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干涉测量是目前成熟的高精度无损检测方法，测量准确度可达纳米量级(rms 值)，但常规单一短波长干涉测量范围易受工作波长制约。双波长干涉术利用两种工作波长的频差信息获取较长合成波长，将测试量程扩展至数微米量级，实现宏观面型与局部微观阶跃结构等多尺度形貌检测的高精度大动态范围同步检测[1−3]。
双波长同步干涉测量中两种波长模式同时工作，数据采集效率高，且生成的低频莫尔条纹更易分辨。双波长莫尔条纹的解调方法主要有迭代法以及特殊移相、主成分分析和傅里叶变换频谱分析等非迭代算法。迭代法需要循环迭代拟合以提升精度，繁琐耗时[4−5]；特殊移相法需设定两种波长，如±π/2、2π等特殊移相量，解耦分离不同波长干涉信息，但其易受移相误差影响[6−8]；主成分法利用不同波长信息的特征值求解，同步解调各自波长信息，但对光强幅度差要求严格[9−11]；传统傅里叶变换法（FT法）处理空域或时域信号频谱，引入载频或设定移相频率，实现频谱分离，间接提取合成波长相位[12]。为直接获取莫尔条纹中低频易分辨包络携有的合成波长信息，莫尔条纹加乘变换法(ATM)将合成波长干涉信息与其他波长干涉信息从相乘耦合转换成非相干叠加形式解耦，引入空域载频[13]或设定移相频率[14]分离提取，但易受干涉图背景分量影响且会引入其他波长信息。基于此，笔者课题组提出了单帧双波长干涉图空域干涉复函数耦合法（CCFC）[15]，避免引入其他频率信息，但同样易受背景分量影响，且需引入空域高载频实现频谱分离。
为抑制背景分量影响与降低频谱分离对空域高载频要求，提出了一种基于时空域共轭复函数耦合的双波长干涉算法（DCD法）。根据时空条纹交叠重构理论[16−17]，利用时域相移与空域载频转换，实现了低载频下频谱分离，降低背景频谱干扰，提取时空域的双波长干涉复函数，经共轭耦合直接获得合成波长干涉图。算法同时设计干涉图组内与相邻组间分别以单波长与合成波长下π/2移相的双重相移策略，以实现合成波长干涉图的直接解调。
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假定双波长同步干涉测试装置的工作波长为$ {\lambda _1} $和$ {\lambda _2} $（$ {\lambda _1} > {\lambda _2} $），若以$ {\lambda _1} $的π/2设置移相量，则第m帧双波长莫尔条纹的光强分布$ {I_m} $为：
$$ \begin{split} {I_m}\left( {x,y} \right) =& a + {b_1}\cos \left[ {{\phi _1}\left( {x,y} \right) + 2\pi {f_{{\text{s}}1}}\left( {m - 1} \right)} \right]{\kern 1pt} {\kern 1pt} {\kern 1pt} + \\& {b_2}\cos \left[ {{\phi _2}\left( {x,y} \right) + 2\pi {f_{{\text{s}}2}}\left( {m - 1} \right)} \right] \end{split} $$ (1) 式中：(x, y)为像素坐标；a为背景光强；$ {b_1} $与$ {b_2} $、$ {\phi _1}\left( {x,y} \right) $与$ {\phi _2}\left( {x,y} \right) $分别表示$ {\lambda _1} $和$ {\lambda _2} $下的光强调制度和相位分布；$ {f_{{\text{s}}1}} $和$ {f_{{\text{s2}}}} $为各自移相频率。
$$ {f_{{\text{s}}1}} = {1 \mathord{\left/ {\vphantom {1 M}} \right. } M},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {f_{{\text{s}}2}} = \left( {{{{\lambda _1}} \mathord{\left/ {\vphantom {{{\lambda _1}} {{\lambda _2}}}} \right. } {{\lambda _2}}}} \right){f_{{\text{s}}1}} = {{\left( {{{{\lambda _1}} \mathord{\left/ {\vphantom {{{\lambda _1}} {{\lambda _2}}}} \right. } {{\lambda _2}}}} \right)} \mathord{\left/ {\vphantom {{\left( {{{{\lambda _1}} \mathord{\left/ {\vphantom {{{\lambda _1}} {{\lambda _2}}}} \right. } {{\lambda _2}}}} \right)} M}} \right. } M} $$ (2) 式中：$ M $表示移相周期2π内干涉图帧数，即当移相量为π/2时，$ M = 4 $。
根据时空条纹交叠重构理论[16−17]，公式(1)中M帧莫尔条纹移相干涉图如图1所示，沿条纹分布x轴方向，数据重新排列组合为同时包含时域和空域信息的双波长时空条纹图（STF）：
$$ \begin{split} {I_{{\text{STF}}}}\left( {x',y} \right) =& a + {b_1}\cos \left[ {{{\phi '}_1}\left( {x',y} \right) + 2\pi {f_{{\text{s}}1}}x'} \right] +\\& {b_2}\cos \left[ {{{\phi '}_2}\left( {x',y} \right) + 2\pi k{f_{{\text{s}}2}}x'} \right] \end{split} $$ (3) 式中：$ x' $为沿x方向扩展后坐标；$ {\phi '_1}\left( {x',y} \right) $和$ {\phi '_2}\left( {x',y} \right) $分别表示两种波长扩展后相位分布。
$$ \left\{ \begin{gathered} \phi {'_1}\left( {{\text{int}}\left( {{{x'}}/{M}} \right),y} \right) = {\phi _1}(x,y) \\ \phi {'_2}\left( {\text{int} \left( {{{x'}}/{M}} \right),y} \right) = {\phi _2}(x,y) \\ \end{gathered} \right. $$ (4) 式中：$ {\text{int}}\left( {} \right) $表示取整函数。公式(3)中，双波长STF经傅里叶变换后频谱分布$ S\left( {{f_{x'}},{f_y}} \right) $为：
$$\begin{split} &\qquad\qquad S\left( {{f_{x'}},{f_y}} \right) = {S_0} + {S_{1, \pm 1}} + {S_{2, \pm 1}} =\\&{S_0} + 0.5{b_1}{\varPhi _{1, \pm 1}}\left( {{f_{x'}} \mp {f_{{\text{s}}1}},{f_y}} \right) + 0.5{b_2}{\varPhi _{2, \pm 1}}\left( {{f_{x'}} \mp {f_{{\text{s}}2}},{f_y}} \right) \end{split} $$ (5) 式中：$ \left( {{f_{x'}},{f_y}} \right) $为频域像素坐标；$ {S_0} $为背景分量频谱；$ {S_{1, \pm 1}} $和$ {S_{2, \pm 1}} $分别为$ {\lambda _1} $和$ {\lambda _2} $的±1级相位谱；$ {\varPhi _{1, \pm 1}} $和$ {\varPhi _{2, \pm 1}} $对应$ {\lambda _1} $和$ {\lambda _2} $下扩展相位函数$\exp \left[ { \pm i{{\phi '}_1}\left( {x',y} \right)} \right]$和$\exp \left[ { \pm i{{\phi '}_2}\left( {x',y} \right)} \right]$的傅里叶变换。由公式(5)可得，在不引入空域载频的情况下，莫尔条纹移相干涉图中时域相移被转换成双波长STF中空域载频。频域$ {S_{1, + 1}} $和$ {S_{2, + 1}} $与$ {S_0} $以及$ {S_{1, - 1}} $和$ {S_{2, - 1}} $之间的距离至少为$ {f_{{\text{s}}1}} $。因频域坐标轴总长度为2π，即当移相量为π/2时，上述频谱距离至少为频域坐标轴总长的1/4。因此，不需空域载频也可实现+1级相位谱与其他频谱的分离。公式(5)中双波长STF频谱进行带通滤波，仅保留两种波长下的+1级相位谱：
$$ \begin{split} & {S_ + }\left( {{f_{x'}},{f_y}} \right) = {S_{1, + 1}} + {S_{2, + 1}}{\kern 1pt} = 0.5{b_1}{\varPhi _{1, \pm 1}}\left( {{f_{x'}} - {f_{{\text{s}}1}},{f_y}} \right) +\\&\qquad\qquad\qquad 0.5{b_2}{\varPhi _{2, \pm 1}}\left( {{f_{x'}} - {f_{{\text{s}}2}},{f_y}} \right) \end{split}$$ (6) 经傅里叶逆变换后，得到时空域干涉复函数：
$$ \begin{split} {C_ + }\left( {x',y} \right) = &0.5{b_1}\exp \left\{ {i\left[ {{\phi _1}\left( {x',y} \right) - 2\pi {f_{{\text{s}}1}}x'} \right]} \right\} +\\& 0.5{b_2}\exp \left\{ {i\left[ {{\phi _2}\left( {x',y} \right) - 2\pi {f_{{\text{s}}2}}x'} \right]} \right\} \end{split} $$ (7) 与公式(7)中共轭的复函数分布可以表示为：
$$ \begin{split} {C_ - }\left( {x',y} \right) =& 0.5{b_1}\exp \left\{ { - i\left[ {{\phi _1}\left( {x',y} \right) - 2\pi {f_{{\text{s}}1}}x'} \right]} \right\} +\\& 0.5{b_2}\exp \left\{ { - i\left[ {{\phi _2}\left( {x',y} \right) - 2\pi {f_{{\text{s}}2}}x'} \right]} \right\} \end{split} $$ (8) 当公式(7)和公式(8)中的一对共轭的时空域干涉复函数相乘耦合后，得到扩展干涉图光强分布为：
$$ \begin{split} {{I'}_{{\kern 1pt} {\text{s}}}}\left( {x',y} \right) =& {C_ + }\left( {x',y} \right) \cdot {C_ - }\left( {x',y} \right) = 0.25\left( {b_1^2 + b_2^2} \right) + 0.5{b_1}{b_2} \cdot\\& \cos \left[ {{\phi _1}\left( {x',y} \right) - {\phi _2}\left( {x',y} \right) - 2\pi \left( {{f_{{\text{s}}1}} - {f_{{\text{s}}2}}} \right)x'} \right]\\[-1pt] \end{split}$$ (9) 根据双波长干涉理论，合成波长$ {\lambda _{\text{s}}} $和其对应的相位分布$ {\phi _{\text{s}}} $表示为：
$$ {\lambda _{\text{s}}} = {{{\lambda _1}{\lambda _2}} \mathord{\left/ {\vphantom {{{\lambda _1}{\lambda _2}} {\left( {{\lambda _1} - {\lambda _2}} \right)}}} \right. } {\left( {{\lambda _1} - {\lambda _2}} \right)}},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\phi _{\text{s}}} = {\phi _1}\left( {x,y} \right) - {\phi _2}\left( {x,y} \right){\kern 1pt} $$ (10) 因此，公式(9)中扩展干涉图写为：
$$ \begin{split} {{I'}_{\text{s}}}\left( {x',y} \right) =& \frac{1}{4}\left( {b_1^2 + b_2^2} \right) + \frac{1}{2}{b_1}{b_2} \cdot\\& \cos \left[ {{\phi _{\text{s}}}\left( {x',y} \right) - 2\pi \frac{{{\lambda _1} - {\lambda _2}}}{{{\lambda _2}}}{f_{{\text{s}}1}}x'} \right] \end{split} $$ (11) 上述合成波长扩展干涉图根据时空条纹交叠重构理论[17−18]依次间隔$ \left( {M - 1} \right) $列，逆向提取，恢复成原始大小的单帧合成波长干涉图：
$$ {I_{\text{s}}}\left( {x,y} \right){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \frac{1}{4}\left( {b_1^2 + b_2^2} \right) + \frac{1}{2}{b_1}{b_2}\cos \left[ {{\phi _{\text{s}}}\left( {x,y} \right)} \right] $$ (12) 
图 1 合成波长干涉图提取算法示意图
Figure 1. Schematic diagram for the extraction of the synthetic-wavelength interferogram
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为进一步恢复合成波长相位，设计了如图2所示的双重移相策略，依次采集多组连续4帧移相干涉图，其同组内部4帧双波长莫尔条纹移相干涉图按单一波长的π/2设置移相量，而相邻组间的莫尔条纹移相干涉图，即前一组的最后一帧与后一组的第一帧之间的移相量，设置为合成波长的π/2移相。在公式(1)的基础上，双重移相策略下的第j组m帧莫尔条纹干涉图光强分布$ {I'_{4\left( {j - 1} \right) + m}} $可描述为：
$$ {I'_{4\left( {j - 1} \right) + m}}\left( {x,y} \right) = a + {b_1}\cos \left[ {{\phi _1}\left( {x,y} \right) + 2\pi {f_{{\text{s}}1}}\left( {m - 1} \right) + 2\pi {f_{\text{s}}} \cdot \left( {j - 1} \right)} \right]{\kern 1pt} {\kern 1pt} {\kern 1pt} + {b_2}\cos \left[ {{\phi _2}\left( {x,y} \right) + 2\pi {f_{{\text{s}}2}}\left( {m - 1} \right) + 2\pi {f_{\text{s}}} \cdot \left( {j - 1} \right)} \right] $$ (13) 式中：$ {f_{\text{s}}} $为合成波长下移相频率。根据1.1节所述，从第j组4帧双波长莫尔条纹移相干涉图提取的单帧合成波长干涉图为：
$$\begin{split} {I_{{\text{s}},j}}\left( {x,y} \right) = &\frac{1}{4}\left( {b_1^2 + b_2^2} \right) + \frac{1}{2}{b_1}{b_2} \cdot\\& \cos \left[ {{\phi _{\text{s}}}\left( {x,y} \right) + 2\pi {f_{\text{s}}}\left( {j - 1} \right)} \right] \end{split} $$ (14) 因相邻组间移相量为合成波长π/2，则$ {f_{\text{s}}} = {1 \mathord{\left/ {\vphantom {1 4}} \right. } 4} $。待测合成波长相位可直接由常规移相算法（如5步法[18]）处理得到：
$$ {\phi _{\text{s}}} = {\arctan }\left\{ {\frac{{2 \cdot \left[ {{I_{{\text{s}},4}}\left( {x,y} \right) - {I_{{\text{s}},2}}\left( {x,y} \right)} \right]}}{{{I_{{\text{s}},1}}\left( {x,y} \right) - 2 \cdot {I_{{\text{s}},3}}\left( {x,y} \right) + {I_{{\text{s}},5}}\left( {x,y} \right)}}} \right\} $$ (15) 
图 2 基于双重移相策略的合成波长相位恢复流程图
Figure 2. Flow chart for the synthetic-wavelength phase retrieval with the double phase shift strategy
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为评价DCD法性能，数值模拟双波长同步干涉装置的工作波长与实际实验装置相同的$ {\lambda _1} = 632.8{\kern 1 pt} {\kern 1 pt} {\text{nm}} $和$ {\lambda _2} = 532{\kern 1 pt} {\kern 1 pt} {\text{nm}} $，其合成波长为$ {\lambda _{\text{s}}} = 3.34{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{μm}} $。峰谷值为74.2 nm的仿真波面分布为$w = 0.008 \times 632.8 \times {{peaks}}\left( {400} \right)$。原始波面上叠加了一个高度1.2 μm的台阶，以突出双波长干涉的测量范围扩展功能。根据公式(13)中光强分布，背景分量与幅度调制均模拟为高斯函数。为直观反映载频影响，条纹数目依据合成波长设置为1，即莫尔条纹包络数目为1。根据DCD法中双重移相策略，仿真生成连续5组4帧双波长莫尔条纹移相干涉图，大小为400 pixel×400 pixel。同组内干涉图间移相量为$ {\lambda _1} $的π/2，而相邻两组间的移相量为$ {\lambda _{\text{s}}} $的π/2。
图3（a）为仿真的第一组内单帧双波长莫尔条纹干涉图，其频谱如图3（b）所示。两种波长下±1级相位谱以及背景谱因原始干涉图中低载频而彼此重叠。为实现+1级相位谱的分离提取，根据DCD算法将同组连续4帧莫尔条纹图数据重新排列得到双波长STF，其频谱如图3（c）所示。因时域相移被转换成空域载频，不需引入高载频即能实现+1级与−1级相位谱以及背景频谱的分离，从而降低了背景分量对信号的干扰。由于同组内π/2相移设定，双波长STF频谱中±1级相位谱分别位于归一化频率轴的(±0.25, 0)坐标附近。
图 3 仿真结果。（a）单帧双波长莫尔条纹干涉图；（b）双波长干涉图的傅里叶变换频谱分布；（c）双波长STF的傅里叶变换频谱分布；（d）提取并恢复到原始尺寸大小的合成波长干涉图；（e）波面恢复的残差分布
Figure 3. Simulated results. (a) One frame initial dual-wavelength interferogram; (b) The corresponding Fourier spectrum for the dual-wavelength interferogram; (c) The Fourier spectrum of the dual-wavelength STF; (d) The obtained synthetic-wavelength interferogram with original size; (e) The residual error for the retrieved wavefront
经带通滤波仅保留+1级相位谱，傅里叶逆变换后获取时空域的双波长干涉复函数，共轭耦合获得$ {\lambda _{\text{s}}} $下扩展干涉图，最后逆向提取恢复到原始尺寸大小的$ {\lambda _{\text{s}}} $下干涉图如图3（d）所示。对比图3（a）和（d），后者条纹分布与前者包络线一致。从每组莫尔条纹干涉图中提取的合成波长干涉图，因相邻组间移相量为$ {\lambda _{\text{s}}} $下π/2，采用5步移相法[18]解调处理，波面恢复残差如图3（e）所示，PV值和RMS值分别优于0.000167$ {\lambda _{\text{s}}} $(0.5569 nm)和0.000027$ {\lambda _{\text{s}}} $(0.0897 nm)。
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同组内莫尔条纹移相干涉图按$ {\lambda _1} $的π/2设置移相量，则$ {\lambda _2} $干涉信息因波长差异性而存在移相偏差，公式(3)中双波长STF可以改写为：
$$ \begin{split} {{I'}_{{\text{STF}}}}\left( {x',y} \right) =& a + {b_1}\cos \left[ {{{\phi '}_1}\left( {x',y} \right) + 2\pi {f_{{\text{s}}1}}x'} \right] + \\&{b_2}\cos \left[ {{{\phi '}_2}\left( {x',y} \right) + 2\pi k{f_{{\text{s}}1}}x' + p\left( {x',y} \right)} \right] \end{split} $$ (16) 式中：$ k = {\text{int}}\left( {{{{\lambda _1}} \mathord{\left/ {\vphantom {{{\lambda _1}} {{\lambda _2}}}} \right. } {{\lambda _2}}}} \right) $为λ1与λ2之比的整数部分；$ p\left( {x',y} \right) $为$ {\lambda _2} $移相偏差引起的误差项。
$$ p\left( {x',y} \right) = \sum\limits_{m = 1}^M {\sum\limits_{n = - \infty }^{ + \infty } {{r_m}} \delta \left( {x' - nM - \left( {m - 1} \right),y} \right)} $$ (17) 式中：n为列数，为方便推导，范围设为$ \left( { - \infty , + \infty } \right) $；$ {r_m} = 2\pi \left( {m - 1} \right) \cdot r \cdot {f_{{\text{s}}1}} $表示第m帧干涉图$ {\lambda _2} $的移相偏差，$ r = \left( {{{{\lambda _1}} \mathord{\left/ {\vphantom {{{\lambda _1}} {{\lambda _2}}}} \right. } {{\lambda _2}}}} \right) - {\text{int}}\left( {{{{\lambda _1}} \mathord{\left/ {\vphantom {{{\lambda _1}} {{\lambda _2}}}} \right. } {{\lambda _2}}}} \right) $为波长之比的小数部分。因两种波长越接近，合成波长越长，即k=1，r值因此非常小，公式（17）中误差项可一阶近似，公式（16）中双波长STF频谱为：
$$ S'\left( {{f_{x'}},{f_y}} \right) = {S_0} + {S_{1, \pm 1}} + {S_{2, \pm 1}} + {E_{2, \pm 1}} $$ (18) 式中：$ {S_{1, \pm 1}} $和$ {S_{2, \pm 1}} $分别表示$ {\lambda _1} $与$ {\lambda _2} $的相位谱；$ {E_{2, \pm 1}} $为波长$ {\lambda _2} $因移相偏差产生的误差谱，即
$$\begin{split} \left\{ \begin{array}{l} {S_{1, \pm 1}} = 0.5{b_1}{\varPhi _{1, \pm 1}}\left( {{f_{x'}} \mp {f_{{\rm{s}}1}},{f_y}} \right) \\ {S_{2, \pm 1}} = 0.5{b_2}{\varPhi _{2, \pm 1}}\left( {{f_{x'}} \mp {f_{{\rm{s1}}}},{f_y}} \right) \\ {E_{2, \pm 1}} = \dfrac{{ \mp 1}}{{iM}}\displaystyle\sum\limits_{m = 1}^M {r_m}\displaystyle\sum\limits_{n = - \infty }^{ + \infty } {\mathrm{exp}} \cdot \\ \qquad\quad\left( { - i2 \pi \left( {m - 1} \right)n{f_{{\rm{s}}1}}}\right) {S_{2, \pm 1}}\left( {{f_{x'}} - n{f_{{\rm{s}}1}},{f_y}} \right) \end{array} \right. \end{split} $$ (19) 公式（19）表明，$ {\lambda _1} $与$ {\lambda _2} $的±1级相位谱$ {S_{1, \pm 1}} $和$ {S_{2, \pm 1}} $彼此重叠位于频谱域坐标$ \left( { \pm {f_{{\text{s1}}}},0} \right) $上，即当同组干涉图移相量为$ {\lambda _1} $的π/2时，±1级相位谱位于归一化频率轴(±0.25,0)坐标附近，与图3（c）中的双波长STF频谱一致。误差谱$ {E_{2, \pm 1}} $则位于频域坐标$ \left( { \pm n{f_{{\text{s1}}}},0} \right) $上。因此，在$ {S_{1, + 1}} $和$ {S_{2, + 1}} $的频谱域坐标$ \left( { + {f_{{\text{s1}}}},0} \right) $附近存在的频谱分布为：
$$ \begin{split} & {S_{ + 1}}\left( {{f_{{\text{s}}1}},{f_y}} \right) = 0.5{b_1}{\varPhi _{1, + 1}}\left( {{f_x} - {f_{{\text{s}}1}},{f_y}} \right) + 0.5{b_2}\left( {1 + {R_1}} \right)\cdot \\&{\varPhi _{2, + 1}}\left( {{f_x} - {f_{{\text{s}}1}},{f_y}} \right) + 0.5{b_2}{R_2} \cdot {\varPhi _{2, - 1}}\left( {{f_x} - {f_{{\text{s}}1}},{f_y}} \right)\\[-1pt] \end{split} $$ (20) $ {R_1} $和$ {R_2} $为公式（17）移相偏差产生的误差因子：
$$ \left\{ \begin{gathered} {R_1} = - \frac{1}{{iM}}\sum\limits_{m = 1}^M {{r_m}} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \\ {R_2} = - \frac{1}{{iM}}\sum\limits_{m = 1}^M {{r_m}\exp \left( { - i4\pi \left( {m - 1} \right){f_{{\text{s}}1}}} \right)} \\ \end{gathered} \right. $$ (21) 公式（20）第2项中$ {R_1} $仅改变$ {S_{2, + 1}} $的幅值大小而不改变其相位，因此不用考虑。第3项中$ {R_2} $因与$ {S_{2, - 1}} $相耦合，当在原始莫尔条纹干涉图中引入沿x轴的空域载频时，公式（20）中两种波长下扩展相位的傅里叶变换将分别变为${\varPhi _{1, \pm 1}}\left( {{f_{x'}} \mp {f_{{\text{s}}1}} \mp {f_{{\text{c}}1}},{f_y}} \right)$和${\varPhi _{2, \pm 1}}\left( {{f_{x'}} \mp {f_{{\text{s}}1}} \mp {f_{{\text{c2}}}},{f_y}} \right)$，其中$ {f_{{\text{c1}}}} $和$ {f_{{\text{c}}2}} $为原始干涉图中$ {\lambda _1} $和$ {\lambda _2} $下沿x轴空域载频，而公式（20）中频谱变为：
$$ \begin{split} {S_{ + 1}}\left( {{f_{{\text{s}}1}},{f_y}} \right) =& 0.5{b_1}{\varPhi _{1, + 1}}\left( {{f_x} - {f_{{\text{s}}1}} - {f_{{\text{c}}1}},{f_y}} \right) + \\&0.5{b_2}\left( {1 + {R_1}} \right){\varPhi _{2, + 1}}\left( {{f_x} - {f_{{\text{s}}1}} - {f_{{\text{c2}}}},{f_y}} \right) +\\& 0.5{b_2}{R_2} \cdot {\varPhi _{2, - 1}}\left( {{f_x} - {f_{{\text{s}}1}} + {f_{{\text{c2}}}},{f_y}} \right) \end{split} $$ (22) 式中，包含$ {R_2} $的第3项与前两项相位谱因空域载频引入而分离，且频谱间距离分别为$ \left( {{f_{{\text{c1}}}} + {f_{{\text{c2}}}}} \right) $和$ 2{f_{{\text{c}}2}} $，方便了后续滤除。
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对DCD算法因分离滤除$ {\lambda _2} $移相偏差的误差谱所需空域载频展开讨论，并与FT法[12]、ATM法[13]、CCFC法[15]等比较。首先，ATM法[13]中会引入其他波长相位谱，其频谱如图4（a）所示。图中${S'_{i, \pm 1}} ( i = 1,2,3,4 )$分别表示$ {\lambda _1} $二阶谐波分量、$ {\lambda _2} $二阶谐波分量、$ {\lambda _1} $和$ {\lambda _2} $的和频分量、以及$ {\lambda _1} $和$ {\lambda _2} $的差频分量（即合成波长）等相位谱分布。为分离提取合成波长相位谱，与其他相位谱间最短距离需满足：
图 4 不同方法频谱分离距离对比。（a） ATM方法；（b） FT方法；（c） CCFC法；（d） DCD方法
Figure 4. Separated spectral distances for different methods. (a) ATM method; (b) FT method; (c) CCFC method; (d) DCD method
$$ {d_{{\text{ATM}}}} = 2\pi \left( {2{f'_{{\text{c}}1}} - {f'_{{\text{c}}4}}} \right) > {\left| {\frac{{\partial {{\phi '}_1}}}{{\partial x}}} \right|_{\max }} + {\left| {\frac{{\partial {\phi _{\text{s}}}}}{{\partial x}}} \right|_{\max }} $$ (23) 式中：$ {f'_{{\text{c}}1}} $和$ {f'_{{\text{c}}4}} $为$ {\lambda _1} $二阶谐波分量和$ {\lambda _{\text{s}}} $沿x轴的空域载频量；$ {\left| {{{\partial \left( {{{\phi '}_1}} \right)} \mathord{\left/ {\vphantom {{\partial \left( {{{\phi '}_1}} \right)} {\partial x}}} \right. } {\partial x}}} \right|_{\max }} $和$ {\left| {{{\partial {\phi _{\text{s}}}} \mathord{\left/ {\vphantom {{\partial {\phi _{\text{s}}}} {\partial x}}} \right. } {\partial x}}} \right|_{\max }} $分别表示$ {\lambda _1} $二阶谐波分量与$ {\lambda _{\text{s}}} $相位分布最高频率值。
其次，FT方法[12]中频谱如图4（b）所示。图中$ {S''_{1, \pm 1}} $和$ {S''_{2, \pm 1}} $分别表示$ {\lambda _1} $和$ {\lambda _2} $相位谱，为分离并提取各自相位谱，其距离需满足：
$$ {d_{{\text{FT}}}} = 2\pi \left( {f'_{{\text{c}}2} - {f'_{{\text{c}}1}}} \right) > {\left| {\frac{{\partial {\phi _1}}}{{\partial x}}} \right|_{\max }} + {\left| {\frac{{\partial {\phi _2}}}{{\partial x}}} \right|_{\max }} $$ (24) 式中：$ {\left| {{{\partial \left( {{\phi _1}} \right)} \mathord{\left/ {\vphantom {{\partial \left( {{\phi _1}} \right)} {\partial x}}} \right. } {\partial x}}} \right|_{\max }} $和$ {\left| {{{\partial {\phi _2}} \mathord{\left/ {\vphantom {{\partial {\phi _2}} {\partial x}}} \right. } {\partial x}}} \right|_{\max }} $分别表示两种波长相位最高频率值；$ {f'_{{\text{c2}}}} $为$ {\lambda _2} $沿x轴空域载频量。
CCFC法[15]中频谱如图4（c）所示，为分离提取+1级相位谱，则频谱间距离需满足：
$$ {d_{{\text{CCFC}}}} = 2\pi \left( {f'_{{\text{c}}1}} \right) > {\left| {\frac{{\partial {\phi _1}}}{{\partial x}}} \right|_{\max }} + {\left| {\frac{{\partial a}}{{\partial x}}} \right|_{\max }} $$ (25) 式中：$ {\left| {{{\partial a} \mathord{\left/ {\vphantom {{\partial a} {\partial x}}} \right. } {\partial x}}} \right|_{\max }} $为背景分量的最高频率值。
DCD法中双波长STF频谱如图4（d）所示，因时域相移转换的高载频，$ {S_0} $、$ {S_{1, + 1}} $和$ {S_{2, + 1}} $已分离，而为分离滤除$ {E_{2, + 1}} $，则距离需满足：
$$ {d_{{\text{DCD}}}} = 2\pi \left( {{f'_{{\text{c}}1}} + {f'_{{\text{c}}2}} } \right) > {\left| {\frac{{\partial {\phi _1}}}{{\partial x}}} \right|_{\max }} + {\left| {\frac{{\partial {\phi _2}}}{{\partial x}}} \right|_{\max }} $$ (26) 上述相位最高频率值与空域载频量相比非常小，因此均采用$ {\left| {{{\partial \phi } \mathord{\left/ {\vphantom {{\partial \phi } {\partial x}}} \right. } {\partial x}}} \right|_{\max }} $统一表示，同时载频量均转换成$ {\lambda _{\text{s}}} $下空域载频量。考虑仿真与实际中$ {\lambda _1} = 632.8{\kern 1 pt} {\kern 1 pt} {\text{nm}} $和$ {\lambda _2} = 532{\kern 1 pt} {\kern 1 pt} {\text{nm}} $，公式（23）~（26）中空域载频条件依次分别为：
$$ \left\{ \begin{gathered} {\text{ATM}}{\kern 1pt} {\kern 1pt} {\text{method:}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{f'_{{\text{c}}4}} > 0.105 \cdot \frac{2}{{2\pi }}{\left| {\frac{{\partial \phi }}{{\partial x}}} \right|_{\max }}{\kern 1pt} {\kern 1pt} {\kern 1pt} \\ {\kern 1pt} {\text{FT}}{\kern 1pt} {\kern 1pt} {\text{method:}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{f'_{{\text{c}}4}} > \frac{2}{{2\pi }}{\left| {\frac{{\partial \phi }}{{\partial x}}} \right|_{\max }}{\kern 1pt} {\kern 1pt} \\ {\text{CCFC}}{\kern 1pt} {\kern 1pt} {\text{method:}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{f'_{{\text{c}}4}} > 0.095 \cdot \frac{2}{{2\pi }}\left( {{{\left| {\frac{{\partial \phi }}{{\partial x}}} \right|}_{\max }} + {{\left| {\frac{{\partial a}}{{\partial x}}} \right|}_{\max }}} \right){\kern 1pt} \\ {\text{DCD}}{\kern 1pt} {\kern 1pt} {\text{method:}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{f'_{{\text{c}}4}} > 0.086 \cdot \frac{2}{{2\pi }}{\left| {\frac{{\partial \phi }}{{\partial x}}} \right|_{\max }} \\ \end{gathered} \right. $$ (27) 此外，如公式（22）中分析，双波长STF频谱域，误差谱幅值与相位谱幅值不同。在相同重叠面积的情况下，幅值相同的两个频谱间分离距离明显大于幅值不同的两个频谱间分离距离。因此，DCD方法中幅值不同的误差谱与相位谱间的分离，实际对空域载频量的要求更小。对于$ {\lambda _1} = 632.8{\kern 1 pt} {\kern 1 pt} {\text{nm}} $和$ {\lambda _2} = 532{\kern 1 pt} {\kern 1 pt} {\text{nm}} $，一个移相周期内$ {r_m} $为$ \left[ {0,{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} 0.094\;7\pi ,{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} 0.189\;5\pi ,{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} 0.284\;2\pi } \right] $。假定两种波长下干涉条纹调制度相等，根据公式（22）中频谱间的幅值关系，推导得$ {\lambda _2} $移相误差谱与距离最近的$ {\lambda _1} $相位谱幅度比值为0.1488。
文中利用两个不同幅值的高斯函数来模拟误差谱与相位谱，估计频谱分离所需的空域载频量，设置幅值比为上述推导的0.1488。当频谱间重叠面积为2%时，不同幅值的分离距离$ d' $与相同幅值的分离距离$ d $之间的比值为0.7313，而当重叠面积为1%时$ {{d'} \mathord{\left/ {\vphantom {{d'} d}} \right. } d} = 0.87 $，因此假设$ {{d'} \mathord{\left/ {\vphantom {{d'} d}} \right. } d} = 0.9 $，公式（27）中DCD方法频谱分离所需的载频量可以表示为：
$$ {f'_{{\text{c}}4}} > 0.077 \cdot \frac{2}{{2\pi }}{\left| {\frac{{\partial \phi }}{{\partial x}}} \right|_{\max }} $$ (28) 与ATM法[13]相比，DCD方法所需的空域载频数值上仅为前者的0.73倍，且后者不引入其他波长干涉信息。而与FT方法[12]相比，DCD方法所需的空域载频数值上仅为前者的0.077倍。与CCFC法[15]相比，DCD方法不仅不受背景分量的影响，且即使不考虑背景分量影响，其所需载频量也仅为前者的0.81倍。
空域载频量的直观反映是条纹数目，图5给出了ATM法[13]、CCFC法[15]、DCD法等三种算法的波面恢复误差随双波长莫尔条纹干涉图中合成波长条纹数目变化的对比结果。当条纹数增加时，即空域载频量增大，频谱间更易分离，三种方法波面恢复误差的PV值和RMS值均逐渐减小。当条纹数小于3时，ATM法[13]因合成波长相位谱与其他波长相位谱重叠而无法分离提取相位。结果表明，在低载频时，DCD方法明显优于其他方法，即使在条纹数为1时，波面恢复仍优于1 nm （PV值）和0.1 nm （RMS值）。
图 5 三种方法的波面解调误差曲线随合成波长条纹数的变化。（a） PV值；（b） RMS值
Figure 5. Demodulated error curve with the variation of fringe number at $ {\lambda _{\text{s}}} $ for three methods. (a) PV; (b) RMS
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为验证DCD法的有效性与可行性，结合实验室所研制的斐索型双波长干涉仪，对阶跃结构高度为7.8 μm的平面台阶样品进行了测试。双波长干涉仪采用压电陶瓷移相方式，工作波长为632.8 nm和532 nm，合成波长达到3.34 μm。实验中根据双重移相策略，采集了五组连续4帧双波长莫尔条纹移相干涉图，同组内干涉图间的移相量为632.8 nm处的π/2，而相邻组干涉图间的移相量为3.34 μm处的π/2。第一组内的单帧莫尔条纹干涉图如图6（a）所示，大小为380 pixel×380 pixel，合成波长下条纹数目为4，其频谱归一化分布如图6（b）所示，在空域高载频下，±1级相位谱与背景分量频谱分离。相应的双波长STF频谱如图6（c）所示。与图7（b）相比，图6（c）双波长STF频谱中，±1级相位谱与背景分量频谱远远分离，且±1级相位谱位于归一化频率轴的(±0.25,0)坐标附近，与组内干涉图的π/2相移设定对应。DCD方法得到的原始尺寸合成波长干涉图如图6（d）所示，其条纹分布明显与图6（a）中包络分布一致。
图 6 台阶样品实验测试数据。（a）双波长干涉图；（b）双波长干涉图的傅里叶变换频谱；（c）双波长STF的傅里叶变换频谱；（d）提取原始尺寸大小的合成波长干涉图
Figure 6. Measured data for the test step in experiment. (a) Dual-wavelength interferogram; (b) Fourier spectrum of the dual-wavelength interferogram; (c) Fourier spectrum of the dual-wavelength STF; (d) The extracted synthetic-wavelength interferogram with original size
图 7 台阶样品在合成波长干涉条纹为4时的检测结果。（a） DCD方法；（b） CCFC方法
Figure 7. Results of the test step with about 4 fringes for the synthetic wavelength. (a) DCD method; (b) CCFC method
样品检测结果如图7（a）所示，其中包括测试台阶的三维分布图、阶跃结构高度分布直方图、以及样品上下表面具体面型分布，同时台阶高度为7.7846 μm。同时，给出了采用CCFC法[15]的检测结果，如图7（b）所示。对比发现， DCD法与CCFC法的检测结果一致，从而验证了DCD法的有效性。
为了进一步验证DCD方法在空域低载频下的有效性和可行性，实验过程中通过调整样品的俯仰倾斜，获得合成波长干涉条纹数目为1时的单帧双波长莫尔条纹干涉图，如图8（a）所示，其相应的频谱归一化分布如图8（b）所示。在空域低载频时，两种波长下的±1级相位谱与背景分量频谱产生重叠而无法分离，因而无法采用CCFC法[15]处理。然而，DCD法中与之对应的双波长STF频谱归一化分布如图8（c）所示。因为双波长STF中时域相移转换成了高载频，两种波长下的±1级相位谱与背景分量频谱仍可以远远分离，最终提取的原始尺寸合成波长干涉图如图8（d）所示。
图 8 低载频时台阶样品的测试数据。（a）双波长干涉图；（b）双波长干涉图的傅里叶变换频谱；（c）双波长STF的傅里叶变换频谱；（d）提取的原始尺寸大小的合成波长干涉图
Figure 8. Measured data for the test step in experiment with lower carrier condition. (a) Dual-wavelength interferogram; (b) Fourier spectrum of the dual-wavelength interferogram; (c) Fourier spectrum of the dual-wavelength STF; (d) The extracted synthetic-wavelength interferogram with original size
空域低载频时，DCD法得到的平面台阶样品最终检测结果如图9所示，其相应的台阶高度为7.7919 μm。对比图7中空域高载频时检测结果，DCD法在不同空域载频时的检测结果保持了一致性，即使在较低载频量时仍能很好地恢复波面。此外，图10为测试样品经白光干涉仪(Veeco NT9100)测量的局部阶跃结构高度分布。实验结果表明，与Veeco NT9100的7.719 μm阶跃结构高度检测结果相比， DCD法即使在合成波长条文数目为1的空域低载频量时，其阶跃结构高度相对误差仍达到0.94%。
图 9 合成波长条纹数目为1时测试台阶样品经DCD方法处理的检测结果
Figure 9. Results of the test step by the proposed DCD method with about 1 fringes for the synthetic wavelength
图 10 实验中台阶样品经白光干涉仪 (Veeco NT9100) 检测的结果
Figure 10. The result of the test step measured by white light interferometer (Veeco NT9100)
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提出了一种基于时空域共轭复函数耦合与双重移相策略，实现双波长同步干涉中低频合成波长干涉条纹直接提取与解调的算法。算法利用时域相移与空域载频的转换，实现了低载频时的频谱分离。通过获取时空域双波长干涉复函数进行共轭耦合，直接提取合成波长干涉条纹，而不引入其他频率分量。双重移相策略的选择则实现了合成波长干涉图利用传统移相算法的直接解调。讨论了不同波长移相偏差对测量结果的影响，同时与其他傅里叶变换方法的空域载频条件进行了对比分析，即使在移相偏差存在的情况下，算法能够以低于其他算法的空域载频要求进行高精度的测量。仿真与实验结果表明，算法有助于实现宏观面型与局部微观阶跃结构等多尺度形貌的大动态范围高精度同步检测，对高精度大偏差形貌光学元件的研制提供了技术支持。
Dual-wavelength interferometric algorithm based on spatial-temporal conjugate complex function coupling
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摘要: 为从双波长莫尔条纹中提取合成波长干涉信息以扩展测试量程，提出了时空域共轭干涉复函数耦合算法。利用干涉图间时域相移与空域载频的转换，低载频时分离频谱以获取时空域复函数，经共轭耦合提取合成波长干涉图，且不引入其他干涉信息。干涉图组内单波长π/2与组间合成波长π/2的双重移相策略，实现了合成波长干涉图提取后的直接解调。与传统空域傅里叶变换方法相比，考虑波长间相移偏差，算法所需载频数值仅为前者的0.077。仿真峰谷值为74.2 nm的波面在莫尔条纹包络数目为1时，恢复偏差峰谷值优于0.5569 nm。实验中，7.8 μm高度台阶样品在低载频时，阶跃高度的相对误差仍优于0.94%。仿真与实验数据验证了算法的可行性，为实现双波长干涉中低频干涉信息的获取提供了新的思路。Abstract:
Objective Traditional single-wavelength interferometry is not suitable to unwrap the correct phase for measuring surface with step or groove, whose depth is larger than half wavelength. Dual-wavelength interferometry (DWI) technique employs an extra wavelength to generate a longer beat-frequency synthetic wavelength (${\lambda }_{{\rm{s}}}$). For synthetic wavelength is much longer than the optical working wavelength, DWI extends the measuring discontinuity limit of interferometry greatly. And DWI could achieve the simultaneous accurate measurements with large dynamic range for the multi-scale morphology characteristics parameters such as the macro profile and local morphology with step. Meanwhile, in the simultaneous dual-wavelength interferometry (SDWI), the two single-wavelength interferogram is captured simultaneously to accelerate the data collection, which is immune to vibration with the advantages of the time saving and higher efficiency. In practice, the dual-wavelength interferogram is usually captured by the monochrome sensor, which is more convenient and economical. And a generated dual-wavelength Moiré fringe pattern appears as the simple incoherent additive superposition of the two corresponding single-wavelength interferogram. The low beat-frequency envelope of the generated fringe pattern indirectly represents the needed synthetic-wavelength information, whose direct extraction is rather arduous. For this purpose, we present a dual-wavelength interferometric algorithm combining with spatial-temporal conjugate complex functions coupling and double phase shift strategy. Methods The method needs to acquire multiple groups of phase-shift dual-wavelength interferogram, and each group consists of four continuous interferogram. The phase shift step among the four frames in each group is required as π/2 at one single wavelength, while π/2 at synthetic wavelength between the adjacent groups by the designed double phase shift strategy (Fig.2). And in dual-wavelength squeezing interferometry for each group, the temporal phase shift in each group is converted into spatial carrier in the generated dual-wavelength STF. Therefore, the +1-order spectral lobes for the two wavelengths could be easily separated from others and filtered in the Fourier spectrum of the generated dual-wavelength STF without extra spatial carrier and elimination of background. After the appropriate band-pass filter and inverse Fourier transform, the single-wavelength interferometric complex functions are obtained. Subsequently, when the conjugate single-wavelength interferometric complex functions are multiplied, the synthetic-wavelength interferometric fringe pattern could be extracted directly (Fig.1). The obtained synthetic-wavelength interferogram from each group with π/2 phase-shift step at ${\lambda }_{{\rm{s}}}$ could be demodulated to retrieve the final synthetic-wavelength phase by the conventional phase-shift algorithm. Results and Discussions Simulations verify that the proposed method has a lower linear carrier requisition than the spatial-domain Fourier transform demodulation theory, which is merely about 0.077 times of latter numerically, even the phase-shift deviation at different wavelength exists (Fig.4). Besides, the feasibility and applicability of the proposed method are verified using simulation and experimental results. Numerical simulation indicates that the demodulated error is better than PV of 0.556 9 nm and RMS of 0.089 7 nm even when the fringe number is 1 at ${\lambda }_{{\rm{s}}}$ (Fig.3). In addition, for the test step in the experiment, the results have validated the effectiveness of the proposed method for the interferogram with lower linear carrier. And the step height deviations for the proposed method are better than 0.94% for the step with the height of 7.8 μm even for one fringe at ${\lambda }_{{\rm{s}}}$ (Fig.9). Conclusions To extract and retrieve the lower frequency synthetic-wavelength interferometric fringe form dual-wavelength Moiré fringe, we present a dual-wavelength interferometric algorithm combining with spatial-temporal conjugate complex functions coupling and double phase shift strategy. Several groups of phase-shift dual-wavelength interferogram are acquired with every contiguous four frames in each group. The temporal phase shift among each group is converted into spatial carrier for the spectral separation with lower spatial carrier. When the obtained spatial-temporal conjugate complex functions are coupling by multiplication, one synthetic-wavelength interferogram could be extracted for each group directly without the introduction of other wavelength interferometric information. For the designed π/2 phase shift at synthetic wavelength between the adjacent groups, the extracted synthetic-wavelength interferogram from every group is demodulated by conventional phase-shift algorithm directly. The proposed method has a lower carrier requisition than the traditional spatial-domain Fourier demodulation theory, which is merely about 0.077 times of the former numerically, even when the phase-shift deviation for different wavelengths exists. -
Key words:
- measurement /
- interferometry /
- Fourier transform /
- dual-wavelength /
- Moiré fringe
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图 1 合成波长干涉图提取算法示意图
Figure 1. Schematic diagram for the extraction of the synthetic-wavelength interferogram
图 2 基于双重移相策略的合成波长相位恢复流程图
Figure 2. Flow chart for the synthetic-wavelength phase retrieval with the double phase shift strategy
图 3 仿真结果。（a）单帧双波长莫尔条纹干涉图；（b）双波长干涉图的傅里叶变换频谱分布；（c）双波长STF的傅里叶变换频谱分布；（d）提取并恢复到原始尺寸大小的合成波长干涉图；（e）波面恢复的残差分布
Figure 3. Simulated results. (a) One frame initial dual-wavelength interferogram; (b) The corresponding Fourier spectrum for the dual-wavelength interferogram; (c) The Fourier spectrum of the dual-wavelength STF; (d) The obtained synthetic-wavelength interferogram with original size; (e) The residual error for the retrieved wavefront
图 4 不同方法频谱分离距离对比。（a） ATM方法；（b） FT方法；（c） CCFC法；（d） DCD方法
Figure 4. Separated spectral distances for different methods. (a) ATM method; (b) FT method; (c) CCFC method; (d) DCD method
图 5 三种方法的波面解调误差曲线随合成波长条纹数的变化。（a） PV值；（b） RMS值
Figure 5. Demodulated error curve with the variation of fringe number at $ {\lambda _{\text{s}}} $ for three methods. (a) PV; (b) RMS
图 6 台阶样品实验测试数据。（a）双波长干涉图；（b）双波长干涉图的傅里叶变换频谱；（c）双波长STF的傅里叶变换频谱；（d）提取原始尺寸大小的合成波长干涉图
Figure 6. Measured data for the test step in experiment. (a) Dual-wavelength interferogram; (b) Fourier spectrum of the dual-wavelength interferogram; (c) Fourier spectrum of the dual-wavelength STF; (d) The extracted synthetic-wavelength interferogram with original size
图 7 台阶样品在合成波长干涉条纹为4时的检测结果。（a） DCD方法；（b） CCFC方法
Figure 7. Results of the test step with about 4 fringes for the synthetic wavelength. (a) DCD method; (b) CCFC method
图 8 低载频时台阶样品的测试数据。（a）双波长干涉图；（b）双波长干涉图的傅里叶变换频谱；（c）双波长STF的傅里叶变换频谱；（d）提取的原始尺寸大小的合成波长干涉图
Figure 8. Measured data for the test step in experiment with lower carrier condition. (a) Dual-wavelength interferogram; (b) Fourier spectrum of the dual-wavelength interferogram; (c) Fourier spectrum of the dual-wavelength STF; (d) The extracted synthetic-wavelength interferogram with original size
图 9 合成波长条纹数目为1时测试台阶样品经DCD方法处理的检测结果
Figure 9. Results of the test step by the proposed DCD method with about 1 fringes for the synthetic wavelength
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