2.1.

Off-Axis Digital Holography

We examine an off-axis digital holography system, the object plane is labeled $(x_0,y_0)$ and the recording plane is $(x,y)$. A beam of wavelength $\lambda$ is divided by a beam splitter into two parts, the wavefront of one part is modulated by the object producing the object beam, and the other part propagates to the recording plane at a known fixed angle to the object beam, as this is the reference beam. The object beam and the reference beam in recording plane are denoted by $O(x,y)$ and $R(x,y)$, respectively. The complex amplitude distribution $U(x,y)$ and the intensity of the hologram $I(x,y)$ in the recording plane can be related as follows:

Fourier spectrum filtering is performed to filter out the zero-order and $-1$ order in the frequency domain, and the filtered hologram $I_f(x,y)$ is obtained by inverse Fourier transforming the $+1$-order part of the spectrum. The complex amplitude distribution of the object wavefront in the object plane is obtained numerically using the angular spectrum algorithm to simulate propagation wave²¹

The phase distribution calculated by inverse tangent function in Eq. (4) is phase wrapped, and the phase takes values between $-\pi$ and $\pi$ (modulo $\pi$). The phase distribution obtained by digital holography can be used to extract the profile of the reflective object after phase aberration correction and phase unwrapping. For reflective object (in air $n=1$), the profile height distribution $h(x_0,y_0)$ is

2.2.

Subpixel Image Registration and Image Fusion

Assuming that the detector has $M\times N$ pixels with a rectangular pixel size of $\mathrm{\Delta }r$, the FOV of the system is $L_x=M\mathrm{\Delta }r$ and $L_y=N\mathrm{\Delta }r$. The methods of expanding the FOV in visible light digital holography include preimaging^22,23 and image fusion.²⁴ The former places an imaging lens in front of the detector, and the focal plane of the detector serves as the virtual recording hologram. The FOV of which is determined by the values of the cross-sectional beam size and the magnification of the imaging systems. In the case of THz imaging, the imaging setup should ideally be compact and lensless to maintain reasonable resolution and signal-to-noise ratio. The latter records the holograms of the different regions of the object then fuses multiple holograms or multiple reconstructed images to expand the FOV of system.

In this paper, a number of holograms were recorded follow shifts of the object horizontally. An image fusion approach to reconstruct the wavefront is proposed. The method consists of three steps: image registration, weighted smoothing, and image fusion.

First, the relative displacement of the reconstructed images is accurately determined by image registration. Assuming the reconstructed amplitude images of neighborhood region $i$ and region $i+1$ are $o_i^{\prime }(x_0,y_0)$ and $o_{i+1}^{\prime }(x_0,y_0)$, respectively. Overlapping subsections are designated two new images $s_i(x_0,y_0)$ and $s_{i+1}(x_0,y_0)$, respectively, having the same size $\xi \times \eta$. The relative displacement of the images is calculated using nonlinear optimization and discrete Fourier transform to within a small fraction of a pixel.²⁵ The image $s_{i+1}(x_0,y_0)$ is considered has a global coordinate translation $(\mathrm{\Delta }x,\mathrm{\Delta }y)$ to the image $s_i(x_0,y_0)$, to within multiplication by an arbitrary constant $\alpha$. A standard FFT-based numerical approach is applied to find the cross correlation peak within a pixel

The normalized root-mean-square (RMS) error metric $E$ is used to quantify the error between $s_i(x_0,y_0)$ and $s_{i+1}(x_0,y_0)$

The smaller the RMS error $E$, the greater the image similarity, the closer the alignment. A matrix multiplication implementation of the two-dimensional discrete Fourier transform is used to refine the initial peak location estimate. Subpixel registration is achieved by searching the cross correlation peak in a $1.5\times 1.5\text{pixel}$ neighborhood (in units of the original pixel). The relative displacement between the reconstructed images $(\mathrm{\Delta }M,\mathrm{\Delta }N)$ can be obtained in this way using the overlap sections.

Then, weighted smoothing is used to reduce the amplitude difference and phase shift caused by any nonuniformity in the uneven distribution of THz beams in the experimental setup. For the reconstructed amplitude images, with overlap sections as the reference, a suitable normalization factor of $o_{i+1}^{\prime }(x_0,y_0)$ is given as

The normalized amplitude distribution of $o_{i+1}^{\prime }(x_0,y_0)$ is then

For the reconstructed phase images, with overlap sections as the reference, the resulting corrected phase shift of ${\phi }_{i+1}^{\prime }(x_0,y_0)$ is given as

The corresponding normalized phase distribution of ${\phi }_{i+1}^{\prime }(x_0,y_0)$ is given as

Finally, bicubic interpolation²⁶ is applied to the resulting the reconstructed images with an upsampling factor of 10, to provide accurate 0.1 pixel spaced sample values. The reconstructed images are expanded in size by zero padding, and image translation with the corresponding displacements is applied to the reconstructed images. Finally, the FOV of the system is successfully expanded by fusing the multiple reconstructed amplitude and phase images.

3.1.

Experimental Setup

An experimental setup for CW THz reflective off-axis digital holography for covered object profile measurement was shown in Fig. 1(a). The THz source was an optical pumped far-IR gas laser (FIRL 295, Edinburgh Instrument). Its central wavelength is $118.83\mu \mathrm{m}$ (2.52 THz), and the maximum power is 500 mW. Two off-axis parabolic mirrors (PM) were used to collimate and expand the terahertz beam. The distribution of the collimated beam was recorded by the pyroelectric detector. And the diameter (full-width at half-maximum) was about 15.4 mm measured by cross-section intensity distribution. The collimated THz wave was divided into two beams by the HRFZ-Si beam splitter, whose splitting ratio at 2.52 THz is 54% for transmission and 46% for reflection. One beam propagated to the recording plane (reference beam) while the other beam illuminates the object producing the object beam. The interference angle was chosen $\sim 0.61\mathrm{rad}$, and the holograms were recorded by a pyroelectric array detector (Pyrocam IV, Ophir Spiricon), which has $320\times 320\text{pixels}$ on an $80\text{-}\mu \mathrm{m}$ pixel pitch.

Fig. 1

(a) CW THz reflective off-axis digital holography experimental setup for covered object measurement, PMs represent off-axis parabolic mirrors, the focal length of PM1 and PM2 are 50.8 and 101.6 mm, respectively, BS is a THz beam splitter. The distance from the object plane to the recording plane is $d$. (b) Photo of the imaged metallic bookmark. (c) The same metallic bookmark covered by a 1.5-mm plate.

The object is a metallic bookmark produced by photo etching [Fig. 1(b)]. It was placed at a distance of 63.0 mm upstream from the detector to ensure adequate resolution. The object plane and the recording plane are not parallel as shown in Fig. 1. In order to verify the imaging capability of the system for covered objects, a 1.5-mm-thick PTFE plate ($n=1.43$) was placed in front of the bookmark [Fig. 1(c)]. This plate can be seen to opaque in the visible band, and the surface of the covered plate is not completely homogeneous due to machining error. The object was placed on a manual linear stage capable of applying up to 25.4 mm of smooth motion, and different regions of the object were illuminated by moving the object.

3.2.

Reconstructed Results of the Single Hologram

The reconstructed results of the metallic bookmark with and without the PTFE plate present were compared. It was found that the preprocessing of 100 frames is required to improve the fringe contrast of the holograms. The holograms for the same region of the object without and with PTFE plate are shown in Figs. 2(a1) and 2(a2). The period of the resulting interference fringe is $4.81\mathrm{lp}/\mathrm{mm}$. As can be seen, the contrast of the hologram is degraded when the PTFE plate is presented because of the inhomogeneity of the covered mask. An autofocus algorithm²⁷ was applied to determine the reconstructed distance. The distance was 63.0 mm in the uncovered case and 63.65 mm in the covered case because of the optical path difference caused by the different refractive index of the air and PTFE. The theoretical lateral resolution of the imaging system is $\delta =\lambda d/N\mathrm{\Delta }r=292.4\mu \mathrm{m}$. An angular spectrum algorithm was used to obtain the amplitude and wrapped phase distributions of the object. The reconstructed amplitude distributions are shown in Figs. 2(b1) and 2(b2), the etched letters can be visually distinguished. It is noted that significant noise is introduced by the inclined plane (the object plane is nonparallel to the recorded plane) and surface roughness of the PTFE plate. The wrapped phase distributions of the object are shown in Figs. 2(c1) and 2(c2). A least square phase unwrapping algorithm²⁸ was used to obtain the continuous phase distributions. With regarded to the phase aberration caused by the intersection angle between object plane and recording plane, the one-step-method²⁹ was applied. The unwrapped phases are shown in Figs. 2(d1) and 2(d2). As can be seen, diffraction at the curve edge of the bookmark reduces the reconstruction quality of the adjacent area.

Fig. 2

The reconstructed results of single region. (a1) and (a2) Holograms without and with PTFE plate. (b1–d1) Reconstructed amplitude, wrapped phase, and unwrapped phase without PTFE plate. (b2–d2) Reconstructed amplitude, wrapped phase, and unwrapped phase with PTFE plate.

In order to show the influence of the PTFE plate on the reconstructed images, we compared the flat regions of the sample without alphabets. Figure 3 shows the amplitude and phase distribution curves taken along the dotted lines shown in Fig. 2. It can be seen that the variation of reconstructed images with the PTFE plate is large than those without the PTFE plate.

Fig. 3

The comparison curves of (a) the reconstructed amplitude and (b) phase distributions in Fig. 2. The red curves in Figs. 3(a) and 3(b) represent the red dotted lines in Figs. 2(b1) and 2(d1), respectively. The blue curves in Figs. 3(a) and 3(b) represent the blue dotted lines in Figs. 2(b2) and 2(d2), respectively.

The profile of the object can be obtained from the unwrapped phase image as indicated in Sec. 2.1. In order to verify the measurement accuracy of the proposed method, cross-sectional profiles are taken along the lines shown in Figs. 2(d1) and 2(d2), the resulting profiles are plotted in Figs. 4(a) and 4(b) for reconstructions with and without the PTFE plate. The object was also directly measured by a three-dimensional (3-D) Surface Profiler (WYKO NT1100, Veeco) giving a depth result of $157.0\mu \mathrm{m}$. The profiles information only can be measured in integral pixels from the phase images. Choosing the same region for comparison, the calculated etching depths of the letter “P” were 161.6 and $150.6\mu \mathrm{m}$ in the uncovered case, and 166.4 and $163.9\mu \mathrm{m}$ in the covered case [see Figs. 4(a) and 4(b)]. The estimated etching depths of the letters are different on the left and right sides because of the phase distortion caused by the nonparallelism between the object plane and the recording plane. The experimental results indicate that the relative error of measuring the profiles of the object by CW THz digital holography are 2.93% and 4.08% in the uncovered case. The relative errors are 5.98% and 4.39% in the covered case, respectively. The accuracy of profiles measurement needs to be further improved. In Figs. 4(c) and 4(d), the 3-D profiles of the bookmark are presented. From the 3-D maps, it can be seen that the upper part and the lower part are not in the same plane, which is consisted with the orientation the system. Moreover, for the flat regions without alphabets, we can see that the surface is undulate when the sample is covered. The experimental results clearly demonstrate that the THz system has the ability to detect the objects behind visually opaque materials such as PTFE.

Fig. 4

(a), (b) Profiles of the object. The red curve and the blue curve represent the experimental data along the cross-sections highlighted in Figs. 2(d1) and 2(d2). (c) 3-D structure of the object without covered. (d) 3-D structure of the covered object.

3.3.

Expanding the Field-of-View by Image Fusion

The width of the metallic bookmark used in the experiment is 34 mm. The illuminating beam size is less than the numerical aperture of the detector because of the limit aperture of beam splitter; therefore, the effective FOV of a single hologram captured using the system is about $320\times 250\text{pixels}$ ($25.6\times 20{\mathrm{mm}}^2$). In order to inspect the complete object, three holograms containing images of different regions of the object were recorded. The experiment was performed so that images with overlap parts appeared in the adjacent holograms. The reconstructed images were fused by the method proposed in Sec. 2.2. The lateral and longitudinal relative displacement between region 2 and region 1 are 83.2 and $-4.6\text{pixels}$, respectively. The lateral and longitudinal relative displacement between region 3 and region 2 are 82.8 and $-6.5\text{pixels}$, respectively. The amplitude and phase distribution of the object after image fusion are shown in Figs. 5(a1) and 5(b1), when the bookmark was not covered. After being covered by the PTFE plate, the amplitude and phase distribution of the bookmark are shown in Figs. 5(a2) and 5(b2). With our proposed method, the FOV of the system is expanded to $25.6\times 35{\mathrm{mm}}^2$, a factor of 1.75 times larger than the initial single hologram value. The intensity of the illuminating THz field is uneven, so the image quality of the reconstructed image after image fusion is degraded.

Fig. 5

The experimental results of expanding the FOV by subpixel image registration and image fusion. (a1), (b1) Reconstructed amplitude and phase of expanding the FOV. The bookmark was not covered. (a2), (b2) Reconstructed amplitude and phase after expanding the FOV. The bookmark was covered behind a 1.5-mm-thick PTFE plate.