We propose a new digital holography method using an ultra-broadband light source and a chromatic phase-shifter. The chromatic phase-shifter gives different frequency shifts for respective spectral frequencies so that the spectrum of the light reflected from the object can be measured to reveal the spectral property of the object, and arbitrary selection of signals in the temporal frequency domain enables single- and multi-wavelength measurements with wide dynamic range. A theoretical analysis, computer simulations, and optical experiments were performed to verify the advantages of the proposed method.
©2012 Optical Society of America
Phase-shifting (PS) is a technique that allows removal of zero-order and twin images [1,2] through multi-exposure holographic recording while shifting the phase of the reference field [3–9]. Digital holography (DH) with PS (PSDH) by an integer fraction of 2π has been proposed. A 2π phase ambiguity that occurs when an illuminated surface has a height difference larger than the wavelength is an inherent problem in interference and digital holographic measurements. One method that has been proposed with the aim of solving this problem is two-wavelength digital holography (TW-DH) [10–13], which has been applied to both coherent DH [10–12] and low-coherence (LC) DH . In this method, the modulo-2π phase distribution must be unwrapped to reconstruct three-dimensional (3D) shape of the object; therefore, if there is still any phase jump greater than π due to steps in the surface of the object, incorrect or ambiguous unwrapping may occur. Another approach is multiple-wavelength (MW) DH, which uses more than two illumination wavelengths to overcome the 2π ambiguity of TW-DH [14–16].
When the TW- or MW-DH is used, the wavelength of the light source, at which the intensity of the reflected light from the object is desired to be highest, is commonly selected to reconstruct the 3D image. This selection is suitable for the monochromatic or homogeneous object in both cases of coherent or low-coherent DH [10–13]. However, the difficulty rises when the measurement is performed with chromatic, inhomogeneous or absorbing object. Because of the spectral property of the object, the spectrum of the reflected light from the object in this case differs from the one of the light source  and this difference is not easy to manage for all points on the surface of the object. In some cases, the selected wavelength may be out of the spectrum of the reflected light from the object and so the hologram corresponding to the selected wavelength is wrongly or not recorded, this causes the 3D image of the object reconstructed with a lot of error.
In this paper, we describe an optical system based on a Michelson interferometer with an ultra-broadband light source and a reference mirror that is moved with constant velocity to modulate the phase of the reference wave. The variation of the interference pattern due to the motion of the reference wave is recorded as holographic images by a fast camera [17–23]. The recorded holographic images are analyzed to reveal the relation between the spectrum of the object wave and the temporal spectrum of the interference pattern so that the spectrum of the light reflected from the object can be measured to reveal the spectral property of the object. Moreover, the system has been shown to work as a chromatic phase-shifter, giving different frequency shifts for respective spectral temporal frequencies of the captured interference pattern, and arbitrary selection of signals in the temporal frequency domain enables intensity image reconstruction, single-wavelength (SW-) and multi-wavelength (MW) measurements with wide dynamic range. First, we examine the requirements for implementing the proposed idea by computer simulation, and then we demonstrate the idea and its advantages using an experimental digital holographic system.
2.1 Dependence of complex amplitude spectrum on light source spectrum and velocity of reference mirror
For a typical setup is shown as in Fig. 1 , the complex amplitudes of the object and reference wave on image sensor of a specific wavelength are respectively expressed by17]. Quantity wb, representing the difference between the angular frequencies of the object wave and the reference wave, is defined asEq. (1) is then expressed as
Because the light source which is used to record holography is broad–band light source, so the interference patterns on sensor image is summing up monochromatic interference patterns of all wavelengths, which is reflected from the object. For simplicity, the spectrum of the light reflected from the object is separated into N sections in which the intensity and interference fringe contrast on a section respectively denoted as Ioi and Mi (1≤i≤N) are considered as constant as shown in Fig. 2 . Therefore, integrating over k in a section yieldsEq. (8), it can be seen that IWL[z(x,y),t] is a function of time; therefore, its temporal Fourier transform can be expressed asEq. (9), it can be seen that the spectrum of FIWL[z(x,y),w] is in the range ofEqs. (11) and (12). For each specific value of f selected in the optical intensity spectrum, the phase information of the object can be extracted as the second term of Eq. (9). On the contrary, when the reference mirror is moved away from the beam splitter (vR<0), the maximum and minimum temporal frequencies of the optical intensity are the absolute values given by Eqs. (11) and (12), and the phase information of the object is identified by the third term of Eq. (9).
An arbitrary temporal frequency f selected from section number i in the spectrum of FIWL[z(x,y),w] is related to the corresponding spectrum of the light source by:Eq. (9), here ki≤k≤ki + 1. Note that z(x,y)w/(2vR) in Eq. (9) can be rewritten as 2z(x,y)k from Eq. (13).
2.2. Sampling complex amplitude
From Eq. (8), when the reference mirror is moved with constant velocity, the optical intensity will be expressed as a function of time. This means that the optical intensity can be sampled by an appropriate camera. The sampled interference patterns are then used to reconstruct the forms of IWL[z(x,y),t] and FIWL[z(x,y),w]. In order to reconstruct the form of IWL[z(x,y),t] exactly, the frame rate of the camera, denoted as fs, should be much larger than two times the maximum frequency of the complex amplitude , which is specified by Eq. (11). This condition can be generalized as:Eq. (14) is satisfied, it is possible to reconstruct the form of FIWL[z(x,y),w] from the sampled holographic images, and the discrete Fourier transform (DFT) of IWL[z(x,y),t] is given byEq. (13) so that the phase of the complex amplitude derived from Eq. (15) can be used to reconstruct a 3D image of the object by using the angular spectrum method [25,26].
2.3. Low-coherence digital holography with selectable wavelengths
From Eqs. (9), (13) and (15), each temporal frequency f selected in the optical intensity spectrum of the interference pattern will identify a complex amplitude of the object. When a single frequency or many different frequencies f are selected, SW- or MW-LCDH can be applied to reconstruct 3D information of the object. The specific case of MW-LCDH adopted here is two-wavelength (TW-) LCDH, and when this method is used, the 3D information of the object will be reconstructed in terms of an equivalent wavelength λe, which is derived from λs1 and λs2 by the following equationEq. (11).
Using TW, a greater value of λe is desired to measure a steeper object shape . To measure an object with a very steep shape, λs1 and λs2 should be very similar. It is assumed that λs1 and λs2 are respectively derived from f1 and f2 by Eq. (11). Because the minimum difference between f1 and f2, denoted as Δfmin, is given byEqs. (17) and (18), a greater value of λemax can be achieved by increasing the velocity of the PZT or expanding the motion range of the PZT to record more samples of the optical intensity.
3. Experimental results
A first experiment was performed to confirm that the spectrum of the light reflected from the object could be measured by using the proposed system. The experiment was carried out with the set-up shown in Fig. 1. A plane mirror was used as the target object. The actual spectrum of the light reflected from the target object is shown in Fig. 2, in which the full width at half maximum (FWHM) ranged from 487 nm to 740 nm. A PZT (PZ 94E, Physik Instrumente) was used to modulate the optical wavefront. The PZT was moved at velocities of 10 μm/s, 20 μm/s, and 30 μm/s. The modulation of the optical intensity of the interference fringes caused by the motion of the PZT was simulated by Eq. (8) and is shown in Figs. 4(a) , 5(a) and 6(a) for the three PZT velocities.
The simulation results were then compared with the experimental ones. In the experiment, the object and reference wave were superimposed on a CMOS image sensor (Redlake; MotionScope M5) with a pixel size of 13.68 μm. The distance from the CMOS sensor to the object mirror was 300 mm. Considering the maximum temporal frequency of the optical intensity of the interference pattern when the reference mirror was moved with a velocity of 30 μm/s, the frame rate of the camera was set at 500 fps. The 500 hologram frames recorded by the camera for 1 s were used to investigate the variation of the optical intensity in time, and the experimental results are shown in Figs. 4(b), 5(b) and 6(b). The actual temporal spectra of the optical intensity were calculated by FFT, and the results are shown in Figs. 7(a) , 7(b), and 7(c). In this case, the frequency step calculated by Eq. (17) was 1 Hz. The FWHM spectrum of the optical intensity corresponding to PZT velocities of 10, 20, and 30 μm/s were 27–41 Hz, 54–82 Hz, and 81–123 Hz, respectively. Based on Eqs. (11) and (12), the FWHM spatial frequencies of the light source were estimated from 487 nm to 740 nm, from 487 nm to 741 nm, and from 487 nm to 740 nm. It can be seen that the spectrum of the light reflected from the object measured in the experiment was approximately the same as that measured with a spectrometer. In this case the, the object is plane mirror so the spectrum of the light reflected from the object and the spectrum of the light source are same. This means the experiment can be used to measure the spectrum of the light source as well.
The second experiment was conducted to examine the effect of errors caused by environmental disturbances or experimental devices (e.g., PZT calibration) on the reconstructed 3D information of the object with different wavelengths. In this experiment SW- and TW-LCDH were applied. The velocity of the PZT was set to 20 μm/s, and the maximum frequency and sampling frequency were estimated using Eqs. (11) and (14), respectively. The frame rate of the CMOS camera was 1000 fps (fs = 1000 Hz). The camera was set to record for 1 s (N = 1000). The frequency step calculated using Eq. (15) was 1 Hz.
Each selected frequency in the optical intensity spectrum specified the wavelength via Eq. (13) and the complex amplitude via Eq. (15). The specific complex amplitude allowed us to use SW-LCDH to reconstruct a 3D image of the object, which was then compared with the original one with a root mean square (RMS) method to evaluate the accuracy. In this case, because the original object was considered to be a perfect flat plane, its center line was simply considered as a horizontal line which can be obtained from the cross-section along the center line of the reconstructed object. The RMS error calculated from the cross-section along the center line of the reconstructed object and the original object for each selected frequency was divided by the corresponding wavelength, and the results in Fig. 8 show that for the wavelength which was selected out of spectrum of the light source, the RMS errors are very larger. Here, because RMS Error is directly proportional to selected wavelength and phase error, so RMS Error divided by the wavelength reflects the phase measurement accuracy of the proposed method.
To measure a steep object, we employed TW-LCDH, in which two different frequencies were selected in the optical intensity spectrum, and the corresponding wavelengths and complex amplitudes were calculated by using Eq. (11) and Eq. (13), respectively. The equivalent wavelength was calculated by Eq. (14). In this experiment, for simplicity, f1 was set to a constant value of 82 Hz, and f2 was varied. The results are shown in Table 1 .
The third experiment was conducted to examine the dependence of the quality of the reconstructed 3D image on the velocity of the PZT, which is directly proportional to the maximum equivalent wavelength according to Eq. (18) when employing TW-LCDH. The experiment was carried out with the CMOS camera at a frame rate of 1000 fps. The optical intensity spectrum was obtained from 1000 frames. The results of employing TW-LCDH are shown in Table 2 . The velocity of the PZT was set at 10 μm/s, 20 μm/s, and 30 μm/s. RMS Error/λemax were also calculated to evaluate the phase errors. The experiment results showed that, as the speed of the PZT was set higher, a larger error occurred. This can be explained by the increasing vibrations with increasing PZT velocity.
The fourth experiment was conducted to verify the effectiveness of the proposed method to measure 3D information of an actual object. A plane mirror tilted by a small angle was used as the target object. The velocity of the PZT was set to 20 μm/s, and the frame rate of the Pro Motion Camera was set to 500 fps (Δt = 1 ms), as in the first experiment. The intensity spectrum of the hologram was also from 54 Hz to 82 Hz.
The TW-LCDH method was applied. Two different frequencies, f1 = 66 Hz and f2 = 64 Hz, were selected in the temporal spectrum of the optical intensity. Two corresponding wavelengths, λ1 = 606 nm and λ2 = 625 nm, were estimated by using Eq. (13), and the equivalent wavelength, λe = 20 μm, was estimated by using Eq. (18). The surface of the object reconstructed by TW-LCDH, a cross-section, and error profile are shown in Figs. 9(a) , 9(b) and 9(c), respectively. The value of RMS error calculated from the error profile was 0.024 μm, which was about 833 times smaller than the equivalent wavelength. Again the frequency of 53 Hz was selected in the intensity spectrum, and SW-LCDH was applied to reconstruct a 3D image of the object with less error, as shown in Fig. 10 . In this case, the profile of the object obtained by TW-LCDH was used to detect and compensate for the 2π ambiguity error of the phase. The value of RMS error calculated from the error profile was 0.009 μm, which was about 84 times smaller than the wavelength.
It is has been proven that each selected frequency in the optical intensity spectrum specified the complex amplitude via Eq. (15). The phase and amplitude information of the complex amplitude corresponding to the selected frequency allow reconstructing the intensity image and 3D image of the object. In the fifth experiment, the tilted mirror was replaced by a color object that was a printed photo of 3 characters R, G and B on the dark background as shown in Fig. 11 . The reference mirror was moved with a velocity of 20 μm/s, the frame rate of the camera was set at 500 fps. The 500 hologram frames recorded by the camera for 1 s. The variations of the optical intensity in time obtained from different color pixels and their spectrum are shown in Figs. 12(a) , 12(b), 12(c), and 12(d), respectively. The FWHM spectrum of the optical intensity corresponding to red, green, blue and background pixels were 56–68 Hz, 72–82 Hz, and 76–86 Hz, and 57-82 Hz showed in Fig. 13 .
For each selected frequency in the optical intensity spectrum, the complex amplitude is specified by Eq. (15). The intensity images corresponding to selected frequencies were shown in Fig. 14 . From the experiment results in Figs. 13 and 14, it can be seen that the proposed method is possible to be used to reveal the spectral property of the target object.
Two different frequencies, f1 = 62 Hz and f2 = 63 Hz corresponding to two wavelength λ1 = 645 nm and λ2 = 635 nm in the red light region, were selected. The equivalent wavelength, λe = 40 μm, was estimated by using Eq. (18). The surface of the object reconstructed by TW-LCDH was shown in Fig. 15(a) . The cross-sections along the areas marked by red, green and blue lines on the surface of the object were respectively shown in Figs. 15(b), 15(c), and 15(d).
The surface of the object was also scanned by a Confocal Microscopy, the cross-section measured by the Confocal Microscopy at the same positions that were marked by red, green and blue lines in Fig. 15(a) were used to evaluate the accuracy of the measurement. The value of RMS error calculated from the error profile along red, green and blue lines were 0.327 μm, 0.724 and 0.963, which were 122, 55 and 41 times smaller than the equivalent wavelength, respectively. Because the selected wavelengths were in the red light region, the intensity of pixels corresponding to the green or blue points on the surface of the object were very weak as shown in Fig. 14. For the green or blue points on the surface of the object, these selected wavelengths were out of spectrum, this explained why RMS errors along green and blue lines were much larger than the RMS error along red one.
4. Conclusion and discussion
We have presented a system that works as a chromatic phase-shifter. A theoretical analysis was described to explain the operating principle of the system. Computer simulation was used to examine the performance, to learn about the requirements of the devices for implementation, and to visualize the experimental results of the proposed system.
The relationship between the spectrum of the light reflected light from the object and the spectrum of the optical intensity allowed us to select an appropriate camera frame rate so that the spectrum of the reflected light could be precisely measured.
Experiments in which both SW- and TW-LCDH were employed showed that the system was also able to perform SW- MW-LCDH for profilometry of the 3D shape of an object with an RMS error much smaller than the wavelength. The selectable wavelength ability of the proposed method is very important in measuring objects with different stepwise surface or inclination angles. For a steep object or an object with high stepwise surface, two close wavelengths should be selected, and for a less steep object or an object with low stepwise surface, two wavelengths with greater separation should be selected.
The experiments performed with the chromatic object showed that the proposed method can be used to reveal the spectral property of the target object. Based on the spectral property, the holographic images of the object are separated into many smaller areas where each small area is consisted of the consecutive pixels that have the similar spectral property. The profile of each object area is reconstructed with the most effective wavelength from the corresponding holographic image areas; the 3D image of the object with less error is then combined from these reconstructed object areas.
The difference between the simulation results and the experimental results can be explained by the error of the PZT and the limited bit depth of the high-speed camera. Vibrations that appeared when the reference mirror was moved are regarded as the main cause of the error in the phase component of the complex amplitude. The error of the phase component of the complex amplitude was directly proportional to the error of the reconstructed image with a proportionality ratio called equivalent wavelength. In fact, a long equivalent wavelength should be used to detect the rough shape of the object, and then a shorter equivalent wavelength or a single wavelength should be used to detect the surface of the object with higher precision. This seems to be difficult with the conventional method, but for the proposed method, it is quite simple. Finally, the proposed system is advantageous over the conventional ones because it is easy to set up, offering cost-effectiveness, with only one light source, and provides a straightforward solution for measuring the surfaces of objects.
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