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
Currently much effort is focussed on the development of digital particle holography, especially to facilitate the three-dimensional study of flow phenomena . The technique involves recording the diffraction pattern of a particle field directly on a CCD . Due to the limited number of pixels, the available space-bandwidth product in digital holography is small when compared to classical holography. As a result, one of the challenges in digital particle holography is to increase the number of particles that can be recorded in a single hologram.
It is not yet fully known how the number of pixels and the number of particles affect the signal-to-noise ratio (SNR) of the numerical reconstruction. Recently it was estimated that the number of particles that could be stored in a digital hologram could not exceed half the number of pixels . This estimate is an upper limit based on the fact that it is impossible to get more measurements of particle position than there are measurements of the (complex) scattered field; it did not involve the physical nature of the holographic recording process. In another study numerical simulations were used to show that the SNR depends on the shadow density, a parameter that relates to the number of particles and the particle size . However, the study did not involve the number of pixels, and as such no quantitative relation between SNR, the number of pixels, and the number of particles was found.
In traditional particle holography the SNR has been thoroughly studied. Meng et al modeled the SNR for in-line holography . In this study the effect of out-of-focus real particle images was neglected, as they expected the virtual particle images to be the biggest source of noise. In a later study relating to off-axis holography, in which the real and virtual images may be separated, Pu and Meng analyzed the SNR due to out-of-focus real particle images . It was shown that the maximum number of particles that can be properly reconstructed from a hologram depends on the hologram’s angular aperture. Meng et al were able to extend the work of Pu and Meng to digital holography . For the first time a quantitative relation between SNR, the number of pixels, and the number of particles was found:
where Isig is the particle intensity, <IN > 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
Prior to deriving a theoretical expression for the SNR we first discuss the numerical propagation of a wavefront. We perform the digital reconstruction using the method as described by Milgram and Li , which is based on the numerical equivalent of an actual optical propagation. If the complex amplitude at the plane of the hologram (z=0) is denoted by ph(x,y;z=0) then for the reconstructed complex amplitude in an arbitrary plane z we find:
where ⊗ denotes convolution and:
Because a convolution is computationally expensive pr is normally found by:
where the Fourier transforms are performed using the Fast Fourier Transform (FFT) algorithm. Although the reconstruction is significantly accelerated, the assumed periodicity inherent to the FFT has an unwanted effect. Information that would normally fall outside the reconstructed area is folded back in. This effect has been illustrated in Fig. 1. This phenomenon is important in digital particle holography. Although the effect may be avoided by zero padding, this can become computationally expensive, especially for large holograms. In our analysis we assume that no zero padding is applied. Whereas in optical holography the intensity of a virtual particle image or a defocused real image tends to zero with increasing distance, in digital holography the energy associated with these images will thus be contained within the holographic reconstruction. In the following section it is shown that the virtual images significantly contribute to the background noise.
3. Theoretical SNR
In holography the signal and the background noise are both formed by coherent light, and therefor there will be mutual interference. This implies that the SNR cannot simply be regarded as the ratio of the signal to the mean noise. Goodman derived an expression for the SNR that considers this interference :
where, similar to Eq. (1), Isig is the intensity of a particle image at an arbitrary location and <IN > 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 [Isig/<IN>]min ≈50 . The requirements on Isig/<IN> 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 [Isig/<IN>]min, and correlation of complex amplitude is inherently very robust .
Due to its importance for the SNR we will focus on the ratio Isig/<IN> in this paper. Once this ratio is known the SNR can be easily found. We start by analyzing the signal intensity Isig , 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 Ap , and the reference wave has real amplitude Ar . It is assumed that Ap 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=(Ar+Ap )2-(Ar - Ap )2=4 Ar Ap , 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=/(), we find that for the modulated intensity due to a single particle we may also write:
At a first glance it may appear strange that the intensity modulation due to a single particle depends on the number of particles. However, the modulated intensity depends on the intensity of the reference beam, which in turn is related to the total scattered intensity from the particles by means of R. Obviously the total scattered intensity depends on N, thereby explaining this perhaps counter-intuitive relation.
Figure 2 shows the recording of an intensity pattern resulting from interference between the reference beam and light from a single particle onto a detector having P pixels. It is easily shown that for the intensity of the n-th pixel we may write In=C+0.5M(1+cos(φn-φo )), where C is a constant, φn =(rn/λ)2π, φo is the reference phase. Without any loss of generality we may choose φo =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 Imax is assigned maximum (unity) transmission. For the amplitude transmission of the n-th pixel we thus find:
The numerical reconstruction is based on the Huygens-Fresnel principle. The center of each pixel on the hologram will serve as a secondary point source; the resulting field in an arbitrary plane can then be found by calculating the mutual interference of all the resulting secondary wavelets. The interference pattern that has been recorded on 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.
For the resulting complex amplitude in O (the original particle location) due to the n-th pixel it can be found that:
where 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:
If a large number of fringes is present in the interference pattern, φn can be considered to be uniformly distributed over the interval [0, 2π]. The summation in Eq. (9) then reduces to:
where it has been assumed that r=rn 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 Isig . Combining Eq. (6) and Eq. (10) yields for the signal intensity:
Because it was assumed that the reconstructed particle image has the size of a single pixel, we find for the power contained within a single particle image:
where Ω is the pixel area.
Thus far we have considered diffraction from a grating due to a single particle. In reality the hologram contains multiple sub-gratings. The -1 diffracted order coming from each of these sub-gratings constructs an isolated real particle image in the previously described way. The +1 (virtual image) and higher diffracted orders all overlap to construct a speckle background. The hologram diffraction efficiency η for all orders other than zero is given by:
where t(x,y) denotes the hologram amplitude transmittance and σt 2 the variance of the amplitude transmission. The variance of the intensity at the time of the recording can be shown to be :
where we have used that the mean signal intensity is given by <IN >=. For the variance of the amplitude transmission we write:
The diffraction efficiency of the hologram is now known. The diffraction efficiency specifies the fraction of all available power that will contribute to the diffracted orders. The total available power in plane F is found when in plane H tn =1 for all n. In this case the total power in plane F is found to be equal to P2ΩA2/r2 . For the total diffracted power in plane F we thus obtain:
In Eq. (12) the power contained in a single particle image was found; for the power that is contained in N particle images (all -1 orders contained in the hologram) thus follows:
For the power contained in the remaining orders, i.e. those forming the background speckle noise, we then write:
Because the hologram has an area PΩ, the mean background intensity is now easily found:
where according to prior assumptions 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
In order to evaluate the validity of Eq. (20) we performed a series of numerical simulations. Using MATLAB a random particle field is generated. This particle field contains N randomly located particles, spread out over a square plane of 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 15wm 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.
During the synthesis of the particle field 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 Isig 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 <IN >.
Once the particle field has been generated the complex amplitude at the hologram due to the random particle field is found using:
where (xj, yj) 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 <Ip >, we find after interference of the particle wave with the reference beam for the intensity at the hologram:
The resulting hologram is stored in computer memory for subsequent analysis. We perform the digital reconstruction using the method described in section 2. The zeroth order beam is filtered out by setting the DC term in the Fourier transform of the hologram to zero.
The simulation was performed using all possible combinations of 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 Isig /<IN > 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 [Isig /<IN >]min≈50 . Clearly our theory and simulations work well below this limit, and as such may be applied to any practical situation.
5. Experimental benchmark
Thus far we have only considered noise due to the virtual images and hologram speckle noise. It was found that the contribution to the background noise due to the virtual images was very similar to the contribution to the background noise due to out-of-focus real particle images as found by Meng 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 Isig /<IN > in digital particle holography:
where κr =0 if all particles lie in a plane, κr =1 if the particles are in a volume having a depth that is much larger than the depth of focus of a particle image, κv =0 if the virtual particle images are spatially separated or numerically suppressed, and κv =1 if the virtual particle images are present in the reconstructed field.
In any practical experimental systems there will be various factors that limit the system from operating at its theoretical maximum performance. Such factors include, but are not limited to, the limited dynamic range of the camera, the non-spherical scattering by a particle, camera noise, stray illumination on the camera, phase-fluctuations in the reference beam, and glare from the particle tank. The ratio of the experimentally obtained ratio Isig /<IN > to the theoretical maximum describes the system performance Π:
The system performance allows to quantitatively compare the performance of systems that operate under different conditions. As such it could form a benchmark for the various systems that are currently under development.
We tested the performance of one of the digital holographic systems in our laboratory. A schematic representation of the setup is shown in Fig. 4. Light from a cw frequency doubled Nd:YAG laser (532 nm) is used to record the hologram. Using a half wave plate and a polarizing beam splitter, the ratio of the reference intensity to the object intensity may be adjusted. Prior to splitting the object beam and the reference beam the laser beam is collimated using a spatial filter and lens 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 κr =1 and κv =1.
Using the acquired digital holograms a series of numerical reconstructions is performed. For each of these reconstructions the center m x m pixels of the hologram are used, where m ranges between 200 and 1000. By determining Isig /<IN > 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.
Based on the information in Fig. 5 we assumed that the particles in the experiment behave as anisotropic scatterers. To incorporate this anisotropic behavior into the numerical reconstruction we constructed an arbitrary scattering function having constant amplitude over a disk with a radius of 100 pixels at the plane of the CCD, after which it linearly falls off until the amplitude becomes zero for a radius of 200 pixels. The assumed scattered amplitude is shown in Fig. 6. When incorporating this scattering behavior in the numerical reconstruction of the experimentally obtained holograms the quality of the reconstructed images is significantly improved for large holograms. Figure 7 shows identical regions of a reconstructed particle field using a hologram of 1000×1000 pixels. In Fig. 7(a) the scattering is assumed isotropic, in Fig. 7(b) the scattering is assumed anisotropic as shown in Fig. 6. The performance of the holographic system is three times higher when anisotropic scattering is assumed.
This example clearly shows how a benchmark test may be used to further improve the performance of a digital holographic system, thereby illustrating the practical significance of the presented theoretical analysis. As mentioned earlier the system performance may also be used to quantitatively compare the performance of digital holographic systems that operate under different conditions.
In this study we have shown that in in-line digital particle holography the SNR due to virtual images and hologram speckle noise is fundamentally limited by the number of particles, the number of pixels, and the reference-to-object intensity ratio. We verified the validity of our theoretical analysis by means of numerical simulation. The theoretical ratio Isig /<IN > was in very good correspondence with the values for Isig /<IN > obtained by numerical simulation of 14400 particle fields.
We found our work to be logically consistent with earlier work that studied background noise due to out-of-focus real particle images . When combining the noise contributions due to both real and virtual particles we presented a practical guide that describes the maximum performance of any digital holographic system. Using this benchmark criterion we tested the performance of a digital holographic system. The results from this benchmark test suggested that the particles scatter anisotropically. When this implied scattering behavior was implemented in the numerical reconstruction, the performance of the digital holographic system increased by a factor three.
We believe that the presented work will benefit many researchers currently working on digital particle holography. The ability to quantitatively analyze the system performance allows for a more systematic way of designing, optimizing, and comparing digital holographic systems.
This work has been funded by the Foundation for Fundamental Research on Matter (FOM), the Netherlands.
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