins 共xm, yn兲=共2mc/D,2nc/D兲, where m, n= ...−2,−1,0,1,2,... The quality and location of the individual aperture on the pupil depends on the scale ratio defined as
K⬅ 共c/D兲 共4兲
for the specific diameter of the effective aperture stop on the DMD in the imaging system. The value of the scale
Fig. 2 Illustration of the binary amplitude transmittance T⬘共x,y兲 for
the normalized circular aperture, which is generated by the DMD.
T共x,y兲 represents a specifically shaped aperture for a conventional
annular apodizer.
ratio K determinates how many resolutions, how many gray levels, and how fast the DMD can dynamically generate the shaped apertures within a specific exposure time.
It is worthwhile to give an example for illustration. If the DMD array is 1024⫻768 with a pixel size of 13.7m2, and the active area is 14.03⫻10.52 mm=147.60 mm2,10
then the number of D is ⬃10.52 mm 共i.e., equal to the width of the active area of the DMD兲 if the effective aper-ture stop is located on the circular area centered at the actual DMD. In the case of K = 0.05, the width of each individual square aperture c is 0.53 mm and equivalent to 38 square pixels with the same amplitude transmittance. There are 10 共i.e., Int关共D/c−1兲/2兴+1兲 gray levels for a specific shaped aperture, including the full bright mode and full dark mode. The current DMD-based system can offer 8 bits or 256 gray levels within a time period of 5.6 ms per primary color.10Thus, the DMD can rapidly generate one shaped aperture with 10 gray levels within the very short exposure time of 0.22 ms共i.e., 5.6⫻10/256兲 in the case of K = 0.05.
The computer program for evaluating Eqs. 共2兲–共4兲 is written in Mathematica software.13 We assumed D = 2 for the simplification and evaluated four cases for the scale ratios K = 0, K = 0.05, K = 0.1, and K = 0.3. The binary am-plitude transmittances of the shaped apertures T
⬘
共x,y兲 are shown in Fig.3. The scale ratio K = 0 stands for the ampli-tude transmittance with a continuous profile. It is evident that the scale level of the binary amplitude transmission at the aperture stop increases with the reduction of scale ratioK, and the distribution of the binary amplitude transmission gets close to the continuous profile if the scale ratio K decreases.
In order to evaluate the relationship between the image performance and the size of the individual square aperture on the normalized pupil, we modified Eqs. 共2兲 and共3兲 to the following: T
⬘
共x,y兲 = E⬘
共x,y兲丢兺
m兺
n T共x,y兲␦冉
x −2ma D冊
␦冉
y − 2na D冊
, 共5兲 0艋 兩m兩,兩n兩 艋 Int冋
D/a − 1 2册
+ 1, 共6兲where 丢 represents the convolution operation. T共x,y兲=1 denotes the amplitude transmittance with a clear aperture. D is the corresponding diameter of the effective aperture stop. c represents the width of each square individual aper-ture generated by DMD in the pupil plane. The parameter a represents the distance between each square individual ap-erture as shown in Fig. 2. ␦关x−共2ma/D兲兴␦关y−共2na/D兲兴 denotes the ␦ function, indicating the location of the indi-vidual aperture in the normalized coordinate on the aperture
stop. E
⬘
共x,y兲=关H共x+c/D兲−H共x−c/D兲兴⫻关H共y+c/D兲− H共y−c/D兲兴 is the binary amplitude transmittance of the individual shaped aperture, which is then scaled and
nor-Fig. 3 Binary amplitude transmittance T⬘共x,y兲 with T共x,y兲=1−共x2+ y2兲 on the normalized pupil in the
condition of D = 2, and scale ratios at共a兲 K=0, 共b兲 K=0.05, 共c兲 K=0.1, and 共d兲 K=0.3, which are generated by DMD.
malized into the pupil coordinate. Int关共D/a−1兲/2兴 is the interpart of 关共D/a−1兲/2兴. H共x+c/D兲, H共x−c/D兲, H共y + c/D兲, and H共y−c/D兲 are the step functions. It is evident that the total aperture function is formed by convolving the individual aperture function with an appropriate array of the␦function, each located at one of the coordinate origins 共xm, yn兲=共2ma/D,2na/D兲, where m, n = . . . −2 ,
−1 , 0 , 1 , 2 , . . . We assumed D = 2 and a = 0.25 in Eqs. 共5兲 and共6兲. Four cases of the binary amplitude transmittances T
⬘
共x,y兲 for c=0.05, c=0.1, c=0.15, and c=0.2 were com-puted and shown in Fig.4. There are nine individual aper-tures along the x- and y-axes within the pupil, respectively. The results show that the individual aperture size on the normalized pupil is shrunk with the width of each square individual aperture 共i.e., the value c兲 generated by DMD. That is equivalent to the term E⬘
共x,y兲 varied with c in Eq. 共5兲.4 Optical Transfer Function of the Pupil with DMD
The OTF is derived from the autocorrelation of the pupil function by using the Hopkins canonical coordinate11and is given by 共s兲 =g共s,0兲 g共0,0兲=
冕
−⬁ ⬁冕
−⬁ ⬁ f共x + s/2,y兲f*共x − s/2,y兲dxdy冕
−⬁ ⬁冕
−⬁ ⬁ f共x,y兲f*共x,y兲dxdy , 共7兲where f共x,y兲 is the pupil function shown in Eq.共1兲, f*共x,y兲 is the complex conjugate of f共x,y兲, and s is defined as the spatial frequency s⬅2FN. Here, F is the f-number of the imaging lens system, is the wavelength, and N is the number of cycles per unit length in the image plane. The denominator of Eq.共7兲is the normalizing factor for making 0共0兲=1. The g共s,0兲 and g共0,0兲 in the OTF for the pupil function f共x,y兲 can then be given by
g共s,0兲 =
冕
−关1 − 共s/2兲2兴1/2 关1 − 共s/2兲2兴1/2冕
−关共1 − y2兲1/2−s/2兴 关共1 − y2兲1/2−s/2兴 T⬘
冉
x + s 2,y冊
· T⬘
冉
x − s 2,y冊
exp关iAx兴dxdy 共8兲 andFig. 4 Binary amplitude transmittance T⬘共x,y兲 with T共x,y兲=1 on the normalized pupil in the condition
of D = 2 and a = 0.25 in different conditions of共a兲 c=0.05, 共b兲 c=0.1, 共c兲 c=0.15, and 共d兲 c=0.2, which are generated by DMD.
g共0,0兲 =
冕
−1 1冕
−共1 − y2兲1/2 共1 − y2兲1/2 关T⬘
共x,y兲兴2dxdy , 共9兲where A = 2k20s. Because the pupil function is an even function, the term of exp共iAx兲 can be reduced to cos关Ax兴 in the integral of Eq.共8兲. Then, g共s,0兲 can be rewritten as g共s,0兲 =
冕
−关1 − 共s/2兲2兴1/2 关1 − 共s/2兲2兴1/2冕
−关共1 − y2兲1/2−s/2兴 关共1 − y2兲1/2−s/2兴 T⬘
冉
x + s 2,y冊
· T⬘
冉
x − s 2,y冊
cos共Ax兲dxdy. 共10兲Equations共9兲and共10兲can be further modified as
g共0,0兲 =
兺
q=−p p再
冕
−共1 − y2兲1/2 共1 − y2兲1/2 关T⬘
共x,y兲兴2dx冎
·⌬y, 共11兲 where y =共1/p兲⫻q,⌬y=共1/p兲. g共s,0兲 =兺
q⬘=−p⬘ p⬘再
冕
−关共1 − y2兲1/2−s/2兴 关共1 − y2兲1/2−s/2兴 T⬘
冉
x + s 2,y冊
· T⬘
冉
x − s2,y
冊
cos共Ax兲dx冎
·⌬y, 共12兲where y =关1−共s/2兲2兴1/2/p
⬘
⫻q⬘
,⌬y=关1−共s/2兲2兴1/2/p⬘
. By replacing the integral in Eqs. 共9兲 and共10兲 with the y-axisFig. 5 Optical transfer functions in an aberration-free system and a defocused system with different
defocused coefficients共a兲20= 0,共b兲20=/,共c兲20= 3/,共d兲20= 5/,共e兲20= 10/, and共f兲
20= 15/, for binary amplitude transmittances of the aperture functions for K = 0.05, K = 0.1, and K
= 0.3, which are generated by the DMD, and for a clear aperture T共x,y兲=1 and one specific shaped aperture T共x,y兲=1−共x2+ y2兲 of a conventional annular apodizer, respectively.
for the summation in Eqs.共11兲and共12兲, an initial setting of p = 100 is made for the number of intervals used to find the value of⌬y=关1−共s/2兲2兴1/2/p
⬘
and⌬y=1/p for g共s,0兲 and g共0,0兲, respectively. Different numbers of y, from −关1 −共s/2兲2兴1/2to关1−共s/2兲2兴1/2, are then used to calculate the OTF.5 Imaging Performance Evaluation
The OTF of the different pupil functions are numerically computed using Mathematica software13 based on Eqs. 共1兲–共4兲,共11兲, and共12兲for binary amplitude transmittances of the aperture functions for K = 0.05, K = 0.1, and K = 0.3, which are shown in Fig.3. The OTF of the aberration-free and defocused system with a clear aperture T共x,y兲=1 共i.e., a uniform-shaped aperture兲 and one specific shaped aper-ture T共x,y兲=1−共x2+ y2兲, as x2+ y2⬉1, and T共x,y兲=0 as x2+ y2⬎1, over the normalized pupil coordinate, which is scaled to make the outer periphery as one unit circle, i.e., x2+ y2ⱕ1, respectively, already investigated theoretically and experimentally in the literature,1 are calculated again here for comparison.
The results for the aberration-free system共i.e.,20= 0兲 is shown in Fig. 5共a兲, and the defocused systems with 20 =/, 3/, 5/, 10/, and 15/are shown in Figs. 5共b兲–5共f兲, respectively. We compared the OTF of the differ-ent scale ratios K to the OTF of a clear aperture T共x,y兲 = 1. For the large values of 20 关e.g., 5/, 10/, and 15/ as shown in Fig. 5共d兲–5共f兲兴, the spatial frequency corresponding to the first zero becomes smaller. Because the spatial frequency of the first zero generally represents the resolution limit of a defocused imaging system, we can take the first zero as defining the degree of focus. The larger degree of focus in the larger value of20commonly represents the longer depth of focus in a defocused system. For the large values of 20, the degree of focus for the aperture with scale ratio K⬍0.3 is larger than for the aper-ture T共x,y兲=1. It is evident that the specific shaped aper-ture, which is generated by the DMD with scale ratio K = 0.3 or less, can then extend the depth of focus compared to a clear aperture in the conventional imaging system. We also compared the OTF of the different scale ratios K to the OTF of one specific shaped aperture T共x,y兲=1−共x2+ y2兲. The OTF value of the former increased and got close to the OTF value of the latter when the scale ratio K decreased gradually, especially in the low spatial frequency region. This shows that the OTFs of the specifically shaped aper-ture, which are generated by the DMD, with scale ratio K = 0.05 or less, can coincide with the OTF of the conven-tional annular apodizer with continuously shaped aperture. To highlight the capability of our approach, we take a spoke pattern to explore imaging performance. Referring to Fig.6, in column A, one could see the images for the clear aperture, while in column B, the images for the specific shaped aperture with scale ratio K = 0.05 are shown. Fur-thermore, Figs. 6共a兲–6共d兲 correspond to the images ob-tained with the defocused coefficients of 20= 0, 20 = 20/,20= 10/, and20= 15/respectively. Com-paring to the images for the specific shaped aperture, the images for the clear aperture show a more critical loss of contrast at high spatial frequencies with larger20. Hence, we can conclude that the image quality will be enhanced as
the specific shaped aperture is used, especially for the large defocus coefficients in an imaging system. In other words, as for a real implementation of the DMD, the specific shaped aperture can extend the depth of focus compared to a clear aperture in the conventional imaging system.
In order to evaluate the relationship between the image performance and the size of the individual square aperture on the normalized pupil, we computed the OTF of the other types of pupil functions based on Eqs. 共1兲, 共5兲, 共11兲, and 共12兲for the binary amplitude transmittances T
⬘
共x,y兲 for c = 0.05, c = 0.1, c = 0.15, and c = 0.2 in the conditions of T共x,y兲=1, D=2 and a=0.25, which is shown in Fig.4. The OTF of the aberration-free system with a clear aperture T共x,y兲=1 was also calculated here for comparison. The OTF’s calculation results for the aberration-free system are shown in Fig. 7. The general tendency of the OTF curve versus the number of the individual aperture within the pu-pil is readily evident. There are nine individual apertures along x- and y-axes within the pupil, respectively. Mean-while, there are nine peaks on the OTF curve at the near-periodic spatial frequencies around 0, 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 1.70, and 1.90. The OTF values of the odd-peak frequencies for different sizes of the individual apertures共i.e., value c兲 remain very similar to the values of the OTFs of the corresponding frequencies for the clear aperture. However, the OTF values of the even-peak fre-quencies can decrease with decrease in the size of the indi-vidual aperture. Therefore, if the size of an indiindi-vidual ap-erture is much less than that of pupil, then OTF values of the even-peak frequencies will almost be zero.Fig. 6 The computer-simulated images of spoke patterns for共A兲 a
clear aperture and共B兲 a specific shaped aperture with the scale ratio
K = 0.05 obtained with different defocused coefficients:共a兲20= 0,
共b兲20= 5/,共c兲20= 10/, and共d兲20= 15/.
6 Conclusions
One programmable apodizer using the DMD and TIR prism system has been applied to incoherent imaging systems. The OTF model semianalytically demonstrated that the proposed programmable apodizer for the specifically shaped aperture generated by a DMD can extend the depth of focus compared to a clear aperture in a defocused sys-tem. It shows that the specifically shaped aperture with scale ratio of K艋0.05 can achieve the same improved im-aging quality as that of the conventional annular apodizer. Meanwhile, the general tendency of the OTF curve versus the number of the individual aperture and the binary ampli-tude transmittances with the discontinuous peak profile within the pupil has been investigated. It is evident that the OTF values of the even-peak frequencies can decrease when the size of the individual aperture decreases.
It is worth noting that the proposed model can rapidly generate one specifically shaped aperture with 10 gray lev-els within the very short exposure time of 0.22 ms in the case of K = 0.05. On the other hand, the TIR prism can make the normal vectors of the object, aperture stop and image planes, respectively, coincide with the optical axis of the optical imaging system for a very compact volume. Further refinement of the shaped aperture design should be able to dynamically provide improved imaging quality for many varied scenes.
Acknowledgments
This work was supported in part by the National Science Council of Taiwan R.O.C. under Project No. 93-2215-E-009-057, and in part by the Ministry of Education and the Academic Top University program at the National Chiao Tung University, Taiwan.
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Chu-Ming Cheng received his BS in
phys-ics from National Sun Yat-sen University in 1995 and MS in electro-optical engineering from National Chiao Tung University in 1997, where he is currently a PhD student in electro-optical engineering. Cheng is also an R&D director in the light-engine R&D De-partment in Young Optics Inc. He has mainly worked on the design and develop-ment of the optical systems in DMD-based projection display technologies for 11 years, and has the 40 issued patents.
Jyh-Long Chern received his BS and MS
in physics from the National Tsing Hua Uni-versity, Hsinchu, Taiwan, in 1984 and 1986, respectively. He received his PhD in optical science from the University of New Mexico, Albuquerque, in 1991. He was a postdoc-toral fellow at the Basic Research Laborato-ries, Nippon Telegram and Telephone Cor-poration, Japan, from April 1991 to August 1992. After his postdoctoral career, he joined the National Sun Yat-sen University, Kaohsiung, Taiwan, in 1992, as an associate professor of physics. In 1995, he became a full professor, and in 1996, he moved to the National Cheng Kung University, Tainan, Taiwan, as a full professor of physics. In August 2002, He joined the faculty of National Chiao-Tung University, where he is currently a professor of electro-optical engineering. -0.2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 s τ (s ) c = 0.05 c = 0.2 c = 0.1 T(x, y) = 1 c = 0.15
Fig. 7 Optical transfer functions in an aberration-free system with a
clear aperture and the binary amplitude transmittances of the aper-ture functions for different conditions of c = 0.05, c = 0.1, c = 0.15, and
