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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 7 — Apr. 4, 2005
  • pp: 2444–2452
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Imaging analysis of digital holography

Lei Xu, Xiaoyuan Peng, Zhixiong Guo, Jianmin Miao, and Anand Asundi  »View Author Affiliations


Optics Express, Vol. 13, Issue 7, pp. 2444-2452 (2005)
http://dx.doi.org/10.1364/OPEX.13.002444


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Abstract

In this study we focus on understanding the system imaging mechanisms given rise to the unique characteristic of discretization in digital holography. Imaging analysis with respect to the system geometries is investigated and the corresponding requirements for reliable holographic imaging are specified. In addition, the imaging capacity of a digital holographic system is analyzed in terms of space-bandwidth product. The impacts due to the discrete features of the CCD sensor that are characterized by the amount of sensitive pixels and the pixel dimension are quantified. The analysis demonstrates the favorable properties of an in-line system arrangement in both the effective field of view and imaging resolution.

© 2005 Optical Society of America

1. Introduction

Digital holography [1

1. U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Measurement Science and Technology 13, R85–R101 (2002). [CrossRef]

] has been attracting intensive research efforts during recent years due to its potentials in automated high-resolution deformation measurement [2

2. J. L. Valin, E. Goncalves, F. Palacios, and J. R. Perez, “Methodology for analysis of displacement using digital holography,” Optics and Lasers in Engineering 43, 99–111 (2005). [CrossRef]

3

3. I. Takahashi, T. Nomura, Y. Morimoto, S. Yoneyama, and M. Fujigaki, “Deformation measurement by digital holographic interferometry,” in Optomechatronic Systems IV, George K. Knopf, eds., Proc. SPIE5264, 206–213 (2003).

] and shape analysis [4

4. J. R. Perez, E. Goncalves, R. De Souza, F. Palacios, M. Muramatsu, J. L. Valin, and R. Gesualdi, “Two-source method in digital holographic contouring,” in 5th Iberoamerican Meeting on Optics and 8th Latin American Meeting on Optics, Lasers, and Their Applications, O. Aristides Marcano and Jose Luis Paz, eds., Proc. SPIE5622, 1422–1427 (2004).

], microscopic imaging and testing [5

5. P. Ferraro, G. Coppola, D. Alfieri, S. De Nicola, A. Finizio, and G. Pierattini, “Recent advancements in digital holographic microscopy and its applications,” in Optical Metrology in Production Engineering, Wolfgang Osten and Mitsuo Takeda, eds., Proc. SPIE5457, 481–491 (2004).

], as well as information transfer, storage and display [6

6. O. Matoba and B. Javidi, “Optical security in data communication and display,” in Optical Information Systems, Bahram Javidi and Demetri Psaltis, eds., Proc. SPIE5202, 68–75 (2003).

], etc. The favorable imaging properties of holography are well utilized in these applications. Furthermore supported by digital recording and numerical evaluation, the imaging performance is advanced in terms of stability tolerance, direct data processing, phase-contrast imaging [7

7. E. Cuche, F. Bevilacqua, and Ch. Depeursinge, “Digital holography for quantitative phase-contrst imaging,” Opt. Lett. 24, 291–293 (1999). [CrossRef]

] and availability of high-quality phase variations. Serving as the basis of these features, analysis of digital holographic imaging is of great importance. In this paper, we study the distinct effects of discretization in digital holography on system imaging. The imaging mechanism and imaging capacity regarding different recording geometries are particularly analyzed. In addition, the impacts induced by a CCD sensor that is characterized by the sensitive pixel amount and pixel dimension to the image formation and quality are quantified.

2. Diffraction and interference analysis

Suppose in the one-dimensional case, a test object has a lateral dimension of LO and a spatial frequency bandwidth of WO . Its space-bandwidth product, denoted as SWO , is then given as SWO =LOWO . In a Fresnel digital holography system, the object beam diffracts at an angle [9

9. Z. L. Yu and G. F. Jin, Optical information processing (Tsinghua University Press, Beijing, 1987).

] α(γO )=λγO , where λ is the working wavelength and γO the spatial frequency of the object. In the hologram plane located at a distance D away from the object, the lateral displacement of the diffraction beam is equal to λDγO . That means that the beam interacting with the object at point xO with a spatial frequency γO will be incident in the hologram plane at the position xH =xO +λDγ O as shown in Fig. 1. The propagation of the object beam can be briefly described in terms of the lateral location and the spatial frequency as

(xo,γo)o(xo+λDγo,γo)H.
(1)
Fig. 1. Propagation sketch of the object beam

In the hologram plane where a CCD sensor is arranged, the superposition of the object wave OH and an in-line reference wave RH produces an intensity distribution that can be expressed by

h(xH)=OH2+RH2+2OHRHcos(2πγOxH).
(2)

Therefore the space-bandwidth product of an in-line hologram is given by

SWH:inline=SWO·(1+λDWOLO).
(3)

Equation (3) represents the requirement on the information capacity of an in-line digital holography system. It should have a space-bandwidth product at least equal to that of the hologram. Otherwise, an object cannot be recovered reliably.

Consider the case of an off-axis arrangement. By offsetting the object at an angle θ along one coordinate axis, say the X-axis, the interference of the object wave with a normally incident reference wave results in a hologram given as

h(xH)=OH2+RH2+2OHRHcos[2π(γO+γθ)xH],
(4)

where, γθ=sinθ/λ is the spatial frequency introduced by the offset angle. It implies that the variation of the SW of an object in the hologram plane occurs in both the lateral dimension and spatial frequency. The procedure can be described as

{xOxH=xO+λDγO,γOγH=[(γO+γθ),0,(γO+γθ)].
(5)

Figure 2 shows the shape sketches of the space-bandwidth product for both of the hologram types. Compared with the more compact feature of an in-line digital hologram, it is seen in the off-axis case that the two image terms shift apart to the central frequencies of ±γθ, respectively. Moreover, their SW shapes are no longer rectangular as that of the original object. Accordingly, CCD sensors that normally have a rectangular SW shape cannot be efficiently utilized.

Mathematically, an off-axis digital hologram has a space-bandwidth product given as

SWH:offaxis=(LO+λDWO)·(WO+2γθ).
(6)

To ensure the spatial frequency components within WO not affected by the quadratic noises, γθ has to satisfy γθ32WO. Therefore, the minimum value of Equation (6) is

SWH:offaxis=4SWO.(1+λDWOLO).
(7)

For a same object, this equation implies that an off-axis holography system must provide a space-bandwidth product that is at least a factor of four greater than that needed in an in-line arrangement so as to include the entire components of SWH .

Fig. 2. Space-bandwidth product of a digital hologram

Comparison of Equation (3) and (7) gives the difference on the requirements for imaging the object information in the two types of system geometries. Originally, the information capacity of an object is an inherent property determined by the spatial dimension and bandwidth of itself. However, when the object is imaged by a digital holography system, constraints are applied to both the object size and the allowable spatial frequency due to the discretization effects. The effective field of view of the system confines the lateral dimension of an object that can be imaged. And the spatial bandwidth that can be transferred reliably is dependent on the system parameters. Therefore, studies on the imaging capacity of a digital holography system are needed by taking into account the different system configurations.

3. Imaging capacity of digital holography systems

In digital holography, whether the information of a test object can be reliably recorded and reconstructed depends on the capability of a system in resolving the micro interference patterns formed by the reference wave and all the point sources over the lateral extension of the object. However, for a CCD sensor that has a limited spatial resolution, the arrangement of the system has to be adjusted accordingly, so as to adapt the resultant fringe spacing to the spatial resolution of the CCD array used.

According to the recording mechanism of Fresnel digital holography [10

10. L. Xu, J. Miao, and A. Asundi, “Properties of digital holography based on in-line configuration,” Opt. Eng. 39, 3214–3219 (2000). [CrossRef]

], it is known that for a test object, there is a minimum recording distance allowed to arrange the object in order to fit the effective field of view of the system. For an in-line system, the effective field of view Lx ×Ly , as show in Fig. 3, is determined by the discrete features of the CCD sensor that are characterized by the pixel amount Nx ×N y and pixel size ΔNx ×ΔNy as

Lx×Ly=(λDΔNxNx·ΔNx)×(λDΔNyNy·ΔNy).
(8)
Fig. 3. Effective field of view of an in-line Fresnel digital holography system

For an off-axis arrangement, since digital imaging of an object has to meet simultaneously both the need of the minimum offset angle and the limitation of the maximum interference angle, the requirement of the recording distance is stricter in comparison with that of an in-line system. The effective field of view is just one-fourth of (Lx ×Ly )in-line.

The distance corresponding to an effective field of view determines the highest achievable lateral resolution of an object with the maximum lateral extension LO inscribed to the rectangular area. It is the spatial bandwidth allowed by the system, which is given below for the two different holographic geometries as

{WO:inline=NLO+N·ΔN,WO:offaxis=N4LO+N·ΔN.
(9)

To evaluate the imaging performance of a digital holography system with the information capacity of an object that can be accepted, the space-bandwidth product in the two system arrangements are

{SWinline=N·LOLO+N·ΔN,SWoffaxis=N·LO4LO+N·ΔN.
(10)
Fig. 4. Imaging capacity of digital holography systems

Equation (10) quantifies the effects given rise to the discretization on digital holographic imaging. The comparison of the two system arrangements is shown in Fig. 4 for the case of N=2048 and ΔN=9µm. In an in-line system, less constraint is applied to the effective field of view, which allows a test object to be imaged with higher resolution, and hence more details can be detected. This property can be utilized for applications in micromeasurement [11

11. L. Xu, X. Peng, J. Miao, and A. Asundi, “Studies of digital microscopic holography with applications to microstructure testing,” Appl. Opt. 40, 5046–5051 (2001). [CrossRef]

12

12. L. Xu, X. Peng, A. Asundi, and J. Miao, “Digital microholointerferometer: development and validation,” Opt. Eng. 42, 2218–2224 (2003). [CrossRef]

], in which imaging quality is essential for evaluating the performance of a metrological system.

4. Discretization effects

The characteristic of discretization in digital holography is introduced by the use of a CCD sensor to sample a hologram with a Nx ×Ny array with sensitive pixels having a finite dimension of ΔNx ×ΔNy .

The size of the sensitive area of a CCD sensor, which is mainly dependent on the amount of the effective pixels Nx ×Ny , plays an important role in Fresnel numerical reconstruction to acquire an image with high lateral resolution. Besides that, Nx ×Ny involves in the determination of the effective field of view of a digital holography system as well. It can be regarded as a system factor evaluating the imaging quality, which can be described below in term of the 2D space-bandwidth product for the two types of recording geometries as

{SWinline=[Nx(Nx·ΔNx)2λD][Ny(Ny·ΔNy)2λD],SWoffaxis=14[Nx(Nx·ΔNx)2λD][Ny(Ny·ΔNy)2λD].
(11)

To show the effect of Nx ×Ny , Fig. 5 illustrates the dependence of SW on the amount of sampling points of a CCD sensor. The involved quantities include λ=532nm and ΔNx ×ΔNy =9µm×9µm. The recording distance D=450mm is shown as an example in the figure, but the general tendency can be observed.

Fig. 5. Effect of sampling amount of CCD sensors

It is seen that the amount of sensitive pixels, to some extent, represents the imaging performance of a digital holographic system. More sampling points help to improve the system resolution and hence achieve higher imaging quality. However, it is noted in the figure that the space-bandwidth product has a maximum for the case of Nx ×Ny =1024×1024 pixels in this example. The reason for it is that a large chip aperture imposes tight constrains to the effective field of view. The integral imaging capacity of the system is degraded in spite of the increase in the lateral resolution. This study reveals the issue of efficient utilization of system capacity. In some applications where just a small recording distance D is allowed, it is possible that only a limited part of a CCD sensor can be used, while the remaining region that does not fulfill the sampling theorem makes no essential contribution to the reconstruction. In an in-line Fresnel system, the effective CCD aperture is given as LS =λD/ΔN-LO .

On the other hand, the simulation of digital sampling of the holograms with discrete points is an idealized situation. In reality, the sampling pulses always have finite area, corresponding to the pixel dimensions. Consider a sample pulse with the extension ΔNx ×ΔNy representing a cell in a CCD chip. The intensity distribution h(xH ,yH ) will then be integrated over the cell. Introducing the rectangular function, the integral over the CCD sensor can be written as

h(xH,yH)=±rect(xHξΔNx)rect(yHηΔNy)h(ξ,η)dξdη
=rect(xHΔNx)rect(yHΔNy)h(xH,yH),
(12)

where ⊗ is the convolution operator. This function is then sampled in the same way as in the ideal case. The reconstructed image wavefield now becomes

U(xI,yI)=(ΔNxΔNy)Sinc(ΔNxxIλD)Sinc(ΔNyyIλD)
·F{h(xH,yH)·exp[jπλD(xH2+yH2)]}
(ΔNxΔNy)comb(ΔNxxIλD)comb((ΔNyyIλD).
(13)

where F{} indicates the 2D Fourier transform.

Fig. 6. Intensity distortion in reconstructed images

It is seen from the equation that the image is modified by a function over the reconstruction area. The influence of the function is on the amplitude term of the wavefield. Therefore, the non-ideal sampling effect induced by the finite pixel size of CCD sensors leads to the distortion of intensity distribution of an image. As shown in the Fig. 6, the intensity at the image corners is reduced to of the values for the ideal case. In practical applications when intensity is concerned, the distortion can be compensated by dividing the respective function from each pixel numerically. Nothing has to be done with it in the interferometric measurements since phase distribution is free from its influence.

The pixel dimension on ΔNx ×ΔNy determines the area of the reconstructed image, as well as the profile of the Sinc function. Studying the distortion gradient of the amplitude over the image area with respect to different pixel size, we obtain the relation curve shown in Fig. 7. It is obvious that CCD sensors having fine sampling pixels help to achieve reconstructed images with more uniform intensity distributions.

Fig. 7. Influence of pixel size on amplitude distortion

5. Conclusions

In conclusion, for the applications of digital holography, especially in the field of micromeasurement, studies on imaging performance are of particular importance. Given rise to the unique characteristic of discretization in digital holography, the system imaging mechanisms are studied in this paper. Through the analysis on the variation of the information capacity of an object during the process of diffraction and interference recording, the imaging requirements, regarding both the in-line and off-axis digital holography geometries, are demonstrated. Moreover, the imaging capacity of a digital holography system is quantitatively evaluated in terms of space-bandwidth product, taking into account the effects due to the discrete features of the CCD sensor that are characterized by the sensitive pixel amount and the pixel size. It is concluded from the analysis that, an in-line system can exhibit better performance, both in terms of a larger effective field of view and a higher imaging resolution.

In studies of the image formation and quality of a digital holography system, the effects introduced by the discretization characteristics of a CCD sensor are discussed. By quantifying the contribution of the amount of sampling pixels to the space-bandwidth product of a system, it is found that, the N, characterizing the CCD chip aperture, describes the information capacity of a system. However, it is worthwhile to mention that imaging performance is an integral quantity of both the effective field of view and imaging resolution. The application of a sensor array needs to coordinate the requirements in both aspects. On the other hand, it is found that the sensitive extension of a sampling pixel contributes to the amplitude distortion of a reconstructed image, in which the four corners are affected most with an intensity decrease to 40.53%.

References and links

1.

U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Measurement Science and Technology 13, R85–R101 (2002). [CrossRef]

2.

J. L. Valin, E. Goncalves, F. Palacios, and J. R. Perez, “Methodology for analysis of displacement using digital holography,” Optics and Lasers in Engineering 43, 99–111 (2005). [CrossRef]

3.

I. Takahashi, T. Nomura, Y. Morimoto, S. Yoneyama, and M. Fujigaki, “Deformation measurement by digital holographic interferometry,” in Optomechatronic Systems IV, George K. Knopf, eds., Proc. SPIE5264, 206–213 (2003).

4.

J. R. Perez, E. Goncalves, R. De Souza, F. Palacios, M. Muramatsu, J. L. Valin, and R. Gesualdi, “Two-source method in digital holographic contouring,” in 5th Iberoamerican Meeting on Optics and 8th Latin American Meeting on Optics, Lasers, and Their Applications, O. Aristides Marcano and Jose Luis Paz, eds., Proc. SPIE5622, 1422–1427 (2004).

5.

P. Ferraro, G. Coppola, D. Alfieri, S. De Nicola, A. Finizio, and G. Pierattini, “Recent advancements in digital holographic microscopy and its applications,” in Optical Metrology in Production Engineering, Wolfgang Osten and Mitsuo Takeda, eds., Proc. SPIE5457, 481–491 (2004).

6.

O. Matoba and B. Javidi, “Optical security in data communication and display,” in Optical Information Systems, Bahram Javidi and Demetri Psaltis, eds., Proc. SPIE5202, 68–75 (2003).

7.

E. Cuche, F. Bevilacqua, and Ch. Depeursinge, “Digital holography for quantitative phase-contrst imaging,” Opt. Lett. 24, 291–293 (1999). [CrossRef]

8.

J. W. Goodman, Introduction to Fourier Optics (The McGraw-Hill Companies, Inc. New York, 1996).

9.

Z. L. Yu and G. F. Jin, Optical information processing (Tsinghua University Press, Beijing, 1987).

10.

L. Xu, J. Miao, and A. Asundi, “Properties of digital holography based on in-line configuration,” Opt. Eng. 39, 3214–3219 (2000). [CrossRef]

11.

L. Xu, X. Peng, J. Miao, and A. Asundi, “Studies of digital microscopic holography with applications to microstructure testing,” Appl. Opt. 40, 5046–5051 (2001). [CrossRef]

12.

L. Xu, X. Peng, A. Asundi, and J. Miao, “Digital microholointerferometer: development and validation,” Opt. Eng. 42, 2218–2224 (2003). [CrossRef]

OCIS Codes
(070.2580) Fourier optics and signal processing : Paraxial wave optics
(090.0090) Holography : Holography
(110.2960) Imaging systems : Image analysis
(120.3940) Instrumentation, measurement, and metrology : Metrology

ToC Category:
Research Papers

History
Original Manuscript: February 11, 2005
Revised Manuscript: March 17, 2005
Published: April 4, 2005

Citation
Lei Xu, Xiaoyuan Peng, Zhixiong Guo, Jianmin Miao, and Anand Asundi, "Imaging analysis of digital holography," Opt. Express 13, 2444-2452 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-7-2444


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References

  1. U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Measurement Science and Technology 13, R85-R101 (2002). [CrossRef]
  2. J. L. Valin, E. Goncalves, F. Palacios and J. R. Perez, �??Methodology for analysis of displacement using digital holography,�?? Optics and Lasers in Engineering 43, 99-111 (2005). [CrossRef]
  3. I. Takahashi, T. Nomura, Y. Morimoto, S. Yoneyama and M. Fujigaki, �??Deformation measurement by digital holographic interferometry,�?? in Optomechatronic Systems IV, George K. Knopf, eds., Proc. SPIE 5264, 206-213 (2003).
  4. J. R. Perez, E. Goncalves, R. De Souza, F. Palacios, M. Muramatsu, J. L. Valin and R. Gesualdi, �??Two-source method in digital holographic contouring,�?? in 5th Iberoamerican Meeting on Optics and 8th Latin American Meeting on Optics, Lasers, and Their Applications, Aristides Marcano O. and Jose Luis Paz, eds., Proc. SPIE 5622, 1422-1427 (2004).
  5. P. Ferraro, G. Coppola, D. Alfieri, S. De Nicola, A. Finizio and G. Pierattini, �??Recent advancements in digital holographic microscopy and its applications,�?? in Optical Metrology in Production Engineering, Wolfgang Osten and Mitsuo Takeda, eds., Proc. SPIE 5457, 481-491 (2004).
  6. O. Matoba and B. Javidi, �??Optical security in data communication and display,�?? in Optical Information Systems, Bahram Javidi and Demetri Psaltis, eds., Proc. SPIE 5202, 68-75 (2003).
  7. E. Cuche, F. Bevilacqua and Ch. Depeursinge, �??Digital holography for quantitative phase-contrst imaging,�?? Opt. Lett. 24, 291-293 (1999). [CrossRef]
  8. J. W. Goodman, Introduction to Fourier Optics (The McGraw-Hill Companies, Inc. New York, 1996).
  9. Z. L. Yu and G. F. Jin, Optical information processing (Tsinghua University Press, Beijing, 1987).
  10. L. Xu, J. Miao and A. Asundi, "Properties of digital holography based on in-line configuration," Opt. Eng. 39, 3214-3219 (2000). [CrossRef]
  11. L. Xu, X. Peng, J. Miao and A. Asundi, "Studies of digital microscopic holography with applications to microstructure testing," Appl. Opt. 40, 5046-5051 (2001). [CrossRef]
  12. L. Xu, X. Peng, A. Asundi, and J. Miao, "Digital microholointerferometer: development and validation," Opt. Eng. 42, 2218-2224 (2003). [CrossRef]

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