# Performance of coherently illuminated multiaperture optical and microwave systems

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Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Order Num ber 8917311 Performance o f coherently illuminated m ulti-aperture optical and microwave system s W atson, Steven M anley, P h .D . The Union for Experimenting Colleges and Universities, 1989 UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PERFORMANCE OF COHERENTLY ILLUMINATED MULTI-APERTURE OPTICAL AND MICROWAVE SYSTEMS DISSERTATION Presented to the Faculty of the Union Graduate School In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY Steven M. Watson, B.S., M.S. Major, USAF February 1989 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents Page List of Figures.................................................. iii Abstract......................................................... vil I. Introduction................................................. II. Theory...................................................... 1 7 A. B. Multi-Aperture Impulse Response :..................... Impulse Response for Concentric Rings of Apertures....... 7 12 III. Results..................................................... 21 A. B. Computer Program......................................... Optical Analysis......................................... 21 21 1. Transmittance Area Comparisons....................... 2. Far-Field Diffraction Patterns....................... 3. Central Lobe Irradiance Comparisons.................. 4. Central Lobe Dimensions.............................. 5. Side Lobe Maxima Comparisons........................ 6. Radial Energy Distributions......................... 22 25 32 35 36 39 Microwave Analysis...................................... 43 1. 2. Far-Field Diffraction Patterns....................... Side Lobe Comparisons Vith and Vithout Central Obscurations........................................ Central Lobe Irradiance Ratios....................... Jitter Analysis..................................... Steerability........................................ Non transmit ting Elements............................ 43 Conclusions................................................. 64 Bibliography..................................................... 67 Appendix A - Computer Program.................................... 71 Appendix B - Experimental Design 85 C. 3. 4. 5. 6. IV. .......................... ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 51 53 57 61 List of Figures Figure Page la. Single large optic................................... lb. Six aperture system withequivalentdiameter ofsingle large optic................................................ 2 2a. Impulse response of a single large optic................... 3 2b. Impulse response of a six aperture system of equivalent diameter.................................................. 3 Field amplitude generated by outer thin ring of thedepicted single large aperture system.............................. 5 Field samplitude generated by the summation of the field amplitudes from the outer and middle rings................. 5 3c. Resultant field amplitude from summation of all three rings. 5 4. Configuration for observing the impulse response for a multi-aperture system..................................... 8 Example of multi-aperture system with an aperture-origin separation of pR and subaperture radius of a ............... 9 3a. 3b. 5. 2 6a. Irradiance pattern for a single aperture of a multi-aperture system.............................................. 13 6b. Six aperture system composed of identical single apertures.. 13 7a. Far-field irradiance pattern for a six aperture system with aperture-origin separation of 2.00a........................ 14 Far-field irradiance pattern for a six aperture system with aperture-origin separation of 3.00a........................ 14 Far-field irradiance pattern for a six aperture system with aperture-origin separation of 4.00a........................ 14 A 19 aperture system illustrating the distance (R) from the origin to the circle on which the centers of each aperture of the outside ring were placed............................ 16 7b. 7c. 8. 9a. 9b. 10a. Seven aperture system comparing the subaperture radii with aref...................................................... 17 19 aperture system comparing the subaperture radii with a *...................................................... ref 17 Single large aperture...................................... 23 iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10b. 10c. lOd. lOe. lOf. 11. 12. 19 aperture system vith an equivalent diameter equal to that of the single large aperture.......................... 23 19 aperture system vith outside ring rotated 6.75° vith respect to inner ring vith an equivalent diameter equal to that of the single large aperture.......................... 23 37 aperture system vith an equivalent diameter equal to that of the single large aperture.......................... 23 61 aperture system vith an equivalent diameter equal to that of the single large aperture.......................... 23 91 aperture system vith an equivalent diameter equal to that of the single large aperture.......................... 23 Transmittance of each multi-aperture system referenced to the single large aperture of equivalent diameter........... 24 Theoretical and experimental far-field diffraction patterns for the single large aperture of normalized radius * aref...................................................... 13. 14. 15. 16. Theoretical and experimental far-field diffraction patterns for the 19 aperture system of total normalized radius aref.................................................... 27 Theoretical and experimental far-field diffraction patterns for the 19 aperture system, vith outside ring of apertures rotated 6.75 vith respect to inner ring, of ........................... total normalized radius - 28 Theoretical and experimental far-field diffraction patterns for the 37 aperture system of total normalized radius « a ,............................................ ref 18. 19a. 19b. 20a. 29 Theoretical and experimental far-field diffraction patterns for the 61 aperture system of total normalized radius aref.................................................... 17. 26 Theoretical and experimental far-field diffraction patterns for the 91 aperture system of total normalized radius * a .................................................... ref 30 31 Comparison of secondary lobe maxima (0 * theoretical values; X * experimental values vith associated uncertainties) 37 Comparison of maximum side lobe values for the 19 rotated aperture system........................................... 38 Comparison of maximum side lobe values for the 19 closepacked aperture system..................................... 38 Field amplitude of outer ring of 19 aperture system........ 40 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20b. Summation of field amplitudes from inner and outer rings of 19 aperture system AO Resultant field amplitude from summation of all 19 apertures................................................. AO Radial Energy Distribution plot for single large aperture, 19, 37, 61, and 91 multi-aperture systems................. A2 Comparison of percentage of irradiance contained in central lobes of the single large aperture, 19, 19R (19 aperture system vith outside ring rotated 6.75 vith respect to inner ring), 37, 61, and 91 aperture systems (0 theoretical values; X « experimental values vith associated uncertainties)................................ A2 22. Arrays for microvave analysis..,........ - A5 23. Single large aperture far-field diffraction pattern in units of dB.............................................. A6 Far-field diffraction patterns for the A(square) element array.................................................... A8 24b. Far-field diffraction 48 24c. Far-field diffraction patterns for the 19 element (rotated) array................................... 20c. 21a. 21b. 24a. 25a. 25b. 26. 27a. 27b. 27c. 28. 29a. patterns for the 7 element array.... y 48 Far-field diffraction patterns generated by the 19 element (rotated) array vithout central obscurations in each subelement.............................................. 49 Far-field diffraction patterns generated by the 19 element (rotated) array vith central obscurations in each subelement.............................................. 49 Comparison of maximum side lobe values (0 - vithout obscuration, X = vith 20X obscuration) for the analyzed arrays.................................................. 50 Far-field diffraction pattern of a 7 element system vithout dephasing....................................... 55 Far-field diffraction pattern of a 7 element system vith the outer ring of elements shifted in phase by X/10 vith respect to thecentral element........................... 55 Far-field diffraction pattern of 7 element system vith tvo elements of the outer ring (separated by one element) dephased by X/10 vith respect to the remaining elements— 55 Far-field diffraction pattern for the 19 element (rotated) array experiencing a 3 a jitter value of .IX............. 56 Side lobe irradiances generated by the multi-element v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29b. 30a. 30b. 31a. 31b. 31c. 31d. 32a. 32b. B-l. systems experiencing jitter............................. 58 Maximum allowable jitter values for each multi-element system which will maintain side lobes equal to or less than that of a perfectly transmitting single antenna 58 Far-field diffraction patterns of the 19 element (rotated) system with the following phase steering magnitudes: O.OX, 0.0X — X, and 0.0X — 2 X........................ 60 Far-field diffraction patterns of the 91 element systems with the following phase steering magnitudes: O.OX, 0.0X — X, and O.OX — 2 X .............................. 60 The central lobe (X) and maximum side lobe (0) gains for various values of phase steering (steering magnitudes) for the 19 element (rotated) system...................... 62 The central lobe (X) and maximum side lobe (0) gains for various values of phase steering (steering magnitudes) for the 37 element system............................... 62 The central lobe (X) and maximum side lobe (0) gains for various values of phase steering (steering magnitudes) for the 61 element system............................... 62 The central lobe (X) and maximum side lobe (0) gains for various values of phase steering (steering magnitudes) for the 91 element system............................... 62 Far-field antenna patterns for the 19 element (rotated) system with a single nonfunctioning element located in the inner ring.......................................... 63 Far-field antenna patterns for the 19 element (rotated) system with a single nonfunctioning element located in the outer ring.......................................... 63 Experimental configuration.............................. 88 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract Many o£ the current designs of multi-aperture optical and microwave systems generate impulse responses which exhibit large side lobe irradiances. Those systems which are designed to minimize side lobe irradiances suffer large increases in the radial dimensions of the central lobes. The purpose of this research was to design and analyze multi-aperture systems which had impulse responses which exhibited side lobe irradiances less than that of the Airy pattern and central lobe widths less than or equal to that of a single large aperture of an equivalent diameter. Multi-aperture systems composed of 19, 37, 61, and 91 apertures satisfied these performance criteria. However, the amount of energy in the central lobes of these multi-aperture systems was less than that of an equivalent single large aperture. Further analysis indicated that central obscurations in each subaperture lowered the maximum side lobe irradiance in the 19 aperture arrays. Random dephasing of the antennas caused degraded performance of the arrays. The ability to steer the arrays using phase variations was investigated. The effects of a nontransmitting antenna in the 19 aperture array vas studied. v ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I. Introduction There are many astrophysical observations that require large diameter optics. The weight and size of these optical systems pose restrictions for transportation into the space environment. this dilemma are multi-aperture optical systems. A possible solution to A six aperture system vith an equivalent diameter of a single large optic is illustrated in Figure 1. Transportation of these systems into orbit would be less restrictive. However, the imaging properties of these systems need to be investigated. Figure 2 illustrates the far-field diffraction patterns of the impulse responses of a single large optic and a six aperture system vith an equivalent diameter^. The multi-aperture system impulse response exhibits relatively large side lobes as compared to the standard Airy pattern. Depending on the configuration of subapertures of the multi-aperture system, the side lobes can become extremely large. The purpose of this research was to design and analyze a multi-aperture system which "performed" as well or better than a single large aperture system of equivalent diameter and allowed a maximum transmittance of irradiance. The performance of an optical system in this research was determined by evaluating the far-field impulse response of each multi-aperture system. was based on two criteria. The performance of the multi-aperture systems The performance was considered good if the central lobe width of the imaged point source was equal to or less than that of the single large aperture of equivalent diameter. Also, good performance was characterized by secondary irradiance maxima which were 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 1. a) Single large optic and b) six aperture system. 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. '1 Fig. 2. Impulse response of a) a single large optic and b) a six aperture system of equivalent diameter. The aperture configuration of each system is depicted in the upper right-hand corner of each plot. less than 0.0175 (Airy pattern secondary maximum)*. Another factor that was considered, in addition to reducing side lobe maxima, was the amount of transmittance that was permitted through the multi-aperture system as compared to a single large aperture of equivalent diameter. Toraldo Di Francia^ examined the reduction of side lobes in proximity to the central lobe while narrowing the central lobe of the far-field diffraction patterns of a single aperture. Be accomplished this by stopping all the light impinging on the aperture except for a thin ring located at the extreme outer edge of the aperture. Adding thin rings at the appropriate locations and with the correct transmittance coefficients on this aperture caused the side lobes to be reduced for an increasingly larger field of view. However, as more thin rings were added, a smaller percentage of irradiance was contained within the central lobe. The remaining irradiance was forced into extremely large side lobes (many times larger than the central lobe) at the edge of the field of view. Figure 3 illustrates the idea of the addition of complex fields generated by concentric rings to reduce side lobes. The single aperture, as depicted in the upper right corner of the figure, has zero transmittance except for three concentric thin rings of equal width and transmittance of 1.00. Figure 3a is the complex field amplitude generated by the outer thin ring. The side lobes exhibit large field amplitudes as compared to the central lobe. Adding the field amplitude from the middle ring to that of the outer ring (Figure 3b) causes the resultant side lobes to decrease in magnitude. The addition of the fields from all the rings (Figure 3c) causes a greater decrease in the side lobe field amplitudes. However, the width of the central lobe produced by the outer ring is less than that of the summation of all of the rings. The research reported here extended the idea of the summation of 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. OUTER RING 05- < -0.5 -10 -5 0 s 10 u a OUTER+MIDDLE+INNER RMGS FIELD AMPUTUDE OUTER + MIDDLE RINGS 0.5- -0.5 -10 -0.5 -5 0 5 10 -1 0 -5 0 5 10 u U Fig. 3. a) Field amplitude generated by outer thin ring of the depicted single large aperture system, b) summation of the field amplitudes from the outer and middle rings, and c) resultant field amplitude from summation of all three rings. 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. complex field amplitudes generated by concentric rings to multi-aperture systems. The approach consisted of designing multi-aperture systems composed of concentric rings of identical circular subapertures which was analogous to the concentric thin rings on a single large aperture. The results of this research indicated that particular designs of multi-aperture systems met and, in some cases, exceeded the performance of a single large aperture system vith an equivalent diameter for the above criteria. These systems were comprised of 19, 37, 61, and 91 circular subapertures. These subapertures were arranged in expanding concentric rings about a common origin. In all cases, the central lobe widths were equal to that of a single large aperture with an equivalent diameter. This was an expected result since the central lobe width of a multi-aperture array is characterized by its largest dimension.^ In addition, the secondary maxima of the impulse responses for each multi-aperture systems was less than that of the Airy pattern. in all cases However, the percentage of irradiance in the central lobes of these systems was less than that of the Airy pattern. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. II. Theory The measures of performance of a multi-aperture system, fur the purposes of this research, were the width of the central lobes and the maximum irradiances of the side lobes of the far-field diffraction patterns. In order to determine the performance of these systems, it was necessary to model and analyze the impulse responses of each multi-aperture system. The theoretical analysis consisted of a derivation of the generalized form of the impulse response for any multi-aperture system. This analysis was taken a step further to derive the complex field amplitude for multiple rings of identical circular apertures. The primary assumption in the analysis was that the aperture systems were illuminated by monochromatic light. Multi-Aperture Impulse Response Figure 4 depicts the configuration used to model a multi-aperture 1 3 system ’ . In this analysis, the field produced by a single point source is propagated from the object plane, through the aperture plane, to the image plane. The subapertures which comprise the multi-aperture systems are identical circular apertures. multi-aperture system. Figure 5 illustrates one form of a This particular arrangement of six apertures illustrates the variables used in the calculation of the impulse response for all of the multi-aperture systems. In this analysis "a" was the radius of each of the subapertures; N was equal to the number of apertures in the system; x n and y n described the location of the centers of the n 1*1 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. / object plane collimating lens aperture lens plane observation (im age) plane Pig. 4. Configuration for observing the impulse response for a multi-aperture system. 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ► 2a Fig. 5. Example of multi-aperture system with an aperture-origin separation of pfl and subaperture radius of a. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. x subaperture; 0n vas the angle in degrees from the x axis; and pR was the distance o£ the n**1 subaperture from the origin of the system and vas expressed in terms of multiples of the subaperture radius. The pupil function of a multi-aperture function can be expressed as the convolution of one of the apertures vith the delta functions vhich describe the location of the centers of each aperture. As a result, the generalized pupil function can be vritten as N P(x,y) - circ[r/a] * } 6(x - Pncos9 n=l ,y - p sin9 ) (1) where circ[r/a] « 1 if r/a is less than or equal to 1, otherwise, it equals 0. The circ function describes a single aperture of the multi-aperture system where a is the radius of the circular aperture and , . r-(x 2+y2 )1/2 The impulse response, h(x^,y^), is defined to be the Fourier 8 1 3 11 12 transform of the exit pupil . Therefore, the impulse response is ' ’ ’ h(x.,y.) = F{P(x,y)} 1 1 N = F{circ[r/al) x F{ ]>S(x _ pncosGn ,y - pnsin0n )} n=l = aJ1 {2na[(xi/Af)2 + (y./Af)2 J1/2} l(x./X£) ,\£.2 ,1/2 + (y,/Af) J N x5expl-i2x (x.p cos© 1 11 + y.p X 11sin© II)/Xf) 1 11 where F{ } denotes the Fourier transform. factors have been discarded. (2) Nonessential proportionality Note that for each delta function, which describes the location of the center of a particular subaperture in the aperture plane, there corresponds a plane wave traveling from the subaperture to the focal point. 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Letting u - 2anXj (3) Xf and v - 2anyi (4) Xf the impulse response can be expressed as h(u,v) = 2na2J1t(u2 + v2)1/2] 7~2 2717? (u + v ) N x 5exPl-i(upncos0_ + vp sin6n)/a] __ 1 n n n n n«l (5> This impulse response can be further simplified by expressing the image coordinates as u = qcost. v » qsin+ (6) where q = the radial coordinate in the image plane and ^ = the angular position of the image plane coordinate. Hence, the impulse response can be expressed as: h(qf$) = 2 — = N 2na J«[q] ^exp[-ip a « nq(cos4cos9n + sin+sinOn )/a] — 2 N 2Jia J1 Iql ^expl-ip qcos(*-0 )/a] — q n=l n n (7) The envelope function of this impulse response is 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2rea2J1 Iq] q (8 ) which is a scaled version of the impulse response for a single subaperture of the multi-aperture system. This envelope function is modulated by cosine fringes which arise from the addition of the complex exponentials which describe the plane waves. The frequency and direction of the cosine fringes is a function of the aperture-origin spacing, pn , and the angular spacing of subapertures, ©n , respectively. Figure 6 illustrates the modulation effect. Figure 6a is the impulse response of any one of the subapertures which comprise the six aperture system depicted in the top right hand corner of Figure 6b. Figure 6b illustrates the modulation of the Airy pattern caused by the addition of subapertures to the single subaperture. It is this modulation which causes the side lobes to increase in irradiance and cause many multi-aperture systems to perform poorly when compared to a single large aperture of an equivalent diameter. Vhen the apertures were moved further apart, the side lobes increased in amplitude. Figures 7a through c illustrate this phenomenon for a six aperture system vith aperture-origin separations of 2.00a, 3.00a, and 4.00a, respectively. Impulse Response for Concentric Rings of Apertures Equation (7) is the generalized expression for the impulse response of any multi-aperture system which contains identical circular apertures. This equation is used to form a subset of equations which describe the impulse responses of multiple rings of identical circular apertures. The multi-aperture systems examined were comprised of concentric rings of identical circular subapertures vith a single subaperture at the 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 6. Irradiance pattern for a) single aperture of a eulti-aperture system and b) six aperture system composed of these identical single apertures. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. oo oo ■n X U oo vT L oo 7 Pig. 7. Far-field irradiance patterns for a six aperture systea with aperture-origin separations of a) 2.00a, b) 3.00a, and c) 4.00a. I origin. The single aperture was surrounded by six apertures which just contacted each other and the center aperture (Figure 8). The radial distance of the center of each of the subapertures comprising the next ring of subapertures was determined by placing an aperture along an axis which passed through the center of the center aperture and an aperture in the six aperture ring. The first aperture, along the horizontal axis, of the next ring just contacted the aperture of the six aperture ring. The distance from the origin to the center of the new aperture (depicted as R on the figure) defined the radial distance of the circle for the placement of the centers of the remaining apertures of the new ring. apertures were symmetrically located along the circle. These All of the multi-aperture systems were constrained to fit inside of a single large aperture of constant radius aref As a result, the subaperture radii of the multi-aperture systems had to be adjusted appropriately. The following is a derivation of a generalized expression for systems of multi-apertures arranged in concentric rings which just fit inside a single large aperture of radius aref To find the expression for the generalized subaperture radii for any concentric ring system, let a ■ the subaperture radius of the apertures which comprised the multi-aperture system. Using geometric arguments and referring to Figures 9a and b, the radius of each subaperture comprising the concentric ring multi-aperture system may be described as: a ■ — ^ ---(2)(rings)+l (9) where rings « number of rings which comprise a multi-aperture system. The radial distance from the origin to the circle on which the centers of the subapertures of a particular ring are placed, pm can be 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 8. A 19 aperture system illustrating the distance (R) from the origin to the circle on vhich the centers of each aperture of the outside ring vere placed. 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 'ref A Fig. 9. a) Seven and b) 19 aperture systems comparing the subaperture radii vith a c. ret I I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. generalized. "m" represents a particular ring beginning with m«l as the ring closest to the origin at a subaperture radius of 2a; m-2 as the next ring with a radial distance of 4a, etc. For any ring, pn may be expressed as: Pi * 2a p2 - 4a « 6a pn = 2raa (10) Substituting equation (9) into (10) yields: Pm " 2maref (2)(rings)+l (11) For each ring, there is an associated number of apertures that can fit without overlap. The number of apertures for each ring can be described by: #ring. * 6 - (1)(6) #ringi - 12 - (2)(6) Kringj - 18 - (3)(6) #ringm * m6 (12) There is also a definite relationship between each ring and the apertures that comprise that ring. Since each larger ring has six more apertures than the preceeding ring, the angular spacing between each aperture in each ring can be expressed as: Bring. = 360°/6 = 60° Bring- = 360°/12 = 30° Bring^ = 360°/18 = 20° Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ©ring m - 360°/m6 ■ 60°/m (13) Therefore, the location of each aperture in a particular ring may be generalized as: ©1 « 0, 0- * 9, + 60/m ©^ « ©J + 29- ©J - ©J + 3©j 0m6 * 9 i + [n>6-i]©2 (14) vhere ©i is the angular location of any single aperture in a particular ring. Substituting equations (9) and (10) into equation (7), the impulse response for each ring is: 0 6 h,(q,+) = 2na J,[q] 2®xPl-i2aqcos(4>-© )/a] --------- n*l q 2 = 2na 6 f J,[ql 2exp[-i2qcos(*-© )) — — y n*l n [(2)(rings)+l] q 2 (15) 12 h2(q»+) * 2na J1[qJ ^exp[-i4aqcos($-©n)/a) -------- n=l q * 2ita 2 12 f J.[q] ^exp[-i4qcos( *-© )] --- * n=l n t(2)(rings)+l] q (16) 2 2m h (q, ♦) = 2 na J.fq] 2«XP [~12mclcos C♦-© )/a] -------- n=l 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2naref Jllq l 2exp[-i2mqcos(*-0n )] (17) I(2)(rings)+1] q which describes the impulse response generated by each ring composed of identical circular subapertures. The frequency of the complex field amplitudes generated by each ring is determined by the argument of each exponential. For the rings that are further from the origin, i.e. as 2m increases, the impulse responses exhibit higher frequencies. However, the rings located in proximity to the origin produce lower frequency field amplitudes. The total impulse response of the multi-aperture system is a summation of the impulse responses (complex field amplitudes) of each of the rings. This coherent summation of impulse responses yields the following complex field amplitude at the image plane: U(q> ♦) = hQ(q,4>) + h^q,*) + h2(q,$) + + ^(q,*) (18) where hg(q,$) is the impulse response for the single aperture located at the origin. Q The irradiance for each diffraction pattern is expressed as: I(q»♦) - U(q,♦)U*(q,♦) (19) Equations (17), (18), and (19) formed the basis for the calculations of the far-field diffraction patterns which, in turn, were used to determine the performance of the concentric ring multi-aperture systems. 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. III. Results The following is a discussion and analysis of the results of the calculated and experimentally obtained diffraction patterns of the concentric ring multi-aperture systems. Using these results, the performance of each system was evaluated and compared to that of a single large aperture of equivalent diameter. COMPUTER PROGRAM The Fortran computer program that was used to compute the diffraction patterns is listed in Appendix A. The first portion of the program computes the diffraction patterns of the envelope function from a point source using the IMSL^ subroutine MMBSJ1 to calculate the required Bessel functions. This was followed by the calculation of the complex exponential functions generated by the various apertures in each system. The final portion of the program finds and stores the maximum side lobe values and computes the radial energy distribution values of each diffraction pattern using IMSL^ subroutine DBCQOU. OPTICAL ANALYSIS The purpose of this research was to design and analyze multi-aperture systems which had impulse responses characterized by maximum side lobe irradiances which were less than that of a single large aperture; i.e. 0.0175. In addition, the resulting diffraction patterns would have central lobe widths which were equal to or less than that of a single large aperture of equivalent diameter. Although not considered one of the 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. performance criteria, the percentage of irradiance in the central lobe of these patterns vas evaluated. Figures 10b through f depict the multi-aperture systems which were designed and analyzed. Each multi-aperture system vas comprised of concentric rings of identical subapertures with a single aperture at the origin of the system. The multi-aperture systems vere comprised of 19, 37, 61, and 91 subapertures, respectively. system. Figure 10b vas the 19 aperture Figure 10c vas the 19 aperture system with the outer ring of 12 apertures rotated 6.75° vith respect to the inner ring (Tschunko and Sheehan^ examined the MTF and radial energy distribution of a similar 19 aperture system). Geometrical constraints dictated the number and location of subapertures which comprised each ring of the various systems. Figures lOd, e, and f vere the 37, 61, and 91 aperture systems, respectively. In each case the radius of each multi-aperture system was equal to that of the single large aperture of Figure 10a. As a result, each of these systems could just fit inside of the single large aperture. Transmittance Area Comparisons Since the amount of energy which passed through each aperture system vas of importance, the transmittance area of each aperture system vas calculated and is depicted in Figure 11. All values vere normalized such that the single large aperture vith an equivalent diameter had 100Z transmittance. Using equation (9) and geometric analysis, the total area of transmittance, T, for each aperture system was: T19 = n(aref) 19 = Tref (-760) (20) 25 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ••••• • • • • • • • • • M M M • • • • • • • • • V&V.V • • • • • • • • • •••••••••• ••••••••••• • • • • • • • • • • • • • • • • • • • Fig. 10. a) Single large aperture, b) 19 aperture system, c) 19 aperture system vith outside ring rotated 6.75 vith respect to inner ring, d) 37 aperture system, e) 61 aperture system, and f) 91 aperture system. All systems have an equivalent diameter equal to that of the single large aperture. 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TRANSMITTANCE AREA COMPARISON 100 H 80 a - £ < a a < 60 a a - o z g 40 - CO z < q : 20 - \ APERTURE SYSTEM Pig. 11. Transmittance of each multi-aperture system referenced to the single large aperture of equivalent diameter. 24 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. T37 * n(are£>2 37 - T ref ('755) (21) * T ref ('753) (22) - T ref (.752) (23) 49 T6i - "<aref>2 ^ 81 T91 - K(aref)2 121 where = *(aref) 2 which is the transmittance area of the single large aperture and T^g, T yj, Tg^, and Tgj are the transmittance areas of the 19, 37, 61, and 91 aperture systems, respectively. The 19 aperture system had the greatest amount of transmittance, 76.OX, vith respect to the single large aperture. Far-Field Diffraction Patterns Figures 12 through 17 are the calculated and experimental diffraction patterns for each of the depicted aperture systems. For each aperture system, the calculated modulus (the square root of the irradiance) of the far-field diffraction pattern (impulse response) is displayed. The modulus is displayed for each aperture system in order to enhance the structure of the side lobes. The upper left-hand corner of these figures represents the aperture system which generated the associated impulse response. The photograph in the upper right-hand corner is the experimental diffraction pattern for the same aperture system. Refer to Appendix B for a description of the experimental configuration used to generate and analyze the diffraction patterns. The maximum central lobe value for each diffraction pattern was normalized to value of 1.00. Since a certain percentage of energy vas 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. }2 ‘i T^eoretical and experimental far-field diffraction patterns for the single large aperture of normalized radius - a . Patterns for ref 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 13. Theoretical and experimental far-field diffraction patterns the 19 aperture system of total normalized radius * aref* 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 14. Theoretical and experimental far-field diffraction patterns for the 19 aperture system, vith outside ring of apertures rotated 6.75 vith respect to inner ring, of total normalized radius » a ref 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ‘ref29 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. Fig. 16. Theoretical and experimental far-field diffraction patterns for the 61 aperture system of total normalized radius - aref* 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 17. Theoretical and experimental far-field diffraction patterns for the 91 aperture system of total normalized radius ■ aref 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. stopped at the multi-aperture systems, a theoretical examination vas conducted to determine the maximum central lobe irradiance values o£ each multi-aperture system. A generalized expression is derived for any system containing concentric rings of identical circular subapertures. Central Lobe Irradiance Comparisons For the single large aperture of radius - aref> the impulse response, assuming unit amplitude plane vave illumination, (referring to equation (17)), is: Urer ,(q) * 2n(aref)2J1(q) rei 1 --- (24) 2 Substituting T re^ for X(arg^) , the value of the impulse response at q 0.0, the maximum central lobe value, is: Uref<°’°> - ^ r e f " Tref <25> 2 This central lobe value of the complex field amplitude is equal to the transmittance area of the single large aperture of radius The irradiance of the central lobe is: Iref(0,0) - Uref(0,0)Uref*(0,0) = (Tref)2 (26) Examining the 19 aperture system comprised of aperturesvith radii equal to that described in equation (9), the complex field amplitude 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (equation (15)) yields: U19(q,*) - 9 2naref ^ 19 Jil<ll I 1 + exp[-i2qcos(*-0n)] q + exp(-i4qcos(^-0n )l (27) Substituting q * 0.0 yields the folloving impulse response: U19(0,0) - it(are£)2(19) 25 - T19 (28) which is the transmittance area for the 19 aperture system. The maximum central lobe irradiance for the 19 aperture system is: I19(0,0) = U19(0,0)U19*(0,0) (29) Substituting equation (30) into (31) yields: I19(0,0) = (T19)2 (30) which demonstrates that the maximum central lobe value for the 19 aperture system is equal to the square of the transmittance area (assuming unit amplitude plane wave illumination). Substituting equation (20) into equation (30) results in: Iig(0,0) = - (Tref)2(.760)2 (Tref)2(.58) (31) The ratio of the central lobe values of the single large aperture and 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 aperture system yields: (32) which is equal to the square of the transmittance value indicated in Figure 11. This analysis can be generalized for any system of multi-aperture systems composed of concentric rings of identical circular apertures. Unit amplitude plane wave illumination is assumed. Combining equations (17) and (18) and expanding, as in equation (27), results in: #sub 1 + exp[-i2qcos($-0n)J + exp[-i4qcos(9-0n)J „ (33) where #sub is the number of subapertures in the system and Tsujj is equal to the transmittance area of any single subaperture. Setting q = ♦ = 0.0, the field amplitude becomes: u#Sub<°-°> • W tsub> (34) ^#sub where T transmittance area of a single subaperture and T transmittance area for the generalized multi-aperture system. The irradiance becomes: 2 (35) I 34 I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vhich indicates that the maximum central lobe irradiance value of any multi-aperture system composed of concentric rings of identical circular subapertures is equal to the square of the transmittance area of that particular system. Using these same arguments, the maximum central lobe irradiances of the other multi-aperture systems examined in this research, as compared to that of a single large aperture of equivalent diameter, equaled the percentage of transmittance area of the respective systems. Central Lobe Dimensions Each of the diffraction patterns vas analyzed to determine the central lobe width and compared to the central lobe width of the single large aperture impulse response. The theoretical and experimental data indicated that the central lobe widths vere equal. Harvey** et. al. arrived at the same conclusions using close-packed synthetic-aperture systems composed of 3, 7, and 19 subapertures vhich had the same equivalent diameter as a single large aperture. The central lobe diameters generated by each aperture system were: D1 °19 s 6.2- .3mm a 6.3- .3mm 6.2- .3mm °19R D37 °61 °91 where » 6.1- .3mm s 6.2- .3mm 6.2- .3mm = central lobe width generated by the single large aperture and 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. °19’ °19R' °37’ °61’ °91 are the central lobe widths generated by the 19, 19 rotated, 37, 61, and 91 aperture systems, respectively. Side Lobe Maxima Comparisons The side lobe patterns and maxima vere also analyzed. The details of the side lobe structure are depicted in the calculated modulus diffraction patterns. As the number of subapertures increased from 19, the structured side lobes, evidenced by the pronounced peaks, moved further from the central lobe. The irradiance of these structured lobes decreased as the number of apertures increased. As the number of subapertures approached 91, the diffraction patterns more closely resembled that of the Airy pattern depicted in Figure 12. The irradiance of the side lobe maxima vere measured for each diffraction pattern (refer to Appendix B for the experimental design). Figure 18 depicts the results of this analysis. In all cases, the side lobe maxima of the multi-aperture systems vere less than the Airy pattern value of 0.0175. For the 19 aperture systems, the maximum side lobe values vere located at the prominent peaks of the structured side lobes. The 19 aperture system vith the outer ring of apertures rotated 6.75° vith respect to the inner ring exhibited the lovest side lobe maxima (the value of 6.75° vas determined empirically). The addition of the fields from the tvo rings of apertures, vith a relative rotation betveen the rings, and the center aperture vas such that the irradiance of the structured side lobes vere reduced. (It should be noted that Harvey^ et. al. examined the far-field diffraction pattern of a 19 subapertrue close-packed synthetic-aperture system as depicted in Figure 19b. Even though this system is similar to the 19 subaperture system examined in this research, 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. SECONDARY LOBE COMPARISON 0.020 LU o z 5 0.015 - < Qd C£ Ld CD 0.010 - 3 oE 2 z o 0.005 - 3 CO 0.000 APERTURE SYSTEM Pig. 18. Comparison of secondary lobe maxima (0 - theoretical values experimental values vith associated uncertainties). 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 00 SIDE LOBE M AX = 0.0136 a SIDE LOBE M AX = 0.0219 b Pig. 19. Comparison of max side lobe values for the a) 19 rotated and b) 19 close-packed aperture systems. the close-packed system yields a secondary maxima of 0.0219. This maxima is 61% greater than that of the 19 aperture system vith the outer ring rotated 6.75°). As the number of apertures increased, the irradiance of the side lobe disk nearest the central lobe increased. For the 37, 61, and 91 aperture systems, the maximum side lobe values vere located in these disks. As is illustrated in Figure 18, as the number of apertures increased, the value of the first side lobe disk approached the value of the first side lobe disk of the Airy pattern. Figures 20a through c illustrate the reason for the reduced maximum side lobe irradiance values for these particular configurations of the subapertures. The 19 aperture system, depicted in the upper right-hand corner, vas used to illustrate this effect. Figure 20a is the calculated field amplitude in the far-field due to the outer ring composed of 12 apertures. Note the large amplitude of the side lobes. Figure 20b represents the addition of the field amplitudes from the outer (12 apertures) and inner (6 apertures) rings. The amplitudes of the side lobes vere substantially reduced compared to those generated by the outer ring alone. apertures. Figure 20c is the field amplitude generated by all 19 The side lobe heights are considerably less than that generated solely by the outer ring. Radial Energy Distributions The next portion of the research consisted of an examination of the percentage of energy contained vithin the central lobe of each of the impulse responses generated by the multi-aperture systems. Figure 21 represents the theoretical results of the calculated radial energy 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. OUTER RING -05 5 0 U s O OUTER+ NCR RNGS 05- -05-1 -0 r -5 0 u b 5 o 19 APERTURES 05- 0 U 5 c Pig. 20. a) Field amplitude of outer ring of 19 aperture system, b) summation of field amplitudes from inner and outer rings of 19 aperture system, and c) resultant field amplitude from summation of all 19 apertures. 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 distributions for the single large aperture and the multi-aperture systems. In each case, the radial energy distributions were computed and normalized for each of the theoretical diffraction patterns in Figures 12 through 17. The results depicted in Figure 21a indicated that as the number of apertures increased, the percentage of irradiance contained in the structured side lobes decreased. The rapid increase of the slope of the radial energy curves, past the location of the first zero, indicated the radial distance where the structured side lobes were located. As depicted in Figure 21a, the structured side lobes for the 19 aperture system, as compared with the other multi-aperture systems, vere closer to the central lobe. This observation can be confirmed by noting structured side lobe locations in Figures 13 through 17. As the number of apertures increased, the percentage of irradiance in the side lobes decreased. As a result, the percentage of irradiance that was contained within the central lobes necessarily increased. This is illustrated in Figure 21a as the percentage of irradiance located from 0.0 radial distance to the location of the first zero. For instance, the radial energy distribution curve for the 91 aperture system closely approximated that of the single large aperture system. However, the curves for the 19, 37, and 61 aperture systems exhibited a larger percentage of irradiance in the side lobes. Figure 21b depicts the theoretical and experimental results of measuring the percentage of irradiance contained within the central lobes of each multi-aperture system. The results indicated that the percentage of irradiance of energy within the central lobe of each of the multi-aperture diffraction patterns was less than that of a single large aperture. Generally, the percentage of irradiance in the central lobes 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. % OF IRRADIANCE IN CENTRAL LOBES RADIAL ENERGY DISTRIBUTION 100-1 lOO-i X 80- 37 80- 60O O z < Q < § 40- 40- C£ cm fs> 20 LOCATION OF FIRST ZERO - 0 8 4 6 2 RADIAL DISTANCE 20 10 -T " \ — i--- r <o~ aperture system Fig* 21. a) Radial energy distribution plot for single large aperture, 19, 37, 61, and 91 aulti-aperture systems, b) Comparison of percentage of irradiance contained in central lobes of the single large aperture, 19, 19R <19 aperture system with outside ring rotated 6.75 with respect to inner ring), 37, 61, and 91 aperture systems (0 - theoretical values; X experimental values vith associated uncertainties). increased as the number of apertures in the multi-aperture systems increased. The percentage of irradiance within the central lobes of the 19 and 37 aperture systems was 73.1% and 73.2X, respectively. percentage for the 61 aperture system increased to 77.5X. The The central lobe of the 91 aperture system contained 82.8Z of the irradiance for the field of view examined. This closely approximated the irradiance contained within the central lobe of the single large aperture (83.8X) with an equivalent diameter. MICROVAVE ANALYSIS The assumption that prevailed in this analysis vas that the multi-aperture systems were illuminated with monochromatic light. This can be seen in the generalization of the image plane coordinates for the impulse responses. The generalizations were: u = 2naxj and v * 2nay^ Af Af (36) where (Xj,y^) were the image plane coordinates, A was the wavelength, and f the distance to the Fourier transform plane (the image plane in this analysis). Using this generalization allows one to utilize this analysis for any wavelength. Far-Field Diffraction Patterns The next portion of this research examined the theoretical diffraction patterns of arrays composed of three to 91 elements (antennas). Far-field calculations and side lobe comparisons were 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. performed for these arrays with and without obscurations. Dephasing of individual elements (jitter) and steerability vere examined for the 19 (rotated), 37, 61, and 91 element systems. Finally, the 19 rotated system vas examined for its far-field performance vith nonfunctioning elements. The arrays that vere examined are depicted in Figure 22. The 19, 19R, 37, 61, and 91 element systems are the same as those examined earlier in this research. It has been assumed that there vas no coupling between the individual elements and that each element vas at least 10 wavelengths in diameter. This last assumption ensures that there are minimal effects on the diffraction patterns due to edge effects^. The far-field diffraction patterns vere calculated using similar algorithms (equations 17, 18, and 19) as those for the optical diffraction patterns. However, since the results of microwave measurements are normally presented in terms of decibels (dB), the normalized irradiance values (I) vere converted using the following equation: IdB = 10 Log10(I) (37) These relative irradiance values were analyzed to determine the performance of the multi-element microwave arrays. Figure 23 is the diffraction pattern for a single large element plotted in units of dB. The first sidelobe is the largest at 17.6 dB below that of the mainlobe (because of the low sampling rate along the U axis, the lobe structure will appear to be jagged. would smooth the lobes). symmetric. A higher sampling rate The single aperture pattern is circularly However, as has been indicated before, the multi-element patterns are not. The succeeding multi-aperture gain patterns are displayed in this format. Representative patterns are shown in Figures 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 element 3 elements 4 elements 4 elements (square) MM V.V 7 elements 19 elements 19 elements (rotated) •••••••••• ••••••••••• •••••••••• 37 elements Fig. 22. 61 elements 91 elements Arrays for nictovave analysis. 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. SINGLE LARGE APERTURE CD -17.6 dB I— (/I -20 LU -30 > Ld -40 H QC ■5 0 5 U (X /D ) Fig. 23. dB. Single large aperture far-field diffraction pattern in 46 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 24a, b, and c. These are the Car-field patterns for the 4(square), 7, and 19 element (rotated) arrays, respectively. Each figure represents a planar cut of the pattern which includes the largest sidelobe. Side Lobe Comparisons With and Without Central Obscurations This portion of the analysis consisted of examining the relative intensities of the side lobes without aperture blocking. of the multi-element arrays with and Blocking simulates the addition of feed horns or Cassegrain secondary reflectors to the central portion of each of the elements that comprised an array (the obscured portion of each aperture has a transmittance = 0.0). As an upper limit, a 20% central obstruction (20% radius of unobstructed element) was chosen as a reasonable upper limit for obscuration due to the feedhorn or secondary reflector. Figures 25a and b depict the far-field diffraction patterns generated by the 19 element (rotated) array with and without the element blockage. It is interesting to note that the central obstructions caused the maximum sidelobe irradiances to decrease to -18.9 dB from the unobstructed value of -18.7 dB (in each case, the central lobe gain was normalized to a value of 0.0 dB. As a result, the side lobe values vere measured relative to the central lobe value generated by each particular array.). The central obstruction modulated the envelope function, generated by a single element of the array, such that the peripheral sidelobes contained within the envelope function vere reduced. However, the sidelobes nearest the mainlobe vere unaffected. Figure 26 is a summary of the results of this analysis for the arrays with and without obscurations. Generally, the central obstructions lowered the maximum sidelobes. However, the 3, 4, 4 square, and 7 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. O -o § <0 4» u CO 9 O' W (8P) AJJSN31M 3Ali\rOM 4> JS u o V) c u 4P (0 a c • O CO «-» CO U t4 <0 u u CO **H C •O 41 ■ 8P ‘AJJSN1LM 3AliV13a -c 41 H H 41 41 *+4 *-s IT3 12 • CO w CO Cm CO o •u ^ w CM ON • H 00 •H ^ OM o aP'AilSNUM 3AliVn3y 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 APS ROWED - 2055 08S0URED 19 APS ROTATED <8 -*>- 16.7 dB - 20- vO -30 -SO -10 u (A/D) -5 u(X/D) a Fiff 25. Far-field diffraction patterns generated by the 19 element (rotated) array a) without and b) with central obscurations in each subeleaent. MAX SIDE LOBE COMPARISON O = WITHOUT OBSCURATIONS X = WITH OBSCURATIONS CD tz (/) LxJ DESIRED MAX SIDE LOBE VALUE (UPPER LIMIT) CtZ Ld CD -15- Q Q i>— O — CO -20 4 NUMBER OF APERTURES X^'vith ? o n K ^ SOn,?f “a*inu“ side lobe values (0 - without obscuration, with 20* obscuration) for the analyzed arrays. 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. aperture systems continued to exhibit side lobe irradiances greater than -17.6 dB. On the other hand, the 37, 61, and 91 aperture systems did not have reduced side lobes but still exhibited side lobes less than -17.6 dB. In these cases, the maximum sidelobe values vere located immediately next to the mainlobes (refer to the diffraction patterns in Figures 15, 16, and 17). As explained above, the central obstructions had no effect on the proximity sidelobes (sidelobes nearest the mainlobe). As a result, the maximum sidelobe values did not change. Central Lobe Irradiance Ratios The next portion of the microwave analysis was an examination of the central peak irradiance ratios of the examined arrays. The central peak ratios vere computed using equation (35) and: w°>°> Iref(0,0) (38) vhere Iref(O,0) is the peak irradiance of the corresponding single large aperture system. Table 1 displays the irradiance ratios for the arrays considered. The 7 element system exhibited the largest central lobe peak irradiance ratio whereas the 4 element array displayed the lovest ratio. 5i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 1 Peak Irradiance Ratios for Multi-Element Arrays AFray. Ilsub(0,0)/Iref(0,0) Single 3 4 4(square) 7 19 37 61 91 1.000 0.417 0.197 0.471 0.605 0.578 0.570 0.567 0.566 The following is a mathematical analysis of the effect on the peak irradiance values for multi-aperture systems with central obscurations. For this analysis, let a - the radius of an element of an array and b = the radius, expressed as a fraction of the element radius, of the central obstruction. The central obstruction has a transmittance of 0.0. The function which describes an array of apertures with central obscurations of radius b is: N F{Circ(r/a) - Circ(r/b)}F{ I S(x-x , y-y )} n«l (39) The impulse response (complex field amplitude) of this system can be expressed as (using equation (7) as a reference): U(q,40 = 2rta2J 1[ql - 2na2b2J ]l[bq] q q N x J exp[-ipnqcos(*-0n)/a] (40) where q = the radial coordinate in the image plane, $ = the angular position of the image plane coordinate, a = radius of the unobscured 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. aperture, and b » radius o£ the central obscuration o£ each aperture. At the location o£ the central lobe maximum, q » ♦ ■ 0.0 vhich yields the field: U(0,0) » [na 2 2 2 - b na ] x [tsubapertures in the system] (41) where the ftsubapertures in the array is equal to the exponential term in equation (40) when the argument is equal to 0.0. 2 If Tgub is substituted for na (the transmittance area of a single aperture of the array), equation (41) becomes: 2 U(0,0) = lTsujj - b Tsujj] x [#subapertures in the array] U(0,0) » ^sub^ “ ^ ) x l#subapertures in the array] 2 (42) Therefore, the irradiance of the central lobe maximum is: 1(0,0) = U(0,0)U*(0,0) ,2... , .2,, .2.2 = (Tsub) (tsubapertures in the array) (1-b ) >> - (Ttsub>2(1 - b2>2 2,, -.2 .4, * <T#sub> (1 - 2b + b > <*3) where T#suj, = (Tsuj))(#subapertures in the array) which is the transmittance area of the unobstructed array. Jitter Analysis The far-field patterns generated by the multi-element arrays are sensitive to the relative dephasing of the array elements 17-22 . To 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. illustrate this dephasing, the outer ring of apertures of the 7 aperture system vas shifted in phase by X/10 with respect to the central element. A close examination of Figures 27a and b indicate that the relative dephasing caused the sidelobe patterns to symmetrically increase in amplitude and shift in position. Dephasing two of the outer ring elements (separated by 1 element) by X/10 with respect to the rest of the array had an adverse affect on the far-field diffraction pattern (Figure 27c). The sidelobe patterns exhibited a marked asymmetry with an increase in amplitude. Calculations vere then performed to analyze the random element-to-element jitter for the 19 (rotated), 37, 61, and 91 element arrays. Jitter, in this context, is defined as the random shifting (phase) of the elements vith respect to each other into and out of the aperture plane. The calculations vere performed using: N UN (u»v) = U(u,v)x Jexp[-ip (ucos0 +vsin0 )/a)exp(i* ) w _ i n n n n n=l (44) where U(u,v) is the impulse response generated by any single subaperture of the array and $n is the phase associated vith each element. The phase values for each element, in terms of X, vere randomly generated using a Gaussian distribution and ranged betveen the folloving values: -0.0125X -0.0250X -0.0375X -0.0500X — — — — +0.0125X +0.0250X +0.0375X +0.0500X vhere the extreme ranges are the 3 a values. Figure 28 illustrates the effect of the jitter on the far-field diffraction pattern on the 19 element (rotated) array. The effect is 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -lA ‘AJJSN31N 3AU.VT3S rfi 8P ‘AUSNliNI BAlJLVBa Pig. 27. Far-field diffraction pattern of a 7 element system a) without dephasing, b) vith the outer ring of elements shifted in phase by X/10 vith respect to the central element, and c) vith two elements of the outer ring (separated by one element) dephased by X/10 vith respect to the remaining elements. o I aP'AUSNUM 3AliVT3a 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. RELATIVE INTENSITY (dB) 19 APS ROTATED: 3 a JITTER = .1A -10- U (V D ) Fig. 28. Far-field diffraction pattern for the 19 element (rotated) array experiencing a 3 a jitter value of .IX. 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. manifested by asymmetry and increased amplitude of the sidelobe patterns. Figure 29a is a compilation of maximum side lobe values generated by the 19 (rotated), 37, 61, and 91 element systems experiencing varying degrees of jitter. For each range of jitter values, the maximum side lobe value vas determined 10 times. lobe irradiances. The RMS value vas computed from these side The x axis reflects the total range of the 3 o values that vere mentioned previously. As seen in Figure 29a, in all cases, the magnitude of the maximum sidelobe increased as the jitter value increased. The 19 element (rotated) system exhibited the lovest side lobes for each jitter value up to a jitter value of approximately .075X. However, it is evident that the rate at vhich the side lobes grew vas inversely proportional to the number of elements in the system. This is apparent in the slopes of the curves, particularly betveen a 3 a jitter values of .050A and .100X. Figure 29b demonstrates the amount of jitter that could be experienced by each system and generate side lobe irradiances vhich vere equal to or less than that of a single perfectly transmitting antenna (-17.6 dB). The 19 element (rotated) system vas able to vithstand .073A maximum jitter and maintain side lobe values equal to -17.6 dB. However, the 91 element system vas able to experience only .048X maximum jitter before it produced side lobes greater than -17.6 dB. Steerability The next portion of this research vas an examination of the ability to steer the central lobes using phase variations of the individual elements of each of the multi-element systems. Crockett and Strange 23 examined the ability to steer the central lobe of a seven element 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CD ■-o5 O Q. C o CD Q. with permission of the copyright owner. Further reproduction prohibited without permission. Jll IL K MAX ALLOWABLE JITTER -I6-1 0.08 ,< 0.07- $ 0.06- QC £ L/l 00 19- 19R £ 0.05- 0.04 1 1 0.0 .025 .050 .075 3a MAXIMUM JITTER (X ) a .100 i 19R 1---- 1 37 61 r 91 APERTURE SYSTEM b Fig. 29. a) Side lobe irradiances generated by the multi-element systems experiencing jitter, b) Maximum allowable jitter values for each multi-element system which will maintain side lobes equal to or less than that of a perfectly transmitting single antenna. close-packed system by smoothly and linearily varying the phase across its face. This researach utilizes the same steering technique where each element vas assigned a discrete phase value. The magnitudes of the phase difference, i.e. the steering magnitudes, across the face of these systems vere: 0.0X 0.0X 0.0X 0.0X — — — — X/2 X 3 X/2 2X where the 0.0X value vas at one edge of the element system and the maximum phase magnitude value is located at the opposite edge. Equation (44) vas used to calculate the far-field antenna patterns for this steering technique. Figures 30a and b illustrate the effects of steering the central lobe of the 19 (rotated) and 91 element systems using the technique described above. The first plot of each series is the ideal far-field diffraction pattern for each system without attempting to phase steer the central lobe. The second plot employs the 0.0X to X phase difference across the face of the multi-element system. Finally, the third plot of each series illustrates the 0.0X to 2X steering magnitude. The dashed curves on the second and third plots are the envelope functions vhich vere generated by a single element of each respective system. These plots indicate that the movement and relative gain of the central lobes vere constrained by the envelopes of each system. In addition, the further the central lobe vas moved from the optical axis, the greater the gain of the side lobes. Since the central lobe dimension of envelope function of the 19 element (rotated) system vas considerably narrower than that of the 91 element system, the central lobe experienced an accelerated gain loss as compared to the 91 system. 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CD ■-o5 O Q. C o CD Q. with permission of the copyright owner. Further reproduction prohibited without permission. 19 APS ROTATED 0 - 2A 0 - A - a 20- I -3 0 -4 0 -50 -O -50 -5 0 5 u(vt>) U(VD) O' 91 APERTURES o 0-2A 0 - A «s> - -3 0 - -O - -5 0 s U(X/D) 10- 20- -3 0 - -3 0 - 8 -so 20 - - _40' -50 -O -5 0 5 n -5 0 -10 -5 U(X/D) Fig. 30. Far-field diffraction patterns of the a) 19 (rotated) and b) 91 element sytsems vith the following phase steering magnitudes: 0.0X, 0.0X — X, and 0.0X — 2X (dashed curve «. envelope function generated by a single antenna of each respective array). In order to compare the performance of the different multi-element systems, the central lobe and maximum side lobe gains vere recorded for each value of steering magnitude. The results of this analysis are contained in Figures 31a, b, c, and d for the 19 (rotated), 37, 61, and 91 multi-element systems, respectively. These plots illustrate that as the number of elements increased, vith a corresponding decrease in the size of the individual elements, the central lobe gain decreased more slovly as the steering magnitude increased. lobe values increased vas less. As a result, the rate at vhich the side This is a result of the increased central lobe vidths of the envelope function generated by the increasingly smaller element dimensions. As explained earlier, these envelope functions constrained the movement of the central lobes. Nontransmitting Elements The final portion of this research examined the performance of the 19 element (rotated) array vith one of the 19 elements nonfunctioning. The far-field antenna patterns vere examined vhen each of the 19 elements had, in turn, zero transmittance. The performance of the system vas determined using the maximum side lobe values. The maximum sidelobe values ranged from -15.85 dB to -15.43 dB vith an RNS value of -15.68 dB. Figures 32a and b illustrate the modulus of the far-field diffraction patterns for the cases of a nonfunctioning element in the inner and outer rings, respectively. the side lobe values. In both cases, the patterns exhibited an increase in The most noticeable effects occurred in the side lobes vhich vere in proximity to the central lobe. 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19R APS - STEERABLfTY 37 APS - STEERABUTY x x -5- -O - 8- - -20 8- I -20 I !-- 1- 0.0 0.5 tO 15 2.0 S1EERNG MAGWTUX (X) STEERNG MACMTUOC (X ) 91 APS - STEERABILITY 61 APS - SnERABUJTY “X -* ---v— ST sr -5 -5 - -20 I 0.0 o!s to 15 10 I1 T 0.0 0 8 f ""1 LO 18 -20 I 0.0 0 8 2.0 tO 18 2.0 STEERNG MAGMDJOC (X ) STtERNG MXGMTUX (X) c Fig. 31. The central lobe (X) and maximum side lobe (0) gains for various values of phase steering (steering magnitudes) for the a) 19 (rotated), b) 37, c) 61, and d) 91 element systems. 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission 1 O' u Fig. 32. Far-field antenna patterns for the 19 element (rotated) system with a single nonfunctioning element located in the a) inner ring and b) outer ring. IV. Conclusions This research examined multi-aperture systems vhich, according to tvo reasonable engineering criteria, could perform as veil as a single large aperture vith an equivalent diameter. The multi-aperture systems examined vere composed of 19, 37, 61, and 91 circular apertures arranged in concentric circles. Due to geometrical considerations, these systems alloved approximately 76% irradiance transmittance as compared to the 100X transmittance of a single large aperture of equivalent diameter. The secondary maxima for each of these systems vas less than that of the single large aperture. The 19 aperture system vith the outside ring rotated 6.75° exhibited the lovest maximum side lobes vith a value of 0.0136. In all cases, the impulse response of the multi-aperture systems had central lobe vidths equal to that of the single large aperture system. Hovever, The percentage of irradiance contained by the central lobes generated by the multi-aperture systems vas less than that of the single large aperture. The 19 and 37 aperture systems exhibited the lovest percentage of central lobe irradiance vhile the 61 and 91 aperture systems exhibited increased central lobe irradiance values. The percentage of irradiance within the central lobe of the 91 aperture system closely approximated that of the single large aperture system. This research also analyzed these multi-element systems in terms of microwave transmitting devices. The far-field patterns vere examined for the effects of central obscurations in each of the subelements, random phase fluctuations (jitter), steerability using a smoothly and linearly 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. phase difference across the face of each system, and nonfunctioning elements. The results indicated that for central obscurations, side lobe values actually were reduced for the 19 and 19 (rotated) element systems when compared to the perfectly transmitting system. Vhen jitter was introduced into each of the systems, the side lobe values increased and the far-field antenna patterns exhibited asymmetry. The 19 element (rotated) system vas able to sustain the greatest amount of jitter and continue to exhibit side lobe values equal to or less than -17.6 dB. The multi-element systems are capable of being phase steered, hovever, the side lobe values began to increase rapidly with a corresponding decrease in central lobe gain. The 91 aperture system exhibited a superior steering ablility when compared to the other multi-element systems. Finally, single nontransmitting elements in the 19 element (rotated) system caused asymmetry in the far-field antenna patterns with increased side lobe maxima. It has been demonstrated that multi-aperture (antenna) systems can exhibit large side lobes. This analysis has demonstrated that there are multi-telescope/antenna designs which actually reduce side lobes below that of a single large telescope (antenna). When used for imaging applications, such as Ladar, Radar, and astrophysical observations, reduction of the side lobes could improve the two point resolution of these systems. The antenna arrays discussed in this reseach could also be utilized for power transmission. In addition to the multi-aperture approach to this research, one could interpret these systems as single large apertures that have been apodised. Reducing the transmittance of certain small portions of the aperture to 0.0 will reduce the side lobe values as compared to that of a single large aperture with no apodization and improve the imaging 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. properties. 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Bibliography 1. Watson, S.M., Mills, J.P., and Rogers, S.K. "Two-Point Resolution Criterion for Multi-Aperture Optical Telescopes," Journal of the Optical Society of America A , Vol. 5, No. 6, (Jun 1988). 2. Shack, R.V., Ramcourt, J.D., and Morrow, H. Six-Element Synthetic Aperture," Applied Optics. 3. Fender, J.S. "Synthetic Apertures: "Effects of Dilution on a 10, 257-259 (1971). An Overview," Synthetic ApertureSystems, Proc. SPIE, 4AO, 2-7 (1983). 4. O'Neill, G.K. "A High Resolution Orbiting Telescope," National Academy of Sciences, Synthetic Aperture Optics, Vol. 2, Woods Hole Summary Study (National Academy of Sciences - National Research Council Advisory Committee to the AFSC), Washington, D.C. (1967). 5. Sintsov V.N. and Zapryagaev, A.F. "Aperture Synthesis in Optics," Usp. Fiz. Nauk 114, 655-676 (December 1974). 6. Kwong, R. "Analytic Expression for the MTF of an Array of Circular Unaberrated Phased Apertures," Applied Optics, Vol. 27, No. 10 (15 May 1988). 7. Born, M. and Wolf, E. Principles of Optics. New York: Pergamon Press, 1980. 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8. Toraldo Di Francia, G. "Super-Gain Antennas and Optical Resolving Power," Suppltemento A1 Vol.IX, Series IX Del Nuovo Cimento, 426, 1952. 9. Goodman, Joseph V. Introduction to Fourier Optics. San Francisco: McGraw-Hill, 1968. 10. IMSL Library. Library. Computer Mathematics and Statistics Fortran Subroutine Houston: IMSL, Inc., 1984. 11. Harvey, J.E., MacFarlane, M.J., and Forgham, J.L. Performance of Ranging Telescopes: "Design and Monolithic Versus Synthetic Aperture," Optical Engineering, Vol. 24, No. 1, 183-188 (Jan/Feb 1985). 12. Tschunko, J.F.A. and Sheehan, P.J. "Aperture Configuration and Imaging Performance," Applied Optics. 10, No. 6. 1432 - 1438, June (1971). 13. Harvey, J.E., MacFarlane, M.J., and Forgham, J.L. Performance of Ranging Telescopes: "Design and Monolithic Versus Synthetic Aperture," Optical Engineering, Vol. 24, No. 1, 183-188 (Jan/Feb 1985). 14. Chow, W.V. "Far-Field Intensity of a Partially Locked Optical Phased Array," Applied Optics. 23, 4332-4338 (1984). 15. Mills, James P., The Effects of Aberrations and Apodisation on the Performance of Coherent Imaging Systems, PhD Dissertation, University of Rochester, New York, 1984. 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16. Taylor, C.A. and Thompson, B.J. "Some Improvements in the Operation of the Optical Diffractometer," J. of Scientific Instruments, 43, 439, 1957. 17. Palma, G.E. and Townsend, S.S. "Performance and Phasing of Multiline Synthetic Apertures," Synthetic Aperture Systems, Proc. SPIE, 440, 68-76 (1983). 18. Vaite, T. and Sun, K. "Physical Understanding of Synthetic Aperture Arrays Via Simple Models," Synthetic Aperture Systems, Proc. SPIE, 440, 52-55 (1983). 19. Crockett, G.A. and Strange, D.A. "Computer Model for Evaluating Synthetic Aperture Propagation," Synthetic Aperture Systems, Proc. SPIE, 440, 77-84 (1983). 20. Harvey, J.E., Silverglate, P.R., and Vissinger, A.B. "Optical Performance of Synthetic Aperture Telescope Configurations," Southwest Conference on Optics, Proc. SPIE, 540, 110-118 (1985). 21. Butts, R.R. "Effects of Piston and Tilt Errors on the Performance of Multiple Mirror Telescopes," Wavefront Distortions in Power Optics, 293, 85-89 (1981). 22. Shellan, J.B. "Phased-Array Performance Degradation Due to Mirror Misfigures, Piston Errors, Jitter, and Polarization Errors," J. Opt. Soc. Am. A, Vol. 2, No. 4, 555-567 (April 1985). 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23. Crockett, G.A. and Strange, D.A. "Computer Model for Evaluating Synthetic Aperture Propagation," Synthetic Aperture Systems, Proc. SPIE, 440, 77-84 (1983). I 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX A COMPUTER PROGRAM 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. o o 5 zc < 3 QC O u. ac UJ o z )-) QC 3 < QC O o o: a. >u 3 *-K o uj a. J tJ h O <<3 VJ 3 O UJ M S tt 3 < J <3 h- QC < D QC U ZU JU J O O Z 1 QCMO< 1 I/) H I Z K X CO UJ *H UJ H 3 IO X X MO h O K W Q U (9 O CM CD Q * w a CM i^ h O lO © X S H S l CMo o • -CM (9 a W Q Q Q U Q CMQ < (M WCM - W VJ ' —QC V * > h >. a Z 2 Q -M x S ^vS Q CM® *9 • Q M CM Q CM 9 CMx o X W Z a CD ph n *tO W ID ® - ^/-n»h U Q a : ox Oa UJ a. * z HI 3 H ~ < < H < < < < < < < £ < < M * - I -J to K < ** 2 s 2 8 3 3 0 O o: k< s< Q. < < < < IS • * z h u h o f f l f f l u f l , « a . o o o . a. o a. a : z | - l - h | - h l - l - h l - y OO QO to * QC QC M o Q- O t - ~ -------- 3 i - 5 2 m lrt 3 m IIII II II II II II II II II II II II IIII II II 3 3 U J L L lU J U J U J U J U J U J U J U iU J U J U J U J U J U J U J UJ HJ I L U - U - U - l L l l . U - U . U . U . U . U . U . UL l i . U . I L LL. a m a a a m a a a a a m m a a * 22222222222222222 U 11 /«v , M ????????????????? “■ r~r-r-»— r - r r - r r r — l— »— r— 3 m e s o a * a " h n n •» 1 0 < o o ' o i s « « H M H M N N N M « N n N N ( i n n » M I. h H H » II H II H M M M „ « j, x 2jZZZ£ZZZZZZZZZZZZ c . c . c . c . 2 ’3 5 2 . 5 2 . 2 2 2 . 2 . 2 . 2 . 2 . j _ UJ lu * £f < ^ 0 uuu j o <_i _ 2 x < Q £ iS . kuj c* oC. C.SS* U. < < UJ u iS S io x ^ 5 z o Z m U JO 30. CD a g £g y * W l. 12 2HO.HU<fi.N ~ UJ II SV5 5 5 ? l « < i 2 f l h S o o o f o „ i * O J ;* O O O Z O 2 2 2 2 ° * ® ® ® ® ® ® ® ® W * i.T . ... q w UJ a: 25 QC I I t - k z f - uj H Z Z UJ UJQ^ Qc > o ae VJ £ X u.k 55SS55555SS5S5555 < < O O O O O O O O O O O O O O O O O y m )m O H Q C<XX n*i»_jpT_T 2 iiiiH iiiiiiiiiiiiiiiim iiiiiiiiii o w w w w w w w w w w w w w w w w w 3 O WShh z » —»»—»»— a uj CL K 2 ph Z X > U O : U I I I l l l l l i M i i j So -U -S zstu >- r v^ _ r . r r vrNp ^ ' ' ^ r vr ‘r'r' h- - I— K K K K K H h - l - | - l - < K H O O < O O O O Q O Q O O O O 0 < < ‘»0 ‘O O H 3 •*H CM CO -* ID «0 f*» ® 0) VJ > CMQc /-X «H CM 6 ^ B -B Q B -O B rM rM ® ® QC A CM '“ ' B (0 UJ < w tO 3 CMW M Jh f f l H ,w < aroc h 3a a ru ; u j O -J J J J J J U J l < < < K 3 UJ LU UJ UJ UJ z O QC QC QC QC QC VJ “> -I U QC J oo O Z M QC hO K VO iiii. QC QC UJUI < /> > hZ <9 O UJ CM UJ S - t o u cm >-c r -> - ^ v O r qc S Z S • UI « X CMX -M ® o > w x w ® VJ < Z K ® - to * h a p -X U J w ^ B -VJ QC Z ® ‘Z w U. 3 3 22222222122221: uj u uj u u i uj uj uj uj u j uj uj uj D C f tK K ft ftk K V ttlK fy iE o u uouu 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. t S „ n_ Si W U o t- ° UJ < Q: < o “■ “ < Qc < o MULTI-APERTURE to QC X UJ > OfN) J D H J O Q 1< o c o O U X o * M H < H QC h Q h UJ UJ UJ 3 X 3 H O * -3 < X > O IU- tO M O 0 .* (0 O HUIQCQC K O O O M QC to -X Z O <01L liflH H K X * Z h < M Z 3 H J UJ UJ O < K x a . u to K M > O h IO < 0 *0 . U JhO J »< < U. UJ V -I o x w 3 02 oc u O IN THE B B*(/)-& Ll B^-CM » M Q V ^ UJ UJ o x UJO X VJ APERTURE X o VJ _ (M H S N OF EACH ® a ® = RADIUS cm 5 — z * I— ID UJ z a CM SUBRAD w Q PARAMETER ^ SYSTEM. S3 r M K Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C PARAMETER ANNRAD r RADIUS OF CENTRAL OBSTRUCTION. C WHEN ANNRAD=SUBRAD, THE ENTIREAPERTURE ISOBSTRUCTED. C C PARAMETER PHASE = THE PHASE, IN RADIANS, OF THE OBSTRUCTING APERTURE. C THETA IS USED IN THE EXPRESSION, EXP(i*th«t«), WHICH C DESCRIBES THE PHASE OF THE OBSTRUCTING APERTURE. C WHEN PHASE =0.0, EXP(i.phase) IS SET C =0.0. THIS ESSENTIALLY REMOVES THE PHASE ANNULUS C FROM THE APERTURE. C C PARAMETER NUM = THE NUMBER OF APERTURES COMPRISING THE SYSTEM. C C PARAMETER TOTRING = TOTAL # RINGS OF APERTURES IN THESYSTEM. THE CENTER C APERTURE IS COUNTED AS A RING. C C PARAMETER NUMAPS = # APERTURES IN THE PARTICULAR RING C CPARAMETER DECREE = THE ANGLE IN DECREES, TO DESCRIBE THE LOCATION OF THE C THE FIRST APERTURE OF THE PARTICULAR RING. THIS CAN C BE THE LOCATION OF ANY SINGLE APERTURE IN THE RING C C PARAMETER DELTADEG = THE ANGULAR SEPARATION BETWEEN EACH APERTURE IN A RING C C PARAMETER DIS = DISTANCE OF THE APERTURES FROM THE ORIGIN IN THE C PARTICULAR RING. ALL APERTURES IN THE RING MUST BE THE C DISTANCE •DIS* FROM THE ORIGIN C C PARAMETER DNORM = NORMALIZATION FACTOR FOR THE FINAL C OUTPUT OF THE INTENSITY C 37 FORMAT(I3,3X,F10.6) 38 FORMAT(F10.6.3X,13) 39 FORMAT(13) 40 FORMAT (F10.6) 41FORMAT(I3,3X,I3,3X,F10.4,3X,F10.4) 42 FORMAT(13,3X,I3,3X,F10.4) 44 FORMAT(I3,3X,F10.4,3X,F10.4) 60 FORMAT(' PERFORMING COHERENT IMAGING') 61 FORMAT(■ PERFORMING INCOHERENT IMAGING’) 66 FORMAT('0VALUE OF SECONDARY SIDE LOBE IN R.H.S. PLANE IS') 60 FORMAT('0VALUE OF SECONOARY SIDE LOBE IN L.H.S. PLANE IS') 61 FORMAT(’ MAX SECONDARY PEAK IS CREATER THAN PRIMARY A PEAK ON R.H.S. OF PLANE’) 82 FORMAT(’ MAX SECONDARY PEAK IS GREATER THAN PRIMARY A PEAK ON L.H.S. OF PLANE’) 63 FORMAT(’ MAX SECONDARY VALUE IS AT (24,24)’) 64 FORMAT(’0POINT SEPARATION = \F10.6) 66 FORMAT(’0 INTENSITY AT (24,24) = ’.F10.6) 66 FORMAT(’0R.H.S. PRIMARY PEAK VALUE = ’,F10.6) 87 FORMAT('0 L .H .S . PRIMARY PEAK VALUE = \F10.6) 70 FORMAT(’0THE IRRADIANCE VALUE USED TO NORMALIZE THE A MAX IRRADIANCE VALUE TO 1.00 IS’) 71 FORMAT('0COMPUTING THE ENERGY UNDER THE DIFFRACTION PATTERN’) 72 FORMAT(’0NO ENERGY CALCULATIONS BEING ACCOMPLISHED’) 73 FORMAT(’0THE MULTIPLICATIVE FACTOR FOR THE FOV = \ F 10.8) 74 FORMAT(’0COMPUTING THE MODULUS’) C PI=3.1416927 IF (COHER.E Q .60)WRITE(•,60) IF (COHER.ER.61)WRITE(•,51) C C COMPUTE THE PHASE TERM FOR THE PHASE ANNULUS Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 TA = CEXP(CMPLX(0.0.PHASE)) IF (PHASE.E Q .0.00)TA s 0.00 C C READ IN VALUES OF THE LOCATION OF THE APERTURES C READ(S,39)TOTRINC NUMI = 1 DO 6000 1=1,TOTRINC READ(6,39)NUMAPS READ(6,40)DIS READ(5,40)DEGREE READ(6,40)DELTADEG IF(NUMI.EQ.1)G0 TO 5003 NUMAPS=NUMI ♦ NUMAPS - 1 5003 CONTINUE DO 6002 I1=NUMI,NUMAPS DEGREE = DEGREE * DELTADEG NX(I1)=C0S(DEGREE*PI/180.0) NY(I1)=SIN(DEGREE*PI/180.0) DIST(II) = DIS PRINT*,I1,DIST(I1).DEGREE IF(II.EQ.NUMAPS)GO TO 5004 6002 CONTINUE 6004 NUMI = II ♦ 1 6000 CONTINUE C C INPUT VALUES OF POINT SEPARATION AND INITIALIZE CENTMIN C 1100 = 0 CENTMIN=40 FMULT = 0 . 4 WRITE(*, 73) FMULT C Ci i i ) i i i ! i i j i i i i i i i ; ; ! ; i | ; i i i i ; | ) { I | ; i i I i i ; | i i I | I i i i i i i i C 'COMMENT'OUT'NEXT LINE IF'NOT ’F I W I N G 2 Di M 'INTENSITY V S ! C POINT SEPARATION PLOTS C C DO 1000 PTSEP=0.0,.6,.05 1100 = 1100 * 1 d I l l l l I l l I I l l I l M l l I l l l l l I I | I l 1i ! ! ! ! I I i i l l !{ I i I I | I I | | | !| | C COMMENT'OUT'NEXT LINE’IF‘FINDING'2 DIM INTENSITY V S . C POINT SEPARATION PLOTS C C PTSEP=0.1936 WRITE(•,64)PTSEP DO 276 1=1,48 DO 280 J=1,48 Y=(J - 24.0)*FMULT X=(I ♦ (24.0 - 2.0 • I))*FMULT IF (X.EQ.0.00000000)X=0.000001 I COMPUTE ARGUMENT FOR FIRST BESSEL FUNCTION (ENVELOPE FUNCTION) Z= SUBRAD • 5QRT((X-PI*PTSEP)**2.0 ♦ Y**2.0) CALL UP MMBSJ1 (IMSL) TO COMPUTE THE BESSEL FUNCTION DUE TO ORIGINAL UNOBSTRUCTED APERTURE FOR POINT SOURCE AT -PTSEP. CALL MMBSJ1 (Z ,IER) E = MMBSJ1(Z,IER) E = (E/Z) • (SUBRAD**2.0) IF (ANNRAD.E Q .0.00)GO TO 200 C C CALL UP MMBSJ1 TO COMPUTE BESSEL FUNCTION FOR THE Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 C C OBSTRUCTING APERTURE FOR POINT SOURCE AT -PTSEP. Z1 = CALL El El (ANNRAD*Z)/SUBRAD MMBSJ1 (Zl.IER) = MMBSJ1 (Zl,IER) = (El/Zl) • (ANNRAD**2.0) C C COMPUTE FUNCTIONS DUE TO LOCATION AND SPACING OF C THE fl APERTURES FOR POINT SOURCE AT -PTSEP. C 200 S=CMPLX(0.0,0.0) DO 100 11=1,NUM XI = -((X»NX(I1)*DIST(I1)) - (NX(II)*PI»DIST(II)*PTSEP)) Y1 = -Y*NY(I1)*DIST(II) Ci i i i i i i vl l-n i i i i i | I i i i i i I i i I i i i i i I I i i i i i i i i i i i i i i i i i i i i i C COMMENT OUT NEXT LINE IF LOOKING AT FIELDS ONLY C C(I1)=CEXP(CMPLX(0.0,XI ♦ Yl)) C CM!!!!!!!!!!! USE THIS NEXT LINE FOR COMPUTING FIELDS ONLY!!!!!!!!!!!! C C G (I1) = COS(Xl +Y1) C 100 S = S G(I1) C C!!!(COMMENT OUT NEXT LINE IF LOOKING AT FIELDS ONLY!!!!!!!! C S2=CABS(S) C C !!!!!!!!!!!USE NEXT LINE ONLY IF LOOKING AT FIELDS!!!!!!!!!!!!!!!!!! Q C S2 = S C C THE FIELD AMPLITUDE DUE TO THE POINT SOURCE LOCATED AT -PTSEP IS: C CONTINUE IF (ANNRAD.EQ.0.00)El=0.00 A(I,J) = (E-(El*(1.00-TA))) • S2 IF (COHER.EQ.S0)GO TO 700 C C THE INTENSITY DUE TO THE POINT SOURCE LOCATED AT -PTSEP IS: C (THIS NEXT LINE IS USED FOR INCOHERENT IMAGING ONLY) C n !!!!!!!!!!!!!<!!!!!!!M !!!!!!!!!!!!!! M !!!!!!!!!!! C C IF WANT TO LOOK AT FIELD ONLY COMMENT OUT NEXT LINE A(I,J) = A (I,J) **2 C C i I I i i i I I I I I i ! i 1 i i i i i I ! i I i ■! i i ! ! I I i i i j ! i i i i i ! M i ! ! i ! ! I i j ! ! j ! ! ! C IF NEED TO OBSERVE THE MODULUSi TAKE COMMENT OFF NEXT 2 LINES C IF NEED TO LOOK AT FIELD ONLY COMMENT OUT NEXT LINE A (I,J) = SQRT(A(I, J)) WRITE(•,74) C 700 CONTINUE C C IF NEED TO OBSERVE MODULUS FOR COHERENT IMAGING,TAKE COMMENT C OFF OF THE NEXT LINE C A (I,J) = ABS (A (I,J) ) IF (PTSEP.EQ.0.00) GO TO 28 C C COMPUTE THE ARGUMENT FOR THE SECOND BESSEL FUNCTION (ENVELOPE FUNCTION) C Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 O' 26 P = SUBRAD • SQRT ((X-»PI*PTSEP) **2 .0 ♦ Y**2.0) C C CALL UP MMBSJ1 (IMSL) TO COMPUTE BESSEL FUNCTION DUE TO C ORIGINAL UNOBSTRUCTED APERTURE FOR POINT SOURCE AT *PTSEP. C CALL MMBSJ1 (P,IER) Fll = MMBSJ1(P,IER) Fll = (FI1/P) • (SUBRAD**2.0) IF (ANNRAD.E Q .0.00)GO TO 3 C C CALL UP MMBSJ1 TO COMPUTE BESSEL FUNCTION FOR THE C OBSTRUCTING APERTURE FOR POINT SOURCE AT +PTSEP. C PI = (ANNRAD • P)/SUBRAD CALL MMBSJ1 (Pi,IER) FI = MMBSJ1 (Pi,IER) FI = (Fl/Pl) • (ANNRA0*«2.0) C C COMPUTE FUNCTION DUE TO THE LOCATION AND SPACING OF THE C SIX APERTURES FOR POINT SOURCE AT »PTSEP. C 3 V = CMPLX(0.0,0.0) DO 125 11=1,NUM X1=-((X.NX(I1).DIST(I1)) ♦ (NX(II)*PI*DIST(11)*PTSEP)) Y1 = -Y*NY(II)*01ST(II) C(I1) = CEXP(CMPLX(0.0,XI + Yl)) 125 V = V ♦ 0(11) V1=CABS(V) C C THE FIELD AMPLITUDE DUE TO THE POINT SOURCE LOCATED AT -»PTSEP IS: C CONTINUE IF (ANNRAD.EQ.0.00)F1=0.00 8(1, J) = (Fll - (FI* (1 .00-TA) )) * VI IF (COHER.EQ.60)CO TO 701 C C THE INTENSITY DUE TO THE POINT SOURcE LOCATED AT +PTSEP IS: C (USE THE NEXT LINE FOR INCOHERENT IMAGING ONLY) C B(I, J) = 8(1, J) **2 701 CONTINUE C C THE TOTAL INTENSITY DUETOBOTHPOINT SOURCES IS(FOR INCOHERENT): C THE TOTAL FIELO AMPLITUDEDUE TO BOTH POINTSOURCESIS (FOR COHERENT): C 28 IF (PTSEP.E Q .0.00)B(I,J)=0.00 C C IF NEED TO LOOK AT FIELD FROM A(I,J) ONLY, TAKE COMMENT OFF OF NEXT C LINE AND PLACE COMMENT ON A(I,J)=0.0 C C B(I,J)=0.0 C C IF NEED TO LOOK AT FIELD FROM B(I,J) ONLY, TAKE COMMENT OFF OF NEXT C LINE AND PLACE COMMENT ON B(I,J)=0.0 C C A(I,J)=0.0 C(I,J) = A (I,J) ♦ B(I,J) IF (COHER.E Q .50)GO TO 702 C C THE INTENSITY IN THE IMAGE PLANE IS (USE FOR INCOHERENT): C INT(I,J)=C(I,J)/DNORM IF (COHER.EQ.61)GO TO 280 C Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 C USE THE NEXT LINE FOR COHERENT IMAGING ONLY. THIS LINE WILL PROVIDE C INTENSITY INFORMATION C C702 INT(I,J) =C (I,J) •*2/DN0RM C! I!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!'!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! C IF NEED TO LOOK AT FIELD AMPLITUDE, PLACE COMMENT ON LINE ABOVE AND TAKE C COMMENT OFF OF THE NEXT LINE. USE FOR MODULUS CALCULATIONS!!!!!!! C 702 INT(I ,J) =C (I,J) /DNORM C 280 CONTINUE 276 CONTINUE C C FIND THE MAXIMUM VALUE OF INTENSITY C H=INT(1,1) DO 10 1=1,48 DO 20 J=1,48 IF (INT(I,J).G T .H)H=INT(I, J) 20 CONTINUE 10 CONTINUE WRITE (*,70) PRINT*,H C C NORMALIZE INTENTSITY TO A MAXIMUM VALUE OF 1.00 C DO 16 1=1,48 DO 17 J=1,48 C C ii i i ; ; I j m | I I ; j i ; | ) I I I I j i I | I i i i | i j ; i j I i ; I ; j j ! I IjI i 'J c COMMENT’OUT NEXT LINE IF'NORMAL12INC VALUES TO A REFERENCE C FROM ANOTHER APERTURE SYSTEM C INT(I,J)=INT(I,J)/H C C iiiiiiiiiiii iii iiiiiiiiiiiiiiii!iiiiiii i iiijilliii C ‘USE ’NEXT'LINE'NORMALIZING'INTENSITY'VALUES T O ’A REFERENCE VALUE C C INT(I,J) = INT(I,J)/8.999998 CNEXT LINE CALCULATES THE DB VALUES. C INT(I,J) = 10.0 • ALOG10(INT(I,J)) C C I I I I 11 I I l l l l I I l I I I 11 I I I ! i I ! I I j I I I i i I M M I i ! i | j I f I i i C NEXT LINE SETS VALUES FOR THRESHOLD VALUES OF INTENSITY C WHEN PLOTTING EXAMPLES,ONLY. COMMENT OUT THIS NEXT LINE C IF PERFORMING ACTUAL CALCULATIONS C C IF(INT(I,J).LT..6)INT(I,J)=.5 C CI !IIIIII|III•IM Ii;IIiiiii!II!!!!!!<!!!!!!!!!!!!!!!1! C j!!!!!!!■!!!!!!!!!!!!!!!!!!!!H !!!!!!!!!!!!!!!!!!!!!! C COMMENT OUT NEXT LINE IF FINDING 2 DIM INTENSITY VS. C POINT SEPARATION PLOTS C WRITE(15,42)I ,J,INT(I,J) 17 CONTINUE 15 CONTINUE 170 = 0 C C i i i i i i i i i I I i ! ! ! ! ! ! ( I ( ( i !| I i | |I i i j i i (i ( i (! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! C THE NEXT’SECTION'COMPUTES'THEENERGY UNDER THE DIFFRACTIONPATTERN C C IF DO NOT WANT TO COMPUTE THE ENERGY, TAKE COMMENT OFF THE NEXT LINE C Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 170 = 70 IF(170.EQ.70)CO TO 8X00 WRITE(*,71) 110 = 0 C INPUT X AND Y DIMENSIONS NNX = 48 NNY = 48 C COMPUTE X AND Y VALUES AND READ THE VALUES OF INT(X,Y) FOR EACH X,Y. DO 6000 II = 1,48 110 = 110 * 1 ^ cs DO 6001 JJ = 1,48 X X (110) = -(110 ♦ (24.0 - 2.0 • 110))*FMULT Y Y (JJ) = (JJ - 24.0)*FMULT F(I10,JJ) = INT(110,JJ) C PRINT*,110,JJ C PRINT.,110,JJ,F(I10,JJ) 6001 CONTINUE 6000 CONTINUE C INPUT THE EXACT DECLARED ROW (X) DIMENSION OF INT(X,Y) IFD=200 C BEGIN THE RADIAL ENERGY CALCULATIONS C FM = FMULT/2.0 C COMPUTE THE VALUE FOR DETERMINING WHETHER TO USE THE DATA C FIELD SURROUNOING THE DATA POINT RADI = SQRT(2.0*(FM..2.0)) •RAD = 0.0 C DETERMINE THE NUMBER OF ITERATIONS(NUMBER OF POINTS TO BE PLOTTED C ON THE CHART. NREF = (YY (48)/FMULT) *2 - 2 C BEGIN THE CALCULATIONS DO 8020 I = 1, NREF 150 = 1 IF(I.E Q .1)GO TO 8200 C COMPUTE THE LENGTH OF THE RADIUS FOR SAMPLING THE DATA FIELD RAD = RAD * (FMULT)/2.0 IF(RAD.G T .Y Y (48))GO TO 8050 DO 8030 110 * 1,48 DO 8040 J10 = 1,48 C COMPUTE THE LOWER LIMIT FOR THE XX COORDINATE AA = XX(I10) - FM C COMPUTE THE UPPER LIMIT FOR THE XX COORDINATE BB = XX(I10) * FM C COMPUTE THE LOWER LIMIT FOR THE YY COORDINATE CC = YY(J10) - FM C COMPUTE THE UPPER LIMIT FOR THE YY COORDINATE DD = Y Y (J10) ♦ FM C COMPUTE THE PRESENT DATA POINT DISTANCE FROM THE ORIGIN (24,24) RADILOC = SQRT(XX(I10)*«2 ♦ YY(J10).*2) C DETERMINE IF ONE IS TO USE THIS DATA POINT LOCATION FOR ENERGY CALC. IF((RADILOC.GT.RADfRADl).OR.(RADILOC.LT.RAD-RADl))GO TO 8040 C COMPUTE THE ENERGY(VOLUME) UNDER THE DIFFRACTION PATTERN AT THIS C PARTICULAR POINT LIMITED BY THE ABOVE LIMITS CALL DBCQDU(F,IFD,X X ,NN X ,Y Y ,NNY,A A ,B B ,C C ,D D ,Q ,W K ,IER) 150 = 150 ♦ 1 C ADD VOLUME ELEMENTS APPLICABLE TO THE PARTICULAR RADIUS OF INTEREST QQA(I50) = QQA(I60 - 1) ♦ Q QQ(I) = QQA(I60) GO TO 8040 8200 QQA(I) =0.0 QQ(I) = 0.0 PRINT*,I ,RAD,QQ(I) GO TO 8020 8040 CONTINUE Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ] 8030 CONTINUE C SUM THE ENERGY VALUES FROM EACH RADIUS TO THE PREVIOUS RADIUS QQ(I) * QQ(I-l) ♦ QQ(I) PRINT*,I ,RAD,QQ(I) 8020 CONTINUE C NORMALIZE THE MAXIMUM VALUE OF ENERGY TO 1.00 QQMAX = QQ(I-l) PRINT*,1-1,QQMAX DO 80S0 I = 1,NREF QQ(I) = QQ(I) /QQMAX C PRINT»,I,QQ(I) WRITE(31,44)I,qq(I) 8060 6100 CONTINUE CONTINUE PRINT* .qQMAX C iIIIIiI IIIiiI I11MiI IiI>iiiI I|I|!If!I||{ |!!! iIiI!| iii! -j o C F I W THE MAX INTENSITY ON r !h !s ! OF PLANE C H1=INT(1,1) DO 11 1=1,24 00 21 J=l,48 IF (1NT(I,J).GT.H1)H1=INT(I,J) IF (HI.Eq.INT(I,J))K1=I IF (HI.Eq.INT(I,J))L1=J 21 CONTINUE 11 CONTINUE C C FIND THE POINT SEPARATION WHEN THE SUMMATION OF THE TWO C PRIMARY PEAKS REACHES A MINIMUM. THIS NEXT PROCEDURE PERFORMS C THIS FUNCTION. CENTMIN IS THE VALUE OF 1100 (POINT SEPARATION) C WHERE THE SUMMATION OF THE TWO PRIMARY PEAKS IS A MIN. c CENT(1100)=INT(24,24) IF((I100.Eq.1) .OR.(I100.OT.CENTMIN))GO TO 1281 DO 1270 160=2,1100 IF(CENT(160).GT .CENT(160-1))CENTMIN=160-1 CONTINUE CONTINUE 1270 1281 C C CHECK IF HAVE A MAX INTENSITY (MAX SECONDARY INTENSITY) C AT INT(24,24) WHEN THE PRIMARY PEAKS HAVE CEASED TO C ADD TO ONE ANOTHER. C IF((I100.GT.CENTMIN).AND.(INT(24,24).CE..999))G0 TO 1806 C C NEXT LINE WILL EXECUTE IF SECONDARY MAX PEAK(NOT C INCLUDING THE POSSIBLE MAX SECONDARY PEAK AT INT(24,24)) C > PRIMARY PEAK C IF (L1.NE.24)C0 TO 1050 PRIM(1100)=H1 GO TO 1100 C C FIND MAX OF PRIMARY PEAK ON R.H.S. OF PLANE WHEN SECONOARY C PEAK (NOT INCLUDING A MAX SECONOARY PEAK LOCATED AT INT(24,24)) C IS GREATER THAN PRIMARY PEAK C 1060 H60=INT(1,24) DO 1061 1=1,24 IF(INT(I,24).GT.H60)H60=INT(1,24) 1061 CONTINUE PRIM(I100)=H60 C WRITE(*,61) C WRITE(*,86)H60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 c » ° C1100 IF ((K1.EQ.24).AND.(L1.EQ.24).AND.(PTSEP.GT.0.0))WRITE(*,61) 1100 CONTINUE C C FIND THE MAX INTENSITY ON L.H.S. OF PLANE C H2=INT(1,1) C 00 12 1=24,46 C DO 22 J=l,48 C IF (INT(I,J).GT.H2)H2=INT(I,J) C IF (H2.EQ.INT(I,J))K2=I C IF (H2 .EQ.INT (I,J) )L2=J C22 CONTINUE C12 CONTINUE C IF (L2.NE.24)GO TO 1150 C GO TO 1200 C C FINO MAX OF PRIMARY PEAK ON L.H.S. OF PLANE WHEN THE C SECONOARY PEAK IS GREATER THAN THE PRIMARY PEAK C C1160 H51=INT(48,24) C DO 1151 1=48,24,-1 C IF(INT(I,24).G T .H61)H51=INT(I,24) C1161 CONTINUE C WRITE(* ,62) C WRITE (•, 87) H61 C C1200 CONTINUE C IF((K2.EQ.24).AND.(L2.EQ.24).AND.(PTSEP.GT.0.0))WRITE(.,62) C 180S CONTINUE C C FIND THE LOCATION OF THE FIRST MIN TO THE RIGHT OF THE C PRIMARY PEAK IN THE RIGHT PLANE C DO 401 I3=K1,1,-1 IF (INT(13-1,LI).G T .INT(13,LI))GO TO 402 401 CONTINUE 402 IMIN1R=I3 C C IIMIN1R IS THE VALUE OF THE MIN C IIMIN1R=INT(13,L1) C C FINO THE LOCATION OF THE FIRST MIN ABOVE THE PRIMARY PEAK C IN THE RIGHT HAND PLANE C DO 403 J3=L1,48 IF (INT(K1,J3*l).GT.INT(K1,J3))GO TO 404 403 CONTINUE 404 JMIN2R=J3 C C FINO THE LOCATION OF THE FIRST MIN TO THE LEFT OF THE PRIMARY C MAX IN THE RIGHT HANO PLANE C 00 405 I3=K1,24 IF (INT(13*1,LI).GT.INT(13,LI))CO TO 406 405 CONTINUE 406 IMIN3R=I3 C C FIND THE LOCATION OF THE FIRST MIN BELOWTHEPRIMARY C MAX IN THE RIGHT HANO PLANE C DO 407 J3=L1,1,-1 IF (INT(K1,J3-1).G T .INT(K1,J3))CO TO 408 407 CONTINUE 408 JMIN4R=J3 C C FIND THE SECONOARY MAX INTENSITY IN THE R.H.S. OF PLANE C H3=INT(1,1) DO 13 1=1,24 DO 23 J=l,48 13=1 J3=J IF (CCI3.LT.IMIN3R).AND.CI3.GT.IMIN1R)) .AND. k ((J3.L T .JMIN2R).AND.(J3.G T .JMIN4R)))GO TO 23 IF (INT(13,J3).C T .H3)GO TO 1800 GO TO 23 1800 H3=INT(13,J3) IRSECMAX=I3 23 CONTINUE 13 CONTINUE IF(PTSEP.E q .0.0)H69=H3 C C IF HAVE A MAX SECONDARY INTENSITY AT INT(24,24) WHEN POINT C SEPARATION IS SUCH THAT PRIMARY PEAKS ARE NOT SUMMING AT INT(24,24) C EXECUTE NEXT LINE C IF((1100.GT.CENTMIN).AND.(INT(24,24).GE..999))GO TO 1810 C C IF THE POINT SEPARATION OF THE TWO POINTS IS SUCH THAT THE PRIMARY C PEAKS ARE SUMMING AT INT(24,24), EXECUTE THE NEXT LINE C IF(1100.LE.CENTMIN)GO TO 1695 GO TO 1598 1810 PRIM(I100)=H3 C WRITE(*,66) C PRINT*,INT(24,24) SEC(1100)=1NT(24,24) GO TO 1S99 C C FIND SECONDARY MAX INTENSITY IN RHS PLANE IF 2 PRIMARY PEAKS C ARE SUMMING AT INT(24,24);I .E .THERE IS A CONTRIBUTION FROM C THE SUMMATION OF THE TWO PRIMARY PEAKS. C C FINO FIRST MIN TO RIGHT OF THE PRIMARY PEAK IN RHS C PLANE IF THE TWO PRIMARY PEAKS ARE SUMMING C 1595 CONTINUE IF((I100.LE.CENTMIN).AND.(INT(24,24).LT..999))IRSECMAX=Kl IF((I100.LE.CENTMIN).AND.(INT(24,24).LT..999))PRIM(I100)=H1 DO 1475 I3=IRSECMAX,1,-1 IF (INT(13-1,LI).GT.INT(13,L1))GO TO 1478 1475 CONTINUE 1478 I2MIN1R=I3 C PRINT*,PRIM(I100) C C FIND THE SECONDARY MAX INTENSITY IN RHS PLANE IF THERE IS A C CONTRIBUTION DUE TO THE SUMMING OF THE PRIMARY PEAKS. C H3=INT(1,1) DO 1700 1=1,24 DO 1705 J=1,48 13=1 J3=J IF(((13.C T .I2MIN1R).AND.(13.L E .24)).AND. k ((J3.LT.JMIN2R).ANO.(J3.GT.JMIN4R)))G0 TO 1705 IF(INT(13,J3).G T .H3)H3=INT(13,J3) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1705 CONTINUE 1700 C 1598 C C CONTINUE CONTINUE WRITE(*,65) PRINT*,H3 SEC(1100)=H3 CONTINUE 1699 C C FIND THE LOCATION OF THE FIRST MIN TO THE RIGHT OF THE PRIMARY C PEAK IN THE LEFT HANO PLANE C C DO 410 I4=K2,24,-1 C IF (INT(I4-1,L2).GT.INT(I4,L2))G0 TO 411 C410 CONTINUE C411 IMIN1L=I4 C C FIND THE LOCATION OF THE FIRSTMIN ABOVE THE C MAX IN THE LEFT HAND PLANE PRIMARY C Co M C DO 412 J4=L2,48 C IF (INT(K2, J4 +1).GT.INT(K2,J4))G0 TO 413 C412 CONTINUE C413 JMIN2L=J4 C C FIND THE LOCATION OF THE FIRSTMIN TO THE LEFT OF THE PRIMARY C MAX IN THE LEFT HANO PLANE C C DO 414 I4=K2,48 C IF (INT(I4+1,L2).GT.INT(I4,L2))GO TO 415 C414 CONTINUE C416 IMIN3L=I4 C CFIND THE LOCATION OF THE FIRST MIN BELOW THE PRIMARY C MAX IN THE LEFT HANO PLANE C C 00 416 J4=L2,1,-1 C IF (INT(K2,J4-1).CT.INT(K2,J4))GO TO 417 C416 CONTINUE C417 JMIN4L=J4 C C FIND THE SECONOARY MAX INTENSITY IN L.H. PLANE C C H4=INT(1,1) C DO 14 1=24,48 C DO 24 J=l,48 C 14=1 C J4=J C IF (((I4.LT.IMIN3L).AND.(14.GT.IMIN1L)) .AND. C k ((J4 .LT.JMIN2L).AND.(J4.GT.JMIN4L)))GO TO 24 C IF (INT(14,J4).GT.H4)H4=INT(14,J4) C24 CONTINUE C14 CONTINUE C WRITE(*,«0) C PRINT*,H4 C IF ( (H3 .T.Q. INT(24 ,24) ) .AND .(H4 .EQ.INT(24 ,24) ) )WRITE(* ,63) C WRITE(*,»S)INT(24,24) C CiI iiiiI iiM iii'i1iI■iiIM I!;!!IM •I{!!!<!!■! !!!!!!!! !!!!!!!! ! C TAKE OFF‘COMMENT’O N 'NEXT LINE IF WANT APERTURE SEPARATION VS C POINT SEPARATION 2-DIM PLOTS C C1000 CONTINUE C Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 C FIND THE LOCATION (POINT SEPARATION) WHERE THE SUMMATION OF THE C TWO PRIMARY PEAKS DECREASES FROM 1.00 C 00 1300 I10=1,CENTMIN IF(CENT(110).LT.1.0)GO TO 1301 CONTINUE INIT=I10 1300 1301 C C WRITE INT(24,24) VALUES FROM EACH POINT SEPARATION WHERE C THE TWO PRIMARY PEAKS SUMMED TOGETHER C DO 1325 110=1,CENTMIN WRITE(18,44)I10,CENT(110) 1325 CONTINUE C C WRITE PRIMARY PEAK VALUES FROM EACH POINT SEPARATION TO A FILE C STARTING FROM X LOCATION INIT C DO 1400 I20=INIT,1100 WRITE(19,44)120,PRIM(120) 1400 CONTINUE C C WRITE SECONDARY MAX VALUE FROM EACH POINT SEPARATION TO A FILE C DO 1450 130=1,1100 IF((130.EQ.1).AND.(PTSEP.EQ.0.0))SEC(130)=H89 WRITE(20,44)130,SEC(130) 1450 CONTINUE C C FIND POINT SEPARATION WHERE THRESHOLD VALUE IS MET OR EXCEEDED CD c w 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025 2030 2035 DO 1960 14=1100,1,-1 IF((SEC(14).GE..1).OR.(CENT(14).GE..1))G0 CONTINUE WRITE(21,38)DIS,14 DO 1970 14=1100,1,-1 IF((SEC(I4).GE..2).OR.(CENT(I4).GE.,2))G0 CONTINUE WRITE(22,36)DIS,14 DO 1980 14=1100,1,-1 IF((SEC(I4).GE..3).OR.(CENT(I4).GE.,3))G0 CONTINUE WRITE(23,38)DIS,14 DO 1990 14=1100,1,-1 IF((SEC(I4).GE..4).OR.(CENT(I4).GE..4))G0 CONTINUE WRITE(24,38)DIS,14 DO 2000 14=1100,1,-1 IF((SEC(14).GE..5).OR.(CENT(14),GE..6))GO CONTINUE WRITE(25,38)DIS,14 DO 2010 14=1100,1,-1 IF((SEC(I4).GE..6) .OR.(CENT(I4).GE.,6))G0 CONTINUE WRITE(26,38)DIS,14 DO 2020 14=1100,1,-1 IF((SEC(14).GE ..7)-OR.(CENT(14).GE..7))GO CONTINUE WRITE(27,38)D1S,14 DO 2030 14=1100,1,-1 IF((SEC(14).GE..8).OR.(CENT(14).CE..8))CO CONTINUE WRITE(28,38)DIS,14 DO 2040 14=1100,1,-1 TO 1965 TO 1975 TO 1985 TO 1995 TO 2005 TO 2015 TO 2025 TO 2035 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 00 F~ IF((SEC(I4).GE..9).OR.(CENT(I4).GE.,9))G0 TO 2046 2040 CONTINUE 2046 WRITE(29,38)DIS,14 C WRITE CENTRAL SLICE OF INTENSITY; I.E. A ONE DIMENSIONAL VIEW. C THE FIRST L00P(J=1,48) VIEWS THE CENTRAL SLICE ALONG THE Y AXIS(X=0.0) C DO 426 J=l,48 WRITE(17,44)J,INT(24, J) 426 CONTINUE C C THIS SLICE VIEWS ALONG THE X AXIS (Y=0.0) C DO 426 1=1,48 WRITE(16,44)I,INT(I,24) 425 CONTINUE CLOSE(UNIT=16) CLOSE(UNIT=18) CLOSE(UNIT=17) CLOSE(UNIT=18) CLOSE(UNIT=19) CLOSE(UNIT=20) CLOSE(UNIT=21) CLOSE(UNIT=22) CLOSE(UNIT=23) CLOSE(UNIT=24) CLOSE(UNIT=25) CLOSE(UNIT=26) CLOSE(UNIT=27) CLOSE(UNIT=28) CLOSE(UNIT=29) CLOSE (UNIT=30) CLOSE (UNIT=31) STOP END APPENDIX B EXPERIMENTAL DESIGN 85 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. Experimental Design Figure B-l is a representation of the experimental configuration used to photograph the diffraction patterns, measure the secondary lobe irradiances, and measure the percentage of irradiance in the central lobes generated by the multi-aperture systems. This configuration was aligned using the methods and techniques described in Mills 8 9 and Taylor . The light source for the experiments was a 50 milliwatt Helium-Neon laser. The laser beam passed through a 10X microscope objective followed by a spatial filter (10 micron pinhole) which functioned as a single point source. The spherical wavefront that was generated by the pinhole passed through absorbing filters. These neutral density filters were used to reduce the irradiance at the detector in order to prevent saturation. filtered spherical wave continued through a doublet. The This doublet functioned as a collimating lens to produce a plane wave. The plane wave passed through the multi-aperture mask (photographic plates). The masks were made in several steps. inscribed on sheets of rubylith. outlines was removed. The aperture designs were The rubylith surrounding the aperture The sheets were placed on a light board and photographed on 2X2 inch glass plates. clear transmitting apertures. The developed plates featured In all cases, the equivalent diameter of all aperture systems vas 0.5 inch. Each photographic mask was placed in a liquid gate; i.e., an optical flat with index of refraction fluid vas placed on each side of the mask, so that aberrations due to the photographic glass could be minimized. After the plane wave passed through the mask, the resulting field vas focused by the next doublet. The resulting irradiance pattern vas 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. magnified by a 20X microscope objective. This image vas then photographed or analyzed using a linear detector array (I-Scan 256 element line scan sensor) and a FND-100 detector. An adjustable iris vas interposed betveen the image plane and the FND-100 in order to stop incremental portions of the diffraction patterns in order to determine the percentage of irradiance in the central lobes. 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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