Calculation on diffraction aperture of cube corner retroreflector

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2 November 10, 008 / Vol., No. 11 / CHINESE OPTICS LETTERS 8 Calculation on iffraction aperture of cube corner retroreflector Song Li (Ó Ø, Bei Tang (», an Hui Zhou ( ï School of Electronic Information, Wuhan University, Wuhan Receive April 8, 008 On the basis of optical property of cube corner retroreflector (CCR, a new perception an calculation approach for iffraction aperture of CCR in two ifferent forms is presente. The relationship between iffraction apertures an incient light with six ifferent combinations of reflection orer an incient angle is establishe. Far-fiel iffraction patterns of CCR uner various incient conitions are also provie. OCIS coes: , , oi: /COL Retroreflector has been wiely use in various laser measurement systems as a cooperative target base on its irect reflecting [1,]. Satellite laser ranging (SLR which is a high-precision space measurement technology comes forth in the 190s. The laser cooperation target in SLR system is laser cube corner retroreflector (CCR or laser retroreflector array which is installe on the surface of the satellite. The orbital altitue of satellite is generally several hunre kilometers to tens of thousans of kilometers. Therefore, the SLR system can be regare as a Fraunhofer iffraction optical system. On the one han, CCRs on the satellite irectionally reflect the laser pulse which comes from the observing station; on the other han, CCRs iffract an reistribute energy of the laser pulse as iffraction aperture. The light fiel istribution iffracte by CCRs is of great significance for the laser ranging system to receive laser pulse echo correctly an to accomplish the ranging function exactly. The most important parameter which influences the far-fiel iffraction characteristics of CCR is the integral region of iffraction aperture. However, previous researches into the far-fiel iffraction of CCR take the iffraction aperture as a whole. We want to explore the eeper reason what makes the light fiel istribution of CCR be taken on like that an why the certain part of light spot is much better than other parts of light spot for the reception. In this paper, the iffraction aperture theory of oblique incience on the CCR is establishe. The CCR consists of three mutually orthogonal planar surfaces an one flat bottom surface. The light which enters the CCR through the bottom surface is reflecte by the three planar surfaces in turn. There are six ifferent reflection orers of light as three reflective surfaces have six ifferent sequences of arrangements []. The reflection orer of light epens on the incient irection an coorinates of light on the bottom surface of CCR. Take the vertex of CCR as the origin an three eges of CCR as X, Y, Z axes to establish the coorinate system, as shown in Fig. 1. Without regar to refraction, for an incient light with any irection (a, b, c, the light irection converts to ( a, b, c after being reflecte by the three reflective surfaces respectively. As shown in Fig., taking the mipoint of bottom surface as the origin o, the bottom surface as x-y plane, an the irection of light which comes out of the CCR after being reflecte by the three reflective surfaces sequently as z-axis, we establish the oxyz coorinate system. Draw a vertical line to one of the bottom surface borers across point O in X-Z plane, with the foot of the perpenicular labele as point T. It is assume that the angle between line OT an z-axis is ϕ, an the angle between line OT an line Oo is ω. These two sets of coorinates are provie with coorinate transformation matrix as [4] M = ( cosϕ sinϕsin ω sinϕcosω 0 cosω sin ω sin ϕ cosϕsin ω cosϕcos ω Fig. 1. Cartesian coorinates of CCR.. (1 Fig.. Schematic iagram of coorinate transformation of CCR /008/ c 008 Chinese Optics Letters

3 84 CHINESE OPTICS LETTERS / Vol., No. 11 / November 10, 008 The coorinate transformations between OXY Z an oxyz coorinate systems go as ( X Y Z = = M ( x y z + 0 ( x y z +, ( where is efine as the right-angle ege length of the CCR. Now we suppose that the unit vector of the incient light in the oxyz coorinate system is ( sin φcosθ, sinφsin θ, cosφ just as shown in Fig. an that in the OXY Z coorinate system is (, a, a. The angle φ of the unit vector is incient angle an the angle θ is azimuth angle. These two sets of unit vector are tie up by the coorinate transformation matrix above. If taking account of refraction, we only nee to multiply the unit vector by a certain factor. It is of the opinion that only lights which are incient on certain area of the bottom surface can be reflecte respectively by those three right-angle planes (planes 1,, an an come out through the bottom surface [5]. As mentione above, there are six ifferent reflection orers of light, which are 1, 1, 1, 1, 1, an 1. It is assume that the intersecting points between light an planes 1,, an are (0, Y 1, Z 1, (X, 0, Z, an (X, Y, 0, respectively, in the OXY Z coorinate system. It is assume that light coming out from the CCR intersects the bottom surface at point (X, Y, Z in the OXY Z coorinate system an point (x, y, z in the oxyz coorinate system. Same as above, the incient light intersects the bottom surface of CCR at point (X, Y, Z in the OXY Z coorinate system an point (x, y, z in the oxyz coorinate system. As it has been figure out that six sets of reflection orer match with six effective iffraction apertures corresponingly, we can get access to effective iffraction aperture of the CCR by fitting these six smaller apertures together []. We will take the reflection orer of 1 as an example to illustrate how to calculate the effective iffraction aperture specifically. Optical property of CCR that is mentione above an knowlege about geometrical optics are applie to obtain a set of equation below: { X X = Y Y a = Z a X + Y + Z =. ( Solve Eq. (, we get X = a1 a1y+x(a+a +a +a Y = a( X Y a 1+a +a + Y. (4 Z = a( X Y +a +a One shoul notice that X, Y, Z are in the OXY Z coorinate system here, an the esire iffraction aperture is on the bottom surface of CCR. It is time to utilize the coorinate transformation matrix to make X, Y, Z convert to x, y, z in the oxyz coorinate system: x = y = z = 0 (a1 a +X (a +a +(a Y +a +a ax +(+a Y +(a a +a +a. (5 Regaring X an Y as attributive variables an x, y as inepenent variables in Eq. (5, then we get { X = a + a x a a y 1 ( a Y = a a x + a +a a y 1 a (a a a. ( Furthermore, we get the relational expressions of (X, Y, 0 with (0, Y 1, Z 1 an (X, 0, Z : X 1 = 0 X = X a1y a Y = 0, Z = ay a Y 1 = a X Y Z 1 = a X. (7 Same as above, we get the relational expression of (x, y, z with (0, Y 1, Z 1 : x = (a1 a +(a Y 1 (+a Z 1 +a +a y = z = 0 (a1+a Y 1 a Z 1+(a a +a +a. (8 The area{ coverage of {(x, y, 0 which { meets the constraints of Y1 0 Z 1 0, X 0 Z 0, X 0 represents the effective iffraction aperture which goes with Y 0 the reflection orer of 1. We shoul also pay attention to the confinement of the CCR s bottom shape on (x, y, z an (x, y, z. As shown in Fig. 4, they are the effective iffraction apertures of reflection Fig.. Schematic iagram of light reflection by CCR. Fig. 4. Effective iffraction aperture of the reflection orer 1. (a Triangular bottom with incient angle of 15 ; (b circular bottom with incient angle of 0.

4 November 10, 008 / Vol., No. 11 / CHINESE OPTICS LETTERS 85 Fig. 5. Effective iffraction aperture of CCR. (a Triangular bottom with incient angle of 15 ; (b circular bottom with incient angle of 0. Fig. 7. Diffraction aperture variation curves with ifferent reflection orers. (a Reflection orer 1 ; (b 1 ; (c 1. Fig.. Effective iffraction aperture of CCR with circular bottom for ifferent values of incient angle φ an azimuth angle θ. (a φ = 5, θ = 0 ; (b φ = 5, θ = 0 ; (c φ = 40, θ = 0 ; ( φ = 40, θ = 0. orer 1 coming along with triangular bottom an circular bottom respectively. Calculating other five effective iffraction apertures an combining these six apertures together, we obtain the effective iffraction aperture of CCR, which is presente in Fig. 5. The above metho is applie to get several pictures of effective iffraction aperture which iffer in the incient angle an azimuth angle. From Fig., it is worth noticing that the area of effective iffraction aperture oes not change with the azimuth angle which only affects the position of iffraction aperture. The effective iffraction aperture rotates an angle of θ compare with the iffraction aperture with azimuth angle 0, so the azimuth angle will have effect on the intensity istribution of CCR s far-fiel iffraction. The area of iffraction aperture ecreases as the incient angle increases, as shown in Fig. 7. The effective iffraction aperture ecreases to zero when the incient angle comes to a threshol value. The transformation rules for the six iffraction apertures coming from six reflection orers of incient light are not ientical, but thanks to symmetrical characteristics, there are three groups of transformation rules. Reflection orers 1 an 1 have the same transformation rule in iffraction aperture, so o 1 an 1, 1 an 1. Three ifferent variation curves of iffraction aperture changing with incient angle are presente in Fig. 7, in which the iffraction apertures are normalize by the effective Fig. 8. Far-fiel iffraction pattern of CCR with the incient angle of 15 an azimuth angles of (a 0 an (b 0. The z axis gives the relative energy magnitue. iffraction aperture when the incient angle of light is zero. From Fig. 7, we can fin that the variation curves of iffraction aperture for the reflection orers 1 an 1 are much more similar compare with the variation curve for the reflection orer 1. The ecreasing rate of variation curve in Fig. 7(c ( 1 is smoother than the other two as the incient angle of light increases. The six iffraction apertures with six reflection orers are equal when the incient angle of light is zero, an they reuce to zero at the same time when the incient angle of light increases to a certain egree. In general, light spot reflecte back from the CCR on the satellite is too large to be receive completely by the receiving telescope on the observing station [7,8]. That is to say, most energy of light reflecte back has to be waste ue to the limite size of the receiving telescope. Uner the best conitions, the telescope receives the part of light spot of which energy ensity is relatively high. From Fig. 7, it can be easily foun that the proportion of iffraction aperture with reflection orer 1 is larger than the other two when the in-

5 8 CHINESE OPTICS LETTERS / Vol., No. 11 / November 10, 008 cient angle is not zero, an the iffraction aperture in this case changes more slowly than the other two. For this reason, the part of light spot which correspons to the reflection orer 1 or 1 is more preferable for the receiving telescope to etect. This can be further verifie by the far-fiel iffraction patterns of CCR shown in Fig. 8. Effective iffraction aperture an computational formula of iffraction presente in Ref. [9] are use to acquire the far-fiel iffraction pattern of CCR which is 90 km away from the observing station on the Earth an of which certain velocity aberration of light is offset. Figure 8 shows the patterns corresponing to incient angle 15 with azimuth angles of 0 an 0. The relative energy magnitue is given with the projectile energy on cube corner retroreflector assume to be a unit value. From Fig. 8, we can arrive at that the far-fiel iffraction pattern changes with the effective iffraction apertures because the effective iffraction aperture varies with incient angle an azimuth angle. We usually calculate the laser energy arriving at the receiving telescope in an approximate way that assumes the energy to be in irect proportion to the specular cross section of CCR [10]. But that is not the case, the receiving energy of ifferent parts in the light spot is ifferent, even the energy of the same part is changing with the incient angle an azimuth angle. We can fin the optimum point for receiving telescope by applying the approach propose in this paper. We have provie the calculation metho of CCR s iffraction aperture with ifferent incience conition an with ifferent reflection orer of light. The iffraction aperture has an influence on the far-fiel iffraction pattern of CCR an also on the reception part of SLR system. The results in this paper are of importance in the esigning of CCR. S. Li s aress is yflisong@public.wh.hb.cn. References 1. Z. Zhang, Z. Cheng, Z. Qin, an J. Zhu, Chinese J. Lasers (in Chinese 4, 94 (007.. H. Chen, X. Ding, Z. Zhong, Z. Xie, an H. Yue, Acta Opt. Sin. (in Chinese 7, 107 (007.. Q. Wan, Y. Guo, X. Wang, B. Sun, C. Lu, an S. Wei, Laser Optoelectron. Prog. (in Chinese 4, (5 0 ( H. Nie, X. Weng, S. Li, an J. Liu, Acta Opt. Sin. (in Chinese, 1470 ( Y. Cai, Z. Fang, G. Chen, an G. T. Chen, Chinese J. Lasers B 9, 49 (000.. H. Chen an J. Tan, J. Optoelectron. Laser (in Chinese 17, 98 ( J. J. Degnan, Millimeter accuracy satellite laser ranging: A review in Contributions of Space Geoesy to Geoynamics Technology, Geoynamics Series, Vol.5 (American Geophysical Union, Washington, 199 pp R. Neubert, in Proceeings of 10th Workshop on Laser Ranging Instrumentation 1 ( H. Zhou, S. Li, Y. Shi, X. Weng, an K. Hu, Opto- Electronic Eng. (in Chinese, (11 5 ( Y. Wang, F. Yang, an W. Chen, Opto-Electronic Eng. (in Chinese 4, (10 5 (007.