Acceptable signal recovery of the band-pass signals typically used in the off-axis digital holography systems is possible in the undersampling conditions. A typical system is considered in which the angle between two beams represents a variable parameter. For the given signal bandwidth and experimental conditions the hologram reconstruction is constrained by the sampling frequency of the array photo-detector. Reconstructions from the undersampled digital holograms are analyzed both theoretically and experimentally. It is shown how increasing the angle values beyond the Nyquist limits leads to repeatedly folding and inverting the reconstructed object image until the fading of the image. The phase point at the image fading and the non-overlapping intervals for correctly preserving the useful information are defined and evaluated. Amplitude distributions are analyzed on the example of the time-averaged holograms acquired for an oscillating membrane. Based on removing the zeroth-order reconstruction term, significant extensions of these intervals are also demonstrated.
©2009 Optical Society of America
In real optical systems, discretization of signals leads to constrains due to the signal space-bandwidth product. According to the sampling theorem, the minimum required sampling frequency must be twice the signal bandwidth in order to avoid the loss of information [1, 2]. Otherwise, undersampling occurs resulting generally in frequency folding or aliasing errors [3–5]. Typical example is digital holography [6, 7], where input information is formed optically while output is generated numerically, and where holograms are images (unlike classical holograms) acquired by the array photo-detectors and processed by computers. On one hand, these images have resolution limitations imposed by the detector properties. On the other hand, the advantages of fast and comfortable hologram acquiring permit versatile applications such as compensation of lens aberrations [8, 9], vibration analysis [10–12], flow analysis [13, 14], microscopic imaging [9, 15], or real-time monitoring . The drawback of possessing a narrow viewing zone imposes the known sampling limitations [17–29]. To avoid these limitations, various techniques have been introduced: (i) enlarging the viewing zone [18, 20, 29], (ii) using the synthetic apertures [19, 22, 25–27], (iii) using diffraction gratings , (iv) padding method , and (v) lateral shear approach . However, with growing application demands, the undersampling experimental conditions in digital holography for acceptable signal recovery must be also considered.
Information retrieval in the undersampling conditions has been generally investigated. For example, by undersampling the interference data and processing the resultant sub-Nyquist interferograms the data acquisition and processing speed have been increased in a scanning white-light interferometer . Similarly, a use of undersampling in the acquisition of NMR signals permits correct reconstruction of images reducing the storage and speed requirements of the data acquisition system . In another example, by reducing the sampling rate in Fourier transform spectroscopy, the scanning time can be appropriately decreased and thus the spectra obtained with good integrity and noise reduction .
In the off-axis digital holography systems, the input bandwidth is defined by the angle between beams (carrier frequency) and the spatial width of the object, while the output reconstruction is constrained by the resolution of the array photo-detector. Thus, in the typical system the experimental parameters are adjusted to satisfy the sampling theorem. Other systems unable to maintain the sampling conditions either due to the configuration constraints or due to changing the sampling conditions during investigation of the fast moving objects, are of interest too. In this paper, we analyze the hologram reconstructions particularly when the sampling theorem is not satisfied introducing the appearance of aliased frequencies. In Sec. 2, the formation of reconstructed image in a quasi-Fourier off-axis digital holography setup and the undersampling effects are described, where one-dimensional case is considered for simplicity. The experimental system, results, and conclusions are presented in Sections 3, 4, and 5, respectively.
2. Theoretical analysis
2.1 Quasi-Fourier setup
where the carrier distance b can be varied along the ξ axis. The scheme of such a setup is shown in Fig. 1, where B denotes the spatial width of s(ξ) and the arrows denote direction of the reference and object wave fronts.
The amplitude field at a distance d from the input plane is calculated according to the Fresnel approximation,
where FT denotes the Fourier transform operator, ζ(x)=exp[i(π/λd)x 2] the diffraction quadratic term, λ the wavelength, and where the constant diffraction term (1/iλd)exp[i(2π/λ)d] is omitted. According to the Eq. (2), the intensity is given by
where the DC(x) and CC(x) denote two diffraction terms, the zero-order diffraction term and the complex conjugate of its preceding term, respectively. Since the hologram reconstruction follows by calculating the modulus of the inverse Fourier transform of the Eq. (3), obviously the reconstructed field consists of the DC term with two object images symmetrically located in respect to the central point of the DC term, i.e.
and, since there are three spatially separated terms, the modulus is
Evidently, the diffraction quadratic term is of no importance in this configuration .
Alternatively, the intensity described by Eq. (3) can be expressed as
where I0(x) denotes the average intensity, the depth of modulation of the cosine grating, K(x) the intensity ratio of the signal and reference beams, ϑ(x) the object phase, uc=b/(λd) the carrier frequency, and φb=-πb 2/(λd) the phase shift (originating from the diffraction quadratic term). Equation (6) demonstrates the main characteristics of the quasi-Fourier digital hologram: (i) there are two symmetrically located reconstructions, (ii) the locations of the reconstructed object images are defined by the carrier frequency uc, (iii) the phase shift φb has no influence on the output.
2.2 Continuous recording and reconstructing (classical holography case)
In the classical holography case, a continuous type of detector of a size L (such as holographic photo-material) is used at P2 yielding the intensity
where the rect function is defined by .
The complex field at the output plane with the coordinate ξ is in the form
which demonstrates, in general, that: (i) the continuous recording and reconstructing impose no limitations to the distance b, and (ii) the finite hologram size impose the resolution limitation in which for small holograms, i.e. L→0, the speckle size, approximately given by (λd/L), becomes large and consequently the smoothing becomes significant.
2.3 Discrete recording and reconstructing (digital holography case)
The intensity captured by a discrete sensor at P2 (such as the CCD sensor) can be described  as:
where Δx denotes the pixel pitch, α the fill factor, N the total number of sensor pixels in one dimension (n), and comb(x/Δx)=∑∞ n=-∞ δ(x-nΔx). To simplify the analysis we neglect the influence of the fill factor, which yields
Basically, Eq. (10) describes a sampling (frequency: us=1/Δx) of a bandpass signal (bandwidth: Wo=B/(λd) centered around the carrier frequency uc. The n-th discrete coordinate value is given by xn=nΔx, while the Nyquist frequency for the system is equal uN=us/2=1/(2Δx).
In this analysis, the variable parameter is the distance of the object from the reference point source (b), and consequently the carrier frequency (uc) and the average angle between the reference and object beams: β=tan-1(b/d). Generally can be written: uc=mus+Δuc, where m is an integer and Δuc=uc-mus the frequency residuum which satisfies the properties: (i) Δuc/us<1 and (ii) Δuc=Δb/(λd), Δb being the carrier distance residuum. Thus, for the discrete case follows: 2πucnΔx=2π(m+Δuc/us)n. Inserting this into Eq. (10) and performing the reconstruction yields:
where kΔξ denote a discrete value of the output coordinate and k ΔbΔξ the location of the reconstructed image of the object. Now, Eq. (11) describes the reconstruction substantially different from the one described by Eq. (8), since the location k ΔbΔξ can differ from the location kbΔξ that would be achieved with an infinite discrete sensor (N→∞).
2.4 Shifting of the reconstructed image of the object
Analysis can be restricted to one half of the output reconstruction frame, i.e. to only one reconstructed image of the object. If we take, for example, right half, then the DC point is located at the left edge of the new frame. Thus, by increasing the distance between the object and the reference point source (b), the reconstructed image at the output of the system is shifting to the right until it reaches the end of the frame (0th interval), then it is folding back (inverted) and shifting toward the frame centrum (1st interval). After reaching the DC point, the image is folding again (doubly inverted=normal) and shifting to the right (2nd interval), etc. Generally, shifting to the right will take place in the even intervals (normal image), while shifting to the left will take place in the odd intervals (inverted image). The overlappings appear during image foldings (at both ends of the frame) and during the image passing across the zeroth diffraction order (at the left end of the frame).
2.5 Non-overlapping intervals
To preserve the complete input object information, we define the non-overlapping intervals for the carrier frequency values (uc):
where WDC≤2W 0 denotes the bandwidth of the zeroth diffraction order. The corresponding intervals for the lateral distance b can be easily calculated from the relation b=uc·λd, while the intervals for the angle between the object and reference beams (β) are:
where ξN=uN·λd is the maximum lateral distance defined by the sampling theorem.
2.6 Non-overlapping intervals applying the subtraction method
The subtraction method was introduced [33, 34] as one amongst several procedures for suppressing the zero-order term in digital holography. The method is based on successive recording and numerical subtracting of two stochastically changed digital holograms of the object in the same physical state. As a result, the hologram reconstruction is generally of lower intensity and with the zero-order term suppressed. The method demonstrated effectiveness also in the time-averaged [10, 34, 35] and interferometric [10, 36] areas of digital holography.
Applying the subtraction method, the non-overlapping intervals for the carrier frequency,
and the corresponding angle intervals,
are thus significantly extended.
2.7 Phase point of the maximum cycle
The phase point ϕ of the maximum cycle, defined as a ratio of the highest spatial frequency resolved by the CCD sensor and the object bandwidth,
shows the phase value in the maximum interval reached by the signal bandwidth.
For experimental measurements we used a krypton-ion laser (wavelength=647 nm) as light source and a CCD sensor (1392×1040 square pixels of 4.65 µm pitch) as light detector. We aligned a quasi-Fourier off-axis setup schematically shown in Fig. 2. A piezoelectric loudspeaker’s membrane (32 mm diameter), shown in Fig. 3, served as an input object. The membrane was mounted on one electronically controlled long driving distance translator. A signal of 4200 Hz frequency and 1 V amplitude was applied to the membrane to induce a stationary vibration mode. Time-averaged holographic interferogram of the membrane yields a fringe pattern that is considered as input object information. Digital holograms are recorded sequentially step by step, increasing each time by 1 mm the distance of the membrane from the reference point source. The exposure time of the CCD sensor was 25 ms. To be able to apply the subtraction method, two digital holograms are captured for each object location, then stored in the computer memory, and later subtracted using a self-written program. These practical steps of the subtraction procedure are simple to apply .
The given experimental parameters of Fig. 2 are: λ=0.647 µm (Krypton ion laser); Δx=4.65 µm and N=1040 pixels (CCD sensor); B=32 mm (membrane diameter); and d=1056 mm (distance between the object and CCD sensor). It is easy to calculate: the sampling frequency, us=215.0 mm-1, the Nyquist frequency (the highest spatial frequency resolved by the CCD sensor), uN=107.5 mm -1, the object bandwidth, WB=46.8 mm -1, and the DC term bandwidth, WDC≈93.6 mm -1. At the Nyquist frequency, the angle is βN=4.0 deg and the lateral distance is tan ξN=d tan (βN)=73.6 mm. For the membrane center to the reference point distance, b, the minimum and maximum values in the zeroth interval are b min=48.0 mm and b max=57.6 mm, respectively, resulting in an useful width of 9.6 mm. Applying the subtraction technique the minimum value becomes b min=16.0 mm, increasing this width to 41.6 mm.
4. Results and discussion
The membrane reconstructions are obtained by: (i) calculating the Fourier transform of the recorded digital holograms, (ii) taking the square modulus of the obtained Fourier transform, and (iii) cutting a window from the center to the edge of the right half of the reconstruction frame containing one object reconstruction. These images are then multiplied (pointwise multiplication) by a constant factor and finally used for composing a movie. The movie (Fig. 4) shows the translation of the membrane along the horizontal line, clearly demonstrating the undersampling concept applied to a band-limited signal in digital holography. Some of these images are presented in Fig. 5. The first three images of each row of Fig. 5 illustrate shifting of the reconstructed image either to the right: upper row (16.0 mm≤b≤57.6 mm) and lower row (163.2 mm≤b≤204.8 mm) or to the left: middle row (89.6 mm≤b≤131.2 mm). The last image in each row demonstrates folding and inverting of the reconstructed image, where the parameter b is: b=73.6 mm (upper row), b=147.2 mm(middle row), and b=220.8 mm (lower row). The contrast of the fringe patterns in Fig. 5, calculated for the zeroth maximum and the first minimum, varies between 0.5 and 0.6 in the upper and middle rows. In the lower row the fringe contrast decreases to the 0.3 for the last image. The lateral resolution shows similar behavior. The pixel dimension of the reconstructed image is equal to 141.5 µm.
The phase point ϕ of the maximum cycle, calculated according to the Eq. (16), is 2.3, which includes a total of four intervals into analysis. The results for the full interval limits (their minimum and maximum values) in terms of the object lateral distance, angle, and frequency are presented in Figs. 6–8. The theoretical values calculated from the parameters of the Sec. 3.2 showed good agreement with the experimentally obtained values. Fig. 6 shows the extension of the full intervals defined by the experimental conditions (or Nyquist frequency). Evidently, the 0th interval is limited by the lateral distance ξN and the angle βN, while the object parameters are not included into this calculations. By introducing the object at a lateral distance b from the optical axis, the non-overlapping intervals can be determined. These intervals are sketched Fig. 7. Evidently, only narrow carrier values are available for the experimental use. By applying the subtraction technique, the carrier values are considerably widened, as demonstrated in Fig. 8.
Basically, the off-axis digital holography systems are the systems with band-pass inputs and outputs constrained with the sampling frequency of the array photo-detector. For the given parameters such as the useful input information and the detector characteristics, the recording conditions involving the carrier frequency are optimized to meet the sampling theorem requirements. In this paper, the effects of increasing the angle between the reference and object beams beyond the Nyquist limits on the hologram reconstruction are analyzed. To describe the repeatedly folding and inverting the reconstructed image of the object until the image fades out, two parameters are introduced: (i) the phase point of the maximum cycle, defined as the ratio of the highest spatial frequency resolved by the array photo-detector and the object bandwidth, and (ii) the non-overlapping intervals for correctly preserving the useful information. It is also demonstrated that the non-overlapping intervals can be significantly extended by applying the subtraction digital holography method. Although the use of the theoretical analysis is demonstrated on an example of time-averaged digital holography, we expect that the main principles could be applied more generally.
This work was supported by Region Alsace through convention “Bourse régional de valorization 2006”, (N° 06/916/270) and Université de Strasbourg, bénéficiere H. Halaq, and by the Croatian Ministry of Science, Education and Sports (project No. 035-0352851-2854).
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