Yeonwoo Jeong, Behnam Tayebi, Jae-Ho Han
Several techniques have been developed to overcome the limitation of sensor bandwidth for 2D signals [1]. Though compressive sensing is an attractive technique that reduces the number of measurements required to record information on a sparse signal basis [2], [3], recording information beyond the Nyquist frequency remains difficult when working with nonsparse signals. Given this constraint, this article focuses on the use of the physical bandwidth of a coherent signal in the complex form instead of its intensity form. The resulting trick combines holographic multiplexing with sampling scheme optimization to obtain the information in a 2D coherent signal from beyond the Nyquist frequency range. The prerequisites for understanding this article are a knowledge of basic algebra and the Fourier transform. Familiarity with holography is also beneficial.
As shown in Figure 1, when two light waves, E1 and E2, approach each other and finally meet at the recording point, P, this situation can be mathematically described by using the principle of superposition of electromagnetic fields and the wave intensity at that point. The principle of superposition in optics states that when multiple waves overlap in a medium, the resultant amplitude is equal to the algebraic sum of the individual wave amplitudes. The intensity is defined as the radiant power density of the light signal detected by a device such as a camera or an eye, and it can be expressed as the time average of the wave amplitude squared. As this article deals only with coherent light, electromagnetic waves can be assumed to maintain a fixed phase displacement over a period of time. Thus, the principle of superposition and the intensity form the basis for recording the interference of coherent light waves and comprise the core of holography [4].
Figure 1. Interference of two light waves and the intensity. Here d is the distances traveled by the beams, ${\omega}$ is the angular velocity, k is the angular wavenumber, and ${\phi}$ represents the phase of the light at time 0. The intensity at point P has the interference term, ${I}_{12}$ other than the individual intensities.
The holography technique was developed to preserve the depth information in an object signal, which cannot be captured by a normal camera [5]. When an object image is captured by an image sensor, the recorded intensity is proportional to the square of the object signal amplitude, causing the phase information of the object signal to be lost. By contrast, holography maintains the phase information using a reference signal. For example, a microscopy technique using holography, as illustrated in Figure 2, can be divided into two setup components: magnification and frequency modulation. Light from the coherent light source is scattered and passed through the object, and the objective lens magnifies the resulting object signal. The reference beam, ER, can be subsequently produced using a grating to duplicate the magnified signal and reorient it to a specific angle. The two signals after grating are physically transformed into the frequency domain at the Fourier plane, while an analog filter removes all information except the center intensity of the duplicated signal to obtain the reference signal, ER. Finally, a second lens is applied to cause the original signal to interfere with the reference signal, with the intensity of this combined signal, $\mid{E}_{R} + {E}_{O}\mid{}^{2}$, recorded by the image sensor. The recorded object signal EO can be represented with its complex magnitude, $\mid{E}_{O}{(}{x},{y}{)}\mid$, and the phase component, ${e}^{{i}{\theta}_{O}\left({x,y}\right)}$, while the magnitude and the phase of the recorded reference signal, ER, are | ER | and ${e}^{i{\theta}_{R}x}$, respectively, where ${\theta}_{R}$ is obtained by dividing ${2}{\pi}$ by the grating period.
Figure 2. The frequency scheme of (a) direct imaging and (b) a single hologram with an optical system.
As indicated by the recorded frequency scheme shown in Figure 2(b), the radius of the physical bandwidth circle describing the object signal intensity is twice that of the object signal amplitude because the Fourier transform of the two multiplied signals is equivalent to the convolution operation between the individual Fourier transformed signals. The phase information of the object is preserved in the bandwidths of the twin sidelobes, expressed as $\widetilde{{U}_{O}}\left({{f}_{x}\pm{\theta},{f}_{y}}\right)$, simply reflecting the original frequency information of the object shifted by $\pm{\theta}$. Therefore, in contrast to the case of direct imaging [Figure. 2(a)], which preserves only the magnitude of the object signal, holography also facilitates the full recovery of the phase information from the object signal by programmatically extracting the bandwidth from one of the sidelobes [6].
According to the Nyquist theorem, distortion can be avoided in digital intensity recording systems by ensuring that the sampling rate used to digitize a signal is at least twice the maximum frequency range denoting the physical bandwidth of the signal. Consequently, when an object signal is captured directly, the frequency range of the sensor bandwidth U0 in Figure 2(a) should be at least four times the frequency range denoting the physical bandwidth of the object signal amplitude kp (i.e., at least twice the frequency range denoting the physical bandwidth of the object signal intensity Kp). As a result, the full sensor bandwidth required to record the signal intensity without distortion should be more than ${16}{(}{4}\,{\times}\,{4}{)}$ times the physical bandwidth of the amplitude, as indicated by the blue box in Figure 2(a); thus, the cost of direct intensity recording is extremely high.
The situation is more adverse when considering the Nyquist theorem for a single hologram. As shown in Figure 2(b), the recorded intensity and its Fourier transform can be respectively expressed as follows: \[{I} = {E}_{R}{E}_{R}^{\ast} + {E}_{O}{E}_{O}^{\ast} + {E}_{O}{E}_{R}^{\ast} + {E}_{R}{E}_{O}^{\ast} \tag{1} \] \begin{align*}{\mathcal{F}}\left({I}\right) & = {\mathcal{F}}\left\{{{E}_{R}{E}_{R}^{\ast}}\right\} + {\mathcal{F}}\left\{{{E}_{O}{E}_{O}^{\ast}}\right\} \\ & \quad + {\mathcal{F}}\left\{{{E}_{O}{E}_{R}^{\ast}}\right\} + {\mathcal{F}}\left\{{{E}_{R}{E}_{O}^{\ast}}\right\}{.} \tag{2}\end{align*}
The frequency information recorded by the sensor includes the amplitudes of the twin sidelobes in the single-hologram scheme shown in Figure 2(b) expressed as the last two terms on the right side of (2). In addition, the direct intensity information of the object, which is represented as the central lobe in Figure 2(b), is expressed as the second term on the right side of (2). After applying the Nyquist theorem, the radii of the sidelobes double, causing the required frequency range for the digitizing sensor in the single hologram scheme, defined as U1 in Figure 2(b), to be at least twice that in the direct-imaging scheme [U0 in Figure 2(a)], and expanding the required sensor bandwidth accordingly. Therefore, this article aimed to introduce a solution to increase the ratio of the recorded amplitude bandwidth to the sensor bandwidth to overcome the inefficiencies otherwise associated with the Nyquist constraint.
Two primary techniques have been developed to improve the available bandwidth with respect to the total frequency area of an image sensor: one based on frequency multiplexing [7] and the other based on sampling scheme optimization [8]. Frequency multiplexing is accomplished by obtaining multiple holograms using one reference beam. With this approach, the recorded intensity of N independent holograms at the image sensor can be expressed as \begin{align*}{I} & = \mid{E}_{R} + {E}_{1} + {E}_{2} + \cdots + {E}_{N}\mid^{2} \tag{3} \\ & = {E}_{R}{E}_{R}^{\ast} + \mathop{\sum}\limits_{{i} = {1}}^{N}{\mathop{\sum}\limits_{{j} = {1}}^{N}{{E}_{i}{E}_{j}^{\ast}}} + \mathop{\sum}\limits_{{i} = {1}}^{N}{{E}_{R}{E}_{i}^{\ast}} \\ & \quad + \mathop{\sum}\limits_{{i} = {1}}^{N}{{E}_{i}{E}_{R}^{\ast}}{.}\end{align*}
Furthermore, the following theoretical constraint is set to ensure that the sum of the N hologram signals is equal to the total object signal: \[{E}_{O} = \mathop{\sum}\limits_{{i} = {1}}^{N}{E}_{i}{.} \tag{4} \]
Thus, when ${N} = {4}$, as illustrated in Figure 3, frequency multiplexing of the holograms can be realized by magnifying and separating the object images into four distinct patches and recording them simultaneously by overlapping the associated signals. Compared to the normal imaging in Figure 3(a), as magnification is inversely proportional to the recorded frequency range, the frequency range required to capture the object decreases according to the magnification of the image, as shown in Figure 3(b). In addition, overlapping the signals enables the image sensor, which uses the same size as for direct imaging, to capture the full area of the object without decreasing the field of view (FOV), i.e., the recorded image area, which is possible because the bandwidth of each signal has been located in a separate area using the holography technique, facilitating the reconstruction of the original object image by extracting each of these signals from the frequency area and combining them together, as shown in Figure 3(c). However, increasing the number of holograms does not necessarily decrease the size of the dead zone in the sensor frequency domain.
Figure 3. Comparison of direct imaging and the method using frequency multiplexing. (a) Direct imaging, (b) direct imaging of the magnified object, and (c) imaging the combined four patches of the object using frequency multiplexing with image reconstruction. FOV: field of view.
Indeed, the sensor bandwidth including the sensor bandwidth for the two holograms in Figure 4(c) is larger than that for direct imaging, as shown in Figure 4(a) (i.e., ${U}_{2}\gt{U}_{0}{)}$, with a considerable portion of the frequency domain remaining unutilized. Figure 4(d) shows a geometrically optimized scheme using a single hologram that exploits the repetitive pattern of the Fourier domain to improve the utilization of the frequency; however, ${U}_{3}\gt{U}_{0}$. Notably, none of the sampling schemes in Figure 4(b)–(d) can record amplitude information beyond the Nyquist constraint, regardless of their utilization of the frequency domain.
Figure 4. 2D frequency schemes of (a) direct imaging, (b) single hologram without optimization, (c) two holograms without optimization, (d) single hologram with optimization, and (e) two holograms with optimization. The sampling schemes are optimized for (f) three, (g) four, (h) 12, and (i) 16 holograms. The physical bandwidth of the intensity and amplitude of the single hologram are shown in brown and gray, respectively, and its digitized intensity and amplitude are shown in violet and black, respectively. The digitized intensity and amplitude of the multiplexed hologram are shown in light and dark green, respectively. SH: single hologram; MH: multiple holograms.
A relationship between bandwidths must be established to optimize the bandwidth available for recording the amplitude of the signal beyond the Nyquist frequency. Here, the required total bandwidth is defined as the sum of the bandwidths of sidelobes of the same size, and each bandwidth in the 2D frequency domain is defined as the square of the length along a single direction of frequency range. Thus, the maximum frequency range along one axis of the total bandwidth ${k}_{d}$ can be written as \[{k}_{d}^{2} = {N}\left({{k}_{d,i}^{2}}\right) \tag{5} \] where ${k}_{d,i}$ represents the maximum frequency range along one axis of sidelobe i and can be expressed by \[{k}_{d,i} = \frac{{k}_{d}}{\sqrt{N}}{.} \tag{6} \]
The relationship between U0 and kd, i can be expressed by \[{U}_{0} = {\gamma}{k}_{d,i} \tag{7} \] where ${\gamma}$ denotes the optimal geometrical factor of the 2D sampling scheme. As kd can be considered the digitized amplitude of the signal, denoted Kp, (7) can be rewritten as \[{U}_{0} = \frac{\gamma}{\sqrt{N}}{K}_{p}\equiv{\mathit{\Gamma}}{K}_{p}{.} \tag{8} \]
Note that (8) relies upon the quantitative relationship between the maximum range along one axis of the sensor bandwidth and the maximum amplitude range of the signal, expressed by ${\gamma}$. Thus, the ratio of ${\gamma}$ to the square root of the number of holograms can be defined as the effective coherent Nyquist factor, ${\mathit{\Gamma}}$, to quantify the efficiency of the sensor bandwidth utilization.
The ideal case in a 1D scheme can be evaluated to provide a logical basis for optimization in a higher dimensional scheme. As shown in Figure 5, assuming that the sidelobes for N holograms are tightly packed in the 1D Fourier domain, the relationship between the maximum frequency range of the sensor, U0,1 d, and that of the digital bandwidth of each sidelobe, kd,i,1 d, can be expressed as \[{U}_{0,1d} = {2}\left({{N} + {1}}\right){k}_{d,i,1d} \tag{9} \]
Figure 5. The ideal case in the 1D scheme.
yielding the optimal geometrical factor ${\gamma}_{1d} = {2}\left({{N} + {1}}\right)$.
Furthermore, as ${K}_{p,1d} = {Nk}_{d,i,1d}$, (9) can be rewritten as \[{U}_{0,1d} = \frac{{2}\left({{N} + {1}}\right)}{N}{K}_{d,i,1d} \tag{10} \] indicating that the effective coherent Nyquist factor for the 1D case, ${\mathit{\Gamma}}_{1d}$, is ${2}\left({{N} + {1}}\right){/}{N}$. Therefore, for a large number of holograms, the single-axis range of the sensor cannot be less than twice that of the single intensity because only an infinite number of holograms $\left({{N}\rightarrow\infty}\right)$ can achieve a ${\mathit{\Gamma}}_{1d}$ value of 2.
The higher dimensional scheme optimization does not exhibit such simple behavior. For example, in contrast to the nonoptimized scheme in Figure 4(c), Figure 4(e) shows the optimal frequency-sampling scheme for a two-hologram technique, which can be easily produced based on the fact that the radius of the central lobe should be twice that of the sidelobes, corresponding to a ${\gamma}$ of 2.71. Accordingly, the relationship between the frequency range along the axis of the digitized amplitude signal and its minimum frequency range on one axis of the sensor bandwidth can be expressed as ${U}_{0} = {1}{.}{92}\left({2.71/\sqrt{2}}\right){K}_{p}$. As the resulting ${\mathit{\Gamma}}$ is less than 2, this optimization achieves the sub-Nyquist condition. Similarly, Figure 4(f) and (g) illustrates the optimal sampling schemes for three and four holograms with ${U}_{0} = {1}{.}{87}{K}_{p}$ and ${U}_{0} = {1}{.}{73}{K}_{p}$, respectively. In an extreme case with 16 tightly packed holograms [Figure 4(i)], the values of ${\gamma}$ and ${\mathit{\Gamma}}$ are 6 and 1.5, respectively. Indeed, all optimized schemes in Figure 4(e)–(i) can be used to design a sub-Nyquist coherent imaging system.
To evaluate the efficiency of the proposed method, the percentage of the sensor occupied by the required bandwidth for digitizing the signal, relative to that occupied by the bandwidth required for the direct imaging, can be expressed as \[{T} = {100}\,{\times}\,{k}_{d}^{2}\frac{\left({\mathop{\sum}\limits_{{i} = {1}}^{N}{k}_{d,i}^{2}}\right)}{{k}_{d,0}^{2}} \tag{11} \] where ${k}_{d,0}$ is ${k}_{d}$ for the scheme of the direct imaging.
Table 1 compares the efficiencies of different sampling schemes when recording frequency information using the same sensor. According to the Nyquist theorem, the metric $T$ should remain less than 25%. However, the values of $T$ for the multiplexed holography based on the sampling schemes presented in Figure 4(c)–(i) are 12.5%, 15.26%, 27.23%, 28.76%, 44.44%, and 33.33%, respectively. Therefore, except for the first four cases listed in Table 1, the frequency information was successfully recorded beyond the Nyquist constraint.
Table 1. Comparison of different sampling schemes.
As mentioned previously, increasing the number of holograms can be inferred to reduce the value of ${\mathit{\Gamma}}$ to less than two. The ideal ${\mathit{\Gamma}}$ for the sampling scheme can be induced from Figure 4(i), and it can be expressed as \[{\mathit{\Gamma}}\approx\sqrt{2}\left({\sqrt{\frac{\left({{N} + {2}}\right)}{N}}}\right){.} \tag{12} \]
Thus, for an infinite number of holograms, ${\mathit{\Gamma}} = \sqrt{2}$, and the maximum value of $T$ is 50%. However, as this value corresponds to an infinite sensor area with an infinite number of holograms, it cannot be achieved owing to the presence of an autocorrelation term and the resolution limits in the frequency domain. Therefore, the lower limit of the effective coherent Nyquist factor is $\sqrt{2}.$
Figure 6 depicts ${\mathit{\Gamma}}$ as a function of the number of captured holograms. The red squares show the optimal value of ${\mathit{\Gamma}}$ manually obtained for the Fourier domain schemes using the single and multiple holograms in Figure 4(d)–(g). The results in Table 1 indicate that an increase in N causes the value of ${\mathit{\Gamma}}$ to decrease toward a lower limit of $\sqrt{2}$, demonstrating that multiplexing the signal and optimizing the sampling scheme significantly improved the quantity of information recorded by the sensor.
Figure 6. Effective coherent Nyquist factor $\left({\mathit{\Gamma}}\right)$ as a function of the number of captured holograms. The red squares (manual) show the optimal value of ${\mathit{\Gamma}}$ for recording one, two, three, and four holograms.
As an example implementation, Figure 7 shows a sub-Nyquist coherent system using four holograms. In the first stage of the optical system, the object signal is magnified and divided into four distinct patches using an objective lens and masks. Although the FOV decreases in inverse proportion to the magnification, in this implementation, all patch signals are represented by holograms and finally gathered at the image sensor. If only one reference beam is used as (3), the sideband cannot be moved independently. Therefore, the light path is split using mirrors and beam splitters to create multiple reference beams with the same number of holograms. Accordingly, the modified intensity expression is given by \begin{align*}{I} & = \mid{E}_{{R}_{1}} + {E}_{{S}_{1}} + {E}_{{R}_{2}} + {E}_{{S}_{2}} \\ & \quad + {E}_{{R}_{3}} + {E}_{{S}_{3}} + {E}_{{R}_{4}} + {E}_{{S}_{4}}\mid^{2} \tag{13} \\ & = \mathop{\sum}\limits_{{i} = {1}}^{4}{\mathop{\sum}\limits_{{j} = {1}}^{4}{{E}_{{R}_{i}}}}{E}_{{R}_{j}}^{\ast} + \mathop{\sum}\limits_{{i} = {1}}^{4}{\mathop{\sum}\limits_{{j} = {1}}^{4}{{E}_{{S}_{i}}}}{E}_{{S}_{j}}^{\ast} \\ & \quad + \mathop{\sum}\limits_{{i} = {1}}^{4}{\mathop{\sum}\limits_{{j} = {1}}^{4}{{E}_{{R}_{i}}}}{E}_{{S}_{j}}^{\ast} + \mathop{\sum}\limits_{{i} = {1}}^{4}{\mathop{\sum}\limits_{{j} = {1}}^{4}{{E}_{{S}_{i}}}}{E}_{{R}_{j}}^{\ast}\end{align*}
Figure 7. An example implementation of the sub-Nyquist coherent system using four holograms. It consists of three parts: magnification, multiplexing with optimization, and recording. The optimized sampling scheme for four holograms can be implemented by physically splitting and combining the magnified object signal.
where ${E}_{{S}_{i}}$ and ${E}_{{R}_{i}}$ represent the object and reference signals, respectively. Each reference and object patch signal can be realized using two gratings and an analog filter. As the grating replicates the incoming light at specified angles and different intensities along a single axis, two gratings with appropriate specifications can set the required modulation frequency in the 2D Fourier domain. In the physical Fourier plane, the signals duplicated by the gratings are passed through analog filters, leaving only two signals. The four zeroth-order signals are preserved as they represent the object signal, while the first-order signals remain as they represent the modulation frequency. Thus, using this technique, four different sets of object patches and their corresponding reference signals can be provided, allowing the sidebands to be located independently.
The first and second terms of (13) represent the bandwidth of the central circle, whereas the third and fourth terms represent the twin sidelobe bandwidths. The required information is contained in the four sidebands, each capturing a patch of the object image. However, according to (13), as the four holograms share the same reference beam and are therefore intermingled, they must be separated. For example, as indicated by \[\mathop{\sum}\limits_{{i} = {1}}^{4}{E}_{{S}_{i}}{E}_{{R}_{1}}^{\ast} = \left({{E}_{{S}_{1}} + {E}_{{S}_{2}} + {E}_{{S}_{3}} + {E}_{{S}_{4}}}\right){E}_{{R}_{1}}^{\ast} \tag{14} \] the bandwidth of the first hologram carried by the first reference signal is colocated with the other three holograms because they are also carried by the first reference signal.
To solve this problem, each set of objects and reference signals must be marked to avoid interference. Therefore, for the optical system in Figure 7, the two light sources with different wavelengths, polarization beam splitters, and dichroic mirrors are used. The dichroic mirror is a specially designed mirror that reflects light in a specific band, and the polarization beam splitter is also a specially designed beam splitter that splits light into two orthogonally polarized beams. The reason for using those components is that two light sources with different wavelengths do not interfere with each other, and neither do two orthogonally polarized beams. Exploiting the light properties, the two dichroic mirrors at the front of the optical system split one light path into two paths having different wavelengths, respectively. Next, the polarization beam splitter before the masks splits each light path again into the two orthogonally polarized beams. Therefore, each reference beam ends up carrying only the corresponding patch signal, and (14) is changed to \begin{align*} & \mathop{\sum}\limits_{{i} = {1}}^{4}{E}_{{S}_{i}}{E}_{{R}_{1}}^{\ast} = {E}_{{S}_{1}}{E}_{{R}_{1}}^{\ast}{,} \\ & \because{E}_{{S}_{2}}{E}_{{R}_{1}}^{\ast} = {E}_{{S}_{3}}{E}_{{R}_{1}}^{\ast} = {E}_{{S}_{4}}{E}_{{R}_{1}}^{\ast} = {0}{.} \tag{15}\end{align*}
In summary, after dividing the magnified object signal into several patches using masks, an optimal scheme can be realized by freely moving the sidebands via the gratings and manipulating the interference by selecting or changing the state of light using multiple light sources and specially designed optical components. Likewise, as the number of holograms increases, the additional components required to avoid interference among holograms, and the space to install them, inevitably increase.
This article presented a trick for realizing sub-Nyquist coherent imaging based on an optimized multiplexing hologram scheme. The Nyquist theorem dictates that the digital imaging bandwidth requires four times the frequency range denoting the physical bandwidth of the object signal amplitude to avoid distortion as the intensity is proportional to the square of the amplitude when using coherent light sources. Thus, the full sensor bandwidth must be at least ${16}{(}{4}\,{\times}\,{4}{)}$ times the bandwidth of the original object signal amplitude to record intensity, wasting significant frequency space. To overcome this limitation, a sub-Nyquist imaging scheme was proposed by exploiting multiplexed holograms with optimization. Several optimized schemes were presented based on two to 16 sub-Nyquist holograms, and a theoretical effective coherent Nyquist factor limit of $\sqrt{2}$ was derived from the ideal cases. Finally, an example implementation of the proposed sub-Nyquist coherent imaging technique was provided using four holograms with two different coherent light sources.
This work was supported by the Ministry of Science and ICT, Korea, under the Information Technology Research Center support program (IITP-2023-RS-2022-00156225) and under the ICT Creative Consilience program (IITP-2023-2020-0-01819) supervised by the Institute for Information and Communications Technology Planning and Evaluation (IITP). Correspondence should be sent to Behnam Tayebi and Jae-Ho Han (corresponding authors). Yeonwoo Jeong and Behnam Tayebi equally contributed for this work.
Yeonwoo Jeong (forresearch4220@gmail.com) received his B.S. degree from the School of Electrical Engineering, Korea University, Seoul, South Korea, in 2017. He is currently pursuing his Ph.D. degree with the Department of Brain and Cognitive Engineering, Korea University, Seoul 02841, South Korea. His research interests include artificial intelligence and signal processing.
Behnam Tayebi (behnamty@gmail.com) received his Ph.D. degree in applied physics and optics from Yonsei University, Seoul, South Korea, in 2015. He was with Korea University as a research professor, and he is currently working as an optical engineering lead at Inscopix, Mountain View, CA 94043 USA. His current research interests include nanoimaging, holography, and signal processing.
Jae-Ho Han (hanjaeho@korea.ac.kr) received his Ph.D. degree in electrical and computer engineering from Johns Hopkins University, Baltimore, MD, USA. He is a full professor with the Department of Brain and Cognitive Engineering, Korea University, Seoul 02841, South Korea. His current research interests include novel optical imaging technologies and image processing for various fields in biomedicine and neuroscience research. He is a Member of IEEE.
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Digital Object Identifier 10.1109/MSP.2023.3310710