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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 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 用常规铣床制造准光镜Daniel Boucher, Jean Burie, Robin Bocquet, and Weidong Chen龚仁华译摘 要：这里描述了一种加工镜子的改进方法，仔细的选择机器的参数和运行的表面参数可以生产出具有足够的15的精度。大量的焦距在50毫米到900毫米之间的球形,抛物线形及椭圆形镜子是使用这种方法加工的。埃里克森已经研究出来了怎么把焦距比提到最高的方法。而这些光学仪器,基本上是用于研制我们的远红外光谱仪。使用相当低的源动力就可以在这类仪器上完成最精确的光线设计。整个系统将被描述成为一个独立的部分，人们将看到,光束传播时的功率损耗已经达到非常低的绝对水平,基本上不会超过1 - 2%。关键词：铣床：离轴角1.介绍在亚毫米波或者是远红外领域，光线的传递比光线的反射的损耗要高出很多。主要是由于传播的介质材料具有较高的吸收作用和制造增透层要求较高。从高斯光束转换方面来说，使用金属反射镜通常是一个很好的解决办法。反射聚焦镜在大功率处理的能力和宽频带操作上有额外的优势。N.R埃里克森在若干年前提出了一个很科学解决方法，他只用一台传统铣床来加工离轴镜。这种方法已经被许多工人在远红外领域中使用，至今仍然很受欢迎。它使光学器件的生产成本适中,而且不影响铣削过程中相关步骤中固有的数值。在本文中,我们应用抛物线原理和椭圆球面镜检查的细节对原来N.R埃里克森的方法提出了一些调整和改进。通过仔细估算误差函数再加以修改得到的结果表明：大尺寸衍射限制的镜子(即焦距比小的镜子)可以比原来工作预期的更容易制造。2.完成圆锥曲线使用N.R埃里克森的方法,确定圆锥曲线铣床的制造的表达式：r(z)=(ztan+S)+R-(z/cos)d ( 1 )下图表示铣床的配置。铣头与常规垂直轴的夹角为（90）。回转台的轴线被称为Z轴，在Z=0的平面上，平面上测得的S长度就是刀具和回转台轴线的距离。d的长度就是在垂直面上铣削轴线与回转台手臂的轴线之间的距离，R表示刀具切削轨道的半径。在任何情况下焦点都位于Z轴上的Z=0点。俯视图光轴投影旋转台轴线旋转台手臂部分切削刀具侧视视图光轴图：铣床机构原理图方程(1)得到的是双重值,但我们可以看到下面只有“-d” 对应一个真正的圆锥形函数。z级数展开大概为零，那么我们要解决的问就是：r(z)=S+(R-d)+(2Stane)z+(d/Rcose)-lz+(d/4 Rcos)z+(d/8 Rcos)z+ .+ e1n2k(Rcose) z ( 2 )其中e=d/(Rcos)，K=1,2,3.n下面这个函数可以与在（2）表达式中圆锥函数的一般表达式相比较：r(z)=eh+(2eh)z+(e-1)z ( 3 )这个表达式用来证实前面提到由N.R埃里克森得出的关于d的表达式的观点。在(3)中，e就是所谓偏心距参数,它的作用为：e=1的时候是抛物线e1的时候是双曲线h确定圆锥曲线准线的位置。式中清楚地表明一个旋转曲面可以被分为一个确定的数和发散函数高次项的总和E。即：E= （ 4 ）其中En=e1n2n( Rcos)z误差函数的收敛性就可以很容易地表示为z Rcose。因此适当的选择参数R,e、D(镜子尺寸)可以减少加工中的误差问题。方程(2)可以用来表示机械参数中所有的圆锥曲线。可以用三个同步的非线性方程解决。e=d/(Rcos2) ( 5 )h=Stan/e ( 6 )eh=S+(R-d) ( 7 )这可以说明这个方法并不只适用于一种解决方案。为了确定E函数的临界值也可以添加一些额外的约束。在简单考虑的基础上可以获得一个理论上的估计值。对于衍射限制的聚焦镜，表面粗糙度的均方值的不得超过/ 50。最大的表面误差要求小于/17，这样才能更好的达到其理想的性能。虽然这个问题不同于目前所说这种情况,表面误差并不是随机分布的，在衍射限制的条件下，可以用等效限制的方法来进行合理的运算。其中最大误差用r来表示。r =r(true conic curve)-r(actual generated curve) ( 8 )屈服于r= E/2r = Ez/2r 其中 E=d/(4Rcos) ( 9 )根据衍射限制的条件我们可以得到：r k/17 或者 E 2f/D （10）其中D是指镜子的直径。(5),(6),(7)和(9)给出了四个方程式，结合机械参数和镜子的参数可以得出：4Eh=( 1 +tan-e)tan/(1 +tan)(tan-e) ( 11)S=eh/tan (12)R=e/(2coseE) (12)d=Recos (14)这个问题就可以被精确地解决了。由给定的方程式可以确定Eo，e和h，的值。同样也可以得到S，R和d的值。方程(13)可以看出，R的值是越大越好。但是在实际情况下它的数值是有限制的,它受到机械本身和机械振动的约束。一般在我们使用的机械系统中R = 100毫米是一个最大的值。在这方面还有另外一点需要留意。我们注意到在当Z=0的时候会获得一个无效的廓形误差,相应的,对于一个抛物线,就是90的离轴镜。假如椭圆表面的无效误差明显的符合相同的值Z=0。我们将会在下面看到更多在这种情况下的详细说明，这代表一个特定的离轴的情况。偏心距参数e用来确定离轴角。在90离轴的情况下不能达到。在他的原著中对于任何加工型材任何离轴的情况的可能性，N.R埃里克森都做出了结论。这项结论并不适用需要达到最高的轮廓精度的情况。3.轴偏离90的抛物面镜子作为第一个例子,我们将讨论抛物面镜子，100毫米90焦距的时候,长轴固定,有限衍射为2500千兆赫。可接受的峰值错误是k / 17,接近6微米。对于抛物面：e=1h=f(1+cos)其中f是镜子的实际焦距，是离轴角。对于f / 5抛物面约束(10)得到：E1.2510由方程(11)可得, 可以取47.5。S，R，d的值就可以通过求解非线性方程(12)，(13)，(14)来确定，可以得到：S=91.5mm，R=74.1mm,d=34.0mm使用这些值可以得到一个误差函数(8)的图.II。这样我们就可以得出结论：这个装置在直径20为毫米，对应f / D = 5的时候将达到衍射极限。r(um)Z(mm)图 II 100毫米抛物面镜子焦距的误差函数4.椭球面镜从ABCD定律来看椭圆球面镜子可以看作是一个简单聚焦元件，其等效焦距f由下面得到：f=f1f2/(f1+f2)fl、f2分别是焦点和椭球面截面的中心点。(图.III)。焦点是总是位于镜子截面的中心。所以对离轴角小于90。偏心距由厦门可以得到：e=sin/(1+cos)这样正如上文所言。fl-f2这对参数完全确定了离轴角的值。参数可表示为：f1=f(1+cos),h=f1/sinE2f1/D现在我讨论椭圆球面镜子的例子。相关的参数：等效焦距f= 100毫米,离轴角近似等于70(即f1 = 134毫米)和f / 5，有限的衍射达到2500千兆赫,如：ED。6讨论性能受到扭曲影响的离轴镜(Ld)或者受到正交偏振影响的离轴镜(Lc):Ld=mtan(/2)/8fLc=mtan(/2)/4f其中m是镜面的光束半径,是镜子离轴角。考虑到镜子光圈和光线截断的影像,要得到耦合效率约99%的一个基本高斯光束需要镜面的直径至少大于光束半径的三倍。我们接下来可以得到：Ld=tan(/2)/128(f/D)Lc=tan(/2)/64(f/D)对于由90的离轴面镜产生的基本高斯光束其离轴损失与焦距比的关系在下面的图.IV中表示。这些损失在f / D3的时候明显可以忽略不计。 损失(%)f/D图.IV镜子离轴损失和f/ D的变化关系7.结论这里描述了一种加工镜子的改进方法，仔细的选择机器的参数和运行的表面参数可以生产出具有足够的15的精度。大量的焦距在50毫米到900毫米之间的球形,抛物线形及椭圆形镜子是使用这种方法加工的。埃里克森已经研究出来了怎么把焦距比提到最高的方法。而这些光学仪器,基本上是用于研制我们的远红外光谱仪。使用相当低的源动力就可以在这类仪器完成最精确的光线设计。整个系统将被描述成为一个独立的部分，人们将看到,光束传播时的功率损耗已经达到非常低的绝对水平,基本上不会超过1 - 2%。8.参考文献：1.N.R
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