Undersampled digital holography - OSA Publishing

Undersampled digital holography Nazif Demoli,1,* Hanan Halaq,2 Kristina Šariri,1 Marc Torzynski,2 and Dalibor Vukicevic2 1 Institute of Physics, Bijenicka cesta 46, PO Box 304, 10001 Zagreb, Croatia Photonics Systems Laboratory, University Louis Pasteur, Bld Sébastien Brant, BP 10413, 67412 Illkirch, France * [email protected]

2

Abstract: 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 OCIS codes: (090.1995) Digital holography; (100.2000) Digital image processing.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9.

10.

11. 12. 13. 14. 15.

E. T. Whittaker, Interpolary Function Theory, Cambridge Tracts in Mathematics and Mathematical Physics, 33 (Cambridge U. P., 1935). C. E. Shannon, "Communication in the presence of noise," Proc. IRE 37, 10-21 (1949). A. Jerri, "The Shannon sampling theorem – its various extensions and applications," Proc. IEEE 65, 1565-1596 (1977). J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996). R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 2000), Chap. 10. U. Schnars and W. P. O. Jüptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002). U. Schnars and W. Jüptner, Digital Holography (Springer-Verlag, 2005). A. Stadelmaier and J. H. Massig, "Compensation of lens aberrations in digital holography," Opt. Lett. 25, 16301632 (2000). L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, "Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram," Appl. Phys. Lett. 90, 041104-1-3 (2007). N. Demoli and D. Vukicevic, "Detection of hidden stationary deformations of vibrating surfaces by use of timeaveraged digital holographic interferometry," Opt. Lett. 29, 1423-2425 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=ol-29-20-2423. G. Pedrini, W. Osten, and M. E. Gusev, "High-speed digital holographic interferometry for vibration measurement," Appl. Opt. 45, 3456-3462 (2006). N. Demoli and I. Demoli, "Dynamic modal characterization of musical instruments using digital holography," Opt. Express 13, 4812-4817 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-13-4812. N. Demoli, D. Vukicevic, and M. Torzynski, "Dynamic digital holographic interferometry with three wavelenths," Opt. Express 11, 767-774 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-7-767. J. M. Desse, P. Picart, and P. Tankam, "Digital three-color holographic interferometry for flow analysis," Opt. Express 16, 5471-5480 (2008). B. Kemper and G. von Bally, "Digital holographic microscopy for live cell applications and technical inspection," Appl. Opt. 47, 52-61 (2008).

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Received 30 Apr 2009; revised 16 Jun 2009; accepted 22 Jun 2009; published 21 Aug 2009

31 August 2009 / Vol. 17, No. 18 / OPTICS EXPRESS 15842

16. N. Demoli, "Real-time monitoring of vibration fringe patterns by optical reconstructing of digital holograms: mode beating detection," Opt. Express 14, 2117-2122 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-6-2117. 17. C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, "Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology," Appl. Opt. 38, 4812-4820 (1999). 18. T. Mishina, F. Okano, and I. Yuyama, "Time-alternating method based on single-sideband holography with half-zone-plate processing for the enlargement of viewing zones," Appl. Opt. 38, 3703-3713 (1999). 19. F. Le Clerc, M. Gross, and L. Collot, "Synthetic-aperture experiment in the visible with on-axis digital heterodyne holography," Opt. Lett. 26, 1550-1552 (2001). 20. T. Mishina, M. Okui, and F. Okano, "Viewing-zone enlargement method for sampled hologram that uses highorder diffraction," Appl. Opt. 41, 1489-1499 (2002). 21. T. Kreis, "Frequency analysis of digital holography," Opt. Eng. 41, 771-778 (2002). 22. J. H. Massig, "Digital off-axis holography with a synthetic aperture," Opt. Lett. 27, 2179-2181 (2002). 23. C. Liu, Z. Liu, F. Bo, Y. Wang, and J. Zhu, "Super-resolution digital holographic imaging method," Appl. Phys. Lett. 81, 3143-3145 (2002). 24. P. Ferraro, D. De Nicola, A. Finizio, G. Coppola, and G. Pierattini, "Recovering image resolution in reconstructing digital off-axis holograms by Fourier-transform method," Appl. Phys. Lett. 85, 2709-2711 (2004). 25. S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, "Synthetic aperture Fourier holographic optical microscopy," Phys. Rev. Lett. 97, 168102-1-4 (2006). 26. V. Mico, Z. Zalevsky, P. Garcia-Martinez, and J. Garcia, "Synthetic aperture superresolution with multiple offaxis holograms," J. Opt. Soc. Am. A 23, 3162-3170 (2006). 27. S. Zhang, "Application of super-resolution image reconstruction to digital holography," EURASIP Journal on Applied Signal Processing 2006, Article ID 90358-1-7 (2006). 28. P. Ferraro, C. Del Core, L. Miccio, S. Grilli, S. De Nicola, A. Finizio, and G. Coppola, "Phase map retrieval in digital holography: avoiding the undersampling effect by lateral shear approach," Opt. Lett. 32, 2233-2235 (2007). 29. Y. Takaki and Y. Hayashi, "Increased horizontal viewing zone angle of a hologram by resolution redistribution of a spatial light modulator," Appl. Opt. 47, 6-11 (2008). 30. P. de Groot and L. Deck, "Three-dimensional imaging by sub-Nyquist sampling of white-light interferograms," Opt. Lett. 18, 1462-1464 (1993). 31. P. Perez and A. Santos, "Undersampling to acquire nuclear magnetic resonance images," Med. Eng. Phys. 26, 523-529 (2004). 32. D. A. Naylor, B. G. Gom, T. R. Fulton, M. K. Tahic, and G. R. Davis, "Increased efficiency through undersampling in Fourier transform spectroscopy," in Fourier Transform Spectroscopy/Hyperspectral Imaging and Sounding of the Environment, Technical Digest (CD) (Optical Society of America, 2005), paper FtuD14, http://www.opticsinfobase.org/abstract.cfm?URI=FTS-2005-FTuD14. 33. N. Demoli, J. Meštrović, and I. Sović, "Subtraction digital holography," Appl. Opt. 42, 798-804 (2003). 34. N. Demoli, D. Vukicevic, and M. Torzynski, "Time-averaged holographic interferometry using subtraction digital holography," Proc. SPIE 5457, 643-650 (2004). 35. N. Demoli, and I. Demoli, "Measuring surface vibrations of musical instruments using an inexpensive digital holography device," Opt. Eng. 44, 0905502-1-3 (2005). 36. N. Demoli, M. Torzynski, and D. Vukicevic, "Enhanced sensitivity digital holographic interferometry," Opt. Express 15, 10672-10680 (2007).

1. Introduction In real optical systems, discretization of signals leads to constrains due to the signal spacebandwidth 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 [16]. 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 [23],

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(iv) padding method [24], and (v) lateral shear approach [28]. 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 [30]. 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 [31]. 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 [32]. 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 We consider a quasi-Fourier digital holography setup [6, 7, 17] in which input amplitude field consists of a reference point source δ (ξ ) with the amplitude R0 and an off-axis signal s (ξ ) , i. e.

ψ (ξ ) = R0δ (ξ ) + s (ξ − b ) ,

(1)

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.

P1 ξ

P 2 x, u

s B

CCD chip b d

β

z

δ Fig. 1. Quasi-Fourier holography configuration. (P1) input plane; (P2) hologram plane.

The amplitude field at a distance d from the input plane is calculated according to the Fresnel approximation, #110829 - $15.00 USD

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ψ ( x ) = ζ ( x ) FT {ψ ( ξ ) ζ ( ξ )} = ζ ( x )  R0 + FT {s (ξ − b ) ζ (ξ )} ,

(2)

where FT denotes the Fourier transform operator, ζ ( x ) = exp i (π λ d ) x 2  the diffraction

quadratic term, λ the wavelength, and where the constant diffraction (1 iλ d ) exp i ( 2π λ ) d  is omitted. According to the Eq. (2), the intensity is given by I ( x ) = DC ( x ) + R0 FT {s (ξ − b ) ζ (ξ )} + CC ( x ) ,

term

(3)

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.

ψ (ξ ) ∝ DC (ξ ) + s (ξ − b ) ζ * ( b ) + s (ξ + b ) ζ ( b ) ,

(4)

and, since there are three spatially separated terms, the modulus is

ψ (ξ ) ∝ DC ( ξ ) + s (ξ − b ) + s (ξ + b ) .

(5)

Evidently, the diffraction quadratic term is of no importance in this configuration [17]. Alternatively, the intensity described by Eq. (3) can be expressed as

{

}

I ( x ) = I 0 ( x ) 1 + V ( x ) cos ϑ ( x ) + 2π uc x + ϕb  ,

(6)

where I 0 ( x ) denotes the average intensity, V ( x ) = 2 K ( x ) 1 + K ( x ) 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 ( λ d ) the carrier frequency, and ϕ b = − π b2 ( λ d ) 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 x I c ( x ) = I ( x ) rect   , L

(7)

 x  1 x ≤ L 2 . where the rect function is defined by rect   =   L  0 otherwise The complex field at the output plane with the coordinate ξ is in the form

ψ c (ξ ) =  DC (ξ ) + s (ξ − b ) ζ * ( b ) + s (ξ + b ) ζ ( b )  ⊗ sinc  L ( λ d ) ξ  ,   −1

(8)

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

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limitation in which for small holograms, i.e. L → 0 , the speckle size, approximately given by ( λ d 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 [16] as:   x   x   x , I d ( x ) =  I ( x ) ⊗ rect    comb   rect    α∆x    ∆x   N ∆x  

(9)

where ∆x denotes the pixel pitch, α the fill factor, N the total number of sensor pixels in one dimension (n), and comb ( x ∆x ) = ∑ n =−∞ δ ( x − n∆x ) . To simplify the analysis we neglect the ∞

influence of the fill factor, which yields  x   x . I d ( x ) = I 0 ( x ) 1 + V ( x ) cos ϑ ( x ) + 2π uc x + ϕb  comb   rect    ∆x   N ∆x 

{

}

(10)

Basically, Eq. (10) describes a sampling (frequency: us = 1 ∆x ) of a bandpass signal (bandwidth: Wo = B ( λ d ) ) centered around the carrier frequency uc . The n-th discrete coordinate value is given by xn = n∆x , while the Nyquist frequency for the system is equal uN = us 2 = 1 ( 2∆x ) . 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 + ∆uc , where m is an integer and ∆uc = uc − mus the frequency residuum which satisfies the properties: (i) ∆uc us < 1 and (ii) ∆uc = ∆b ( λ d ) , ∆b being the carrier distance residuum. Thus, for the discrete case follows: 2π uc n∆x = 2π ( m + ∆uc us ) n . Inserting this into Eq. (10) and performing the reconstruction yields: ψ d ( k ∆ξ ) ∝  DC ( k ∆ξ ) + s ( k ∆ξ − k∆b ∆ξ ) ζ * ( k∆b ∆ξ ) + s ( k ∆ξ + k∆b ∆ξ ) ζ ( k∆b ∆ξ ) 

⊗sinc  N ∆x ( λ d ) k ∆ξ  ⊗ comb  ∆x ( λ d ) k ∆ξ  ,     −1

−1

(11)

where k ∆ξ denote a discrete value of the output coordinate and k∆b ∆ξ 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∆b ∆ξ can differ from the location kb ∆ξ 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

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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 ): us Wo   us 1 i 2 + 2 (WDC + Wo ) , ( i + 1) 2 − 2  , i = 0, 2, 4, ... ( even )  

(12a)

us 1  us Wo  i 2 + 2 , ( i + 1) 2 − 2 (WDC + Wo ) , i = 1, 3, 5, ... ( odd )  

(12b)

where WDC ≤ 2W0 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 ⋅ λ d , while the intervals for the angle between the object and reference beams ( β ) are:  −1  3B  B −1   tan  iξ N + 2d  , tan  ( i + 1) ξ N − 2d    

   , i = 0, 2, 4, ... ( even ) 

 −1  B  3B   −1   tan  iξ N + 2d  , tan  ( i + 1) ξ N − 2d   , i = 1, 3, 5, ... ( odd )     

(13a)

(13b)

where ξ N = uN ⋅ λ d 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, us Wo   us 1 i 2 + 2 Wo , ( i + 1) 2 − 2  , i = 0, 2, 4, ... ( even )  

(14a)

us 1   us Wo i 2 + 2 , ( i + 1) 2 − 2 Wo  , i = 1, 3, 5, ... ( odd )  

(14b)

and the corresponding angle intervals,  −1  B  B  −1   tan  iξ N + 2d  , tan  ( i + 1) ξ N − 2d   , i = 0, 2, 4, ... ( even )     

(15a)

 −1  B  B −1   tan  iξ N + 2d  , tan  ( i + 1) ξ N − 2d    

(15b)

   , i = 1, 3, 5, ... ( odd ) 

are thus significantly extended.

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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,

φ=

λd 2∆xB

,

(16)

shows the phase value in the maximum interval reached by the signal bandwidth. 3. Experiment

3.1 Setup 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 [33].

M ML L1

M VA

LASER BS

M M

Ob.

M SF

L2

CCD

Fig. 2. Scheme of the experimental setup. (VA) variable attenuator; (M) mirrors; (BS) beam splitter; (SF) spatial filter; (ML) micro lens; (L1) cylindrical lens; (L2) lens; (Ob.) object.

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Fig. 3. Photograph of the membrane.

3.2 Parameters The given experimental parameters of Fig. 2 are: λ = 0.647 µ m (Krypton ion laser); ∆x = 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

β N = 4.0 deg and the lateral distance is ξ N = d tan ( β N ) = 73.6 mm . For the membrane center to the reference point distance, b, the minimum and maximum values in the zeroth interval are bmin = 48.0 mm and bmax = 57.6 mm , respectively, resulting in an useful width of 9.6 mm. Applying the subtraction technique the minimum value becomes bmin = 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 #110829 - $15.00 USD

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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 ξ N and the angle β N , 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.

Fig. 4. Effects of increasing lateral distance of the oscillating membrane (Media 1).

Fig. 5. The reconstruction images demonstrating the 0th interval (upper row), 1th interval (middle row), and 2nd interval (lower row). The red arrow denotes the shift direction, while the minus sign denotes inversion.

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P1

P2

ξ (mm)

-1

u (mm )

294.4

430.0

rd

3 interval

220.8

322.5

+ nd

2 interval

147.2

15.6

215.0

o

o

st

11.8

1 interval

73.6

8.0

o

107.5

+ th

0 interval 0.0

4.0

o

d = 1056 mm

0.0

Fig. 6. Limits of the full interval defined by the experimental conditions.

P1

P2

ξ (mm)

-1

u (mm )

294.4

430.0

246.4 236.8

360.7 346.6

220.8

322.5

204.8 195.2

299.8 285.7

147.2

215.0 145.2 131.1

99.2 89.6

73.6

107.5 84.3 70.3

57.6 48.0

0.0

d = 1056 mm

0.0

Fig. 7. Limits for the non-overlapping intervals obtained by introducing the object (a membrane of a diameter 32 mm) into analysis.

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P1

P2

ξ (mm)

294.4

430.0

278.4

407.5

236.8

346.6

220.8

322.5

204.8

299.8

163.2

238.9

147.2

215.0

131.2

192.0

89.6

131.1

73.6

107.5

57.6

84.3

16.0

23.2

0.0

-1

u (mm )

d = 1056 mm

0.0

Fig. 8. Same as in Fig. 7, but with the use of the subtraction method.

5. Conclusion

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. Acknowledgments

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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(C) 2009 OSA

