Holographic volume displacement calculations via multiwavelength digital holography L. Williams,1,* P. P. Banerjee,1 G. Nehmetallah,2 and S. Praharaj3 1
Electro-Optics Program, University of Dayton, Dayton, Ohio 45469, USA
2
Electrical Engineering and Computer Science, Catholic University of America, Washington DC 20064, USA 3
DMS Technology, Inc., 2905 Westcorp Boulevard SW, Suite 220, Huntsville, Alabama 35805, USA *Corresponding author: [email protected] Received 17 September 2013; accepted 27 January 2014; posted 31 January 2014 (Doc. ID 197901); published 5 March 2014
In this work multiwavelength digital holography is applied to calculate the volume displacement of various topographic surface features. To accurately measure the volume displacement of macroscopic features, long synthetic wavelengths up to several millimeters are generated using tunable IR laser sources. Practical methods of implementation are considered, including geometric effects of both Michelson and Mach–Zehnder recording configurations and error due to wavelength selection. © 2014 Optical Society of America OCIS codes: (090.1995) Digital holography; (090.2880) Holographic interferometry. http://dx.doi.org/10.1364/AO.53.001597
1. Introduction
Holographic topography is the measurement of surface shape by means of holographic recording and reconstruction. This is generally known as holographic contouring when performed with analog holography, and can be performed using multiwavelength digital holographic interferometry (DHI) [1–3]. When recorded digitally, the phase contours may be postprocessed to yield the object height to within 1∕1000 of the synthetic wavelength under ideal conditions [3]. However, in many practical recording configurations (i.e., in the presence of noise, vibration, laser instability) the realizable depth resolution is limited to about 1∕100 of the synthetic wavelength. Digital holographic topography is sometimes known as holographic profilometry, owing to the frequent practice of analyzing only a cross section of a given topography. Multiwavelength digital holography (MDH) has been explored extensively in the past to quantify 1559-128X/14/081597-07$15.00/0 © 2014 Optical Society of America
surface topography and displacement measurements for both fixed objects and time-varying objects [1,3–5]. Typically, the difference between the two wavelengths, Δλ, is of the order of 8–40 nm, which yields synthetic wavelengths Λ of approximately 30–10 μm, respectively. Under the assumption that topographic resolution is of the order of 1∕100 of Λ, this allows feature heights of the order of 1–3 μm to be resolved reasonably well [4,5]. However, some work has also been performed using much longer synthetic wavelengths to measure millimeter-scale features, while nanometer-scale measurements can be performed using phase-shifting DH [3,6]. The proposed method extends the applicability of MDH to measure surface topography on the scale of several millimeters using very long synthetic wavelengths with the intent of calculating the total volume displacement of various surface features. Additionally, 3D surface maps and 2D contour plots of topographic features are generated from the MDH data for a well-characterized test object. To the best of our knowledge, MDH has not been used to perform volume displacement measurements 10 March 2014 / Vol. 53, No. 8 / APPLIED OPTICS
1597
1
Synthetic Wavelength Variation as λ1 nears λ2 = 766nm
10
0
Synthetic Wavelength (cm)
as proposed, although volumetric calculations have been performed using Moiré topography [7,8]. This paper begins with a brief review of MDH, with some discussion of the geometric relationship between synthetic phase accumulation and feature height, as well as pertinent sources of error when using long synthetic wavelengths. Next, the volume calculation method is described and the process illustrated with a brief simulation, followed by experimental results, a brief conclusion, and discussion of future work.
10
-1
10
-2
10
2. Long Synthetic Wavelength Digital Holography
The principles governing MDH and synthetic wavelength generation have been widely cited in the literature, and therefore are only briefly summarized here [1,3,5,9]. MDH requires two holograms to be recorded at two separate wavelengths, λ1 and λ2 , and the phase difference to be subsequently computed. The phase φλ1;2 ξ; η of each hologram individually is determined by φλ1;2 ξ; η arctan
Im Γ1;2 ξ; η ; Re Γ1;2 ξ; η
-3
10
765
766
767
768
769
770
771
772
773
774
775
776
Variable Wavelength λ1 (nm)
Fig. 1. Synthetic wavelength as a function of the variable wavelength λ1 , with λ2 766 nm.
(1)
where Γ1;2 ξ; η is the complex valued reconstruction for holograms 1 and 2, respectively. The phase difference Δφ is given by Δφ
φλ1 − φλ2 φλ1 − φλ2 2π
if φλ1 ≥ φλ2 ; if φλ1 < φλ2 :
(2)
Here, Δφ denotes the phase of the beat wavelength between λ1 and λ2 , known as the synthetic wavelength Λ, which is given by Λ
λ1 · λ2 : jλ1 − λ2 j
(3)
The resulting 2D phase map generated by twowavelength phase subtraction yields surface topography with longitudinal feature resolution roughly on the order of Λ, which are viewed as fringe contours of constant elevation distributed across the object surface. Thus, proper choice of λ1 and λ2 , such that λ1 ∼ λ2 , will yield very long synthetic wavelengths, as illustrated in Fig. 1. The amount of synthetic phase accumulation is proportional to the object surface feature height and the illumination angle θ, as shown in Fig. 2, which is typically either 0° or 45°, depending upon implementation of either Michelson or Mach– Zehnder recording geometries, respectively. Thus, the 2D phase map of the surface may be translated to the true object height htrue by using the relation htrue
Λ · Δφ · cos θ : 4π
(4)
The digital interferogram resulting from phase subtraction is a 2D map of the wrapped phase, exhibiting 1598
APPLIED OPTICS / Vol. 53, No. 8 / 10 March 2014
Fig. 2. Length of phase accumulation in relation to object height, htrue , and incident angle.
modulo 2π fringe spacing of the synthetic wavelength, Λ, which must be unwrapped to yield the absolute synthetic phase. This is generally accomplished using one of several phase unwrapping algorithms. The resulting unwrapped phase map can be scaled to represent physical distance via Eq. (4), and plotted in either 2D or 3D using software such as MATLAB. For this work, the phase unwrapping max-flow/min-cut (PUMA) MATLAB algorithm is adopted. This method is based upon the graph-cut technique developed by Bioucas-Dias and Valadao [10]. The PUMA algorithm has been chosen for its ready availability and good performance in the presence of significant phase noise. It should be noted that the spatial heterodyne technique may also be employed to capture both wavelength measurements in a single composite holographic exposure [3,11]. This is accomplished by introducing a different angular tilt to the λ1 and λ2 reference beams. These angular tilts in the spatial domain introduce linear phase shifts in the frequency domain of the recorded composite hologram. When reconstructed, the different phase shifts result in spatially separated object locations in the image that correspond to their respective λ1 and λ2
s 2 2 σ λ1 σ λ2 σΛ Λ : 2 λ1 λ22
recordings. One of these reconstructed objects may be cropped and digitally overlaid upon the other to perform the required phase subtraction.
2
This wavelength selection error can lead to significant uncertainty in the synthetic wavelength as λ1 approaches λ2, as shown in Fig. 3. Thus, for short synthetic wavelengths, any single topographic measurement can be assumed to have reasonably high accuracy, limited primarily by shot and coherence noise [12]. However, wavelength selection error quickly begins to dominate at long synthetic wavelengths. It is interesting to note that the percent error, which is proportional to σ Λ ∕Λ, increases linearly with Λ. The measured λ1 and λ2 error distributions are approximately Gaussian; therefore multiple measurements of the same object may be averaged to yield a more accurate result. For this work multiple measurements at different Λ values were used to compute the mean depth and volume displacement for various sample surfaces.
3. Long Synthetic Wavelength Error
As λ1 approaches λ2, the synthetic wavelength Λ becomes infinite (see Fig. 1). However, in practice λ1 and λ2 cannot be made equal due to various sources of wavelength error, including spectral broadening, wavelength drift, and mode competition. Thus, a key challenge is in generating the highly stable, yet subtly different wavelengths required for such long synthetic wavelengths. Previous work has demonstrated the applicability of using Fabry– Perot étalons to perform mode selection/stabilization at the laser output [9]. For this work, however, two semiconductor lasers are used, one fixed at 766.00 nm (λ2 ), and the other (λ1 ) tunable from 764.00 to 781.00 nm. The spectral accuracy, stability, and repeatability of both lasers have been characterized via spectrometer to reveal an expected error in λ1 and λ2 of approximately 0.0085 nm and 0.0092 nm, respectively. Each wavelength is generated from a separate laser source; therefore λ1 and λ2 are independent variables with zero covariance. The normalized covariance of λ1 and λ2 was experimentally verified to be less than 10−5 , as determined by 50 spectrometer measurements. In this case, the expected error in Λ due to wavelength selection can be determined by the well-known relationship σ 2Λ
∂Λ ∂λ1
2
σ 2λ1
∂Λ ∂λ2
2
σ 2λ2 ;
4. Holographic Volume Calculation
Volume displacement calculations can be readily determined via holographic topography by first calculating the unwrapped/scaled 3D phase map, as previously described, then defining a reference surface bounding both the lateral extent and either the upper or lower surface of the desired volume. The unwrapped phase image is then numerically integrated over this lateral extent, and subtracted from the reference volume. The difference yields the volume displacement of interest, which is given by Z Z Volume ϕΛ dA ρdS;
(5)
A
where σ Λ is the expected error in Λ, and σ λ1;2 is the error in λ1;2 . This error is found to increase quadratically with Λ, as given by
(7)
A
where A denotes the laterally bounded surface, ϕΛ is the unwrapped/scaled synthetic wavelength phase image, and ρ is the reference surface. The sign of
Λ Error as λ1 Varies
100
(6)
Λ Error
6
120
70
Error in Λ (cm) (solid)
Percent Error in Λ
80
60 50 40 30 20
5
100
4
80
3
60
2
40
1
20
Percent Error in Λ (dashed)
90
10 0 766
0 766.05
766.1
766.15
766.2
Variable Wavelength λ1 (nm)
(a)
766.25
766.3
0 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Synthetic Wavelength Λ (cm)
(b)
Fig. 3. Percent error in Λ as a function of (a) λ1 and (b) Λ, for the measured values of σ λ1 0.0085 nm and σ λ2 0.0092 nm. 10 March 2014 / Vol. 53, No. 8 / APPLIED OPTICS
1599
(a)
(b)
100 90 80
100
100
90
90
80
70
55 50
80 45 70
70 60
40
60 60
50
50
35
40
30
40
50
30
40
30
20
30
20
25
10
20 10
20
30
40
50
60
70
80
90
100
(c)
20
10 10
20
30
40
50
60
70
80
90
100
15
(d)
Fig. 4. Illustration of holographic volume calculation. (a) The unwrapped phase surface (e.g., an asymmetric Gaussian surface) with the region of integration bounding the area of interest, (b) the reference volume, which is then subtracted from the volume found in (a) to yield the volume of the Gaussian cap only, (c) contour map of the tilted surface, and (d) contour map determined from (c) after numerical flattening, i.e., reducing the average “bias” incline to zero.
Fig. 5. Modified Michelson configuration (a) allowing for object illumination at normal incidence with a tilted reference wave and Mach–Zehnder configuration (b) with 45° object illumination angle. 1600
APPLIED OPTICS / Vol. 53, No. 8 / 10 March 2014
the second integral in Eq. (7) is determined by the placement of the reference surface either above or below the ϕΛ surface. This process is simulated in Figs. 4(a) and 4(b) using a tilted asymmetric Gaussian surface. The unwrapped phase image can also be used to generate a contour map describing the surface, either before or after numerical flattening, which is of interest in several practical applications.
In practice the reconstructed surface is often tilted, as shown in Fig. 4(a), since the longitudinal axis (z axis) of the hologram surface is defined by the CCD plane normal. Therefore, if the tilt angle between the surface normal of the sample and the z axis is larger than a few degrees then numerical flattening can aid in generating more intuitive contour maps, although the volume calculation remains unaffected.
Fig. 6. Illustration of the MDH volume calculation process for Λ 1.869 mm (color online). (a) Photograph of the dent in an aluminum test surface, (b) one of the reconstructed holograms, (c) the wrapped 2D phase map after phase subtraction, (d) the unwrapped 2D phase map via PUMA algorithm, (e) distance-scaled 3D topographic map, (f) contour map illustrating topography, (g) 3D topographic map including the reference surface (red circular area), and (h) the reference volume without topographic map. 10 March 2014 / Vol. 53, No. 8 / APPLIED OPTICS
1601
5. Experimental Results
In this work, a series of dents in the surface of an aluminum plate is characterized via long synthetic wavelength topography, and the volume displacement is calculated, using a variable (λ1 765–781 nm) and fixed IR source (λ2 766.00 nm). A Lumenera (Model LU120M) camera with 1024 × 1024 pixels and of 6.7 μm pixel size is used to record the holograms. Using the modified Michelson configuration shown in Fig. 5(a), two holograms of each sample are captured, one at each wavelength (λ1 , λ2 ), and the MDH reconstruction process is performed as previously described for a given synthetic wavelength. The modified Michelson configuration shown in Fig. 5 is used to illuminate the target at normal incidence. This eliminates the “shadow effect” due to the 45° illumination angle in the Mach–Zehnder configuration [Fig. 5(b)], which may cause parts of the sample to be poorly illuminated. Figures 6(a)–6(h) illustrate the MDH reconstruction process for one of the aluminum samples, including the volume displacement calculation for the depressed portion of the dent. The mean of eight measurements, spanning Λ 1.869 mm to Λ 7.930 mm, yields a volume of 37.31 mm3 0.95 mm3 . The holographically measured mean depth of the dent is 1.63 mm 0.05 mm. This compares favorably with the depth measurement performed via caliper, which is 1.57 mm 0.13 mm. It should be noted that the relatively large error of the caliper measurement is due to performing repeated measurements on the sample, which exhibits a rather pitted and rough surface, reducing the absolute repeatability of each measurement. Additionally, the feature depth is measured from the planar surface of the aluminum sample, which is below the “ridge” of material surrounding the dent, as illustrated in Fig. 6(g). Because the aluminum dent is approximately spherical, it is possible to compare the measured volume to that analytically calculated for a spherical cap of equal size. Assuming the mean calipermeasured depth and radius of 1.57 and 3.80 mm, respectively, the volume of the spherical cap is 37.63 mm3 . Although this comparison is not expected to be exact (since the aluminum surface is rough) it does reveal that the holographic calculation matches closely with the nearest analytical comparison available. It should be noted that the wrapped phase map typically exhibits a great deal of noise, as seen in Fig. 6(c), which is not eliminated by the PUMA algorithm [10]. Indeed, PUMA was initially chosen because of its ability to perform well in the presence of such noise. Thus, for subsequent noise reduction the unwrapped phase maps have been subjected to 7 × 7 pixel median filtering prior to volume calculation or contour mapping. Topographic and volumetric measurements have been performed for a variety of additional objects, consisting primarily of additional depressions in 1602
APPLIED OPTICS / Vol. 53, No. 8 / 10 March 2014
an aluminum surface, with similar accuracy. For approximately spherical test features, the resulting accuracy was typically within a few percent of the analytically calculated volume (assuming a spherical cap). These measurements have also been repeated using the single-exposure spatial heterodyne method briefly discussed earlier. The spatial heterodyne method has been generally successful, although the resulting error tends to be larger due to ambiguity in pixel matching when the two object reconstructions are overlaid prior to phase subtraction. In general, such pixel matching is only accurate to within 1∕2 pixel, which causes additional phase aberration/noise in the reconstruction. 6. Conclusion
In conclusion, MDH can be effectively applied to accurately calculate the volume displacement of various topographic surface features, at the millimeter scale, using long synthetic wavelengths. However, at long synthetic wavelengths, the individual wavelength uncertainties in λ1 and λ2 (i.e., wavelength selection error) can lead to significant uncertainty in Λ. This difficulty can be overcome by computing the mean of several displacement measurements to yield a more accurate result. Several practical methods of implementation have been examined, including geometric scaling based upon illumination angle, phase unwrapping in the presence of noise, and noise suppression, with this method’s effectiveness verified by comparison to both physical measurement and analytical calculation with excellent agreement. Additionally, this method is currently being extended to the holographic microscopy regime using short synthetic wavelengths to perform volume displacement calculations and submicrometer topographic measurements for microelectromechanicalsystems (MEMS) metrology with depth resolution comparable to that of white-light interferometry. References 1. U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer, 2010). 2. J. Goodman, Introduction to Fourier Optics (Roberts, 2005). 3. T. Kreis, Handbook of Holographic Interferometry (Wiley, 2005). 4. D. Abdelsalam, R. Magnusson, and D. Kim, “Single-shot, dual-wavelength digital holography based on polarizing separation,” Appl. Opt. 50, 3360–3368 (2011). 5. C. Mann, P. Bingham, V. Paquit, and K. Tobin, “Quantitative phase imaging by three-wavelength digital holography,” Opt. Express 16, 9753–9764 (2008). 6. Y. Morimoto, T. Matui, M. Fujigaki, and N. Kawagishi, “Subnanometer displacement measurement by averaging of phase difference in windowed digital holographic interferometry,” Opt. Eng. 46, 025603 (2007). 7. J. Dirckx and W. Decraemer, “Deformation measurements of the human tympanic membrane under static pressure using automated moire topography,” Proc. SPIE 1429, 34–38 (1991). 8. S. Xenofos and C. Jones, “Theoretical aspects and practical applications of Moire topography,” Phys. Med. Biol. 24, 250–261 (1979).
9. E. Barbosa, E. Lima, M. Gesualdi, and M. Muramatsu, “Enhanced multiwavelength holographic profilometry by laser mode selection,” Opt. Eng. 46, 075601 (2007). 10. J. Bioucas-Dias and G. Valadao, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007). 11. J. Haus, B. Dapore, N. Miller, P. Banerjee, G. Nehmetallah, P. Powers, and P. McManamon, “Instantaneously captured
images using multiwavelength digital holography,” Proc. SPIE 8493, 84930W (2012). 12. J. Kuhn, T. Colomb, F. Montfort, F. Charriere, Y. Emery, E. Cuche, P. Marquet, and C. Despeursinge, “Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express 15, 7231–7242 (2007).
10 March 2014 / Vol. 53, No. 8 / APPLIED OPTICS
1603
Electro-Optics Program, University of Dayton, Dayton, Ohio 45469, USA
2
Electrical Engineering and Computer Science, Catholic University of America, Washington DC 20064, USA 3
DMS Technology, Inc., 2905 Westcorp Boulevard SW, Suite 220, Huntsville, Alabama 35805, USA *Corresponding author: [email protected] Received 17 September 2013; accepted 27 January 2014; posted 31 January 2014 (Doc. ID 197901); published 5 March 2014
In this work multiwavelength digital holography is applied to calculate the volume displacement of various topographic surface features. To accurately measure the volume displacement of macroscopic features, long synthetic wavelengths up to several millimeters are generated using tunable IR laser sources. Practical methods of implementation are considered, including geometric effects of both Michelson and Mach–Zehnder recording configurations and error due to wavelength selection. © 2014 Optical Society of America OCIS codes: (090.1995) Digital holography; (090.2880) Holographic interferometry. http://dx.doi.org/10.1364/AO.53.001597
1. Introduction
Holographic topography is the measurement of surface shape by means of holographic recording and reconstruction. This is generally known as holographic contouring when performed with analog holography, and can be performed using multiwavelength digital holographic interferometry (DHI) [1–3]. When recorded digitally, the phase contours may be postprocessed to yield the object height to within 1∕1000 of the synthetic wavelength under ideal conditions [3]. However, in many practical recording configurations (i.e., in the presence of noise, vibration, laser instability) the realizable depth resolution is limited to about 1∕100 of the synthetic wavelength. Digital holographic topography is sometimes known as holographic profilometry, owing to the frequent practice of analyzing only a cross section of a given topography. Multiwavelength digital holography (MDH) has been explored extensively in the past to quantify 1559-128X/14/081597-07$15.00/0 © 2014 Optical Society of America
surface topography and displacement measurements for both fixed objects and time-varying objects [1,3–5]. Typically, the difference between the two wavelengths, Δλ, is of the order of 8–40 nm, which yields synthetic wavelengths Λ of approximately 30–10 μm, respectively. Under the assumption that topographic resolution is of the order of 1∕100 of Λ, this allows feature heights of the order of 1–3 μm to be resolved reasonably well [4,5]. However, some work has also been performed using much longer synthetic wavelengths to measure millimeter-scale features, while nanometer-scale measurements can be performed using phase-shifting DH [3,6]. The proposed method extends the applicability of MDH to measure surface topography on the scale of several millimeters using very long synthetic wavelengths with the intent of calculating the total volume displacement of various surface features. Additionally, 3D surface maps and 2D contour plots of topographic features are generated from the MDH data for a well-characterized test object. To the best of our knowledge, MDH has not been used to perform volume displacement measurements 10 March 2014 / Vol. 53, No. 8 / APPLIED OPTICS
1597
1
Synthetic Wavelength Variation as λ1 nears λ2 = 766nm
10
0
Synthetic Wavelength (cm)
as proposed, although volumetric calculations have been performed using Moiré topography [7,8]. This paper begins with a brief review of MDH, with some discussion of the geometric relationship between synthetic phase accumulation and feature height, as well as pertinent sources of error when using long synthetic wavelengths. Next, the volume calculation method is described and the process illustrated with a brief simulation, followed by experimental results, a brief conclusion, and discussion of future work.
10
-1
10
-2
10
2. Long Synthetic Wavelength Digital Holography
The principles governing MDH and synthetic wavelength generation have been widely cited in the literature, and therefore are only briefly summarized here [1,3,5,9]. MDH requires two holograms to be recorded at two separate wavelengths, λ1 and λ2 , and the phase difference to be subsequently computed. The phase φλ1;2 ξ; η of each hologram individually is determined by φλ1;2 ξ; η arctan
Im Γ1;2 ξ; η ; Re Γ1;2 ξ; η
-3
10
765
766
767
768
769
770
771
772
773
774
775
776
Variable Wavelength λ1 (nm)
Fig. 1. Synthetic wavelength as a function of the variable wavelength λ1 , with λ2 766 nm.
(1)
where Γ1;2 ξ; η is the complex valued reconstruction for holograms 1 and 2, respectively. The phase difference Δφ is given by Δφ
φλ1 − φλ2 φλ1 − φλ2 2π
if φλ1 ≥ φλ2 ; if φλ1 < φλ2 :
(2)
Here, Δφ denotes the phase of the beat wavelength between λ1 and λ2 , known as the synthetic wavelength Λ, which is given by Λ
λ1 · λ2 : jλ1 − λ2 j
(3)
The resulting 2D phase map generated by twowavelength phase subtraction yields surface topography with longitudinal feature resolution roughly on the order of Λ, which are viewed as fringe contours of constant elevation distributed across the object surface. Thus, proper choice of λ1 and λ2 , such that λ1 ∼ λ2 , will yield very long synthetic wavelengths, as illustrated in Fig. 1. The amount of synthetic phase accumulation is proportional to the object surface feature height and the illumination angle θ, as shown in Fig. 2, which is typically either 0° or 45°, depending upon implementation of either Michelson or Mach– Zehnder recording geometries, respectively. Thus, the 2D phase map of the surface may be translated to the true object height htrue by using the relation htrue
Λ · Δφ · cos θ : 4π
(4)
The digital interferogram resulting from phase subtraction is a 2D map of the wrapped phase, exhibiting 1598
APPLIED OPTICS / Vol. 53, No. 8 / 10 March 2014
Fig. 2. Length of phase accumulation in relation to object height, htrue , and incident angle.
modulo 2π fringe spacing of the synthetic wavelength, Λ, which must be unwrapped to yield the absolute synthetic phase. This is generally accomplished using one of several phase unwrapping algorithms. The resulting unwrapped phase map can be scaled to represent physical distance via Eq. (4), and plotted in either 2D or 3D using software such as MATLAB. For this work, the phase unwrapping max-flow/min-cut (PUMA) MATLAB algorithm is adopted. This method is based upon the graph-cut technique developed by Bioucas-Dias and Valadao [10]. The PUMA algorithm has been chosen for its ready availability and good performance in the presence of significant phase noise. It should be noted that the spatial heterodyne technique may also be employed to capture both wavelength measurements in a single composite holographic exposure [3,11]. This is accomplished by introducing a different angular tilt to the λ1 and λ2 reference beams. These angular tilts in the spatial domain introduce linear phase shifts in the frequency domain of the recorded composite hologram. When reconstructed, the different phase shifts result in spatially separated object locations in the image that correspond to their respective λ1 and λ2
s 2 2 σ λ1 σ λ2 σΛ Λ : 2 λ1 λ22
recordings. One of these reconstructed objects may be cropped and digitally overlaid upon the other to perform the required phase subtraction.
2
This wavelength selection error can lead to significant uncertainty in the synthetic wavelength as λ1 approaches λ2, as shown in Fig. 3. Thus, for short synthetic wavelengths, any single topographic measurement can be assumed to have reasonably high accuracy, limited primarily by shot and coherence noise [12]. However, wavelength selection error quickly begins to dominate at long synthetic wavelengths. It is interesting to note that the percent error, which is proportional to σ Λ ∕Λ, increases linearly with Λ. The measured λ1 and λ2 error distributions are approximately Gaussian; therefore multiple measurements of the same object may be averaged to yield a more accurate result. For this work multiple measurements at different Λ values were used to compute the mean depth and volume displacement for various sample surfaces.
3. Long Synthetic Wavelength Error
As λ1 approaches λ2, the synthetic wavelength Λ becomes infinite (see Fig. 1). However, in practice λ1 and λ2 cannot be made equal due to various sources of wavelength error, including spectral broadening, wavelength drift, and mode competition. Thus, a key challenge is in generating the highly stable, yet subtly different wavelengths required for such long synthetic wavelengths. Previous work has demonstrated the applicability of using Fabry– Perot étalons to perform mode selection/stabilization at the laser output [9]. For this work, however, two semiconductor lasers are used, one fixed at 766.00 nm (λ2 ), and the other (λ1 ) tunable from 764.00 to 781.00 nm. The spectral accuracy, stability, and repeatability of both lasers have been characterized via spectrometer to reveal an expected error in λ1 and λ2 of approximately 0.0085 nm and 0.0092 nm, respectively. Each wavelength is generated from a separate laser source; therefore λ1 and λ2 are independent variables with zero covariance. The normalized covariance of λ1 and λ2 was experimentally verified to be less than 10−5 , as determined by 50 spectrometer measurements. In this case, the expected error in Λ due to wavelength selection can be determined by the well-known relationship σ 2Λ
∂Λ ∂λ1
2
σ 2λ1
∂Λ ∂λ2
2
σ 2λ2 ;
4. Holographic Volume Calculation
Volume displacement calculations can be readily determined via holographic topography by first calculating the unwrapped/scaled 3D phase map, as previously described, then defining a reference surface bounding both the lateral extent and either the upper or lower surface of the desired volume. The unwrapped phase image is then numerically integrated over this lateral extent, and subtracted from the reference volume. The difference yields the volume displacement of interest, which is given by Z Z Volume ϕΛ dA ρdS;
(5)
A
where σ Λ is the expected error in Λ, and σ λ1;2 is the error in λ1;2 . This error is found to increase quadratically with Λ, as given by
(7)
A
where A denotes the laterally bounded surface, ϕΛ is the unwrapped/scaled synthetic wavelength phase image, and ρ is the reference surface. The sign of
Λ Error as λ1 Varies
100
(6)
Λ Error
6
120
70
Error in Λ (cm) (solid)
Percent Error in Λ
80
60 50 40 30 20
5
100
4
80
3
60
2
40
1
20
Percent Error in Λ (dashed)
90
10 0 766
0 766.05
766.1
766.15
766.2
Variable Wavelength λ1 (nm)
(a)
766.25
766.3
0 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Synthetic Wavelength Λ (cm)
(b)
Fig. 3. Percent error in Λ as a function of (a) λ1 and (b) Λ, for the measured values of σ λ1 0.0085 nm and σ λ2 0.0092 nm. 10 March 2014 / Vol. 53, No. 8 / APPLIED OPTICS
1599
(a)
(b)
100 90 80
100
100
90
90
80
70
55 50
80 45 70
70 60
40
60 60
50
50
35
40
30
40
50
30
40
30
20
30
20
25
10
20 10
20
30
40
50
60
70
80
90
100
(c)
20
10 10
20
30
40
50
60
70
80
90
100
15
(d)
Fig. 4. Illustration of holographic volume calculation. (a) The unwrapped phase surface (e.g., an asymmetric Gaussian surface) with the region of integration bounding the area of interest, (b) the reference volume, which is then subtracted from the volume found in (a) to yield the volume of the Gaussian cap only, (c) contour map of the tilted surface, and (d) contour map determined from (c) after numerical flattening, i.e., reducing the average “bias” incline to zero.
Fig. 5. Modified Michelson configuration (a) allowing for object illumination at normal incidence with a tilted reference wave and Mach–Zehnder configuration (b) with 45° object illumination angle. 1600
APPLIED OPTICS / Vol. 53, No. 8 / 10 March 2014
the second integral in Eq. (7) is determined by the placement of the reference surface either above or below the ϕΛ surface. This process is simulated in Figs. 4(a) and 4(b) using a tilted asymmetric Gaussian surface. The unwrapped phase image can also be used to generate a contour map describing the surface, either before or after numerical flattening, which is of interest in several practical applications.
In practice the reconstructed surface is often tilted, as shown in Fig. 4(a), since the longitudinal axis (z axis) of the hologram surface is defined by the CCD plane normal. Therefore, if the tilt angle between the surface normal of the sample and the z axis is larger than a few degrees then numerical flattening can aid in generating more intuitive contour maps, although the volume calculation remains unaffected.
Fig. 6. Illustration of the MDH volume calculation process for Λ 1.869 mm (color online). (a) Photograph of the dent in an aluminum test surface, (b) one of the reconstructed holograms, (c) the wrapped 2D phase map after phase subtraction, (d) the unwrapped 2D phase map via PUMA algorithm, (e) distance-scaled 3D topographic map, (f) contour map illustrating topography, (g) 3D topographic map including the reference surface (red circular area), and (h) the reference volume without topographic map. 10 March 2014 / Vol. 53, No. 8 / APPLIED OPTICS
1601
5. Experimental Results
In this work, a series of dents in the surface of an aluminum plate is characterized via long synthetic wavelength topography, and the volume displacement is calculated, using a variable (λ1 765–781 nm) and fixed IR source (λ2 766.00 nm). A Lumenera (Model LU120M) camera with 1024 × 1024 pixels and of 6.7 μm pixel size is used to record the holograms. Using the modified Michelson configuration shown in Fig. 5(a), two holograms of each sample are captured, one at each wavelength (λ1 , λ2 ), and the MDH reconstruction process is performed as previously described for a given synthetic wavelength. The modified Michelson configuration shown in Fig. 5 is used to illuminate the target at normal incidence. This eliminates the “shadow effect” due to the 45° illumination angle in the Mach–Zehnder configuration [Fig. 5(b)], which may cause parts of the sample to be poorly illuminated. Figures 6(a)–6(h) illustrate the MDH reconstruction process for one of the aluminum samples, including the volume displacement calculation for the depressed portion of the dent. The mean of eight measurements, spanning Λ 1.869 mm to Λ 7.930 mm, yields a volume of 37.31 mm3 0.95 mm3 . The holographically measured mean depth of the dent is 1.63 mm 0.05 mm. This compares favorably with the depth measurement performed via caliper, which is 1.57 mm 0.13 mm. It should be noted that the relatively large error of the caliper measurement is due to performing repeated measurements on the sample, which exhibits a rather pitted and rough surface, reducing the absolute repeatability of each measurement. Additionally, the feature depth is measured from the planar surface of the aluminum sample, which is below the “ridge” of material surrounding the dent, as illustrated in Fig. 6(g). Because the aluminum dent is approximately spherical, it is possible to compare the measured volume to that analytically calculated for a spherical cap of equal size. Assuming the mean calipermeasured depth and radius of 1.57 and 3.80 mm, respectively, the volume of the spherical cap is 37.63 mm3 . Although this comparison is not expected to be exact (since the aluminum surface is rough) it does reveal that the holographic calculation matches closely with the nearest analytical comparison available. It should be noted that the wrapped phase map typically exhibits a great deal of noise, as seen in Fig. 6(c), which is not eliminated by the PUMA algorithm [10]. Indeed, PUMA was initially chosen because of its ability to perform well in the presence of such noise. Thus, for subsequent noise reduction the unwrapped phase maps have been subjected to 7 × 7 pixel median filtering prior to volume calculation or contour mapping. Topographic and volumetric measurements have been performed for a variety of additional objects, consisting primarily of additional depressions in 1602
APPLIED OPTICS / Vol. 53, No. 8 / 10 March 2014
an aluminum surface, with similar accuracy. For approximately spherical test features, the resulting accuracy was typically within a few percent of the analytically calculated volume (assuming a spherical cap). These measurements have also been repeated using the single-exposure spatial heterodyne method briefly discussed earlier. The spatial heterodyne method has been generally successful, although the resulting error tends to be larger due to ambiguity in pixel matching when the two object reconstructions are overlaid prior to phase subtraction. In general, such pixel matching is only accurate to within 1∕2 pixel, which causes additional phase aberration/noise in the reconstruction. 6. Conclusion
In conclusion, MDH can be effectively applied to accurately calculate the volume displacement of various topographic surface features, at the millimeter scale, using long synthetic wavelengths. However, at long synthetic wavelengths, the individual wavelength uncertainties in λ1 and λ2 (i.e., wavelength selection error) can lead to significant uncertainty in Λ. This difficulty can be overcome by computing the mean of several displacement measurements to yield a more accurate result. Several practical methods of implementation have been examined, including geometric scaling based upon illumination angle, phase unwrapping in the presence of noise, and noise suppression, with this method’s effectiveness verified by comparison to both physical measurement and analytical calculation with excellent agreement. Additionally, this method is currently being extended to the holographic microscopy regime using short synthetic wavelengths to perform volume displacement calculations and submicrometer topographic measurements for microelectromechanicalsystems (MEMS) metrology with depth resolution comparable to that of white-light interferometry. References 1. U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer, 2010). 2. J. Goodman, Introduction to Fourier Optics (Roberts, 2005). 3. T. Kreis, Handbook of Holographic Interferometry (Wiley, 2005). 4. D. Abdelsalam, R. Magnusson, and D. Kim, “Single-shot, dual-wavelength digital holography based on polarizing separation,” Appl. Opt. 50, 3360–3368 (2011). 5. C. Mann, P. Bingham, V. Paquit, and K. Tobin, “Quantitative phase imaging by three-wavelength digital holography,” Opt. Express 16, 9753–9764 (2008). 6. Y. Morimoto, T. Matui, M. Fujigaki, and N. Kawagishi, “Subnanometer displacement measurement by averaging of phase difference in windowed digital holographic interferometry,” Opt. Eng. 46, 025603 (2007). 7. J. Dirckx and W. Decraemer, “Deformation measurements of the human tympanic membrane under static pressure using automated moire topography,” Proc. SPIE 1429, 34–38 (1991). 8. S. Xenofos and C. Jones, “Theoretical aspects and practical applications of Moire topography,” Phys. Med. Biol. 24, 250–261 (1979).
9. E. Barbosa, E. Lima, M. Gesualdi, and M. Muramatsu, “Enhanced multiwavelength holographic profilometry by laser mode selection,” Opt. Eng. 46, 075601 (2007). 10. J. Bioucas-Dias and G. Valadao, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007). 11. J. Haus, B. Dapore, N. Miller, P. Banerjee, G. Nehmetallah, P. Powers, and P. McManamon, “Instantaneously captured
images using multiwavelength digital holography,” Proc. SPIE 8493, 84930W (2012). 12. J. Kuhn, T. Colomb, F. Montfort, F. Charriere, Y. Emery, E. Cuche, P. Marquet, and C. Despeursinge, “Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express 15, 7231–7242 (2007).
10 March 2014 / Vol. 53, No. 8 / APPLIED OPTICS
1603
