IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 A New Design of Fractal Optical Modulation A. A .Mohammad , K.H. Harby,T. A. K. Al-Aish Department of Physics ,College of Education Ibn Al-Haitham,University of Baghdad Received in : 23, May , 2010 Accepted in : 23,March, 2011 Abstract In this paper it was designed a new fractal optical modulation by using a new iteration of fractal function, the result was analyzed by MTF evaluation, and it compared with results of normal optical modulation. The normal and fractal optical modulator is a circular disc which has a radius R=9cm, both of them consist of twenty sectors, ten sectors are opaque and the other ten sectors are transmitted for the light. The fractal optical modulator contains two patterns, the pattern two can be used to detect the target, and pattern one can be used to lock the target The best similarity of MTF behavior for normal and fractal Reticle was evaluating the power transparent depends on the size of the laser spot and the size of the sector, where the proportionality between them is directly. Keywords: Fractal Optical Modulator, Chopping frequency, The Modulation Transfer Function MTF Introduction There are many electro- optical tracking systems using the optical package of electromagnetic radiation spectrum which can be various types used to cover many of the civilian and military applications, these systems are classified into two types depending on the nature of work, which are (passive-mode & Active mode). The main part in the electro- op tical tracking systems which are used to determine the target locating is optical modulation disk (Reticle) [1]. Reticle produces forms of modulation that allows various instruments to differentiate objects or targets from their backgrounds and to produce appropriate signals that make possible a variety of applications, from measurement to guidance [2]. The basic principles of Reticle The optical modulation disk is the often used in the electro- optical tracking systems as optical filter for background discrimination. The design and movement of the Reticle is to enhance the object and suppress the background. The detect a point source in its environment refers to the efficiency of the Reticle, as shown in Fig (1) [3]. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 The Reticle pattern is determined by the requirement of the Reticle system. Fig(2) shows the simple Reticle consists of 12 sectors, 6 transparent and 6 opaque. The Reticle is rotating and the optical system slowly scans the scene from the left to the right. When the point object is in the field of view of the optical system, the Reticle pattern will generate a modulated output detector system signal consisting of square wave with frequency corresponding to the spinning rate times the number of transparent sectors. The output signal will be close to square wave as long as the point source fits within one sector. As the object becomes increasingly extended, the output signal becomes distorted, less modulated, and the modulation will become increasingly reduced as shown in Fig(2), the point source results in a square wave, while the extended cloud results in a signal with no or very little modulation. Each Reticle has aperture scanning which is adapted to it’s application [4, 5]. Reticle Design In this paper, two models has been designed for Reticle; the first design is normal way, so as to compare the results obtained from this model with the results of the second model, which was designed by using Fractal Function, a new technique of it. Normal Modulator Design The normal optical modulator is a circular disc which has a radius R, which assumes the number of sector is (twenty sectors), ten sectors are opaque and the other ten sectors are transmitted for the light as shown in Fig (3). One may consider these ten sectors also as opaque for the other regions of electro -magnetic wave spectrum. By assuming the incident light is a perpendicular to the modulator which is moveable in a circular form. Hence the light beam will make discrete circles according to the number of sectors. Therefor the resultant will be a circumference of the circle. Fractal Modulator Design It is non-linear deterministic equations can self-generate irregular outputs. This system can be simulated when behavior is linear or nearly non-linear. When it increases, though smooth on short time scales, random and unpredictable behavior can be seen over longer periods. Let (H(x),h(d)) be a metric space, and let .: xxf  be a function. Let xs  ,then:-     }.:{ sxxfsf  The function f is one-to-one. If xyx , , and    yfxf  , so yx  ,then the matrix space can be given by the equation:- TAA ` …………… (1) Where A is a point in initial area. A' is a new point in under matrix operation (T) The matrix (T) is given by:-        dc ba T …………. (2) The transformation (W) in Euclidean plane can be given by:-[6] IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011    fdycxebyaxyxW  ,, ………… (3) The points a, b, c, and d define rotation and scaling operations to be applied to the point and are called affine transformation. The e and f points define a translation to be applied to the point. The transformation (W) can be defined in this formula [7, 8, 9]:-                            f e y x dc ba y x WxW ………… (4) Or   TAxxW  ………… (5) Where:- Ax = the matrix             y x dc ba T= the horizontal vector       f e By using this concept and IFS kit program, we have designed optical modulator as shown in Fig(10). this optical modulator consists of two pattern circles. Each circle is divided into ten transparents and ten opaque sectors (q). The first pattern, is (inner pattern) designed in a circle with a radius of 0.1 cm, the maximum distance of this pattern is equal to 3 cm (from whole disc) , as shown in Fig (4 ). After conducting the operations of scaling, rotation and iteration (for many times) it has been got the pattern as shown in Fig (5) and Fig (6), while Table 1 represents the data of the first pattern. The second pattern (outer pattern) is designed in an equilateral triangle of side length 0.1 cm within the last third of the disk, the maximum points of this pattern is equal to 9 cm, where we left a blank space in the middle 3 cm in length, i.e., starting from a distance of 6 cm from the first pattern, As shown in Fig (7).After (many times) of conducting the operations of scaling, rotation and iteration the result as shown in Fig.8 and Fig (9), while Table 2 represents the data of the second pattern. Modulation Transfer Function MTF The modulation transfer function is, as the name suggests, 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. Fig (11) Illustrates the black and white bars in row A of the test pattern below. This pattern consists of totally black bars on a totally white background. If we assign the number 255 to the totally white areas, and 0 to the totally black areas, and we plot a line profile of the test pattern, we get the graph shown in C. The regions at 0 correspond to the black lines; the regions at 255 correspond to the white lines If it been taken a line profile of Image Pattern B, the Graph D above it will be the result. For the widest spaced set of black and white bars, the plot goes between 0 and 255. This corresponds to the performance of the lens when it recors low frequency detail. For the next set of patterns we can see that the plot no longer reaches either 255 or 0. The modulation in Target A is no longer faithfully reproduced in Image B. The formal definition of MTF [10]: MTF = (maximum intensity - minimum intensity )/(maximum intensity + minimum intensity) IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 For the first pattern group, the MTF is 1. For the second pattern set, the MTF can be calculated to be 0.8. For the third set, the MTF is 0.5, and for the fourth set, the MTF is 0.1. If there were any finer patterns, with narrower black and white bars, the MTF would be 0 and the image of the pattern would be a uniform gray level, represented on the plot by a straight line at a value of 127. 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 "resolution" of the lens. In this case, the final pattern set with an MTF of 0.1 would be classified as "just resolved" by this lens. The Implementation Result and Discussion Obtained the results of this work through the establishment of a special program named "Disk op tical modulator" using the language visual basic 6 contains many parameters and as shown in Fig (12) When calculating the frequency has been converted to units (Rev / s), as well as for angular velocity w, The Law of frequency is given by 2/wfr  ………. (6) qfrfc  ..……. (7) Where fc chopping Frequency, fr rotation Frequency and q number of sectors. To calculate the MTF we used the following law        BABA BABA MTF    …………… (9) Where A the amplitude of increasing frequency and B the amplitude of incident frequency (assumed 0.5 mm) see Fig (13) The results that were obtained based on a number of information assumed as shown in Table 3. First, we may draw the relationship between the rotation frequency and Chopping frequency with number of sector depending on data in Table(4), we got the curve shown in Fig(14) Fig(14) shows that both frequencies have become a sine function oscillating between zero and maximum value, and since the Reticle contains ten sections, therefore, the maximum value of chopping frequency greater than the maximum value of the rotation frequency of ten times and that is identical with the eq(7). Table 5 shows the method of calculating the radius of the Fractal that contains ten sections and every section contains ten circles as described in Fig (15) .So this circle design was applied to other 9 remained circles. Note: it has been taken one of the ten sections and it was the first section of the origin point (x0, y0 = 2.7, 0), and the ten points are distributing as follow, with the note that e = 2.7, f = 0 and taking into account the negative sign. R= Y=cx0+dy0+f X=ax0+by0+e ………….….. (10) Where R radius of sub circle IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 Power transparent from the Reticle disk If we assume the use of a source of laser-energy 650000 watt/m2, with spot size 0.3 mm, a large part of the energy of this package will lost as a result of the processes of reflection and absorption as they pass in the transparent disk with a permeability of τr =0.9, since reticle consists of ten sections of the window area of each section Sn (are shown in Table 6), the power transparent P of each sector is given by the p =Qr Sn τr ……………………. (11) So the movement of any section in a circular motion takes approximately 0.0001 seconds (for the disc consists of 10 sections of dark does not allow passage the power and 10 section window allows passage power) that would lead to cut the signal on an ongoing basis every 0.0001 seconds as shown in Table (7) and Figs (16, 17), which represent the relationship between power transparent and the time, also it shows, that the power transparent is directly proportional with size of sector. Then we evaluate the modulation transfer function MTF by using equation (9) for normal and fractal reticle as shown in Table8 To explain the Table 8 we draw the relationship between the frequency and radius we get the curve graph shown in Figs (18, 19, 20) that give the frequency decreases with increasing radius in Normal and Fractal Reticle. By drawing the relationship between MTF and chopping frequency fc, It gives the behavior of MTF repeat itself, and remains similar in both two models in the case of fractal Reticle,(see Fig.21) while Fig(22) shows that the MTF curve is less dramatically with increasing frequency . Conclusions 1- The rotation frequency is inversely proportional to the radius of rotation, in the case of the Normal Reticle note frequency less steadily with increasing radius in the beginning, but at radii large (at the end of the disk) we note a decrease of gradual frequency is similar to the decrease that was obtained in the case of Fractal Reticle 2-The MTF is inversely proportional with the rotational frequency 3-The best similarity of MTF behavior for normal and fractal Retile was at the end of the l Reticle 4- Power transparent depends on the size of the laser spot and the size of the sector, while the proportionality between them is directly 5- The type of supposed optical modulator can be defined by using the suitable spot size. 6- Pattern two of fractal reticle can be used to detect the target by using large size of spot size, and pattern one can be used to lock the target by using smaller size than spot size. 7- It is possible to design multi propos of modulator depending on multi patterns. References 1. Reyad, N. A. (2004), Design Study on Laser Guidance System Employing an Optical Reticle, PH.D Thesis, Al- Rasheed College of engineering. 2. Marvin, K. S. (2001), Bandwidth-Efficient Digital Modulation with Application to Deep-Space Communications, Book, California Institute of Technology. 3. Harry, L. V. (2001), Detection, Estimation, and Modulation Theory, ISBNs: 0-471- 10793-X (Paperback); 0-471-22109-0 (Electronic) George Mason University, New York. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 4. Biberman ,L. M. (1966) Reticles in Electro-Optical devices, ergamon Press, London. 5. Tektronix and Tek,(2009), Digital Modulation Fundamentals, www.tektronix.com. 6. Fuqin, X. (2000), Digital Modulation Techniques, Book ,p10-20, Library of Congress Cataloging-in-Publication Data, USA. 7. Fadl, W. (2004), Design Optical Modulator by Using Fractal Function Geometry, MSc Thesis, Al-Mustansiryah University. 8. Mandelbrot, B.B. (1982), The Fractal Geometry Of Nature, W.H. Freeman and Co, New York. 9. Thair, A.A.(2002), Fractal Image Synthesis by Iterated Function System, MSc Thesis, University of Baghdad. 10. Ahmed, S.A. (2008), Calculation of MTF for Optical Disk Modulator by Using Fractal Function, MSc Thesis, University of Technology. Table (1): The data of the first pattern Table (2): The data of the second pattern IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 Table(3): The results of normal and Fractal Reticle disk State Normal Reticle Fractal Reticle Pattern 1 Pattern2 radius 0.09 m 0.03 m 0.09 m Time 0.002 sec 0.002 sec 0.002 sec Number of sector 20 20 20 spot size of laser 0.5 mm 2 0.5mm 2 0.5mm 2 Angle of sector 18 degree 18 degree 18 degree Circumference 0.5652 m 0.1884 m 0.5652 m Area of disk 0.025434 m2 0.002826 m2 0.025434 m2 Angular velocity 1744.44 rad/sec 5233.33 rad/sec 1744.44 rad/sec Rotational frequency 277.77 rad/sec 833.33 rad/sec 277.77 rad/sec Chopping frequency 2777.7 rad/sec 8333.3 rad/sec 2777.7 rad/sec Table(4): The relation between number of sector and frequency No. of sector Normal Reticle Fractal Reticle Rotation frequency Chopping frequency Pattern 1 Pattern 2 Rotation frequency Chopping frequency Rotation frequency Chopping frequency 1 0 0 0 0 0 0 2 277.77 2777.7 833.33 8333.3 277.77 2777.7 3 0 0 0 0 0 0 4 277.77 2777.7 833.33 8333.3 277.77 2777.7 5 0 0 0 0 0 0 6 277.77 2777.7 833.33 8333.3 277.77 2777.7 7 0 0 0 0 0 0 8 277.77 2777.7 833.33 8333.3 277.77 2777.7 9 0 0 0 0 0 0 10 277.77 2777.7 833.33 8333.3 277.77 2777.7 11 0 0 0 0 0 0 12 277.77 2777.7 833.33 8333.3 277.77 2777.7 13 0 0 0 0 0 0 14 277.77 2777.7 833.33 8333.3 277.77 2777.7 15 0 0 0 0 0 0 16 277.77 2777.7 833.33 8333.3 277.77 2777.7 17 0 0 0 0 0 0 18 277.77 2777.7 833.33 8333.3 277.77 2777.7 19 0 0 0 0 0 0 20 277.77 2777.7 833.33 8333.3 277.77 2777.7 IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 Table(5): Shows the method of calculating the radius of the Fractal Pattern 2 Pattern 1 R Y X R Y X 9 0 9 3 0 3 8.8580 0.5292 8.8281 2.94798 0.1764 2.9427 8.4217 0.8559 8.3781 2.80723 0.2853 2.7927 7.8840 0.8559 7.8219 2.62286 0.2853 2.6073 7.3908 0.5292 7.3719 2.46362 0.1764 2.4573 7.2 0 7.2 2.4 0 2.4 7.3908 -7.3719 2.46362 -2.4573 7.8840 -7.8219 2.62286 -2.6073 8.4217 -8.3781 2.80723 -2.7927 8.8580 -8.8281 2.94798 -2.9427 Table (6): Data of sub sector for normal and fractal reticle State Normal Reticle Fractal Reticle Pattern 1 Pattern2 radius of sub circle 0.09 m 0.0003 m 0.0009 m Time 0.002 sec 0.002 sec 0.002 sec Number of sector 20 20 20 spot size of laser 0.5 mm 2 0.5mm 2 0.5 mm 2 Angle of sector 18 degree 18 degree 18 degree Circumference sub sector 0.02826 m 0.001884m 0.005652 m Area of sub sector 0.0012717 m 2 282 x 10 -9 m 2 2543 x 10 -9 m 2 Power transparent 3.7197225 watt 0.00004133025 0.00037197225 IBN AL- HAITHAM J. FOR PURE & APPL. 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VOL.24 (3) 2011 Table (7): The power transparent of Reticle disk No. of sector Time in sec Normal reticle Pattern 1 fractal reticle Pattern 2 fractal reticle Power transparent in Power transparent in Power transparent in 1 0.0001 0 0 0 2 0.0002 3.7197225 0.00004133025 0.00037197225 3 0.0003 0 0 0 4 0.0004 3.7197225 0.00004133025 0.00037197225 5 0.0005 0 0 0 6 0.0006 3.7197225 0.00004133025 0.00037197225 7 0.0007 0 0 0 8 0.0008 3.7197225 0.00004133025 0.00037197225 9 0.0009 0 0 0 10 0.001 3.7197225 0.00004133025 0.00037197225 11 0.0011 0 0 0 12 0.0012 3.7197225 0.00004133025 0.00037197225 13 0.0013 0 0 0 14 0.0014 3.7197225 0.00004133025 0.00037197225 15 0.0015 0 0 0 16 0.0016 3.7197225 0.00004133025 0.00037197225 17 0.0017 0 0 0 18 0.0018 3.7197225 0.00004133025 0.00037197225 19 0.0019 0 0 0 20 0.002 3.7197225 0.00004133025 0.00037197225 Table(8 ):The MTf of Normal and fractal Reticle Normal Reticle Fractal Reticle Inner Pattern Outer Pattern R Fc MTF R Fc MTF R Fc MTF 0.009 27777.77 0.06 0.03 8333.33 0.2 0.09 2777.77 0.6 0.018 13888.88 0.12 0.02906 8602.89 0.193 0.08720 2866.97 0.581 0.027 9259.25 0.18 0.026477 9442.15 0.176 0.0794 3148.61 0.529 0.036 6944.44 0.24 0.02287 10931.35 0.152 0.0686 3644.31 0.457 0.045 5555.55 0.3 0.01946 12846.86 0.129 0.05840 4280.82 0.389 0.054 4629.62 0.36 0.018 13888.88 0.12 0.054 4629.62 0.36 0.063 3968.25 0.42 0.01946 12846.86 0.129 0.05840 4280.82 0.389 0.072 3472.22 0.48 0.02287 10931.35 0.152 0.0686 3644.31 0.457 0.081 3086.41 0.54 0.026477 9442.15 0.176 0.0794 3148.61 0.529 0.09 2777.77 0.6 0.02906 8602.89 0.193 0.08720 2866.97 0.581 IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 Fig(1): simple Reticle optical system Fig(2): The Reticle system scans the scene the incoming radiation is modulated. Fig (3) The Normal optical modulator Fig(4): The initial shape of the first pattern Fig(5): The first pattern after 1 iteration Fig(6): The first pattern after 10 iteration IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 Fig(7): The initial shape of the second pattern Fig(8): The second pattern after 1 iteration Fig(9): The second pattern after 10 iteration Fig (10): The fractal optical modulator IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 Fig. (11) [A] original test pattern [B] image of the test pattern [C] line profile of the original test pattern where 255=white and 0=black[D] line profile of the image of the test pattern where 255=white and 0= black Fig.(12) The Disk optical modulator Program Fig. (13): The shape of wave Fig.( 14) :The relation between No. of sector versus frequency IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 rmax rmin Fig. (15): The minimum and maximum radius of sub sector Fig. (16): The relationship between power transparent and the time for Normal Reticle Fig .(17): The relationship between power transparent and the time for fractal Reticle Fig (18) : Normal Reticle :The frequency decreases with increasing radius Fig. (19) : Fractal Reticle(pattern 1) :The frequency decreases with increasing radius IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 Fig. (20): Fractal Reticle(pattern 2) :The frequency decreases with increasing radius Fig. (21): The MTF versus fc with spot size 0.0005(Normal Reticle) Fig. (22): The MTF versus fc with spot size 0.0005(fractal Reticle) 2011) 3( 24مجلة ابن الھیثم للعلوم الصرفة والتطبیقیة المجلد تصمیم جدید للتضمین البصري الكسوري ثائر عبد الكریم خلیل العایش،خالد هالل حربي،عبد الرزاق عبد السالم محمد د ،كلیة التربیة ابن الهیثم ،قسم الفیزیاء جامعة بغدا 2010 ،ایار،23:استلم البحث في 2011اذار، ، 23 :قبل البحث في الخالصة تكرار جدید للدالة الكسوریة ، وتم عمالالتضمین البصري الكسوري باست في هذه البحث وضع تصمیم جدید لقرص .تحلیل النتیجة عن طریق حساب دالة االنتقال الضمني ، وذلك بالمقارنة مع نتائج قرص التضمین البصري العادي ، وكل منهما 9cmنصف قطر یساوي يوقرصا التضمین البصري العادي والكسوري عبارة عن قرص دائري ذ .عشرة منها مضیئة واالخرى مظلمة، "امقطع 20یتكون من نموذج االول االو للكشف عن الهدف ، عملیستاآلخر وذج مننموذجین االأوقرص التضمین الكسوري یحتوي على .للقفل على الهدف عملیست كسوري كان عند نه . یة القرص العادياوافضل تشابه لسلوك دالة االنتقال الضمني بین قرصا التضمین االعتیادي وال وان التناسب بینهما .كذلك فأن القدرة النافذة من القرصین تعتمد على حجم بقعة اللیزر الساقطة وحجم المقطع الكسوري .طردي التضمین البصري الكسوري ، تردد القطع، دالة االنتقال الضمني: مفتاحیةالكلمات ال