
Here, we introduce the principle of DWT with an example in holographic imaging. Consider an interference between a plane wave and a point source, which is the simplest inline hologram. The intensity pattern is
$$\begin{split} {I}_{0}\left(x,y:z\right)=&{I}_{a}+{I}_{b}cos\left({k}_{0}\frac{{x}^{2}+{y}^{2}}{2z}\right) \\=&{I}_{a}+{I}_{b}cos\left(\frac{\pi \left({x}^{2}+{y}^{2}\right)}{{\lambda }_{0}z}\right) \end{split} $$ (1) where z is the distance from the point source to the measurement plane, and
$ {k}_{0}=2\pi /{\lambda }_{0} $ is the propagation constant with$ {\lambda }_{0} $ being the wavelength. Assume now, that there are two wavelengths$ {\lambda }_{1,2} $ making this interference pattern, then the respective intensity patterns are$$ {I}_{1,2}\left(x,y:z\right)={I}_{a}+{I}_{b}cos\left(\frac{{k}_{1,2}\left({x}^{2}+{y}^{2}\right)}{2z}\right) $$ (2) where
$ {k}_{1,2}=2\pi /{\lambda }_{1,2} $ and the constants$ {I}_{a},{I}_{b} $ have been left unchanged for the two wavelengths for simplicity. If these two intensity patterns are added, the sum of the cosines yields$$\begin{split}& cos\left(\frac{{k}_{1}\left({x}^{2}+{y}^{2}\right)}{2z}\right)+cos\left(\frac{{k}_{2}\left({x}^{2}+{y}^{2}\right)}{2z}\right)\\=&2cos\left(\frac{\left({k}_{1}+{k}_{2}\right)\left({x}^{2}+{y}^{2}\right)}{2z}\right)cos\left(\frac{\left({k}_{1}{k}_{2}\right)\left({x}^{2}+{y}^{2}\right)}{2z}\right) \end{split}$$ (3) The addition and subtraction of the propagation constants can be recast in the form
$ {k}_{1}\pm {k}_{2}= 2\pi ({{\lambda }_{2}\pm {\lambda }_{1}})/({{\lambda }_{1}{\lambda }_{2}}) $ . Of relevance here is the second term, expressible as$ cos\left({\pi \left({x}^{2}+{y}^{2}\right)}/({\Lambda z})\right) $ , where$$ \Lambda =\frac{{\lambda }_{1}{\lambda }_{2}}{{\lambda }_{2}{\lambda }_{1}} $$ (4) In other words, it is as if the interference pattern is now being recorded by a wavelength which is neither
$ {\lambda }_{1} $ nor$ {\lambda }_{2} $ , but a synthetic or artificially created wavelength Λ, which, typically, is expected to be much larger than$ {\lambda }_{1,2} $ . The resulting advantage is that there are far less number of Newton’s rings.In the first demonstration of dualwavelength holography (DWH), the fringe pattern obtained from light passing through an optical element such as a lens (i.e., a phase object) under test, using a wavelength
$ {\lambda }_{1} $ in an interferometer such as the MachZehnder type as shown in Fig. 1, was photographed. This photographic recording of the fringe pattern (hologram) was then developed and replaced in the interferometer in the exact position it occupied during exposure, and it was illuminated with the fringe pattern obtained by testing the optical element using a different wavelength$ {\lambda }_{2} $ . The moiré pattern obtained is identical to the interferogram that would have been obtained if the optical element were tested using a synthetic wavelength Λ as defined in Eq. 4. A simple way to see this is to note that if the first hologram is “read” using the interfering waves from the second wavelength, note that$ cos\left({{k}_{1}\left({x}^{2}+{y}^{2}\right)}/({2z})\right) $ multiplied by the diverging wave from the point source$ exp\left(j{k}_{2}{\left({x}^{2}+{y}^{2}\right)}/({2z})\right) $ contains a term$ exp\left(j{({k}_{1}k}_{2})({{x}^{2}+{y}^{2}})/({2z})\right) $ , where j denotes the square root of 1 in engineering convention. The intensity pattern comprising the interferogram contains$ cos\left({({k}_{1}k}_{2})({{x}^{2}+{y}^{2}})/({2z})\right) $ , which is similar to the second term on the RHS of Eq. 3, and therefore can be expressed in terms of the synthetic wavelength given in Eq. 4.Fig. 1 MachZehnder configuration for hologram recording using wavelength
$ {\lambda }_{1} $ , followed by reconstruction using wavelength$ {\lambda }_{2} $ , yielding the interferogram or moire pattern generated using the synthetic wavelength Λ. This schematic is adapted from the original figure in Wyant^{7}.DWH involves finding the difference between a fringe pattern recorded at one instant of time and a fringe pattern existing at some later instant of time. If the two fringe patterns are different for reasons other than wavelength change, e.g., air turbulence, incorrect results result. The effect of air turbulence can be reduced by recording the two interferograms resulting from the two wavelengths simultaneously. When this interferogram (hologram) is illuminated with a plane wave, spatially filtered, and reimaged, one obtains an interferogram identical to that obtained using the first method of DWH described above. Since both fringe patterns are recorded simultaneously, and dispersion is small, air turbulence is essentially the same as if a long wavelength light source were used in the interferometer.
With the advent of digital recording, DWH could be modified so that two holograms are either simultaneously or sequentially recorded using wavelengths
$ {\lambda }_{1,2} $ , followed by numerically processing the phase data after reconstruction. The methodology of DH is slightly different from conventional holography. While in analog holography the interference pattern is recorded on a film and developed, in DH the intensity is recorded in a digital camera sensor. The hologram is numerically reconstructed using the laws of Fresnel propagation (or backpropagation), depending on whether the virtual or real image is being reconstructed.Consider the hologram of a point source (see Eq. 1), which is slightly modified to take into account any angle that may be introduced between the reference and the object:
$$ h(x,y){\propto I}_{0}\left(x,y:z\right)={I}_{a}+{I}_{b}cos\left({k}_{0}\frac{{x}^{2}+{y}^{2}}{2z}\right)cos\left(Kx\right) $$ (5) where
$ K={k}_{0}sin\theta $ , with θ representing the angle in the$ xz $ plane between the reference and the object waves. If this is “illuminated” by a plane wave and back propagated, the optical field after a distance$ –z $ of backpropagation can be expressed as$$\begin{split} \Gamma\left(x,y\right)=&h\left(x,y\right)* g\left(x,y;z\right)\propto h\left(x,y\right)\\&*\exp\left(+j{k}_{0}\frac{{x}^{2}+{y}^{2}}{2z}\right) \end{split}$$ (6) where
$ g\left(x,y;z\right) $ is the impulse response for propagation, * denotes the convolution operator. This convolution can be computed with a single Fourier transform method:$$ \begin{split} {\rm{\Gamma }}\left( {x,y} \right) &= \mathop \int\int \limits_{  \infty }^\infty h\left( {x',y'} \right)exp \left( { + j{k_0}\frac{{{{\left( {x  x'} \right)}^2} + {{\left( {y  y'} \right)}^2}}}{{2z}}} \right)dx'dy'\\ \propto & \mathop \int\int \limits_{  \infty }^\infty h\left( {x',y'} \right)exp \left( {j{k_0}\frac{{{{x'}^2} + {{y'}^2}}}{{2z}}} \right)\\&exp \left( {j\left( {\frac{{{k_0}}}{z}} \right)\left( {xx' + yy'} \right)} \right)dx'dy'\\ =& {F_{x,y}}{\left. {\left[ {h\left( {x',y'} \right)exp \left( {j{k_0}\frac{{{{x'}^2} + {{y'}^2}}}{{2z}}} \right)} \right]} \right_{{k_{x,y}} = {k_0}\left( {x,y} \right)/z}} \end{split}$$ (7) In Eq. 7,
$ {F}_{x,y} $ is the Fourier transform operator. For the simple hologram function stated above, it can be shown that after propagating a distance$ –z $ , one of the terms in$ h\left(x,y\right) $ will yield the (virtual) image of the point source, with the other terms corresponding to the dc (zeroth order) and the defocused (real or twin) image of the object. If there is an angle between the reference and the object while recording the hologram, the different components will be spatially separated. Alternatively,$ h\left(x,y\right) $ can be Fourier transformed, filtered around the spatial frequency K, and centered before backpropagation; this removes the dc (zerothorder) and the twinimage components.From the reconstructed complex field
$ \Gamma \left(x,y\right) $ in Eq. 7, the intensity and phase of the object can be obtained using$$ I\left(x,y\right)={\left\Gamma \left(x,y\right)\right}^{2} $$ (8) $$ \phi \left(x,y\right)=arctan\left(\frac{Im\left[\Gamma \left(x,y\right)\right]}{Re\left[\Gamma \left(x,y\right)\right]}\right) $$ (9) The phase
$ \phi (x,y) $ is proportional to the object surface feature height and the illumination angle θ, theoretically,$ 0\le \theta < 90^{\circ} $ ^{102,103}. The reconstructed phase in Eq. 9 is a 2D map of the wrapped phase, exhibiting modulo$ 2\pi $ fringe spacing of the wavelength λ, which must be unwrapped to yield the absolute phase. The resulting unwrapped phase is the phase profile for a phase object, and plotted in either 2D or 3D. A commonly used unwrapping technique is the phaseunwrapping maxflow/mincut (PUMA) algorithm, based on the graphcut technique developed by BioucasDias and Valadao^{104}. The PUMA algorithm is commonly chosen for its ready availability and good performance in the presence of significant phase noise.Just as light passing through a phase object acquires additional phase, light reflected off a contoured object also acquires phase, and can be similarly studied, as shown schematically in Fig. 2. Holographic topography is the measurement of surface shape by means of holographic recording and reconstruction. When recorded digitally, the phase contours may be postprocessed to yield the object height up to 1/100th of the synthetic wavelength^{105}. Digital holographic topography is a more general 2D case of holographic profilometry, where only a crosssection of a given topography can be extracted, and avoids the need for scanning.
Fig. 2 Schematic showing that light transmitted through a phase object as in a or reflected from a surface as in b acquire phase profiles, characteristic of the refractive index profile or shape of the phase object as in a or the topography of the surface as in b, respectively.
DWDH has been explored extensively to quantify surface topography and displacement measurements for both fixed and timevarying objects^{30,63,74,102,103,105}. By choosing different wavelengths, unambiguous phase unwrapping can be achieved to resolve different scales of depth jumps. Typically, the difference between the two wavelengths,
$ \Delta \lambda =\left{\lambda }_{2}{\lambda }_{1}\right $ , is of the order of$ 1050\;{{nm}} $ , which yields synthetic wavelengths Λ of approximately$ 3010\;\mu m $ , respectively. Under the assumption that topographic resolution is of the order of 1/100 of Λ, this allows feature heights of the order of$ 13\;{{\mu m}} $ to be resolved reasonably well^{63}. However, considerable work has also been performed using much longer synthetic wavelengths to measure mm or even cmscale features^{106}. Proper choice of$ {\lambda }_{1},{\lambda }_{2} $ , such that$ {\lambda }_{1}~{\lambda }_{2} $ , yields very long synthetic wavelengths. For instance, using acoustooptics and RF frequency of ~1 GHz, synthetic wavelengths in the order of tens of cm can be achieved using a 514 nm laser source^{78}.For large height variations, typically tens to hundreds of wavelengths, the height variation corresponds to several multiples of
$ 2\pi $ of phase. Due to the$ arctan $ function used in the phase reconstruction process (Eq. 9), the phase information is wrapped in the modulo of 2π, which creates a 2πambiguity when unwrapping the phase. This limits the maximum absolute distance measurement along axial direction within$ {\lambda }/{2n} $ , where λ is the wavelength and n is the refractive index. Large height variation situations can be improved by using the DWT by carefully selecting two different wavelengths, using them to record and reconstruct holograms separately, and generating a synthetic wavelength to reconstruct the combined hologram. The synthetic wavelength is usually much larger than the optical wavelength, thus, it can resolve larger height variations. The single wavelength technique (SWT) is not an ideal solution to very small height variations^{105,107}. For small variations, use of the DWT together with fringe subdivision, nanometer scale axial resolution is possible^{64}.The essence of DWDH is to have two wavelengths accumulating different phases in a given distance and generate the beat pattern from phase difference. The principles governing DWDH and synthetic wavelength generation^{73,105,107} are briefly summarized here. DWDH requires two holograms to be recorded at two separate wavelengths,
$ {\lambda }_{1,2} $ , and the phase difference to be subsequently computed. For$ {\lambda }_{1} $ , the phase difference of two adjacent fringes can be expressed as$ {\phi }_{1}=({2\pi }/{{\lambda }_{1}})\Delta d $ , where$ \Delta d $ is axial height separation. Need to mention that the illumination angle and geometry of interferometer have been taken into consideration in$ \Delta d $ . Similarly, for$ {\lambda }_{2} $ ,$ {\phi }_{2}=({2\pi }/{{\lambda }_{2}})\Delta d $ . The phase difference can be expressed as the height variation multiplied by a synthetic wavenumber K corresponding to a synthetic wavelength Λ:$$ {\phi }_{1}{\phi }_{2}=\frac{2\pi }{\dfrac{{\lambda }_{1}{\lambda }_{2}}{{\lambda }_{1}{\lambda }_{2}}}\Delta d=\frac{2\pi }{\Lambda }\Delta d $$ (10) Thus, the synthetic wavelength originates from the phase accumulation of coherent light wave, which is dependent on the wavelength or the longitudinal spatial frequency of the light source. With a given height jump, the phase accumulated by different wavelengths are different. The subtraction of the two phases yields a beat frequency or equivalently a beat (or synthetic) wavelength that can extend the axial measurement range.
Typically, two holograms for two different wavelengths are required to be captured sequentially or by two cameras. The two holograms are reconstructed separately to get two complex fields of the object at the reconstruction plane. Note that due to different wavelengths used in the holograms, a pixel mismatch problem exists. To calculate the phase difference map for synthetic requires a pixeltopixel subtraction. The mismatch comes from the wavelength dependence of the pixel size in the reconstruction plane^{103},
$$ \Delta {x}_{{r}_{i}}=\frac{{\lambda }_{i}\Delta d}{N\Delta x}i=1,2 $$ (11) where
$ \Delta {x}_{{r}_{i}} $ is the reconstruction pixel size, N is the number of the pixels along xaxis,$ \Delta x $ is the pixel pitch of the camera, i is the index of the wavelengths. The reconstruction pixel size is linearly related to the wavelengths. In Eq. 11, the height$ \Delta d $ and the pixel pitch cannot be changed for a fixed system. To achieve pixel mismatch correction, in other words,$ \Delta {x}_{{r}_{1}}=\Delta {x}_{{r}_{2}} $ ; the only way is to increase or decrease the number of pixels. A typical way to match the pixels is zeropadding the longer wavelength hologram before reconstruction; and after reconstruction, zeropadding the shorter wavelength hologram or cropping the longer wavelength hologram^{16,78}. The padsize is the number of zero elements to be added symmetrically to each edge of the hologram, rounded to the nearest integer value, and is given by$$ {\rm{padsize}}=round\left[\frac{N}{2}\left(\frac{{\lambda }_{1}}{{\lambda }_{2}}1\right)\right] $$ (12) assuming
$ {\lambda }_{1} > {\lambda }_{2} $ . After pixel matching, the two phase maps are pixelwise subtracted according to the recipe:$$ \Delta {\rm{\Phi }}=\left\{\begin{array}{l}{\phi }_{1}{\phi }_{2}\quad if\quad {\phi }_{1} > {\phi }_{2}\\ {\phi }_{1}{\phi }_{2}+2\pi\quad if \quad {\phi }_{1}\le {\phi }_{2}\end{array}\right. $$ (13) The phase difference map
$ \Delta {\rm{\Phi }} $ is equivalent to the phase reconstructed from a hologram recorded by the synthetic wavelength. Then upon phase unwrapping and phasetoheight conversion ($ \Delta d={\Delta {\rm{\Phi }}}/{2\pi }{\rm{\Lambda}} $ ), the 3D topography of the surface can be deduced.The principle shown above is simple and straightforward. This is to give readers, especially for those who beginning research in this field, a general idea of how to implement DWT in an application. Note that the zero padding is needed to match the scaling effect from different wavelengths or for chromatic aberration compensation. There are also many other techniques or methods in DWDH reconstruction, such as transfer function approach^{102}, ShenWang convolution method^{108}, chirp Fourier transform^{109}, etc. These methods still need “zeropadding” due to sampling requirement of fast Fourier transform algorithm. However, this is different from the zeropadding mentioned in Eq. 12. Also, the holographic imaging is only one of the methods which can implement DWT. These methods including interferometry, CDI, ptychography, Fourier ptychography, FPP, will be introduced in the next section of this paper.
It is worth mentioning that phaseshifting techniques are also commonly adopted in DWDH^{110,111}. By modulating the reference path with several constant phase shifts, a phaseshifting algorithm can be applied to retrieve the phases of static objects and remove the twin image, zerothorder components, with tradeoff in extra captures of holograms or interferograms.
The other direction of DWT is to use the sum of two phase maps. The sum of the phase map, similarly can generate a wavelength
$ {{\rm{\Lambda}}}_{+}={{\lambda }_{1}{\lambda }_{2}}/({{\lambda }_{1}+{\lambda }_{2}}) $ smaller than each of the wavelengths. Noted that when$ {\lambda }_{1}\approx {\lambda }_{2} $ , the$ {{\rm{\Lambda}}}_{+}\approx {{\lambda }_{1}}/{2} $ . Together with the synthetic wavelengths generated from the phase difference (Eq. 4),$ {{\rm{\Lambda}}}_{+} $ can be used to retrieve the phase with high accuracy or low noise performance, in the meantime, still can achieve the extended height/phase measurement^{43}. To retrieve the phase, a twosensitivity temporal phaseunwrapping technique has been developed^{112}. The essence of the algorithm lies in the formula:$$ {{\tilde \phi}}_{S}=G{{\tilde \phi}}_{D}+W\left[{\phi }_{S}G{{\tilde \phi}}_{D}\right] $$ (14) where
$ \left\{{\phi }_{D},{\phi }_{S}G{{\tilde \phi}}_{D}\right\}\in \left(\pi ,\pi \right]; $ $ {{\tilde \phi}}_{S} $ and$ {{\tilde \phi}}_{D} $ are unwrapped phases of phase sum and phase difference wavelengths ($ {{\rm{\Lambda}}}_{+},{\rm{\Lambda}} $ ), respectively;$ G={{\rm{\Lambda}}}/{{{\rm{\Lambda}}}_{+}} $ ;$ {\phi }_{S} $ is the wrapped phase of the phase sum wavelength;$ W\left[x\right]={\rm{arg}}\left\{{e}^{jx}\right\} $ is a wrapping function. From Eq. 14, the unwrapped phase can be obtained. Need to mention that even though the technique is named as a phase unwrapping technique, the phase map resulting from Eq. 14 may still need extra phase unwrapping step, when the phase/height jump is larger than the synthetic wavelength Λ. Despite this, the twosensitivity temporal phaseunwrapping technique can possibly achieve a better signaltonoise ratio (SNR) than the traditional phase difference mathematical framework. There is a SNR gain of$ {G}^{2} $ between phase sum and phase difference^{43}. 
The DWT shows great advantage in avoiding phase unwrapping ambiguity, achieving subwavelength longitudinal resolution imaging with fringe subdivision, singleshot imaging, and adapting to various coherent/incoherent imaging techniques. However, it still has limitations. Here, we analyze:
(a) the limit of maximum resolvable phase or height based on selection of system parameters (Itoh condition),
(b) choice of optimum detector size based on truncation ratio, which is the ratio of the object size to the detector size (truncation condition), and
(c) possible noise sources degrading the reconstruction results.
To systematically investigate the first two conditions mentioned above, the following assumptions are made:
(1) to quantify the performance improvement due to dualwavelength measurements, the phase retrieval algorithms are assumed to be accurate. For this analysis, the PUMA algorithm developed by BioucasDias and Valadao^{104} is used; however, any phase unwrapping algorithm could have been used. In actual experiments, the final performance has dependency on both phase retrieval and phase unwrapping algorithms;
(2) for any wavelength
$ {\lambda }_{1} $ , an object of size$ {L}_{O} $ and a detector placed at a distance of Z away from the object (Fig. 8). The object can be transparent (Fig. 8a) so that it can be placed directly in the path of a collimated beam to collect the diffracted light (represented by the deformed wavefronts), or reflective (Fig. 8b) so that a beam splitter can be used to illuminate it and then collect the diffracted light. The assumption is:$ Z\gg {Z}_{F} $ , where the Fraunhofer distance,$ {Z}_{F}=2{{L}_{O}}^{2}/{\lambda }_{1} $ . For a detector of size$ {L}_{D} $ placed in the farfield region$ (Z\gg {Z}_{F}) $ , the minimum detectable feature size of the object is,$ \Delta {x}_{O}=({{\lambda }_{1}Z})/({N\Delta {x}_{D}} )$ (from Eq. 11). Here,$ {L}_{D}={\Delta x}_{D}N $ is the detector size,$ {\Delta x}_{D} $ is the pixel pitch of the detector and N is the number of pixel along xaxis. Note that this can also be valid in near field using Eq. 7. For simplicity, the simulations below are performed assuming farfield. Since features smaller than a pixel also cannot be detected (due to aliasing),$ {\Delta x}_{O} $ is equivalent to pixel size in object plane.Fig. 8 A simplified setup that records the diffraction pattern off of a a transparent (phase) object and b reflective object (with no absorption) illuminated with a collimated beam. For both cases the objecttodetector distance Z is such that
$ Z\gg {Z}_{F} $ . The reflective setup uses a beam splitter to illuminate the object. The features on the objects are arbitrarily chosen.As long as the above assumptions are satisfied, the following two conditions must be met for successful unwrapping using synthetic wavelength.

The Nyquist limit^{137}, also known in literature as the Itoh condition^{104,138}, states that the maximum phase slope (i.e. change of phase per pixel) that can be successfully unwrapped is
$$ {{\tilde S}}_{Itoh}=\pi \left[\frac{rad}{pixel}\right] $$ (15) A π radian phase can be achieved by propagation of a distance
$ {\lambda }_{1}/2 $ along the zaxis. Meanwhile, Eq. 11 provides the effective pixel pitch on the object plane. For singlewavelength measurement, this becomes$$ {{S}_{Itoh,{{\lambda }_{1}}}}=\frac{{}^{{{\lambda }_{1}}}\diagup{}_{2}\;}{{}^{Z*{{\lambda }_{1}}}\diagup{}_{{{L}_{D}}}\;}=\frac{{{L}_{D}}}{2Z}\left[ \frac{\mu m}{\mu m} \right] $$ (16) The maximum phase slope anywhere in the object plane must be smaller than the Itoh phase slope, i.e.
$ {{S}_{max}\le S}_{Itoh}. $ For a given optical system design, the Itoh slope,$ {S}_{Itoh} $ (in units of$ {\mu m}/{\mu m} $ ), serves as a system parameter since it can be determined in terms of wavelengths, detector size, objecttodetector distance etc. A higher$ {S}_{Itoh} $ corresponds to a more capable optical system that can handle surfaces with a relatively sharper height jump.From Eq. 16,
$ {S}_{Itoh,{\lambda }_{1}} $ has no dependency on the wavelength used for measurement. The benefit of using a larger$ {\lambda }_{1} $ could be a better phase resolution in the zaxis. However, this benefit is counteracted by the larger pixel size in the transverse plane. In the farfield, the diffraction pattern is the Fourier transform of the object phase distribution. Experimentally, only a portion of the diffraction pattern can be collected due to the finite size of the detector. The process of omitting the higher spatial frequencies (at the outskirts of the diffraction pattern) is called “apodization”. From Eq. 16, it is apparent that both a larger detector size$ {L}_{D} $ and a smaller objecttodetector distance Z (assuming$ Z\gg {Z}_{F} $ still holds) reduce apodization and cause a higher value of$ {S}_{Itoh,{\lambda }_{1}} $ .DWT uses a synthetic wavelength Λ (Eq. 4). For this case, the new Itoh condition becomes
$$ {{S}_{Itoh,\Lambda }}=\frac{{}^{\Lambda }\diagup{}_{2}\;}{{}^{Z*{{\lambda }_{2}}}\diagup{}_{{{L}_{D}}}\;}=\frac{{{L}_{D}}}{2Z}\frac{{{\lambda }_{1}}}{{{\lambda }_{1}}~{{\lambda }_{2}}}\left[ \frac{\mu m}{\mu m} \right] $$ (17) For a given detector size and distance, DWT improves the maximum detectable slope limit by a factor of
$ {{\lambda }_{1}}/({{\lambda }_{1}{\lambda }_{2}} )$ . In other words, a larger$ {\lambda }_{1} $ and, correspondingly, an increased zaxis resolution, give a higher value of$ {S}_{Itoh,\Lambda } $ . Also, closer separation of$ {\lambda }_{1} $ and$ {\lambda }_{2} $ leads to a larger$ {S}_{Itoh,\Lambda } $ . This improvement is illustrated in the following example, typical of a realistic experiment.Assume that a detector is placed
$ Z=100\;mm $ away from a phase object of size$ {L}_{O} $ (see Fig. 8). The detector has$ 512\;\times \;512 $ pixels and the pixel pitch is$ 5.2\;{\rm{\mu m}} $ . An object size of$ {L}_{ox},{L}_{oy}=2\;{\rm{\mu m}} $ is used and the two wavelengths used for measurement are$ 0.501\;{\rm{\mu m}} $ and$ 0.496\;{\rm{\mu m}} $ . For this scenario, the Fraunhofer distance$ {Z}_{F}=15.96\;{\rm{\mu m}} $ is much smaller compared to the total propagation distance Z. Thus, for SWT,$ {S}_{Itoh,{\lambda }_{1}}=0.13312 $ and for DWT,$ {S}_{Itoh,\Lambda }=1.33386 $ . The object comprises of a Gaussian height profile$ {h}_{in}\left(x,y\right)= $ $ {h}_{0}{e}^{({{x}^{2}+{y}^{2}})/{{{\sigma }_{0}}^{2}}} $ with a peak height,$ {h}_{0}=10\;\mu m $ , and a width$ {\sigma }_{0}=400\;\mu m $ as in Fig. 9a. The truncation ratio T is defined asFig. 9 a True height profile for a reflective object at the object plane. It is a Gaussian height profile with peak height
$ {h}_{0}=10\;\mu m $ , and width$ {\sigma }_{0}=400\;\mu m $ . The black dotted line denotes the waist of the Gaussian where the maximum slope$ {S}_{max} $ is located. b For single wavelength measurement,$ SNR\;{\rm{vs}}\;{S}_{max} $ . For$ {{S}_{max} > S}_{Itoh,{\lambda }_{1}} $ , the$ SNR $ drops. c When$ {{S}_{max} > S}_{Itoh} $ , the phase unwrapping fails. d The$ SNR\;{\rm{vs}}\;{S}_{max} $ with dual wavelength measurement. Under the condition$ {{S}_{max} > S}_{Itoh,\Lambda } $ ,$ SNR $ drops. e When$ {{S}_{max} < S}_{Itoh} $ , the phase unwrapping was successful.$$ T=\frac{{\sigma }_{x}}{{L}_{ox}} $$ (18) For these parameters of the optical system, the performance of PUMA is simulated. In a physical experiment, the diffraction pattern would be collected by a detector. Since the detector has finite size, the object plane is effectively pixelated, according to Eq. 11. The pixel size depends on the illumination wavelength. Analogous to the experiment, the object plane is sampled to match the effective pixelation. A resampling algorithm is used to achieve slightly different pixelations at two wavelengths. Experimentally, using some phase retrieval algorithms, a wrapped phase profile on the object plane can be recovered. For simulation purpose, a modulo
$ 2\pi $ function is used to wrap the resampled input phase profile. Then, PUMA is used to unwrap the phase. Using appropriate wavelength, the phase profile is converted to a height profile and compared against the input height profile. The SNR is defined using the relation$$ SNR=10\log_{10}\left(\frac{{h}_{in,rms}}{{h}_{err,rms}}\right) $$ (19) where,
$ {h}_{in,rms} $ is the rootmeansquared (RMS) height of the object and$ {h}_{err,rms} $ is the RMS error between actual height$ {h}_{in}(x,y) $ and measured height$ {h}_{out}(x,y) $ :$$ {h}_{in,RMS}={\left(\frac{1}{{L}_{ox}{L}_{oy}}{\iint }_{x=0,y=0}^{{x=L}_{x},{y=L}_{y}}{\left({h}_{in}\left(x,y\right)\right)}^{2}dxdy\right)}^{1/2} $$ (20) $$ {h}_{err,RMS}={\left(\frac{1}{{L}_{ox}{L}_{oy}}{\iint }_{x=0,y=0}^{{x=L}_{x},{y=L}_{y}}{(h}_{out}\left(x,y\right){{h}_{in}(x,y))}^{2}dxdy\right)}^{1/2} $$ (21) For a given truncation ratio (
$ T=0.2 $ for this case), the peak of the height profile is varied and the maximum slope of that height profile$ {S}_{max} $ (i.e. the slope at the inflection point which is$ 1{\sigma }_{0} $ away from the peak location) is determined. For each peak height, the SNR is also calculated using Eqs. (1921). As predicted by the Itoh condition, the SNR drops when$ {{S}_{max} > S}_{Itoh} $ . Fig. 9b demonstrates these phenomena for single wavelength measurement. For one input height profile (Fig. 9a), the measured height profile is shown in Fig. 9c. The measurement fails (as indicated by a very low SNR in Fig. 9b) due to$ {{S}_{max} > S}_{Itoh,{\lambda }_{1}} $ . Then, the measurement is repeated using dual wavelengths. This increases the$ {S}_{Itoh} $ limit to$ {S}_{Itoh,\Lambda }=1.33386 $ . Fig. 9d indicates that the SNR keeps at a higher value until$ {S}_{Itoh,\Lambda } $ . Fig. 9e shows the measured height profile with high SNR. This example illustrates how the dual wavelength measurement pushes the$ {S}_{Itoh} $ limit higher and makes a high SNR measurement possible for a height profile that clearly has low SNR for a single wavelength measurement. 
For a successful phase unwrapping (
$ SNR > 10\;dB $ ), the Itoh condition (i.e.$ {{S}_{max} < S}_{Itoh} $ ) is a necessary but not sufficient condition. The truncation ratio T also needs to be within certain range. From Fig. 10, the$ SNR $ peaks within the range$ 0.05 < T < 0.5 $ . For an optical system (i.e. given$ {S}_{Itoh} $ limit) and certain input object size, a smaller T produces less resolved object plane causing lower$ SNR $ . On the other hand, for the same system and input object size, a larger T introduces apodization artifacts such as ripples at the edge of the measured object height. The range of T will be further investigated in the future.Fig. 10 The
$ SNR\;{\rm{vs}}\;{S}_{max} $ with dual wavelength measurement for different truncation ratios a T = 0.05, 0.10, 0.15 and 0.20 b T = 0.3, 0.4 and 0.5.Fig. 11 shows an actual and computed height profile when the two conditions (i.e. the Itoh condition and truncation condition) are met. Four Gaussian height profiles are placed in an array to mimic a physical object with surface variations. The output
$ SNR $ is ~12 dB for this simulation which represents successful unwrapping. 
Compared to SWT, DWT usually has a noisier performance^{30,68}, since using the synthetic wavelength imposes an amplification to the noise. The error sources are briefly summarized here. Possible reasons for noises are wavelength drifts (linewidth of each of the illumination wavelength), chromatic aberrations, dispersion, speckle noise, speckle decorrelation noises, retrace errors, and air turbulence. The noise sources also become limiting factors to achieve very large unambiguous distance measurements experimentally. Currently, the reported achievable synthetic wavelengths are in the order of tens of centimeters^{78,114}.
The stability of laser central wavelengths may not directly affect the noise performance, however, this effects the accuracy of the synthetic wavelength’s calculation^{139}. The accuracy of the synthetic wavelength becomes essential especially for large synthetic wavelengths. Fabry–Perot etalons have been used to perform mode selection/stabilization at the laser output^{89}. The expected error
$ {\sigma }_{\Lambda } $ in Λ due to wavelength selection can be determined using^{73}$$ {{\sigma }_{\Lambda }}^{2}={\left(\frac{\partial \Lambda }{\partial {\lambda }_{1}}\right)}^{2}{{\sigma }_{{\lambda }_{1}}}^{2}+{\left(\frac{\partial \Lambda }{\partial {\lambda }_{2}}\right)}^{2}{{\sigma }_{{\lambda }_{2}}}^{2} $$ (22) where
$ {\sigma }_{{\lambda }_{1},{\lambda }_{2}} $ denote the errors in$ {\lambda }_{1},{\lambda }_{2} $ . This yields$$ {\left({\sigma }_{\Lambda }/{\Lambda }^{2}\right)}^{2}={\left({\sigma }_{{\lambda }_{1}}/{{\lambda }_{1}}^{2}\right)}^{2}+{\left({\sigma }_{{\lambda }_{2}}/{{\lambda }_{2}}^{2}\right)}^{2} $$ (23) indicating that the error increase quadratically with Λ, This wavelength selection error can lead to significant uncertainty in the synthetic wavelength as
$ {\lambda }_{1}\to {\lambda }_{2} $ . Thus, for short synthetic wavelengths, any single phase (height) measurement can be assumed to have reasonably high accuracy, limited primarily by shot and coherence noise^{66}. However, wavelength drift errors quickly begin to dominate at long synthetic wavelengths. The fractional error, which is proportional to$ {\sigma }_{\Lambda }/\Lambda $ , increases linearly with Λ.For contribution to errors from the laser linewidth, since DWT usually has a small separation between different wavelengths, each of the illuminating lasers is supposed to have a narrow bandwidth. For SWT, the phase corresponding to a height h is given by
$ {\phi }_{\lambda }={2\pi h}/{\lambda } $ . Hence, the magnitude of error in phase is$ \left{\delta \phi }_{\lambda }\right={2\pi h}/{{\lambda }^{2}}\left\delta \lambda \right $ . Therefore, fractional error in measurement of phase is$ {\left{\delta \phi }_{\lambda }\right}/{{\phi }_{\lambda }}={({2\pi h}/{{\lambda }^{2}}})/{({2\pi h}/{\lambda })}\left\delta \lambda \right={\left\delta \lambda \right}/{\lambda } $ , where$ \left\delta \lambda \right $ may arise from the linewidth of the laser. In DWT, the phase corresponding to a height h is given by$ {\phi }_{\Lambda }={2\pi h}/{\Lambda } $ . Hence,$ \left{d\phi }_{\Lambda }\right={2\pi h}/{{\Lambda }^{2}}\left\delta \Lambda \right $ . The fractional error in measurement of phase is$ {\left{\delta \phi }_{\Lambda }\right}/{{\phi }_{\Lambda }}={\left\delta \Lambda \right}/{\Lambda } $ . We need to reexpress this in terms of$ \left\delta \lambda \right $ for a fair comparison with the result for SWT. Now$ \Lambda ={{\lambda }_{1}{\lambda }_{2}}/({\left{\lambda }_{1}{\lambda }_{2}\right})\approx {{\lambda }^{2}}/{\Delta \lambda } $ , as two similar wavelengths are commonly used. Then,$ d\Lambda =\left(({2\lambda \Delta \lambda {\lambda }^{2}({d\Delta \lambda }/{d\lambda }}))/{{\left(\Delta \lambda \right)}^{2}}\right)d\lambda $ , and the magnitude of the maximum error is$ \left\delta {\rm{\Lambda}}\right=(({2\lambda \Delta \lambda +{\lambda }^{2}\left{d\Delta \lambda }/{d\lambda }\right})/ $ $ {{\left(\Delta \lambda \right)}^{2}})\left\delta \lambda \right $ . Therefore,$ {\left{\delta \phi }_{\Lambda }\right}/{{\phi }_{\Lambda }}={\left\delta \Lambda \right}/{\Lambda }=2{\left\delta \lambda \right}/{\lambda }+ $ $ \left{d\Delta \lambda }/{d\lambda }\right({\left\delta \lambda \right}/{\Delta \lambda }) $ . Note that$ \left{d\Delta \lambda }/{d\lambda }\right \sim 1 $ , since the change in$ \Delta \lambda =\left{\lambda }_{1}{\lambda }_{2}\right $ is equal to the change in either wavelength$ {\lambda }_{1,2} $ . The ratio of the fractional error of DWT to that of SWT is$$ \frac{{Error}_{DWT}}{{Error}_{SWT}}=\frac{\dfrac{\left{\delta \phi }_{\Lambda }\right}{{\phi }_{\Lambda }}}{\dfrac{\left{\delta \phi }_{\lambda }\right}{{\phi }_{\lambda }}}=\frac{2\dfrac{\left\delta \lambda \right}{\lambda }+\dfrac{\left\delta \lambda \right}{\Delta \lambda }}{\dfrac{\left\delta \lambda \right}{\lambda }}=2+\frac{\lambda }{\Delta \lambda } $$ (24) Using
$ \lambda =500\;nm $ ,$ \Delta \lambda =20\;nm $ , Eq. 24 shows that the error in DWT due to laser linewidth is at least 25 times more than the error in SWT.The usage of more than one wavelength is usually accompanied with chromatic aberrations and dispersions in the optical system. These factors have also become the noise sources of DWT. Strategies have been proposed to correct chromatic aberrations. One way is to convert the wavelengthdependent phase term into the scaling effect in the lateral plane. As discussed in the “Principle of DWT” section, the pixel size at the reconstruction plane is rescaled by Eq. 11 and then matched for pixeltopixel subtraction with zeropadding (Eq. 12). The advantage of this method is it is convenient and easy to determine, but may suffer from the imperfection of pixel match. As shown in Eq. 12, due to the discrete nature of pixels, the calculation results of padsize need to be rounded to the nearest integer. This rounding process also contribute to the noise. This can be reduced by carefully selecting the number of pixels of the phase maps and also the wavelengths to make
$ {N}/{2}\left({{\lambda }_{1}}/{{\lambda }_{2}}1\right) $ close to an integer in sacrifice of imposing another constrain to the wavelength selection. Thanks to the advancement of signal processing techniques, the matching issue can be resolved by rescaling the pixel size with chirped scaling algorithms and fast Fourier transform^{140,141}. The other way to deal with the chromatic aberration is to digitally refocus the optical field of the corresponding wavelength to the focused plane^{18}. The nature of chromatic aberration is that the focal plane is shifted along optical axis for different wavelengths. This can be interpreted as a spherical aberration is added to each of the wavelength. Since the intensity and phase of the complex field can be obtained by all the quantitative phase imaging methods, the digital propagation can be performed to refocus the complex optical field to the focal plane. This technique is easy to implement and free of pixel matching, but determining when the object is in focus is tricky and may vary among human visual identifications or different infocus evaluation metrics^{142,143}. The dispersion errors, as the other wavelengthdependent error source caused by the dispersion of optical components (e.g. beam splitter), can be monitored and managed by numerical algorithms mentioned above or by modeling the static dispersion of the optical components and compensating with numerical calculations^{15}.Speckle noise exists in all coherent imaging techniques. Due to the noise amplification shortcoming of DWT, the speckle noise has been significantly amplified in DWT, especially for large synthetic wavelengths. Thus, to mitigate the speckle noise effect is of great importance. A common way to reduce the speckle noise in SWT is to take multiple shots and average them, or equivalently, using a rotating diffuser whose spinning period is much smaller than the camera’s exposure time. For DWT, averaging two phase difference maps is possible^{84}. Other methods to reduce speckle noise in SWT^{144} can also be applied to DWT. As we discussed the speckle noise here, it is worth mentioning that the DWT can be applied to both diffuse or specular reflecting surfaces. We have used DWT with reference beams using diffuse surfaces such as screwheads, dents, fingerprints, etc. using synthetic wavelengths in the order of tens, hundreds and thousands of microns^{73,75,145}, partly diffuse and partly specular surfaces such as silicon wafer samples with photoresist features coated with aluminum using synthetic wavelengths in the order of microns^{145}, of as well as for (specular) reflecting surfaces, such as mirrors separated by distances in the order of centimeters using synthetic wavelengths in the order of tens of centimeters (achieved using acoustooptic modulators at 1 Ghz for frequency shifting)^{78}.
Speaking of diffuse surfaces, the speckle decorrelation noise is common during diffuse surface measurement. There are a few possible origins of the speckle decorrelation noise: the modifications at the object surface between two measurements (e.g. mechanical shifts, heating), laser wavelength changes, reduced number of camera pixels, defocusing in reconstruction (mostly from inaccurate propagation distance), saturation of camera pixels in hologram recording, insufficient bitdepth of the camera (shot noise induced) and so on^{146150}. For DWT, the major cause of speckle decorrelation noise in diffuse surface measurement is from recording two holograms with two different wavelengths. Possible ways of reducing the speckle decorrelation noises are: averaging over multiple captures^{148}, or digital filtering. Numerous digital filtering techniques have been proposed to reduce the speckle decorrelation noise: spatial filtering (viz., moving average filter, medium filter, Gaussian filter), Wiener filter, adaptive filter^{151}, wavelet approaches^{152,153}, and so on. An exhaustive performance analysis and comparison among these digital filter techniques (a number of 34 algorithms) in reducing speckle decorrelation noise have been made quantitatively by Montresor and Picart^{150}. Retrace errors, existing in specular surface testing, are due to presence of nonnull aspherical surface during interferometric testing, and specially in systems like the traditional interferometers, where the test surface is at a focal distance past the focus of a perfect focusing lens. This type of noise can be calibrated by ray tracing the testing system model^{19,154}. Air turbulence is another source of noise. The turbulence in air can lead to the inhomogeneous distribution of refractive index in space. This nonuniform distribution will add different phase maps to the object and reference beam phases. Currently, no perfect solution has been proposed to solve this issue. One potential way to compensate for air turbulence is to store a hologram of the turbulence and read it out with the optical field from the object under investigation that passes through the same turbulence^{155}. Other correction methods using sharpness metrics have also been employed in DH and in holographic aperture ladar^{156,157}. The other way can possibly be using adaptive optics methods to compensate the turbulence induced phase in realtime^{158161}. Attempts have been made to compensate for errors introduced due to nonuniform and random refractive index changes of air with post processing of data from holographic interferometry measurements. One approach, which uses a purely temporal interference filter, reduced the noise from the refractive index errors from air disturbance over a rather limited spatial region^{162}. A more complete spatiotemporal filter requires that the spatiotemporal statistics of the noise from the air can be measured^{163}. The method has been applied to the case of locally homogenous/isotropic refractive index fluctuations in a wind tunnel.
A review of the dualwavelength technique for phase imaging and 3D topography
 Light: Advanced Manufacturing 3, Article number: (2022)
 Received: 01 September 2021
 Revised: 28 January 2022
 Accepted: 15 February 2022 Published online: 02 April 2022
doi: https://doi.org/10.37188/lam.2022.017
Abstract:
Optically transmissive and reflective objects may have varying surface profiles, which translate to arbitrary phase profiles for light either transmitted through or reflected from the object. For highthroughput applications, resolving arbitrary phases and absolute heights is a key problem. To extend the ability of measuring absolute phase jumps in existing 3D imaging techniques, the dualwavelength concept, proposed in late 1800s, has been developed in the last few decades. By adopting an extra wavelength in measurements, a synthetic wavelength, usually larger than each of the single wavelengths, can be simulated to extract large phases or height variations from micronlevel to tens of centimeters scale. We review a brief history of the developments in the dualwavelength technique and present the methodology of this technique for using the phase difference and/or the phase sum. Various applications of the dualwavelength technique are discussed, including height feature extraction from micron scale to centimeter scale in holography and interferometry, singleshot dualwavelength digital holography for highspeed imaging, nanometer height resolution with fringe subdivision method, and applications in other novel phase imaging techniques and optical modalities. The noise sources for dualwavelength techniques for phase imaging and 3D topography are discussed, and potential ways to reduce or remove the noise are mentioned.
Research Summary
Dual Wavelength Technique: a convenient way to measure large optical phases
Using optical illumination at two different wavelengths, a synthetic wavelength can be simulated to extract large phases, or height variations from microns to centimeters. This is because optically transmissive/reflective objects may have varying surface profiles, which may translate to arbitrary optical phase profiles. The dual wavelength technique can be adapted to coherent diffractive imaging, ptychography, nonlineofsight imaging, etc. A summary of the performance of the dual wavelength technique is presented, including error sources from the sampling limit (Itoh condition), wavelength drift, chromatic aberration, dispersion, speckle, retrace errors, and air turbulence.
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