汽車觀后鏡刮水及清洗系統(tǒng)設(shè)計(jì),汽車觀后鏡刮水及清洗系統(tǒng)設(shè)計(jì),汽車,觀后鏡刮水,清洗,系統(tǒng),設(shè)計(jì) International Journal of Infrared and Millimeter Waves, Vol. 13, No. 9, 1992 QUASI-OPTICAL MIRRORS MADE BY A CONVENTIONAL MILLING MACHINE Daniel Boucher, Jean Burie, Robin Bocquet, and Weidong Chen Laboratoire de Spectroscopie Hertzienne Universitd des Sciences et Technologies de Lille 59655 Villeneuve dAscq, France Received June 1, 1992 Introduction In the submillimeter-wave or far-infrared domain, transmissive optics have significantly higher losses than reflective optics. It results from the relatively high absorption of dielectric materials and from the difficulty of manufacturing anti-reflection layers. For Gaussian beam transformation, metal reflector mirrors provide usually a better solution. Reflective focusing mirrors offer additional advantages of high power handling capability and broad band operation. N. R. ERICKSON presented some years ago a very elegant method needing only a conventional milling machine to cut off-axis mirrors 1. This method has been exploited by a lot of workers in the far-infrared field and remains very popular. It allows the development of optical components at moderate cost and is free from the step effect inherently associated to numerical milling processes. In this paper, we present some modifications and corrections to the original ERICKSONs method. Applications to off-axis parabolic and ellipsoidal mirrors are examined in details. By a careful estimation of the error function and some modifications to the method, it is shown that diffraction-limited mirrors of larger size (i.e. of lower focal ratio) than expected in the original work can easily be manufactured. Realisation of the conic section Using ERICKSONs notations, the conical section generated by a milling machine is described by: r2(z)=(ztan0+S)2+R2-(z/cosO)2l/2+d 2 ( 1 ) 1395 0195-9271/92/0900-1395506.50/0 9 1992 Plenum Publishing Corporation 1396 Boucher et al. Fig. I represents the milling machine configuration . The mill head is tipped from usual vertical axis by an angle (90 The axis of the rotary table is defined as the Z axis; the distance between the plane of the cutter and the rotary table axis is measured as S in the Z=0 plane; d is the distance between the vertical plane containing the mill axis and the axis of the rotary table arm, and R is the radius of the cutter orbit. In any case the focal point is located at Z=0 on the z axis. Z z=o 90 . mill axis cutter rr piece to cut side view rotary table arm rotarg table axis mill axis projection | i 1 top view Fig. I Schematic of the milling machine setup Equation (1) is double valued, but as will be seen below only -d corresponds to a true conical function. The z series expansion around zero point is: r2(z)=S2+(R-d)2+(2Stane)z+(d/Rcos2e)-lz2+ (d/4 R3cos4O)z4+(d/8 R5cos6O)z6+ . + e21n2k25/16(Rcose)2(k-1)-1z2 k (2) where e2=d/(Rcos2e) and k=2, 3 . n. This series can be compared with the general expression of conical functions expressed in the focal representation 2: Quasi.Oplical Mirrors 1397 r2(z)=e2h2+(2e2h)z+(e2-1)z 2 (3) This expression confirms the previously mentionned observation done by ERICKSON relative to the sign of d. In (3), e is the so called excentricity parameter, the value of which is: e=l for a parabola el for a hyperbola and h fixes the position of the conic curve directrix. It clearly appears that a surface of revolution can be cutted with an accuracy limited by the sum E of higher order terms in the development function, i.e.: oo E= /_,En (4) n=2 where En=e21n2n25/16( Rcos0)2(n-1 )-1 z 2 n The convergence of this error function can be easily demonstrated for zRcose. So the machining error can be minimized by a proper choice of parameters R, e and D (mirror dimension). Any conic section can be fitted to equation (2) which is expressed in mechanical parameters. Three simultaneous nonlinear equations have to be solved. e2=d/(Rcos2e) ( 5 ) h=Stane/e 2 ( 6 ) e2h2=S2+(R-d) 2 (7) As an evidence this system does not admit a unique solution. An additional constraint can be added in order to fix a threshold value for the E function. A realistic estimation can be obtained on the basis of simple considerations. For diffraction-limited focusing mirrors, the rms surface roughness must be less than ./50 3. A peak surface error of less than X/17 is then required to closely approach ideal performance 4. Although the problem is different in the present case, where errors are not randomly distributed, an equivalent limit can reasonably situate the regime of operation in diffraction-limited conditions. The peak error Ar is then described in the form: 1398 Boucher et aL yielding Ar=lr(true conic curve)-r(actual generated curve)l (8) Ar = E/2r = Eoz4/2r where Eo=d/(4R3cos40) (9) According to the condition for diffraction-limited operation we have: Armax k/17 or EO 2.f/D 4 (10) where D is the mirror diameter. (5), (6), (7) and (9) give four equations combining mechanical parameters and mirrors parameters: 4 Eoh 2=( 1 +tan2O-e2)2tan20/(1 +tan2O)(tan20-e2) (11) S=e2h/tane ( 1 2 ) R=e/(2coseqEo) ( 1 3) d=Re2cos2e ( 1 4 ) This system can be exactly solved. For given Eo, e and h, O can be determined, then S, R, and d so do. Equation (13) reveals that R has to be taken as large as possible. In practice its value will be limited by mechanical and vibrational constraints. In our mechanical system R=100 mm is a maximum value. At this point another observation has to be done. We note that a null profile error is obtained for z=0 only, corresponding, for a parabola, to 90 off-axis mirrors. In case of elliptic surfaces the null error obviously corresponds to the same value z=0. As will be shown in more details below, it leads to a particular off-axis situation. The excentricity parameter e fixes the off- axis angle. The 90 off-axis situation cannot be reached. In his original work ERICKSON concluded on the possibility of machining profiles for any off-axis situation. This conclusion does not apply when the best achievable profile accuracy is needed. 90 off-axis paraboloidal mirrors As a first example we shall discuss the case of a paraboloidal mirror, 100 mm focal length, 90 off-axis, diffraction-limited up to 2500 GHz. The acceptable peak error will be k/17, close to 6 p.m. For a paraboloid: e=l and h=f(l+cos) where f is the effective focal length of the mirror and its off-axis angle. For a f/5 paraboloid the constraint (10) leads to: Quasi-Optical Mirrors 1399 E01.2510 -4 From equation (11), O must be taken equal to 47.5 S, R, d can then be determined by solving equations (12), (13), (14): S=91.5 mm, R=74.1 mm, and d=34.0 mm Using these values, a representation of the error function (8) is given by Fig. II. We can conclude that the device will be diffraction limited up to a diameter of 20 mm corresponding to f/D=5. Ar ( Lm) 8 I I I -10 0 10 Z (ram) Fig. II Error function for the 100 mm focal length parabolic mirror off-axis ellipsoidal mirrors From the ABCD law an ellipsoidal mirror can be treated as a simple focusing element with an equivalent focal length f given by: f=flf2/(fl+f2) fl, f2 are respectively the distance between focal points and the center of the cutted section of ellipsoidal surface (Fig. III). The focus is always located facing the center of the mirror to be cutted. So the off-axis angle is smaller than 90 The excentricity is given by: e=sin/(l+cos) 1400 Boucher et aL and where as previously stated, the couple of parameters fl-f2 totally defines the off-axis angle . The parameters can be expressed as: fl=f(l+cos), h=fl/sin, with Eo2M1/D 4 We shall now discuss the example of an ellipsoidal mirror with close parameters: equivalent focal length f=100 mm, approximative off- axis angle equal to 70 (i.e. f1=134 mm) with f/5 and diffraction-limited up to 2500 GHz, i.e. EoD. 1401 Discussion Performance of off-axis mirrors are affected by distortion (Ld) and cross-polarization (Lc) losses /5: Ld=e) m 2tan2(/2)/8f 2 Lc= rn 2tan2 (/2)/4 f 2 where m is the beam radius at mirror surface and is the mirror off- axis angle. Considering effects of mirror aperture and beam truncation, a coupling efficiency of about 99% for a fundamental Gaussian beam requires a mirror diameter at least three times larger than the beam radius 6. We have then: Ld=tan2(/2)/128(f/D) 2 Lc=tan2 (/2)/64(f/D) 2 The off-axis losses versus focal ratio are illustrated in Fig. IV for a fundamental Gaussian beam transformed by a 90 off-axis mirror. These losses obviously appear as negligible for f/D3. losses (%) 2. 1.5. .5. 0 0 2 ! i 4 6 f/D Fig. IV Variation in mirror off-axis losses with f/D 1402 Boucher et aL Conclusion A modified method for machining off-axis mirrors has been described. By a careful choice of machine parameters, revolution surface can be generated with a sufficient accuracy for ;L15 llm. A large set of spherical, paraboloidal and ellipsoidal mirrors, whose focal lengths are comprised between 50 mm and 900 mm, have been machined using the method. Focal ratio improving the uppest limit expected by ERICKSON have been manufactured. These optical devices have essentially been used in the development of our far-infrared heterodyne spectrometer. Due to the rather low source power achievable in these kind of intruments the greatest care has to be taken in the design of optical lines. The whole system will be described in a separate paper. It will be seen that powers losses in beam propagation have reached very low absolute levels, rarely exceeding 1 or 2%. Rfrences 1 N.R. ERICKSON, Off-axis mirror made using a conventional milling machine, Appl. Opt., 18, 956-957, 1979 2 G. GIRARD and A. LENTIN, Gomtrie/Mcanique, Hachette, 1964 3 P.F. GOLDSMITH, Quasi-optical techniques at millimeter and submillimeter wavelengths, Infrared and millimeter waves, 6, ch.5, 1982 4 J. RUZE, Antenna tolerance theory-A review, IEEE Prec., 54, 633-640, 1966 5 J.A. MURPHY, Distortion of a simple Gaussian beam on reflection from off-axis ellipsoidal mirrors, Int. J. Infrared and Millimeter Waves, 8, 1165-1187, 1987 6 J. LESURF, Millimetre-Wave Optics, Devices & Systems, Adam Hilger, Bristol and New York, 1990