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

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References

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    [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), <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]
  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).
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  6. Y. Pu and H. Meng, "Intrinsic speckle noise in off-axis particle holography," J. Opt. Soc. Am. A 21, 1221-1230 (2004).
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  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).
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  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).
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. Opt. Soc. Am. A (1)

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]

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Meas. Sci. Tech. (1)

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]

Meas. Sci. Technol. (2)

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

K. D. Hinsch, �??Holographic particle image velocimetry,�?? Meas. Sci. Technol. 13, R61-R72 (2002).
[CrossRef]

Opt. Expr. (1)

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]

Proc. R. Soc. Lond. A (1)

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]

Other (1)

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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Figures (7)

Fig. 1.
Fig. 1.

Schematic representation of the numerical reconstruction of the hologram located in z=0. The reconstructed image contains a diagonal bar. The folding due to usage of the FFT is clearly illustrated in the plane z=z3.

Fig. 2.
Fig. 2.

Schematic drawing of the interference pattern formed due to interference between light coming from a single particle located at position O in the plane F and a plane reference wave. The resulting hologram in the plane H is recorded on a CCD having P pixels. The distance from the particle to the n-th pixel is rn . During the numerical reconstruction the plane F is also composed of P pixels.

Fig. 3.
Fig. 3.

Simulated SNR as a function of theoretical SNR for all possible combinations of N=[50, 250, 500, 1000], P=[10000, 22500, 40000], and R=[1, 2, 5, 10, 100, 1000]. Each point represents the average obtained from 200 simulations. The dashed line represents the practical threshold to distinguish particles [7].

Fig. 4.
Fig. 4.

Experimental setup for recording an in-line digital hologram of a particle field. HW: half wave plate; SF: spatial filter; Lx: lens; PBS: polarizing beam splitter; BD: beam dump; IM: image of particle field; P: polarizer; M: mirror.

Fig. 5.
Fig. 5.

The system performance Π as a function of varying hologram size (m x m pixels). The solid line has been added to assist the eye (not a fit).

Fig. 6.
Fig. 6.

Assumed anisotropic scattering amplitude footprint.

Fig. 7.
Fig. 7.

Region of reconstructed particle field using a hologram of 1000×1000 pixels. In (a) the particle is assumed to scatter isotropically (Π=0.05), in (b) the scattering is assumed anisotropic as shown in Fig. 6 (Π=0.15).

Equations (30)

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I sig I N = π P 4 N ,
p r ( x , y ; z ) = p h ( x , y ; z = 0 ) n ( x , y ; z ) ,
n ( x , y ; z ) = exp [ i k x 2 + y 2 + z 2 ] x 2 + y 2 + z 2 .
p r ( x , y ; z ) = 𝔉 1 [ 𝔉 p h ( x , y ; z = 0 ) · 𝔉 n ( x , y ; z ) ] ,
SNR = I sig I N 1 + 2 I sig I N ,
M = 4 R N A p 2 .
t n = I n I max .
A O n = A r n t n e i φ n
= A r n ( C + 0.5 M ( 1 + cos ( φ n ) ) ) I max e i φ n ,
A O = n = 1 P A r n ( C + 0.5 M ( 1 + cos ( φ n ) ) ) I max e i φ n .
A O = APM 4 r I max .
I sig = A 2 r 2 I max 2 P 2 RNA p 4 .
𝓟 sig = A 2 r 2 I max 2 P 2 RNA p 4 Ω ,
m 0 η m = t ( x , y ) 2 ¯ t ( x , y ) ¯ 2
σ t 2 ,
σ I 2 = I N 2 ( 1 + 2 R ) = N 2 A p 4 ( 1 + 2 R ) ,
σ t 2 = N 2 A p 4 ( 1 + 2 R ) I max 2 .
𝓟 d = P 2 Ω A 2 r 2 σ t 2
= P 2 Ω A 2 N 2 A p 4 ( 1 + 2 R ) r 2 I max 2 .
𝓟 1 = N 𝓟 sig
= P 2 Ω A 2 N 2 A p 4 R r 2 I max 2 .
𝓟 N = 𝓟 d 𝓟 1
= P 2 Ω A 2 N 2 A p 4 ( 1 + R ) r 2 I max 2 .
I N = 𝓟 N P Ω
= P A 2 N 2 A p 4 ( 1 + R ) r 2 I max 2 .
I sig I N = P R N ( 1 + R ) ,
U ( x , y ) = j = 1 N 1 exp [ i k ( x x j ) 2 + ( y y j ) 2 + z 2 ] ,
I ( x , y ) = U ( x , y ) + R I p 2 .
[ I sig I N ] max = P N R ( 1 + κ r R + κ v R ) ,
Π = [ I sig I n ] exp [ I sig I n ] max .

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