
For illustration of basic processes of fourwavelength MWDH in Fig. 1, the object consists of a pair of glass plates with resolution target patterns with the step size between them measured to be 1.54 mm. As detailed in Materials and Methods, a set of four holograms are acquired by phaseshifting digital holography using four lasers with wavelengths
$ {\lambda _1} = 0.639\;616\;\mu {\rm{m}} $ ,$ {\lambda _2} = $ $ 0.639\;595 \; \mu {\rm{m}} $ ,$ {\lambda _3} = 0.639\;539 \; \mu {\rm{m}} $ , and$ {\lambda _4} = 0.638\;929 \; \mu {\rm{m}} $ . Fig. 1a, b display the first two phase profiles$ {\Phi _1}\left({x,y} \right) $ and$ {\Phi _2}\left({x,y} \right) $ . Because of the minute difference of 0.000 025$ \mu {\rm{m}} $ between the two wavelengths, any difference between the two phase profiles is not easily discernable. Take the difference between the two phase profiles, modulo$ 2\pi $ , and rescale with the synthetic wavelength${\Lambda _{12}} = {\lambda _1}{\lambda _2}/\left {{\lambda _1}  {\lambda _2}} \right $ , to obtain the height profile$ {Z_{12}} $ . The threedimensional shape of the object then becomes apparent, shown in Fig. 1c, as well as the step between the two glass plates in Fig. 2a. The set of parameters used in this experiment is tabulated in Table 1.Fig. 1 Basic process of optical phase unwrapping by MWDH using four wavelengths, tabulated in Table 1. The object is a stack of two resolution target surfaces, with field of view
$ 24 \times 24\,{\rm{m}}{{\rm{m}}^2}. $ a$ {\Phi _1}\left({x,y} \right) $ ; b$ {\Phi _2}\left({x,y} \right) $ ; c$ {Z_{12}}\left({x,y} \right) $ ; d$ {Z_{12 \cdots 3}}\left({x,y} \right) $ ; e$ {Z_{12 \cdots 4}}\left({x,y} \right) $ ; f$ {Z_{{\rm{proc}}}}\left({x,y} \right) $ . See Fig. 2 for indication of the zscale. See text for details.Fig. 2 Graphs of crosssections of a
$ {Z_{12}} $ ; b$ {Z_{12 \cdots 3}} $ ; c$ {Z_{12 \cdots 4}} $ ; d$ {Z_{{\rm{proc}}}} $ , through the yellow line indicated in Fig. 1a. The vertical scales, in$ \mu {\rm{m}} $ , are multiplied by a factor 0.5 to account for the reflection geometry. The horizontal scale is the pixel index. A$ 3 \times 3 $ median filter is applied to$ {Z_{{\rm{12}} \cdots {\rm{4}}}} $ to obtain$ {Z_{{\rm{proc}}}} $ . Measured height noise is tabulated in Table 1. The measured step height is 1.55 mm, comparable to the 1.54 mm nominal glass thickness.n $ {\lambda _n},\,\mu {\rm{m}} $ $ {\Delta _{1n}},\,\mu {\rm{m}} $ $ {\Lambda _{1n}},\,\mu {\rm{m}} $ $ \delta {Z_{1n}},\,\mu {\rm{m}} $ 1 0.639 616 2 0.639 595 0.000 021 19 749.176 919.138 3 0.639 539 0.000 077 5 301.693 564.925 4 0.638 929 0.000 687 594.695 593.199 proc 101.721 Table 1. Parameter used in Fig. 1.
$ {\lambda _n} $ : individual wavelength;$ {\Delta _{1n}} $ : difference wavelength;$ {\Lambda _{1n}} $ : synthetic wavelength;$ \delta {Z_{1n}} $ : noise in$ {Z_{1n}} $ ; ‘proc’: final processed profile.The difference of the two phase profiles corresponds to another phase profile with very large synthetic wavelength
$ {\Lambda _{12}} = 19\; 749 \; \mu {\rm{m}} $ , amplifying the wavelength by a factor of$ {\Lambda _{12}}/{\lambda _1} \approx 3 \times {10^4} $ . The difference phase map has the same level of phase noise, other than a factor of order one, as the initial phase maps, meaning that with the amplification of effective wavelength, the noise in the height profile is also amplified by the same factor. In Fig. 2a, the noise in the height profile is measured to be$ \delta {Z_{12}} = 919.1 \; \mu {\rm{m}} $ , and the 3D profile in Fig. 1c shows significant amount of rough and spiky textures on the surfaces. Significant reduction in noise is achieved by iterative stitching of additional difference phase profiles$ {Z_{13}} $ and$ {Z_{14}} $ with shorter synthetic wavelengths$ {\Lambda _{13}} = 5\;302 \;\mu {\rm{m}} $ and$ {\Lambda _{14}} = 595 \; \mu {\rm{m}} $ , respectively. The stitching process maintains the overall height range to$ {\Lambda _{12}} $ while reducing the noise to the proportionately lower levels of$ {Z_{13}} $ and$ {Z_{14}} $ , whose measured noise in Fig. 2d, e are$ \delta {Z_{13}} = 564.9 \;\mu {\rm{m}} $ and$ \delta {Z_{14}} = 593.2 \;\mu {\rm{m}} $ , respectively. The last stitching operation with$ {Z_{14}} $ has no apparent improvement in the measured noise because the reduction in synthetic wavelength is too rapid. The necessary condition is given in the Materials and Methods section, but with the given set of laser wavelengths, there is no flexibility in the choice of wavelengths. On the other hand, as we will see, the generation of effective wavelength can be made very flexible by varying the object illumination angle instead. A common and effective method of reducing spiky noise is using median filters. A mild median filter with$ 3 \times 3 $ window is applied to the last profile$ {Z_{12 \cdots 4}} $ of the iterative series to obtain the final processed profile$ {Z_{{\rm{proc}}}} $ in Fig. 1f and Fig. 2d, where the corresponding noise is reduced to$ \delta {Z_{{\rm{proc}}}} = 101.7 \;\mu {\rm{m}} $ . The measured step height in$ {Z_{{\rm{proc}}}} $ is 1.55 mm, consistent with the nominal thickness of the glass plate.In Fig. 3, 4, a similar set of parameters is used to image the surface of a US quarter dollar coin, placed on a mirrorlike surface. The synthetic wavelengths used were
$ {\Lambda _{12}} = 28\; 213 \;\mu {\rm{m}} $ ,$ {\Lambda _{13}} = 5\; 302\;\mu {\rm{m}} $ ,$ {\Lambda _{14}} = 595 \;\mu {\rm{m}} $ . The singlewavelength phase profiles in Fig. 3a, b have no discernable features, because of the random granular structure of the metallic surface. Yet the surface profile$ {Z_{12}} $ from the difference profile does reveal the overall features of the surface pattern. The iterative processing with ‘stitching’ is applied as above, and in order to improve the image quality, a$ 3\; \times \;3 $ median filter was applied to each step of$ {Z_{12}} $ ,$ {Z_{123}} $ , and$ {Z_{12 \cdots 4}} $ . The noise level, measured over a small smooth area of the coin surface, is reduced from$ \delta {Z_{12}} = 385.0 \; \mu {\rm{m}} $ to$ \delta {Z_{14}} = 3.8 \; \mu {\rm{m}} $ . It is seen that final processed images with just a few micrometer noise is achieved with relatively mild application of median filters.Fig. 3 Example of optical phase unwrapping by MWDH. The object is a US quarter dollar coin, with field of view
$ 24 \times 24\;{\rm{m}}{{\rm{m}}^2} $ . A set of four wavelengths are used, such that$ {\Lambda _{12}} = 28\; 213 \; \mu {\rm{m}} $ ,$ {\Lambda _{13}} = 4 \;328 \; \mu {\rm{m}} $ , and$ {\Lambda _{14}} = 485 \; \mu {\rm{m}} $ . a$ {\Phi _1}\left({x,y} \right) $ ; b$ {\Phi _2}\left({x,y} \right) $ ; c$ {Z_{12}}\left({x,y} \right) $ ; d$ {Z_{12 \cdots 3}}\left({x,y} \right) $ ; e$ {Z_{12 \cdots 4}}\left({x,y} \right) $ ; f$ {Z_{{\rm{proc}}}}\left({x,y} \right) $ . See Fig. 4 for indication of the zscale.Fig. 4 Graphs of crosssections of a
$ {Z_{12}} $ ; b$ {Z_{12 \cdots 3}} $ ; c$ {Z_{12 \cdots 4}} $ ; d$ {Z_{{\rm{proc}}}} $ , through the yellow line indicated in Fig. 3a. The vertical scales, in$ \mu {\rm{m}} $ , are multiplied by a factor 0.5 to account for the reflection geometry. The horizontal scale is the pixel index. A$ 3 \times 3 $ median filter is applied to each of$ {Z_{12}} \sim{Z_{12 \cdots 4}} $ and to$ {Z_{{\rm{proc}}}} $ . Measured height noise varies as$ \delta {Z_{12}} = 385.0 \; \mu {\rm{m}} $ ,$ \delta {Z_{12 \cdots 3}} = 22.6 \; \mu {\rm{m}} $ ,$ \delta {Z_{12 \cdots 4}} = 3.8 \; \mu {\rm{m}} $ , and$ \delta {Z_{{\rm{proc}}}} = 3.2 \; \mu {\rm{m}} $ .The process of MADH is illustrated with an example in Fig. 5, 6, where the object consists of a set of three mirrorlike surfaces with nominal step heights 1.54 mm and 0.94 mm between them. A set of eight holograms are acquired, using a single HeNe laser of
$ {\lambda _0} = 0.632\;8 \; \mu {\rm{m}} $ , at illumination angles$ {\theta _n} = {\theta _0} + {\Delta _n} $ , tabulated in Table 2. In this experiment, an alternate version of the optical apparatus was used, where the reference beam is stationary and only the object illumination is scanned. The starting position$ {\theta _0} = {18^ \circ } $ corresponds to the case of the object beam coinciding with the reference beam direction, while$ {\Delta _n} $ are precisely controlled by the rotation stage with$ {0.000\;25^ \circ } $ resolution. The first two phase profiles$ {\Phi _1} $ and$ {\Phi _2} $ are displayed in Fig. 5a, b. The patterns indicate slight relative tilt between the three surfaces. The angular step$ {\theta _2}  {\theta _1} = {0.0014^ \circ } $ between the first two holograms determines the synthetic wavelength$ {\Lambda _{12}} = 7\; 545 \;\mu {\rm{m}} $ , with the effective wavelengths$ {\lambda _n} = {\lambda _0}\cos {\theta _n} = 0.601\; 829 \;\mu {\rm{m}} $ and$0.601\; 781\; \mu {\rm{m}} $ , for$ n = 1$ and 2, repectively. Additional angular steps$ {\theta _n}  {\theta _1} $ , which increase by a factor of two in each step in this example, lead to synthetic wavelengths$ {\Lambda _{1n}} $ that decrease by a factor of$ \alpha \approx 0.50 $ , each step for$ n > 2 $ . The ‘stitching’ process is iterably applied with the series of profiles$ {Z_{13}}, \cdots,{Z_{18}} $ . The noise levels$ {\delta _{1n}} $ of the ‘stitched’ maps$ {Z_{12 \cdots n}} $ , in Fig. 5, 6 are measured as the standard deviation of a small area near the center of the frame. They progressively decrease by a factor close to α each step, while maintaining the overall height range$ {\Lambda _{12}} $ . The surface profiles rendered as 3D plots in Figs. 5c−f show obvious enhancement of image quality between$ {Z_{12}} $ and$ {Z_{12 \cdots 8}} $ , as the noise decreases from$ \delta {Z_{12}} = 127.4 \; \mu {\rm{m}} $ to$ \delta {Z_{18}} = 3.5 \;\mu {\rm{m}} $ , with$ \delta {Z_{18}}/\delta {Z_{12}} = 0.027\;5 = {\alpha ^{5.18}}$ , consistent with the expected factor of$ {\alpha ^{n  2}} $ . In this case, the noise reduction is quite complete throughout the surface except at the steps because of the beveled edges reflecting insufficient light towards the camera. With application of a mild$ 3 \times 3 $ median filter on the last stitched profile$ {Z_{12 \cdots 8}} $ , a final noise level of$ \delta {Z_{{\rm{proc}}}} = 2.0 \; \mu {\rm{m}} $ is achieved over a height range of$ {\Lambda _{12}} = 3\; 758 \; \mu {\rm{m}} $ .Fig. 5 Basic process of optical phase unwrapping by MADH using eight illumination angles for eight effective wavelengths, tabulated in Table 2. The object is a stack of three mirrorlike surfaces, with field of view
$ 9.87 \times 5.55\,{\rm{m}}{{\rm{m}}^2} $ . a$ {\Phi _1}\left({x,y} \right) $ ; b$ {\Phi _2}\left({x,y} \right) $ ; c$ {Z_{12}}\left({x,y} \right) $ ; d$ {Z_{12 \cdots 3}}\left({x,y} \right) $ ; e$ {Z_{12 \cdots 8}}\left({x,y} \right) $ ; f$ {Z_{{\rm{proc}}}}\left({x,y} \right) $ . Plots for$ {Z_{12 \cdots 4}} $ through$ {Z_{12 \cdots 7}} $ are not displayed. See text for details. See Fig. 6 for indication of the zscale.Fig. 6 Graphs of crosssections of a
$ {Z_{12}} $ ; b$ {Z_{12 \cdots 3}} $ ; c$ {Z_{12 \cdots 8}} $ ; d$ {Z_{{\rm{proc}}}} $ , through the yellow line indicated in Fig. 5a. The vertical scales, in$ \mu {\rm{m}} $ , are multiplied by a factor 0.5 to account for the reflection geometry. The horizontal scale is the pixel index. Graphs for$ {Z_{12 \cdots 4}} $ through$ {Z_{12 \cdots 7}} $ are omitted. A$ 3 \times 3 $ median filter is applied to$ {Z_{{\rm{12}} \cdots {\rm{8}}}} $ to obtain$ {Z_{{\rm{proc}}}} $ . Measured height noise is tabulated in Table 2. The measured step heights are 1.56 mm and 0.97 mm, comparable to the nominal glass thicknesses of 1.54 mm and 0.95 mm.n ${\lambda _n},\,\mu {\rm{m}}$ ${\Delta _{1n}},\,\mu {\rm{m}}$ ${\Lambda _{1n}},\,\mu {\rm{m}}$ $\delta {Z_{1n}},\,\mu {\rm{m}}$ 1 0.601 829 2 0.601 781 0.000 048 7 545.193 127.426 3 0.601 733 0.000 096 3 772.296 76.897 4 0.601 637 0.000 192 1 885.847 35.985 5 0.601 445 0.000 384 942.623 20.029 6 0.601 060 0.000 769 470.397 6.961 7 0.600281 0.001548 233.376 4.057 8 0.598697 0.003 132 115.043 3.507 proc 2.018 Table 2. Parameter used in Fig. 5.
${\lambda _n}$ : individual wavelength;${\Delta _{1n}}$ : difference wavelength;${\Lambda _{1n}}$ : synthetic wavelength;$\delta {Z_{1n}}$ : noise in${Z_{1n}}$ ; ‘proc’: final processed profile.Fig. 7, 8 is an example surface profile of a coin (U.S. penny) placed on another (U.S. quarter). Here a set of eight holograms are acquired using the stepping factor
$ \alpha = 0.50 $ for the synthetic wavelengths ranging from$ {\Lambda _{12}} = 2 \;911.5 \;\mu {\rm{m}} $ to$ {\Lambda _{18}} = 44.1 \;\mu {\rm{m}} $ . Because of the microscopically diffuse surface of the coin, as well as the slopes on the boundaries of the letters and relief features, there is significant noise in the raw profile$ {Z_{12}} $ , shown in Fig. 7c, 8a. The relatively small numerical aperture of the system, approx. 0.1, is not able to collect enough light from these sloped boundary surfaces for accurate phase information. The same stitching procedure as above is applied using the series$ {Z_{13}}, \cdots,{Z_{18}} $ , but after each step a$ 3\; \times\; 3 $ median filter is applied. This produced reasonably clean surface profile of the coin without noticeable degradation or loss of lateral resolution. The measured noise reduced from$ \delta {Z_{12}} = 147.3 \; \mu {\rm{m}} $ to$ \delta {Z_{18}} = 10.4 \; \mu {\rm{m}} $ .Fig. 7 Example of optical phase unwrapping by MADH. The object is a coin, US penny, placed on top of another, a US quarter, with field of view
$ 9.87 \times 5.55\,{\rm{m}}{{\rm{m}}^2} $ . A set of eight wavelengths are used, such that$ {\Lambda _{1n}} = [2\; 911.5, 1\; 455.5, 727.5, 363.5, 180.9, 89.7, 44.1] \;\mu {\rm{m}} $ for$ n = 2,\,3,\, \cdots,\,8 $ . a$ {\Phi _1}\left({x,y} \right) $ ; b$ {\Phi _2}\left({x,y} \right) $ ; c$ {Z_{12}}\left({x,y} \right) $ ; d$ {Z_{12 \cdots 3}}\left({x,y} \right) $ ; e$ {Z_{12 \cdots 8}}\left({x,y} \right) $ ; f$ {Z_{{\rm{proc}}}}\left({x,y} \right) $ . See Fig. 8 for indication of the zscale.Fig. 8 Graphs of crosssections of a
$ {Z_{12}} $ ; b$ {Z_{12 \cdots 3}} $ ; c$ {Z_{12 \cdots 8}} $ ; d$ {Z_{{\rm{proc}}}} $ , through the yellow line indicated in Fig. 7a. The vertical scales, in$ \mu {\rm{m}} $ , are multiplied by a factor 0.5 to account for the reflection geometry. The horizontal scale is the pixel index. A$ 3 \times 3 $ median filter is applied to each of$ {Z_{12}} \sim{Z_{12 \cdots 8}} $ and to$ {Z_{{\rm{proc}}}} $ . Measured height noise varies as$ \delta {Z_{12 \cdots n}} = [147.3, 29.5, 27.2, 23.2, 15.5, 13.9, 10.4] \; \mu {\rm{m}} $ for$ n = 2,\,3,\, \cdots,\,8 $ , and$ \delta {Z_{{\rm{proc}}}} = 9.0 \; \mu {\rm{m}} $ .As described in Section 4, there is a limit on how fast the iterative noise reduction can progress, i.e. how small the stepping factor α can be. For larger amount of initial noise, the stepping factor cannot be chosen too small, for otherwise the iterative noise reduction can become incomplete. To illustrate, in Fig. 9, the same coin surface data as in Fig. 8 is used, but instead of stitching all of the profiles
$ {Z_{13}}, \cdots,{Z_{18}} $ on to$ {Z_{12}} $ , only the partial set of$ {Z_{12}} $ ,$ {Z_{15}} $ , and$ {Z_{18}} $ is used for stitching. Here,$ Z\left({12} \right) $ ,$ Z\left({12 + 15} \right) $ , and$ Z\left({12 + 15 + 18} \right) $ , denote$ {Z_{12}} $ , it stitched with$ {Z_{15}} $ , and the last stitched with$ {Z_{18}} $ , respectively. This in effect reduces the stepping factor to$ \alpha = 0.125$ . Otherwise the same set of processing procedure is applied as with Fig. 8, including the$ 3 \times 3 $ median filter at each stitching step. The stepping factor is too small and the iteration is too fast, and significant amount of noise escapes the reduction procedure, the final noise level being 34$ \mu {\rm{m}} $ for$ \alpha = 0.125$ , compared to 9.0$ \mu {\rm{m}} $ for$ \alpha = 0.50$ .Fig. 9 Effect of iteration being too rapid with
$ \alpha = 0.125 $ . Graphs of crosssections of a$ Z\left({12} \right) $ ; b$ Z\left({12 + 15} \right) $ ; c$ Z\left({12 + 15 + 18} \right) $ ; d$ {Z_{{\rm{proc}}}} $ . The vertical scales, in$ \mu {\rm{m}} $ , are multiplied by a factor 0.5 to account for the reflection geometry. The horizontal scale is the pixel index. A$ 3 \times 3 $ median filter is applied to each profile. Measured height noise varies as$ \delta {Z_{12}} = 147.3 \; \mu {\rm{m}} $ ,$ \delta {Z_{12 + 15}} = 58.7 \; \mu {\rm{m}} $ ,$ \delta {Z_{12 + 18}} = 42.2 \; \mu {\rm{m}} $ , and$ \delta {Z_{{\rm{proc}}}} = 34.1 \; \mu {\rm{m}} $ . See text for details.A few more examples of surface profiles are given in Fig. 10, by optical phase unwrapping with MWDH or MADH. Imaging parameters of the examples are summarized in Table 3.
Fig. 10 Examples of MWDH and MADH images. Imaging parameters are tabulated in Table 3.
Fig object method N_{λ} FOV, µm × µm Λ_{12}, µm Λ_{1N}, µm a PCB MWDH 4 30 500 × 20 352 13 636.2 242.9 b PCB MWDH 4 21 200 × 14 160 18 595.0 243.0 c PCB MWDH 4 21 200 × 14 160 18 595.0 243.0 d brass relief plaque MWDH 4 30 500 × 20 352 13 636.2 241.1 e coin, US dime MADH 8 9 870 × 5 552 1 455.8 22.1 f three mirrorlike surfaces MADH 4 18 432 × 18 432 19 378.6 2 212.3 Table 3. Parameter used in Fig. 10.
${N_\lambda }$ : number of wavelengths;${\Lambda _{12}}$ : first synthetic wavelength;${\Lambda _{1N}}$ : last synthetic wavelength. 
The basic process of multiwavelength optical phase unwrapping is illustrated with simulations in Fig. 11. Suppose the object is an inclined plane of height
$ {a_h} = 15 \; \mu {\rm{m}} $ , depicted with the function$ h\left(x \right) $ in Fig. 11a, and a set of holograms are acquired using a series of wavelengthsFig. 11 Simulation of 2wavelength optical phase unwrapping. a
$ h\left(x \right) $ ; b$ {Z_1}\left(x \right) $ ; c$ {Z_2}\left(x \right) $ ; d$ {Z_{12}}\left(x \right) $ . The vertical scales are in$ \mu {\rm{m}} $ . The horizontal scale is the pixel index. Height of the incline is 15$ \mu {\rm{m}} $ . Assumed wavelengths are$ {\lambda _1} = 0.635 \; \mu {\rm{m}} $ ,$ {\lambda _2} = 0.600 \; \mu {\rm{m}} $ , so that$ {\Lambda _{12}} = 10.886 \; \mu {\rm{m}} $ . Phase noise of$ \varepsilon = 3 \%$ results in height noise$ \delta {Z_{12}} = 0.462 \; \mu {\rm{m}} $ .$$ \left\{ {{\lambda _n}} \right\} = {\lambda _1},{\lambda _2},{\lambda _3}, \cdots $$ (1) The complex optical fields
$$ {E_n}\left({x,y} \right) = {a_n}\left({x,y} \right) \cdot \exp \left[ {i{\Phi _n}\left({x,y} \right)} \right] $$ (2) are computed from the captured holographic interference patterns, where
$ {a_n}\left({x,y} \right) $ and$ {\Phi _n}\left({x,y} \right) $ are the amplitude and phase profiles of the optical field, respectively. In offaxis holography, the phase is encoded as a modulation of the highspatialfrequency interference fringe patterns in a single camera frame, whereas in phaseshifting digital holography, three or more camera frames are combined while the phase of reference field is shifted. If the acquired optical field resulted from reflection from or transmission through a structured surface or layer, the phase profile$ {\Phi _n}\left({x,y} \right) $ is directly proportional to the profile of the optical path length$ Z\left({x,y} \right) $ $$ {\Phi _n}\left({x,y} \right) = Z\left({x,y} \right) \cdot \frac{{2\pi }}{{{\lambda _n}}} $$ (3) The wrapped phase problem arises from the fact that the optical phase is defined only up to modulo
$ 2\pi $ , and the derived surface height profile$$ {Z_n}\left({x,y} \right) = {\Phi _n}\left({x,y} \right) \cdot \frac{{{\lambda _n}}}{{2\pi }} $$ (4) is limited to modulo
$ {\lambda _n} $ . In Fig. 11b the height profile$ {Z_1} $ acquired using a wavelength$ {\lambda _1} = 0.635 \; \mu {\rm{m}} $ has a sawtooth shape of height equal to the wavelength. A phase noise of standard deviation$ 2\pi \cdot \varepsilon $ , with$ \varepsilon = 3\% $ , is added in the phase profile to simulate random phase noise between hologram acquisitions.Twowavelength optical phase unwrapping is based on taking the conjugate product of two optical fields acquired with two slightly different wavelengths,
$ {\lambda _1} $ and$ {\lambda _2} $ , as$$ {E_{12}}\left({x,y} \right) = {E_1} \cdot E_2^* = {a_{12}} \cdot \exp \left[ {i{\Phi _{12}}\left({x,y} \right)} \right] $$ (5) where
$$ {\Phi _{12}} = {\rm{mod}} \left({{\Phi _1}  {\Phi _2} + \pi,2\pi } \right)  \pi $$ (6) The particular modulo operation is necessary when the phase angle of a complex number is defined in the range
$ \left[ {  \pi,\pi } \right] $ . The difference phase$ {\Phi _{12}} $ is equivalent to the phase of an effective, or synthetic, wavelength$ {\Lambda _{12}} $ , which can be much larger than the original wavelengths:$$ {\Phi _{12}} = 2\pi \frac{Z}{{{\lambda _1}}}  2\pi \frac{Z}{{{\lambda _1}}} = 2\pi \frac{Z}{{{\Lambda _{12}}}} $$ (7) where
$$ {\Lambda _{12}} = \frac{{{\lambda _1}{\lambda _2}}}{{\left {{\lambda _1}  {\lambda _2}} \right}} $$ (8) The optical path derived from the difference phase can range up to the synthetic wavelength,
$$ {Z_{12}}\left({x,y} \right) = {\Phi _{12}} \cdot \frac{{{\Lambda _{12}}}}{{2\pi }} $$ (9) as illustrated in Fig. 11d with
$ {\lambda _1} = 0.635\;0 \;\mu {\rm{m}} $ and$ {\lambda _2} = 0.600\;0 \; \mu {\rm{m}} $ , so that$ {\Lambda _{12}} = 10.886 \;\mu {\rm{m}} $ , and the height profile$ h\left(x \right) $ is properly represented up to the maximum$ {\Lambda _{12}} $ .By choosing two closely spaced wavelengths
$ {\lambda _1} $ and$ {\lambda _2} $ , the synthetic wavelength$ {\Lambda _{12}} $ can be made as large as necessary to cover the maximum surface height of the object to be profiled. On the other hand, the extension of the synthetic wavelength comes with amplification of noise in the height profile. For suppose the phase noise in$ {\Phi _1} $ is a fraction of$ 2\pi $ ,$$ \delta {\Phi _n} = 2\pi \cdot \varepsilon $$ (10) where ε is a dimensionless random function that includes both spatial and frametoframe variations. The corresponding noise in the derived optical path is
$$ \delta {Z_n} = \delta {\Phi _n} \cdot \frac{{{\lambda _n}}}{{2\pi }} = \varepsilon \cdot {\lambda _n} $$ (11) which is approximately
$ \delta {Z_1} = 0.019\; \mu {\rm{m}} $ with the assumed values of$ {\lambda _1} = 0.635 \; \mu {\rm{m}} $ and$ \varepsilon = 3\%$ . The noise in the optical height derived from the difference phase and synthetic wavelength is, with$ \delta {\Phi _{12}} \approx \sqrt 2 \delta {\Phi _1} $ ,$$ \delta {Z_{12}} = \delta {\Phi _{12}} \cdot \frac{{{\Lambda _{12}}}}{{2\pi }} = \sqrt 2 \varepsilon \cdot {\Lambda _{12}} $$ (12) and the noise in the optical height is amplified in proportion to the amplification in wavelength, to approx.
$ \delta {Z_{12}} \approx 0.462\;\mu {\rm{m}} $ in Fig. 11d. The amplification in wavelength is accompanied by similar amplification in noise.The noise can be reduced if a shorter synthetic wavelength,
$ {\Lambda _{13}} $ is used, with a larger difference$ {\lambda _1}  {\lambda _3} $ . The simulation in Fig. 12 uses$ {\lambda _3} = 0.530\;0\; \mu {\rm{m}} $ , so that$ {\Lambda _{13}} = 3.205 \; \mu {\rm{m}} $ . Start from$ {Z_{12}}\left({x,y} \right) $ with the range$ {\Lambda _{12}} = 10.886 \; \mu {\rm{m}} $ , Fig. 12a, and make integer steps of$ {\Lambda _{13}} $ , Fig. 12b. Then, ‘stitch’ the steps with the new profile$ {Z_{13}}\left({x,y} \right) $ , Fig. 12c. That is,Fig. 12 Simulation of 3wavelength optical phase unwrapping. a
$ {Z_{12}} $ ; b$ {Y_{12}} = {\rm{round}}\left( {{Z_{12}}/{\Lambda _{13}}} \right) \cdot {\Lambda _{13}} $ ; c$ {Z_{13}} $ ; d$ {Z'_{12 \cdots 3}} = {Y_{12}} + {Z_{13}} $ ; e$ {Z_{12 \cdots 3}} $ . The vertical scales are in$ \mu {\rm{m}} $ . The horizontal scale is the pixel index. Assumed third wavelength is$ {\lambda _3} = 0.530 \; \mu {\rm{m}} $ , so that$ {\Lambda _{13}} = 3.205 \; \mu {\rm{m}} $ . Final height noise is reduced to$ \delta {Z_{12 \cdots 3}} = 0.136 \; \mu {\rm{m}} $ .$$ {Z_{12 \cdots 3}}\left({x,y} \right) = {\rm{round}}\left({\frac{{{Z_{12}}\left({x,y} \right)}}{{{\Lambda _{13}}}}} \right) \cdot {\Lambda _{13}} + {Z_{13}}\left({x,y} \right) $$ (13) where
$ {\rm{round}}\left({} \right) $ represents integer rounding operation. If the phase noise is included in the above expression,$$ \begin{split} {Z_{12 \cdots 3}} + \delta {Z_{12 \cdots 3}} =& {\rm{round}}\left({\frac{{{Z_{12}} + \delta {Z_{12 \cdots 3}}}}{{{\Lambda _{13}}}}} \right) \cdot {\Lambda _{13}} + \left({{Z_{13}} + \delta {Z_{13}}} \right) \\ =& {Z_{12 \cdots 3}} + \Delta \cdot {\Lambda _{13}} + \delta {Z_{13}} \\ =& {Z_{12 \cdots 3}} + \Delta \cdot {\Lambda _{13}} + \sqrt 2 \varepsilon {\Lambda _{13}} \\ \end{split} $$ (14) Here
$ \Delta \cdot {\Lambda _{13}} $ represents spikes of height$ {\Lambda _{13}} $ scattered at positions near the step boundaries of$ {\rm{round}}\left({} \right) $ , as evident in Fig. 12d. These spikes can be suppressed by adding or subtracting$ {\Lambda _{13}} $ wherever the absolute difference$ \left {{Z_{12}}  {Z_{12 \cdots 3}}} \right $ is comparable to$ {\Lambda _{13}} $ . This process is valid with the requirement$$ \frac{{\delta {Z_{12}}}}{{{\Lambda _{13}}}} \approx \sqrt 2 \varepsilon \cdot \frac{{{\Lambda _{12}}}}{{{\Lambda _{13}}}} \ll 1 $$ (15) and
$ {\Lambda _{13}} $ should be chosen large enough compared to$ \sqrt 2 \varepsilon \cdot {\Lambda _{12}} $ , or$$ \alpha = {\Lambda _{13}}/{\Lambda _{12}} \gg \sqrt 2 \varepsilon $$ (16) The overall noise is then reduced from
$ \sqrt 2 \varepsilon \cdot {\Lambda _{12}} \approx $ $ 0.462\; \mu {\rm{m}} $ to$ \sqrt 2 \varepsilon \cdot {\Lambda _{13}} \approx 0.136 \; \mu {\rm{m}} $ , as demonstrated in Fig. 12e.In the above simulation example, the synthetic wavelength is extended to over 10
$ \mu {\rm{m}} $ , which can be suitable for profiling microscopically structured surfaces. On the other hand, for applications requiring profiling of macroscopic surfaces with millimeters of height range, the large amplification of wavelength entails similarly large noise. In order to reduce the large noise, it is then necessary to extend the iterative series using a larger number of wavelengths. Thus, another difference phase profile$ {\Phi _{14}}\left({x,y} \right) $ with synthetic wavelength$ {\Lambda _{14}} < {\Lambda _{13}} $ is used to ‘stitch’ the new profile$ {Z_{14}}\left({x,y} \right) $ on to the previous synthesized profile$ {Z_{12 \cdots 3}}\left({x,y} \right) $ . If we again let$ {\Lambda _{14}} = \alpha {\Lambda _{13}} $ , the noise level for the new profile$ {Z_{12 \cdots 4}} $ is$ \delta {Z_{12 \cdots 4}} = \sqrt 2 \varepsilon {\Lambda _{14}} = \sqrt 2 \varepsilon \cdot {\alpha ^2}{\Lambda _{12}} $ . The process can continue until$ \delta {Z_{12 \cdots n}} = \varepsilon \cdot {\Lambda _{1n}} = \varepsilon \cdot {\alpha ^{\left({n  2} \right)}}{\Lambda _{12}} $ reaches desired height resolution. In the simulation of Fig. 13, the height range is now$ {\Lambda _{12}} = 2\; 000.0 \; \mu {\rm{m}} $ with$ {\lambda _1} = 0.635\; 000 \; \mu {\rm{m}} $ and$ {\lambda _2} = 0.634\; 798 \; \mu {\rm{m}} $ , or$ {\Delta _{12}} = {\lambda _1}  {\lambda _2} = 0.000 \;202 \; \mu {\rm{m}} $ . Three more wavelengths are used with difference wavelengths$ {\Delta _{13}} = 0.000\; 805 \; \mu {\rm{m}} $ ,$ {\Delta _{14}} = 0.003\; 209\; \mu {\rm{m}} $ , and$ {\Delta _{15}} = 0.012\; 646 \; \mu {\rm{m}} $ for synthetic wavelengths$ {\Lambda _{13}} = 500.0 \; \mu {\rm{m}} $ ,$ {\Lambda _{14}} = 125.0\; \mu {\rm{m}} $ , and$ {\Lambda _{15}} = 31.25 \; \mu {\rm{m}} $ , respectively, with a fixed stepping factor$ \alpha = 0.25$ for the series. Assuming phase noise with$ \varepsilon = 3\%$ , the initial noise 84.9$ \mu {\rm{m}} $ in$ {Z_{12}} $ is progressively reduced to 21.2$ \mu {\rm{m}} $ , 5.3$ \mu {\rm{m}} $ , and 1.3$ \mu {\rm{m}} $ in$ {Z_{13}} $ ,$ {Z_{14}} $ , and$ {Z_{15}} $ , respectively.Fig. 13 Simulation of iterative reduction of noise for large height range
$ {\Lambda _{12}} = 2 000.0 \; \mu {\rm{m}} $ , using a series of five wavelengths. a$ {Z_{12}}\left(x \right) $ ; b$ {Z_{12 \cdots 3}}\left(x \right) $ ; c$ {Z_{12 \cdots 4}}\left(x \right) $ ; d$ {Z_{12 \cdots 5}}\left(x \right) $ . The vertical scales are in$ \mu {\rm{m}} $ . The horizontal scale is the pixel index. With a stepping factor$ \alpha = 0.25 $ , the synthetic wavlengths are set to$ {\Lambda _{1n}} = [2\; 000.0, $ $ 500.0, 125.0, 31.25] \; \mu {\rm{m}} $ , for$ n = 2,\, \cdots,\,5 $ , which can be obtained from the wavelengths$ {\lambda _n} = [0.635\; 000, 0.634\; 798, 0.634\;195, 0.631\;791, $ $ 0.62\; 354] \; \mu {\rm{m}} $ , for$ n = 1,\,2,\, \cdots,\,5 $ . With the assumed phase noise of$ \varepsilon = 3\% $ , the height noise progressively dcreases as$ \delta {Z_{12 \cdots n}} = [77.7, 20.5, 5.3, 1.2] \; \mu {\rm{m}} $ for$ n = 2,\, \cdots,\,5 $ .If the iteration is made to run too fast with the stepping factor α too small, the noise reduction by stitching as described above can be incomplete and significant amount of noise can remain after the procedure. In Fig. 14a, the noise is increased to
$ \varepsilon = 7\%$ while using the same set of wavelengths$ {\lambda _1}, \cdots,{\lambda _5} $ , with$ \alpha = 0.25$ , as the above example. The condition$ \varepsilon \ll \alpha $ is not sufficiently strong that there are many spikes$ \Delta \cdot {\Lambda _{1n}} $ in resulting profiles$ {Z_{12 \cdots n}} $ . In order to improve the noise figure, we slow the iteration using a larger value of$ \alpha = 0.50$ , and increase the number of wavelengths to eight, so that the final synthetic wavlengths in the two series are the same:$ {\Lambda _{18}} = 31.25 \; \mu {\rm{m}} $ with$ \alpha = 0.25$ and$ {\Lambda _{15}} = 31.25 \; \mu {\rm{m}} $ with$ \alpha = 0.50$ . In Fig. 14b, the stitching process is applied to the series of eight wavelengths to obtain the profiles$ {Z_{12}},\,{Z_{12 \cdots 3}},\, \cdots, $ $ \,{Z_{12 \cdots 8}} $ . The final noise level close to$ \delta {Z_{12 \cdots 8}} = 2.19 \;\mu {\rm{m}} $ is achieved without significant number of spike. This simulation result suggests that large height or noisy surfaces may be profiled using slower progression of$ {\Lambda _{1n}} $ , i.e. larger α, and larger number of wavelengths.Fig. 14 Effect of stepping factor α for large noise profiles, with
$ \varepsilon = 7\% $ . For the column a, the same stepping factor$ \alpha = 0.25 $ is used as in Fig. 13. The measured height noise is$ \delta {Z_{12 \cdots n}} = [110.0, 37.2, 51.6, 50.6] \; \mu {\rm{m}} $ for$ n = 2,\, \cdots,\,5 $ . For the column b, a slower stepping factor$ \alpha = 0.50 $ is used with eight wavelengths, so that the final synthetic wavelength$ {\Lambda _{18}} = 31.25 \; \mu {\rm{m}} $ is the same as$ {\Lambda _{15}} $ in a). Graphs for$ {Z_{12 \cdots 5}} \sim{Z_{12 \cdots 7}} $ are omitted. The vertical scales are in$ \mu {\rm{m}} $ . The horizontal scale is the pixel index. Now, the measured height noise is$ \delta {Z_{12 \cdots n}} = [99.2, 51.5, 35.5, $ $ 13.8, 11.5, 2.9, 2.2] \; \mu {\rm{m}} $ for$ n = 2,\, \cdots,\,8 $ . That is, the final noise value of$ \delta {Z_{12 \cdots 8}} = 2.2\,\mu {\rm{m}} $ obtains for$ \alpha = 0.50 $ , instead of$ \delta {Z_{12 \cdots 5}} = 50.6\,\mu {\rm{m}} $ for$ \alpha = 0.25 $ .The ‘stitching’ process is rather sensative to a few different types of errors and these need to be addressed carefully. The process requires that each of the phase profiles
$ {\Phi _n}\left({x,y} \right) $ is referenced to an identical global phase, which is never the case due to random variations in the laser, optical, and mechanical systems. This is remedied by fixing the difference phase$ {\Phi _{1n}}\left({{x_0},{y_0}} \right) $ at a suitable, relatively noisefree, reference point to be zero.Another source of error is that the synthetic wavelength is inversely proportionale to a small difference wavelength. In MWDH, the laser wavelengths drift with temperature and other factors. Although a high precision wave meter is available, the current setup does not incorporate tight monitoring and control of it. In MADH, the angular position is more closely monitored but the precision is limited and any mechanical instability may affect it. In any case, at least in the current setup, it is necessary, during postprocessing, to verify and adjust the accuracy of synthetic wavelengths. This is carried out by scanning the synthetic wavelength over a range and looking for optimal values for minimizing the noise spikes.

For example, to achieve a height range of
$ {\Lambda _{12}} = 2.0$ mm near$ \lambda = 0.635\;\mu {\rm{m}} $ , the required wavelength difference is$ \Delta \lambda \approx 0.2$ nm. And to achieve$ \mu {\rm{m}} $ level height resolution, several wavelengths are needed. For such applications, acquiring and maintaining multiple laser sources with precisiontuned wavelengths quickly becomes prohibitively difficult. The cost and complexity of the optical system can be significantly reduced if, instead of multiple lasers, a single laser is used and the multiple effective wavelengths are generated by changing the illumination angle of the object. The variation of the optical phase through a given height h of the object is$ 2\pi h/\lambda \cos \theta $ , which can be described as illumination from a fixed direction with effective wavelength$$ {\lambda _\theta } = \lambda \cdot \cos \theta $$ (17) Multiple effective wavelengths
$ {\lambda _n} $ can be generated simply by tuning the illumination angle to various positions$ {\theta _n} $ . The synthetic wavelength between the 1st and nth effective wavelengths is then$$ {\Lambda _{1n}} = \frac{{{\lambda _1}{\lambda _n}}}{{\left {{\lambda _1}  {\lambda _n}} \right}} = \lambda \cdot \frac{{\cos {\theta _1}\cos {\theta _n}}}{{\left {\cos {\theta _1}  \cos {\theta _n}} \right}} \approx \lambda \cdot \frac{{\cos {\theta _1}\cos {\theta _n}}}{{\Delta \theta \cdot \sin \theta }} $$ (18) The rest of the theoretical description of MADH is in exact parallel with that of MWDH, but there are significant differences in practical considerations of the two methods. The MWDH requires multiple lasers or tunable lasers as well as a means to combine and switch between them, whereas the MADH requires one good laser and a means for fine control of illumination angle. To achieve 2.0 mm height range as above, the required angle shift is
$ \Delta \theta = {0.070^ \circ } $ , when$ \theta = {15^ \circ } $ , indicating the level of angular precision needed. The MADH is mostly immune from chromatic aberration but can be affected by spherical or higher order geometric aberrations due to the movement of the illumination across optical apertures. In order to keep$ {\Lambda _{1n}} $ becoming too large, or the angular step$ \Delta \theta $ too small, the angle θ needs to be somewhat away from zero,$ \theta \sim {15^ \circ } $ for example. This can cause shadowing or obscuration of the object surface when the surface has sharp or tall features. 
A schematic of one version of the optical system is depicted in Fig. 15, based on Michelson interferometer configuration. It can accommodate several operating modes, including multiwavelength or multiangle holography, as well as offaxis or phaseshifting holography. The fiber coupler FC introduces laser light into the system. Typical power input at FC is a few mW. Expanded and collimated beam is reflected by the mirror MI. The confocal lens combination L1 (f1 = 100 mm) and L2 (f2 = 200 mm) images the rotating mirror to the object plane, which is in turn imaged on to the camera plane by the lens L4 and a camera lens LC. The reference mirror MR is also conjugate with the camera plane, and is mounted on a piezo actuator for phase shifting. The polarizing beamsplitter PBS and quarter wave plates QO and QR together with the polarizer P controls and optimizes intensity ratio between the object and reference beams. The lenses L2 and L4 and quarter wave plates and polarizer, as well as the polarizing beam splitter are 2 inches in diameter or width. Typical field of view of the object plane is 10 ~ 30 mm.
Fig. 15 Schematic of the MWDH/MADH apparatus. FC: fiber coupler; MO: microscope objective; L3: collimating lens; MI: illumination mirror on digital rotation stage; L1 & L2: confocal lens pair to image MI to object and reference planes; PBS: polarizing beam splitter; QO & QR: quarter wave plates; MR: piezomounted reference mirror; L4 & LC: camera imaging lens; P: linear polarizer; cam: camera.
For multiwavelength experiments, a set of four lasers (Lasos DPSS) are used with nominal wavelengths
$ {\lambda _1} = 639.617$ nm,$ {\lambda _2} = 639.592$ nm,$ {\lambda _3} = 639.416$ nm, and$ {\lambda _4} = 637.928$ nm. These lasers are fibercoupled into a fourchannel fiber switch (Leoni), before being coupled into the interferometer. All fibers are polarization preserving single mode. The laser wavelengths do drift with changing temperature, and a high finesse wavemeter (Toptica) with 0.1 pm resolution was used to measure the wavelengths in real time before calculating synthetic wavelengths and other parameters.For multiangle experiments, a 30 mW HeNe laser fibercoupled into the interferometer is deflected by the illumination mirror MI mounted on a motorized rotation stage (Thorlabs DDR25), which has encoder resolution of 0.00025 deg/count. The starting angle
$ {\theta _0} = {18^\circ } $ is set with manual estimate using a ruler. Instead of trying to measure this angle with any better precision, the synthetic wavelength can be calibrated according to the known step size of an object. By placing both the object and reference planes conjugate to the camera, it is ensured that the relative angle of incidence between the two beams does not change with the rotation of the mirror.In another configuration, based on a modified MachZhender interferometer, only the object beam is tilted while the reference beam remains stationary. This produces a slope in the holographic phase profile, but can be easily compensated for numerically in reconstruction. The reference mirror MR can be tilted by an appropriate amount for offaxis digital holography, or piezoshifted for phaseshifting digital holography.
Typically, a complexvalued hologram is acquired by combining several camera frames of interference intensity while the reference mirror is piezoshifted, per standard procedures of phaseshifting digital holography. Then the source wavelength is switched in MWDH or the MI mirror angle is switched in MADH. A series of such holograms are acquired, which are then postprocessed to construct the 3D surface profile of the object. Pythonbased programs are used for most of hardware control, image acquisition, and holographic image processing. Some parts of the software system also use LabVIEW and Matlab.
Phase microscopy and surface profilometry by digital holography
 Light: Advanced Manufacturing 3, Article number: (2022)
 Received: 17 September 2021
 Revised: 08 February 2022
 Accepted: 18 February 2022 Published online: 06 May 2022
doi: https://doi.org/10.37188/lam.2022.019
Abstract:
Quantitative phase microscopy by digital holography is a good candidate for highspeed, high precision profilometry. Multiwavelength optical phase unwrapping avoids difficulties of numerical unwrapping methods, and can generate surface topographic images with large axial range and high axial resolution. But the large axial range is accompanied by proportionately large noise. An iterative process utilizing holograms acquired with a series of wavelengths is shown to be effective in reducing the noise to a few micrometers even over the axial range of several millimeters. An alternate approach with shifting of illumination angle, instead of using multiple laser sources, provides multiple effective wavelengths from a single laser, greatly simplifying the system complexity and providing great flexibility in the wavelength selection. Experiments are performed demonstrating the basic processes of multiwavelength digital holography (MWDH) and multiangle digital holography (MADH). Example images are presented for surface profiles of various types of surface structures. The methods have potential for versatile, high performance surface profilometry, with compact optical system and straightforward processing algorithms.
Research Summary
Holographic surface profiling: fast highresolution 3D surface topography
Current technology in surface profilometry is limited by mechanical scanning or insufficient resolution. A new type of 3D surface profile topography is based on digital holography where the amplitude and phase of light reflected by the surface is acquired in a set of holograms, which are then numerically processed by the computer. The holograms acquired using multiple wavelengths or multiple angles of illumination are combined in an iterative process to yield surface profiles with several millimeters of range and with a few micrometers of height resolution. The methods have potential for versatile, high performance surface profilometry, with compact optical system and straightforward processing algorithm.
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