基于波数域相位补偿的白光干涉测量

White-Light Interferometry Based on Phase Compensation in Wavenumber Domain

白光扫描干涉测量中,频域分析(FDA)方法通过波数域的相位解包裹和线性拟合得到斜率来计算物体表面形貌,干涉仪中的振动噪声和背景噪声会给斜率测量带来误差,从而影响FDA方法的测量精度。研究发现,波数域相位线性拟合后的常数项能够有效补偿噪声对FDA方法的影响,波数域相位补偿(PCWD)方法通过对波数域的线性相位作傅里叶逆变换,得到空域的振幅和相位分布,以距离振幅最高点最近的相位零点测量光滑反射镜表面形貌。仿真结果表明:存在振动噪声时,PCWD方法有90%的概率补偿FDA测量结果中平均8.9 nm的误差,也有10%的概率给FDA测量结果额外引入平均1.9 nm的误差;存在背景噪声时,FDA与PCWD方法的平均测量误差分别为4.5 nm和0.6 nm。实验结果表明:FDA方法测量的反射镜表面结构的起伏小于25 nm,重复率为7.4 nm;PCWD方法测量的反射镜表面结构的起伏小于4 nm,重复率为1.1 nm。综上所述,PCWD方法在FDA方法的基础上进一步提升了测量精度。

Objective Interferometry is commonly used in industrial measurement because of its high accuracy,high speed,and noninvasive nature.Because the light with a single wavelength causes the limit of phase ambiguity,the white-light laser is used as a light source to provide a large measurement region and a better noise resistance.For white-light scanning interferometry,frequency-domain analysis(FDA)is a common method for obtaining the surface profile of an object from the slope of the phase,where the phase has been unwrapped and linearly fitted in the wavenumber domain.However,the measurement accuracy of FDA will decrease when the vibration and background noises change the slope of the linear-fitted phase.Thus,phase compensation in wavenumber domain(PCWD)is proposed to eliminate the effect of the vibration and background noises of FDA in this study.PCWD improves the measurement accuracy based on the FDA with simple signal processing.We assume that PCWD can be helpful in industrial measurement because of its high accuracy and stability.Methods The noise changes the slope of the linear-fitted phase in the wavenumber domain for FDA.Surprisingly,the constant term of the linear-fitted phase in the wavenumber domain compensates for the FDA noise-induced changes in the slope of the linear-fitted phase.The FDA and PCWD propose two parameters for determining the measurement as z_(s)=-α_(1)/4πand z_(p)[Eq.(5)],respectively.The difference is that the z_(s)is determined by the slope of linear-fitted phase α_(1),whereas the z_(p)is determined by the α_(1) and the constant term of linear-fitted phase α_(0).If the noise-affected α_(0) and α_(1) have the opposite trend,z_(p)[Eq.(5)]will be more accurate than z_(s).However,do the α_(0)and α_(1)always have the opposite trend?Because we are unsure about this,a simulation is performed to analyze the following questions:1)Does the z_(p)provide a higher measurement accuracy than z_(s)?2)Is it possible for both α_(0) and α_(1) to have the same noise-affected change trend,resulting in a larger measurement error for z_(p)?The FDA and PCWD are also used to measure a smooth mirror in the experiment.Thus,the roughness and repeatability could be used to estimate the FDA and PCWD because the ideal measurement result should be a smooth and continuous surface.Results and Discussions In a simulation,100000groups of vibration noiseφ_(n)(z)are added to the interference signal to obtain the z_(s) and z_(p).The position of the object is z_(0)=0μm.The value of z_(s)is distributed in the region of[-0.04μm,0.04μm];the value of z_(p)is distributed in the region of[-0.01μm,0.01μm](Fig.5).Thus,the z_(p) provides a higher measurement accuracy than the z_(s).Further,the change in α_(0)and α_(1)caused by vibration noiseφn(z)is analyzed[Fig.6(a)].The variation of α_(0) and α_(1)is denoted as theΔα_(0) andΔα_(1).If the trend of α_(0) and α_(1)is different,Δα_(0)×Δα_(1)<0.Fig.6(a)shows that the 90%data satisfiedΔα_(0)×Δα_(1)<0.If the trend of α_(0)and α_(1)is the same,Δα_(0)×Δα_(1)>0.There are 10%data satisfiedΔα_(0)×Δα_(1)>0.Based on Eq.(5),there is a 90%possibility that z_(p) is more accurate than z_(s).Simultaneously,z_(p)has a 10%possibility of being less accurate than z_(s).To analyze the character of α_(0),the influence of α_(0)on z_(p)is discussed[Fig.6(b)].There is a 90%possibility for z_(p)to decrease the average error of 8.9nm and a 10%possibility for z_(p)to increase the average error of 1.9nm.To conclude,the z_(p)can provide better measurement accuracy based on the entire surface than the z_(s).In addition,10groups of the background noise φ_(b) (z)obtained from the experiment are added to the interference signal to acquire the values of|z_(s)|and|z_(p)|(Table 1).The average values of|z_(s)|and|z_(p)|are 4.5and 0.7nm,respectively;the repeatability of|z_(s)|and|z_(p)|are 3.0and 0.5nm,respectively.Thus,the z_(p)has a better background noise resistance.The experimental result shows that the variation of the surface profile obtained from z_(s)and z_(p)is less than 25and 4nm,respectively;their repeatability are 7.4and 1.1nm,respectively.This result shows that PCWD provides good measurement accuracy and stability,which is consistent with the simulation results.Conclusions In the present study,PCWD is proposed to measure the surface profile based on FDA.PCWD uses the constant value of the linear-fitted phase in the wavenumber domain to compensate for the noise-induced error in the slope of the linear-fitted phase.Further,a simulation is performed to study the characteristics of PCWD.Unexpectedly,some kinds of noise will increase the measurement error for PCWD.However,the negative effect cannot deny the good performance of PCWD in measuring the entire surface.The experimental result also agrees with the simulation,which proves that PCWD has better measurement accuracy and stability.Our study provides the whitelight scanning interferometer with a method for determining the surface profile based on FDA with good noise resistance and helps the white-light scanning interferometer adapt to poor environments.