IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 A New Design of Fractal Optical Modulation A. A .Mohammad , K.H. Harby,T. A. K. Al-Aish Departme nt of Physics ,College of Education Ibn Al-Haitham,Unive rsity of Baghdad Received in : 23, May , 2010 Accepte d in : 23,March, 2011 Abstract In this p ap er it was designed a new fr actal op tical modulation by using a new iteration of fractal function, the result was analyzed by MT F evaluation, and it comp ared with results of normal optical modulation. The normal and fractal op tical modulator is a circular disc which has a radius R=9cm, both of them consist of twenty sectors, ten sectors are op aque and the other ten se ctors are transmitted for the light. The fractal op tical modulator contains two p atterns, the pattern two can be used to detect the target, and p attern one can be used to lock the target The best similarity of MTF behavior for normal and fractal Reticle was evaluating the p ower transp arent dep ends on the size of the laser sp ot and the size of the sector, where the p roportionality between them is directly. Keywords: Fractal Op tical M odulator, Chopp ing frequency, The M odulation Transfer Function M TF Introduction There are many electro- op tical tracking sy st ems using the op tical p ackage of electromagn etic radiation sp ectrum which can be various typ es used to cover many of the civilian and military applications, these sy stems are classified into t wo typ es dep ending on the nature of work, which are (p assive-mode & Active mode). The main p art in the electro- op tical tracking sy st ems which are used to determine the target locating is op tical modulation disk (Reticle) [1]. Reticle p roduces forms of modulation that allows various instruments t o differentiate objects or targets from their backgrounds and to p roduce approp riate signals that make p ossible a variety of app lications, from measure ment to guidan ce [2]. The basi c principles of Reticle The optical modulation disk is the often used in the electro- op tical tracking sy stems as optical filter for background discrimination. The design and movement of the Reticle is to enhance t he object and sup p ress t he background. The detect a p oint source in its env ironment refers t o the efficiency of the Reticle, as shown in Fig (1) [3]. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 The Reticle p att ern is determined by the requirement of the Reticle sy st em. Fig(2) shows the simple Reticle consists of 12 se ctors, 6 transp arent and 6 op aque. The Reticle is rotating and the op tical sy st em slowly scans the scene from the left to the right. When the p oint object is in the field of view of the op tical sy st em, the Reticle p att ern will generate a modulated outp ut detector syst em signal consisting of square wave with frequency corresp onding to the sp inning rate times the number of transp arent sectors. The output signal will be close to square wave as long as the p oint source fits within one sector. As t he object becomes incr easin gly extended, the output signal beco mes 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 r esults in a si gnal with no or very little modulation. Each Reticle has ap erture scannin g which is adap ted to it’s app lication [4, 5]. Reticle Design In this p ap er, two models has been designed for Reticle; the first design is normal way , so as to comp are the results obtained from this model with the results of the second model, which was desi gn ed by using Fractal Function, a new t echnique of it. Normal Modul ator Design The normal op tical modulator is a circular disc which has a radius R, which assumes the number of sector is (twenty sectors), ten sectors are op aque and the other ten sectors are transmitted for the light as shown in Fig (3). One may consider these ten sectors also as op aque for the other regions of electro -magnetic wave sp ectrum. By assuming the incident light is a p erpendicular 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 Modul ator Desi gn It is non-linear deterministic equations can self-generate irregular outp uts. T his sy st em can be simulated when behavior is linear or nearly non-linear. When it increases, though smooth on short time scales, random and unp redictable behavior can be seen over longer periods. Let (H(x),h(d)) be a metric sp ace, 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 sp ace can be given by the equation:- TAA ` …………… (1) Where A is a point in initial area. A' is a new point in under matrix op eration (T) The matrix (T) is given by :-        dc ba T …………. (2) The transformation (W) in Euclidean p lane can be given by :-[6] IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011    fdycxebyaxyxW  ,, ………… (3) The points a, b, c, and d define rotation and scaling op erations to be ap p lied to t he p oint and are called affine transformation. The e and f points define a translation to be app lied to the p oint. T he 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 p rogram, we have designed op tical modulator as shown in Fig(10). this op tical modulator consists of two p att ern circles. Each circle is divided into ten transp arents and ten opaque sectors (q). The first p att ern, is (inner p att ern) designed in a circle with a radius of 0.1 cm, the maximum distance of this p att ern is equal to 3 cm (from whole disc) , as shown in Fig (4 ). Aft er conducting the op erations of scaling, rotation and iteration (for many times) it has been got the p att ern as shown in Fig (5) and Fig (6), while Table 1 represents the data of the first p att ern. The second p att ern (outer p att ern) is designed in an equilateral triangle of side length 0.1 cm within the last third of the disk, the maximum p oints of this p att ern is equal to 9 cm, where we left a blank sp ace in the middle 3 cm in length, i.e., st arting from a dist ance of 6 cm from the first p att ern, As shown in Fig (7).Aft er (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 patt ern. Modul ation 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 ima ge. In other words, it measures how faithfully the lens reproduces (or transfers) detail from the object to the image p roduced by the lens. Fig (11) Illustrates the black and white bars in row A of the test p attern below. This p attern 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 tot ally black ar eas, and we p lot a line profile of the test pattern, we get the graph shown in C. The regions at 0 corresp ond to the black lines; the regions at 255 corresp ond to the white lines If it been taken a line p rofile of I mage Pattern B, the Grap h D above it will be the result. For the widest sp aced set of black and white bars, the p lot goes b etween 0 and 255. This corresp onds t o the performance of the lens when it recors low frequency detail. For t he next set of p att erns we can see that the plot no longer reaches either 255 or 0. The modu lation in Target A is no longer faithfully reproduced in Image B. The formal defin ition of MT F [10]: MT F = (maximum intensity - minimum intensity )/(maximum intensity + minimum intensity ) IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 For t he first p att ern group , the M TF is 1. For t he second p att ern set, the M TF can be calculated to be 0.8. For t he third set, the M TF is 0.5, and for the fourth set, t he M TF is 0.1. If there were any finer p att erns, with narrower black and white bars, the M TF would be 0 and the image of the p att ern would be a uniform gray level, represented on the p lot by a st raight line at a value of 127. The point at which y ou can no longer see any variation in the image is the point at which the M TF is zero, and that's the definition of the "resolution" of the lens. In this case, the final p att ern set with an M TF of 0.1 would be classified as "just resolved" by this lens. The Implementation Result and Discussion Obt ained the results of this work through the establishment of a sp ecial p rogram named "Disk op tical modulator" usin g the lan guage visual basi c 6 contains many p arameters and as shown in Fig (12) When calculating the frequency has been converted to units (Rev / s), as well as for angu lar velocity w, The Law of frequency is given by 2/wfr  ………. (6) qfrfc  ..……. (7) Where fc chopp ing Frequency , fr rotation Frequency and q number of sectors. To calculate the M TF we used the following law        BABA BABA MTF    …………… (9) Where A the amp litude of incr easin g frequency and B the amp litude of incident frequen cy (assumed 0.5 mm) see Fi g (13) The results t hat were obtained based on a numb er of infor mation assumed as shown in Table 3. First , we may draw the relationship between the rotation frequency and Chopp ing frequen cy with number of sector dep ending on data in Table(4), we got the curve shown in Fi g(14) Fig(14) shows that both frequencies have become a sine function oscillatin g between zero and maximum value, and since the Reticle contains t en sections, therefore, the maximum value of chopp ing frequen cy 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 Fi g (15) .So this cir cle des ign was app lied to ot her 9 remained cir cles. Not e: it has been taken one of the ten sections and it was t he first section of the origin p oint (x0, y 0 = 2.7, 0), and the ten points are d istributing as follow, with the note that e = 2.7, f = 0 and taking into account the negative sign. R= Y=c x0+dy 0+f X=ax0+by 0+e ………….….. (10) Where R rad ius of sub circle IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 Power transparent from the Reticle disk If we assume the use of a sourc e of laser-energy 650000 watt/m 2 , with sp ot size 0.3 mm, a lar ge p art of the energy of this p ackage will lost as a result of the p rocesses of reflection and absorp tion as they p ass in the transp arent disk with a permeability of τr =0.9, since reticle consists of ten sections of the window area of each se ction Sn (are shown in Table 6), the p ower transp arent P of each sector is given by the p =Qr Sn τr ……………………. (11) So the movement of any section in a circular motion takes app roximately 0.0001 seconds (for the disc consists of 10 sections of d ark does not allow p assage the p ower and 10 section window allows p assage p ower) that would lead to cut t he signal on an ongo ing basis every 0.0001 seconds as shown in Table (7) and Fi gs (16, 17), whi ch represent the relationship between p ower transp arent and the time, also it shows, t hat the power transp arent is directly p rop ortional with size of sector. T hen we evalu ate the modulation transfer function MT F by using equation (9) for normal and fr actal reticle as shown in Table8 To exp lain the Table 8 we dr aw the relationship between the frequency and radius we get the curve graph shown in Figs (18, 19, 20) that give the frequency decreases with increasin g r adius in Normal and Fractal Reticle. By drawing the relationship between MTF and chopp ing frequency fc, It gives the behavior of M TF rep eat itself, and remains similar in both two models in the case of fractal Reticle,(see Fi g.21) whi le Fi g(22) shows that the M TF curve is less dramatically with increasin g fr equency . Conclusions 1- The rotation frequency is inversely p rop ortional to t he radius of rotation, in the case of the Normal Reticle note frequency less st eadily 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 M TF is inversely p rop ortional with t he rotational frequency 3-The best similarity of M TF behavior for normal and fractal Retile was at the end of the l Reticle 4- Power transp arent depends on the size of the laser sp ot and the size of the sector, while the p rop ortionality between them is directly 5- The ty p e of sup p osed op tical modulator can be defined by using the suitable sp ot size. 6- Pattern two of fractal reticle can be used to detect t he target by using large size of sp ot size, and patt ern one can be used to lock the target by using smaller size than sp ot size. 7- It is p ossible to design multi prop os of modulator depending on multi patt erns. Re ferences 1. Reyad, N. A. (2004), Design Study on Laser Guidance Sy st em Emp loy ing an Opt ical Reticle, PH.D Thesis, Al- Rasheed College of engineering. 2. M arvin, K. S. (2001), Bandwidth-Efficient Di gital M odulation with App lication to Deep-Sp ace Communications, Book, California Institut e of Technology . 3. Harry , L. V. (2001), Detection, Estimation, and M odulation Theory , ISBNs: 0-471- 10793-X (Pap erback); 0-471-22109-0 (Electronic) Geor ge M ason University , New York. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 4. Biberman , L. M . (1966) Reticles in Electro-Opt ical devices, ergamon Press, London. 5. Tektronix and Tek,(2009), Digital M odulation Fundamentals, www.tektronix.com. 6. Fuqin, X. (2000), Digital M odulation Techniques, Book ,p 10-20, Library of Congress Catalogin g-in-Publication Data, USA. 7. Fadl, W. (2004), Desi gn Op tical M odulator by Using Fractal Function Geometry , M Sc Thesis, Al-M ustansiry ah University. 8. M andelbrot, B.B. (1982), The Fractal Geometry Of Nature, W.H. Freeman and Co, New York. 9. Thair, A.A.(2002), Fractal Ima ge Sy nthesis by Iterated Function Sy st em, M Sc Thesis, University of Baghdad. 10. Ahmed, S.A. (2008), Calculation of M TF for Op tical Disk M odulator by Using Fractal Function, M Sc 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. S CI. VOL.24 ( 3) 2011 Table(3): The results of normal and Fractal Reticle disk S tate Normal Reticle Fractal Reti cle 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 sp ot size of laser 0.5 mm 2 0.5mm 2 0.5mm 2 Angle of se ctor 18 degr ee 18 degr ee 18 degr ee Circumfer ence 0.5652 m 0.1884 m 0.5652 m Area of disk 0.025434 m 2 0.002826 m 2 0.025434 m 2 Angular v elocity 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 Chopp ing frequency 2777.7 rad/sec 8333.3 rad/sec 2777.7 rad/sec Table(4): The relation between number of sector and freque ncy No. of se ctor Normal Reticle Fractal Reti cle 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 27 77.7 83 3.33 83 33.3 27 7.77 2777.7 3 0 0 0 0 0 0 4 277.77 27 77.7 83 3.33 83 33.3 27 7.77 2777.7 5 0 0 0 0 0 0 6 277.77 27 77.7 83 3.33 83 33.3 27 7.77 2777.7 7 0 0 0 0 0 0 8 277.77 27 77.7 83 3.33 83 33.3 27 7.77 2777.7 9 0 0 0 0 0 0 10 277.77 27 77.7 83 3.33 83 33.3 27 7.77 2777.7 11 0 0 0 0 0 0 12 277.77 27 77.7 83 3.33 83 33.3 27 7.77 2777.7 13 0 0 0 0 0 0 14 277.77 27 77.7 83 3.33 83 33.3 27 7.77 2777.7 15 0 0 0 0 0 0 16 277.77 27 77.7 83 3.33 83 33.3 27 7.77 2777.7 17 0 0 0 0 0 0 18 277.77 27 77.7 83 3.33 83 33.3 27 7.77 2777.7 19 0 0 0 0 0 0 20 277.77 27 77.7 83 3.33 83 33.3 27 7.77 2777.7 IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 Table(5): S hows 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 se ctor for normal and fractal reti cle 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 sp ot size of laser 0.5 mm 2 0.5mm 2 0.5 mm 2 Angle of se ctor 18 degr ee 18 degr ee 18 degr ee Circumfer ence 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 transp arent 3.7197225 watt 0.00004133025 0.00037197225 IBN AL- HAITHAM J. FOR PURE & APPL. S CI. 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 transp arent in Power transp arent in Power transp arent 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 Patt ern Out er Pattern R Fc MT F R Fc MT F R Fc MT F 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. S CI. VOL.24 ( 3) 2011 Fig(1): simple Reticle optical syste m Fig(2): The Reticle syste m 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. S CI. VOL.24 ( 3) 2011 Fig(7): The initial shape of the second pattern Fig(8): The se cond pattern after 1 iteration Fig(9): The se cond pattern after 10 iteration Fig (10): The fractal optical modul ator IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 Fig. (11) [A] original test pattern [B] image of the test pattern [C] line profile of the origi nal test pattern where 255=whi te and 0=black[D] line profile of the image of the test pattern whe re 255=whi te 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. S CI. VOL.24 ( 3) 2011 rm ax rm in Fig. (15): The minimum and maximum radius of sub se ctor 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. S CI. 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 جامعة بغداد ،كلیة التربیة ابن الهیثم ،قسم الفیزیاء 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