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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 d'Ascq, 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 ERICKSON's 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 ERICKSON's notations, the conical section generated by a milling machine is described by: r2(z)=(ztan0+S)2+{\[R2-(z/cosO)2\]l/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+\[(d/Rcos2e)-l\]z2+ (d/4 R3cos4O)z4+(d/8 R5cos6O)z6+ .... + e21n2{\[k25/16(Rcose)2(k-1)\]-1}z2 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=e21n2{\[n25/16( Rcos0)2(n-1 )\]-1 }z 2 n The convergence of this error function can be easily demonstrated for z
3. 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 ;L>15 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%. R~f~rences \[1\] N.R. ERICKSON, "Off-axis mirror made using a conventional milling machine", Appl. Opt., 18, 956-957, 1979 \[2\] G. GIRARD and A. LENTIN, "G~om~trie/M~canique", 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