## Influence of virtual images on the signal-to-noise ratio in digital in-line particle holography

Optics Express, Vol. 13, Issue 7, pp. 2578-2589 (2005)

http://dx.doi.org/10.1364/OPEX.13.002578

Acrobat PDF (539 KB)

### Abstract

A theoretical analysis describing the dependence of the signal-to-noise ratio (SNR) on the number of pixels and the number of particles is presented for in-line digital particle holography. The validity of the theory is verified by means of numerical simulation. Based on the theory we present a practical performance benchmark for digital holographic systems. Using this benchmark we improve the performance of an experimental holographic system by a factor three. We demonstrate that the ability to quantitatively analyze the system performance allows for a more systematic way of designing, optimizing, and comparing digital holographic systems.

© 2005 Optical Society of America

## 1. Introduction

1. K. D. Hinsch, “Holographic particle image velocimetry,” Meas. Sci. Technol. **13**, R61–R72 (2002). [CrossRef]

2. U. Schnars and W. Juptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. **33**, 179–181 (1994). [CrossRef] [PubMed]

3. J. M. Coupland, “Holographic particle image velocimetry: signal recovery from under-sampled CCD data,” Meas. Sci. Technol. **15**, 711–717 (2004). [CrossRef]

4. M. Malek, D. Allano, S. Coetmellec, and D. Lebrun, “Digital in-line holography: influence of the shadow density on particle field extraction,” Opt. Expr. **12**, 2270–2279 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2270 [CrossRef]

*et al*modeled the SNR for in-line holography [5

5. H. Meng, W. L. Anderson, F. Hussain, and D. Liu, “Intrinsic speckle noise in in-line particle holography,” J. Opt. Soc. Am. A **10**, 2046–2058 (1993). [CrossRef]

6. Y. Pu and H. Meng, “Intrinsic speckle noise in off-axis particle holography,” J. Opt. Soc. Am. A **21**, 1221–1230 (2004). [CrossRef]

*et al*were able to extend the work of Pu and Meng to digital holography [7

7. H. Meng, G. Pan, Y. Pu, and S. H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Tech. **15**, 673–685 (2004). [CrossRef]

*I*

_{sig}is the particle intensity, <

*I*

_{N}> is the mean speckle noise intensity,

*P*is the number of pixels, and

*N*is the number of particles. In their analysis Meng

*et al*ignored the effects due to virtual particle images and hologram speckle noise. Although the waves associated with these two noise sources may be spatially separated in optical off-axis holography, doing so in digital in-line holography is not trivial. In this study we present an analysis of the SNR based on the properties of the digital in-line holographic process. We relate the SNR in in-line digital particle holography to the number of pixels, the number of particles, and the ratio between reference and object intensity. We will do this for point particles that all lie in the same plane. By doing so we have isolated the problem from the already studied effect due to out-of-focus real particle images. Also we assume that the holograms are recorded using a CCD with infinite dynamic range.

## 2. Numerical propagation

8. J. H. Milgram and W. Li, “Computational Reconstruction of Images from Holograms,” Appl. Opt. **41**, 853–864 (2002). [CrossRef] [PubMed]

*z*=0) is denoted by

*p*

_{h}

*(x,y;z=0)*then for the reconstructed complex amplitude in an arbitrary plane

*z*we find:

*p*

_{r}is normally found by:

## 3. Theoretical SNR

9. J. W. Goodman, “Film grain noise in wavefront-reconstruction imaging,” J. Opt. Soc. Am. **57**, 493–502 (1967). [CrossRef] [PubMed]

*I*

_{sig}is the intensity of a particle image at an arbitrary location and <

*I*

_{N}> is the mean intensity of the noise around this location. If a SNR of 5 is needed to distinguish particle images from the background it is found that [

*I*

_{sig}

*/<I*

_{N}

*>]*

_{min}≈50 [7

7. H. Meng, G. Pan, Y. Pu, and S. H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Tech. **15**, 673–685 (2004). [CrossRef]

*I*

_{sig}

*/<I*

_{N}

*>*largely depend on the type of processing that is used to analyze the reconstructed particle field. Correlation type processing is recommended to maintain low values for [

*I*

_{sig}

*/<I*

_{N}

*>*]

*min*, and correlation of complex amplitude is inherently very robust [10

10. J. M. Coupland and N. A. Halliwell, “Holographic displacement measurements in fluid and solid mechanics: immunity to aberrations by optical correlation processing,” Proc. R. Soc. Lond. A **453**, 1053–1066 (1997). [CrossRef]

*I*

_{sig}

*/<I*

_{N}

*>*in this paper. Once this ratio is known the SNR can be easily found. We start by analyzing the signal intensity

*I*

_{sig}, and consider all particles to behave as point source scatterers, emitting a spherical wavefront. At the plane of the CCD the light scattered from the particle has real amplitude

*A*

_{p}, and the reference wave has real amplitude

*A*

_{r}. It is assumed that

*A*

_{p}is constant over the CCD; this is valid when the CCD size is small compared to the distance between the CCD and the particle field. Using for the intensity of a wave that

*I=A*

^{2}, we find that the modulated intensity due to a single particle at the time of recording is

*M*=(

*A*

_{r}

*+A*

_{p})

^{2}-(

*A*

_{r}

*- A*

_{p})

^{2}=4

*A*

_{r}

*A*

_{p}, where for now it has been assumed that each particle interferes only with the reference beam. If we denote the total number of particles by

*N*, and express the ratio between the reference beam intensity and the total object intensity as

*R*=

*P*pixels. It is easily shown that for the intensity of the

*n*-th pixel we may write

*I*

_{n}

*=C*+0.5

*M*(1+

*cos*(

*φ*

_{n}

*-φ*

_{o})), where

*C*is a constant,

*φ*

_{n}=(

*r*

_{n}

*/λ*)2

*π, φ*

_{o}is the reference phase. Without any loss of generality we may choose

*φ*

_{o}=0. The interference pattern is recorded using a digital camera, and stored for later processing. During the numerical reconstruction the recorded intensity pattern is used as the hologram amplitude transmission. The location of the hologram with the maximum recorded intensity

*I*

_{max}is assigned maximum (unity) transmission. For the amplitude transmission of the

*n*-th pixel we thus find:

*P*pixels in the plane

*H*will now be used to reconstruct the complex amplitude, and thereby the intensity, in

*P*pixels on plane

*F*around

*O*. The numerical aperture (NA) of the hologram is assumed sufficiently high to allow for the reconstructed particle image to be contained in a single pixel. The next section of this paper will discuss when this assumption is valid.

*O*(the original particle location) due to the

*n*-th pixel it can be found that:

*A*is the amplitude of a secondary point source at unit distance from the source. Because the reconstruction is carried out within the computer all quantities may be expressed in arbitrary units. From Eq. (8) follows for the reconstructed amplitude in

*O*due to all pixels:

*φ*

_{n}can be considered to be uniformly distributed over the interval [0, 2π]. The summation in Eq. (9) then reduces to:

*r=r*

_{n}is constant for all

*n*(extent of hologram small compared to reconstruction distance). Obviously the reconstructed intensity in

*O*corresponds to the intensity of the reconstructed particle, and therefor forms the signal intensity

*I*

_{sig}. Combining Eq. (6) and Eq. (10) yields for the signal intensity:

*Ω*is the pixel area.

*η*for all orders other than zero is given by:

*t(x,y)*denotes the hologram amplitude transmittance and

*σ*

_{t}2 the variance of the amplitude transmission. The variance of the intensity at the time of the recording can be shown to be [11]:

*I*

_{N}>=

*F*is found when in plane

*H t*

_{n}=1 for all

*n*. In this case the total power in plane

*F*is found to be equal to

*P*

^{2}

*ΩA*

^{2}

*/r*

^{2}. For the total diffracted power in plane

*F*we thus obtain:

*N*particle images (all -1 orders contained in the hologram) thus follows:

*PΩ*, the mean background intensity is now easily found:

*P*≫0,

*N*≫0, and

*R*≥1. When comparing our result (Eq. (20)) with that of Meng

*et al*(Eq. (1)) a great similarity in the expression for Isig/<IN> is found. Essentially Meng

*et al*considered all the energy associated with the real particle images to constitute a noise background. Because the energy contained in the virtual images should be equal to the energy contained in the real images, also the noise associated with the virtual images should be similar. Our study also reveals the dependence of the hologram speckle noise on the reference intensity ratio R.

## 4. Numerical simulation

*P=m*x

*m*pixels, each pixel having width

*w*. The wavelength is

*w*/15, this corresponds to 532 nm for an 8 µm pixel, which are typical values for the illumination wavelength and CCD pixel dimension. The particle field is positioned at a distance 15

*wm*from the hologram. At this distance the interference pattern due to a particle in the center of the field still meets the Nyquist sampling criterion. Furthermore this NA allows the particles to be reconstructed as point source particles because the pixels surrounding the actual particle location lie in or beyond the first null of the Airy disk.

*N*-1 particles are positioned randomly. However, these particles are not allowed to be placed within a 30-by-30 pixel box located in the center of the particle field. This box is reserved to perform the signal and noise measurements. To measure the signal intensity a particle is placed in the center of the box. After the hologram has been synthesized and the numerical reconstruction has been performed the intensity of this particle

*I*

_{sig}can be measured, and the particle image is guaranteed free from overlap with other images. In a next measurement the box remains empty. After the numerical reconstruction has been performed the mean intensity of this empty box then gives an accurate measure for the background intensity <

*I*

_{N}>.

*(x*

_{j}

*, y*

_{j}

*)*is the location of the

*j*-th particle, and z is the distance between the hologram and the particle field. If the mean intensity due to the particle field is denoted by <

*I*

_{p}>, we find after interference of the particle wave with the reference beam for the intensity at the hologram:

*N*=[50, 250, 500, 1000],

*P*=[10000, 22500, 40000], and

*R*=[1, 2, 5, 10, 100, 1000]. For every combination of these parameters 200 synthetic holograms were created and reconstructed; 100 to determine the signal and 100 to determine the noise. It turned out that 100 realizations are sufficient to reach converged statistics. The results obtained after simulation of 14400 particle fields are shown in Fig. 3. In this figure the numerically obtained values for

*I*

_{sig}/<

*I*

_{N}> are plotted against the theoretical expectation, and as such the data is expected to fall on the solid line. Clearly there is very good correspondence between the theory and the simulation, thereby supporting the validity of Eq. (20). The dashed line in Fig. 3 indicates the practical rule of thumb that [

*I*

_{sig}/<

*I*

_{N}>]

*min*≈50 [7

7. H. Meng, G. Pan, Y. Pu, and S. H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Tech. **15**, 673–685 (2004). [CrossRef]

## 5. Experimental benchmark

*et al*. This is in line with our expectations when one assumes real and virtual images to contain equal amounts of energy. When combining the work of Meng et al with our current study we are able to construct a practical guide for the maximum value of

*I*

_{sig}/<

*I*

_{N}> in digital particle holography:

*κ*

_{r}=0 if all particles lie in a plane,

*κ*

_{r}=1 if the particles are in a volume having a depth that is much larger than the depth of focus of a particle image,

*κ*

_{v}=0 if the virtual particle images are spatially separated or numerically suppressed, and

*κ*

_{v}=1 if the virtual particle images are present in the reconstructed field.

*I*

_{sig}/<

*I*

_{N}> to the theoretical maximum describes the system performance

*Π*:

*L1*. The object beam then illuminates 500 particles with a 60 µm diameter that are fixed in a resin-filled tank. Using lenses

*L2*and

*L3*and a beam dump, the undiffracted light is filtered. At the second polarizing beam splitter the object beam and reference beam are recombined. Using a polarizer that is oriented at 45° with respect to the horizontal axis the object and reference beams are given identical polarization. The interference pattern that is formed is recorded using a 12-bit CCD camera with 1024×1024 pixels (Imager Intense, LaVision). It was determined that

*R*=3. Because both out of focus real particle images and virtual images will be present in the numerical reconstruction we choose

*κ*

_{r}=1 and

*κ*

_{v}=1.

*m*x

*m*pixels of the hologram are used, where m ranges between 200 and 1000. By determining

*I*

_{sig}/<

*I*

_{N}> of the digitally reconstructed images the performance of the system is benchmarked for varying hologram size. In Fig. 5 it can be seen that for a hologram size of 300×300 pixels the system performs nearly at 40% of its theoretical maximum. With increasing hologram size the performance drops to

*Π*=0.05 for a hologram of 1000×1000 pixels. This drop in performance may be due to various effects. Thus far, the particle scattering has been assumed isotropic. The drop in performance for

*m*>300 could be an indication that the scattered light at the plane of the CCD due to a single particle is confined to an area having a diameter of roughly 300 pixels, rather than covering the entire CCD array as would be expected in the case of isotropic scattering. The drop in performance may also be due to phase-fluctuations in the reference beam that become more significant over a larger area.

## 6. Conclusion

*I*

_{sig}/<

*I*

_{N}> was in very good correspondence with the values for

*I*

_{sig}/<

*I*

_{N}> obtained by numerical simulation of 14400 particle fields.

**15**, 673–685 (2004). [CrossRef]

## Acknowledgments

## References and Links

1. | K. D. Hinsch, “Holographic particle image velocimetry,” Meas. Sci. Technol. |

2. | U. Schnars and W. Juptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. |

3. | J. M. Coupland, “Holographic particle image velocimetry: signal recovery from under-sampled CCD data,” Meas. Sci. Technol. |

4. | M. Malek, D. Allano, S. Coetmellec, and D. Lebrun, “Digital in-line holography: influence of the shadow density on particle field extraction,” Opt. Expr. |

5. | H. Meng, W. L. Anderson, F. Hussain, and D. Liu, “Intrinsic speckle noise in in-line particle holography,” J. Opt. Soc. Am. A |

6. | Y. Pu and H. Meng, “Intrinsic speckle noise in off-axis particle holography,” J. Opt. Soc. Am. A |

7. | H. Meng, G. Pan, Y. Pu, and S. H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Tech. |

8. | J. H. Milgram and W. Li, “Computational Reconstruction of Images from Holograms,” Appl. Opt. |

9. | J. W. Goodman, “Film grain noise in wavefront-reconstruction imaging,” J. Opt. Soc. Am. |

10. | J. M. Coupland and N. A. Halliwell, “Holographic displacement measurements in fluid and solid mechanics: immunity to aberrations by optical correlation processing,” Proc. R. Soc. Lond. A |

11. | J. W. Goodman, “Statistical properties of laser speckle patterns,” |

**OCIS Codes**

(090.0090) Holography : Holography

(090.1760) Holography : Computer holography

(100.2000) Image processing : Digital image processing

(100.3010) Image processing : Image reconstruction techniques

(100.6890) Image processing : Three-dimensional image processing

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 2, 2005

Revised Manuscript: March 21, 2005

Published: April 4, 2005

**Citation**

W. Koek, N. Bhattacharya, J. Braat, T. Ooms, and J. Westerweel, "Influence of virtual images on the signal-to-noise ratio in digital in-line particle holography," Opt. Express **13**, 2578-2589 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-7-2578

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### References

- K. D. Hinsch, �??Holographic particle image velocimetry,�?? Meas. Sci. Technol. 13, R61-R72 (2002). [CrossRef]
- U. Schnars and W. Juptner, �??Direct recording of holograms by a CCD target and numerical reconstruction,�?? Appl. Opt. 33, 179-181 (1994). [CrossRef] [PubMed]
- J. M. Coupland, �??Holographic particle image velocimetry: signal recovery from under-sampled CCD data,�?? Meas. Sci. Technol. 15, 711-717 (2004). [CrossRef]
- M. Malek, D. Allano, S. Coetmellec, and D. Lebrun, �??Digital in-line holography: influence of the shadow density on particle field extraction,�?? Opt. Expr. 12, 2270-2279 (2004), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2270">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2270<a/> [CrossRef]
- H. Meng, W. L. Anderson, F. Hussain, and D. Liu, 'Intrinsic speckle noise in in-line particle holography,' J. Opt. Soc. Am. A 10, 2046-2058 (1993). [CrossRef]
- Y. Pu and H. Meng, "Intrinsic speckle noise in off-axis particle holography," J. Opt. Soc. Am. A 21, 1221-1230 (2004). [CrossRef]
- H. Meng, G. Pan, Y. Pu, and S. H. Woodward, �??Holographic particle image velocimetry: from film to digital recording,�?? Meas. Sci. Tech. 15, 673-685 (2004). [CrossRef]
- J. H. Milgram and W. Li, "Computational Reconstruction of Images from Holograms," Appl. Opt. 41, 853-864 (2002). [CrossRef] [PubMed]
- J. W. Goodman, �??Film grain noise in wavefront-reconstruction imaging,�?? J. Opt. Soc. Am. 57, 493-502 (1967). [CrossRef] [PubMed]
- J. M. Coupland and N. A. Halliwell, �??Holographic displacement measurements in fluid and solid mechanics: immunity to aberrations by optical correlation processing,�?? Proc. R. Soc. Lond. A 453, 1053-1066 (1997). [CrossRef]
- J. W. Goodman, �??Statistical properties of laser speckle patterns,�?? in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), Chap. 2.

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