Influence of detector noise in holographic imaging with limited photon flux

I. S. Wahyutama; G. K. Tadesse; A. Tünnermann; J. Limpert; J. Rothhardt

doi:10.1364/OE.24.022013

1. Introduction

Lensless imaging schemes, such as coherent diffractive imaging and related techniques, with extreme ultra-violet (XUV) light and X-rays have seen spectacular advances during the recent years. Powerful light sources such as synchrotrons and free-electron lasers (FELs) enabled imaging of nanoscale objects with unprecedented spatial [1] and temporal resolution [2]. Due to the rapid progress in development of ultrafast lasers, coherent XUV sources also became available in many laboratories and on a table-top. Today, single-shot imaging [3] and 22 nm resolution [4] have been demonstrated with laser based light sources. In addition, high power fiber lasers-driven high harmonic sources recently enabled real-time and sub-wavelength lensless imaging [7]. Further scaling towards shorter wavelengths [5] and higher photon flux is therefore to be expected [6,7]. In contrast to coherent diffractive imaging, which requires complicated iterative phase-retrieval algorithms to reconstruct the sample in amplitude and phase, holographic imaging provides a way of direct and robust object reconstructions by incorporating reference waves. Indeed, recent studies of ultrafast demagnetization dynamics utilized holographic imaging, due to its robust and simple way of image reconstruction [2]. However, so far the resolution of holographic imaging with table-top sources was limited to > 50 nm [8,9], which is particularly due to the limited photon flux.

There are two, commonly implemented holographic imaging schemes, namely Fourier-Transform-Holography (FTH) [10–16] and HERALDO [9,17–20]. In FTH, the sources for the reference wave are small pinholes placed around the sample. The size of the reference holes necessarily limits the photon flux of the reference waves and hence reduces the fringe contrast. HERALDO overcomes this limitation in FTH due to the small-sized reference by expanding the reference aperture to have certain shape [9,18,19].

As in any imaging system, consideration about the noise always comes to mind whenever a high image quality is desired. Our work here is devoted for this purpose namely - to study the sensitivity of FTH and HERALDO in the presence of noise. To the best of our knowledge, there have been only few works which go along this line of investigating the noise sensitivity in holographic imaging [19,21,22]. The first theoretical treatment of the noise in the diffraction pattern in holographic imaging was demonstrated in [19]. The authors limit their consideration to shot noise only which is a type of noise associated with the uncertainty as a consequence of the quantum nature of the incoming photons on the detector (for this reason, the detection condition in which only the shot noise is present is also called “photon-limited”). The comparison of the shot noise response of FTH and HERALDO reveals that HERALDO is less sensitive to noise than FTH under photon-limited conditions. In practice, other types of noise which are independent of intensity are also present, e.g. dark current and readout noise of the detector [23]. Recent experiments devoted to studying the noise responses between FTH and HERALDO are reported in [21,22]. In [21], it is reported that the noise level in reconstructions for HERALDO are very high in cases of limited photon flux. By applying a low-pass filter to the diffraction pattern, the noise has been suppressed successfully, however, at the expense of lowered resolution. While in [22], by employing broadband illumination spanning the visible wavelengths they were able to obtain frequency resolved reconstructions and after rescaling the reconstructions from each frequency, the average of these reconstructions was calculated in order to improve the SNR.

In the first portion of this manuscript, we provide a numerical comparison on the noise statistics between FTH and HERALDO under the presence of shot-noise as well as a constant noise floor that is typical to photon-flux limited experiments. We found that the choice of the numerical differentiation method to be used in reconstructing through HERALDO, despite looking trivial at the first glance, actually has a big impact on the reconstruction quality, especially when more than one derivative operators are involved. In the last part, we provide a thorough comparison of the resolution and SNR in the object space for FTH and various HERALDO implementations.

2. Response of FTH and HERALDO to uniform noise

The thermal and readout noises from a detector in an imaging experiment are present in all pixels of the camera and are not dependent on the signal intensity. We refer to the combined noise from these types of noise sources as uniform noise in this work. For the sake of convenience, we assume that these noise sources impose a variation in the measured electrons from the chip in a uniform manner across the entire pixel array. To numerically investigate the effect of this uniform noise, we present reconstructions of three kinds of objects designed for use in FTH and HERALDO schemes in Fig. 1. The main sample to be imaged is the “IAP” structure. For FTH, this structure is surrounded by two pinholes (Fig. 1(a)) while for HERALDO a line (Fig. 1(d)) or a rectangle (Fig. 1(g)) serve as reference. Each pinhole in Fig. 1(a) has a size of one pixel, as for the reference line in Fig. 1(d) it is one pixel in width. Note that the number of references can be freely chosen as long as all of them are coherently illuminated and the minimum separation conditions are fulfilled. Using more than one references allows one to improve the final SNR by averaging the reconstructions from each reference structures [9]. In this study, we chose to use at most only two references (the one in FTH) because our aim is mainly to investigate the nature of the noise subject to the various reconstruction methods necessary in each holographic imaging technique, for which at least one reference is sufficient. In each of the three objects, the separation between the main sample and the reference must be such that the distance between the right-most point in the main sample and the reference structure is at least equal to the horizontal span of the main sample. In this sense, in Fig. 1(g) the reference structures which will give rise to the useful reconstructions are the two right corners of the rectangle.

Fig. 1 Numerical comparison of the response of FTH and HERALDO to a uniform noise in the detector plane. The first column shows the object for (a) FTH (d) HERALDO with line reference and (g) HERALDO with rectangular reference. The second column shows the diffraction patterns, after adding noise, from the corresponding objects. The corresponding reconstructions are shown in the third column.

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The objects are assumed to be illuminated by a plane wave with an amplitude equal to 0.4 (for the object size being considered here, this value corresponds to the highest intensity in the order of 10⁶ counts in the camera, which is typical in some CDI experiments [7,9]). In particular, in Fig. 1(g), only the right corners satisfy the separation conditions. Therefore the reconstructions in this case will be taken from these locations. To simulate the effect of noise, a random noise with an intensity-independent probability distribution constant throughout the detector is added to the diffraction patterns of each object. The detector noise is simulated by a random variable having uniform probability distribution between 0 and 100 counts (corresponding to a variance of about 832, this value is based on observations of some experimentally recorded background noise data and we believe that this value is representative for typical experiments). The reconstructions are then obtained by running the corresponding algorithmic procedures for each scheme. For FTH, the reconstruction is just the inverse transform of the diffraction pattern. While in HERALDO, the diffraction patterns are multiplied with the appropriate polynomials before being transformed back to yield the reconstructions. The noisy diffraction patterns and the reconstructions are shown in the second and third columns of Fig. 1. One point to mention here is that in order for the analysis to be wavelength-independent, we make pixels as the unit of lengths throughout this manuscript.

In calculating the reconstructions for HERALDO, i.e. second and third rows, we multiply the diffraction patterns with the appropriate linear polynomials which corresponds to the required derivatives in the object plane [18]. The result is, as we can see in Fig. 1(f) and 1(i), a dramatic increase of the noise in the reconstructions which is strongest for the rectangular reference. This enhanced noise effect has its root in the reconstruction method being implemented, namely the multiplication with linear polynomials. It can be significantly reduced, especially in the case of a rectangular reference, when the derivative of the autocorrelation was chosen instead, as will be shown in subsection 3. The following sections of the manuscript deals with the theoretical treatment of this effect.

The treatment can be initiated by writing the expression of the noisy diffraction pattern,

I [j, k] = I_{0} [j, k] + η [j, k]

η [j, k] = η_{0} [j, k] + η_{m}

where η_m is a constant, η₀[j, k] is a random, intensity independent noise distribution having zero mean and variance

σ_{I}^{2}

(hence the mean and variance of η[j, k] are equal to η_m and

σ_{I}^{2}

respectively). I[j, k] and I₀[j, k] are the noisy and noise-free diffraction patterns respectively. η[j, k] can have any form of distribution so long as this distribution is the same for all values of j and k. The autocorrelation function of η[j, k] satisfies

〈 η [j, k] η [j^{'}, k^{'}] 〉 = σ_{I}^{2} δ_{j j^{'}} δ_{k k^{'}} + η_{m}^{2}

(from here on, angle bracket always denotes ensemble average). Note that since I[j, k] and I₀[j, k] are diffraction patterns, they must be real, imposing η[j, k], η₀[j, k], and η_m to be real as well. Throughout this manuscript, the indices [j, k] will be used in the Fourier plane, e.g. in the diffraction pattern, with j representing the horizontal direction and k the vertical direction. In the object plane, the indices [m, n] are used where m represents the horizontal direction and n the vertical direction. The pixel array is assumed to be square containing M² pixels, so that each of the indices satisfies j, k, m, n = 0, 1, 2, …, M−1.

In each of the following subsections, we will provide the calculations of the noise statistics in reconstructions of various holographic schemes with the goal of obtaining the expressions of noise variances. In the end, we will also incorporate shot-noise and summarize the findings by comparing the SNR from the different schemes.

2.1. FTH

In FTH, the reconstruction process is obtained by applying the inverse Fourier transform on the diffraction pattern. Numerically, this is usually done by computing the inverse fast Fourier transform (IFFT). Denoting the noisy reconstruction (autocorrelation) as R[m, n], this process mathematically looks like

\begin{array}{l} R [m, n] = IFFT [I [j, k]] = IFFT [I_{0} [j, k] + η [j, k]] \\ = R_{0} [m, n] + \frac{1}{M^{2}} \sum_{j k}^{M - 1} η [j, k] \exp (\frac{2 π i}{M} (j m + k n)) \end{array}

where R₀[m, n] = IFFT[I₀[j, k]] is the noise-free autocorrelation. A sum notation with double index

\sum_{j k}^{M - 1}

means a double summation over j and k with each of them starts from 0 until M − 1. The measured diffraction patterns are usually centered in the camera pixel array, and since the above summations over j and k always start from (j, k) = (0, 0), which corresponds to the upper left corner pixel in the camera, the diffraction pattern I[j, k] is not centered relative to its indexing. Therefore, Eq. (2) will result in a linear phase ramp in R[m, n]. To eliminate this phase modulation, another phase factor with opposite sign must be inserted in the expression. Thus,

\begin{array}{l} R [m, n] \Rightarrow \exp (\frac{- i 2 π}{M} (j_{c} n + k_{c} m)) R [m, n] \\ = R_{0} [m, n] + \frac{1}{M^{2}} \sum_{j k}^{M - 1} η [j, k] \exp (\frac{2 π i}{M} (J m + K n)) \end{array}

where, for simplicity, we retain the notation for R₀[m, n], J = j − j_c, and K = k − k_c (from this point on, whenever J and K appear, summation over j_c and k_c are implied). Here, (j_c, k_c) = ((M − 1)/2, (M − 1)/2) is the central pixel in the diffraction pattern. We will also assume that all holographic separation conditions have been met.

In all derivations that follow, the central pixel (m, n) = (0, 0) will be excluded from the calculation because the desired reconstructions will not encompass this location. We will derive the variance of the real part of R[m, n], while the variance of the imaginary part can be easily computed in a similar manner. The real part of R[m, n] is

\begin{array}{l} ℜ [R [m, n]] = ℜ [R_{0} [m, n]] + \frac{1}{M^{2}} \sum_{j k}^{M - 1} η [j, k] \cos (\frac{2 π}{M} (J m + K n)) \\ = ℜ [R_{0} [m, n]] + ℜ [E [m, n]] \end{array}

where E[m, n] is the (complex) noise in the reconstruction. The two terms in the last line above are uncorrelated as can be checked by calculating the covariance between them, which should be equal to zero. The covariance C_AB between two random variables A and B is given by

C_{A B} = 〈 A B 〉 - 〈 A 〉 〈 B 〉

[24]. It follows then that we can apply the sum law of random variables, which states that the variance of the sum of two uncorrelated random variables is equal to the sum of the variances of each random variable [25]. The variance of ℜ[R₀[m,n]] is zero as this is a deterministic quantity, which means the variance of ℜ[R[m,n]] is equal to the variance of ℜ[E[m,n]]. The ensemble average of ℜ[E[m,n]] is easily seen to be zero

〈 ℜ [E [m, n]] 〉 = \frac{η_{m}}{M^{2}} \sum_{j k}^{M - 1} \cos (\frac{2 π}{M} (J m + K n)) = 0

In the last step, we have incorporated the fact that the summation is equal to the real part of (23) (see appendix A), therefore it must be zero. In order to calculate 〈ℜ[E[m, n]]²〉, the autocorrelation relation

〈 η [j, k] η [j^{'}, k^{'}] 〉 = σ_{I}^{2} δ_{j j^{'}} δ_{k k^{'}} + η_{m}^{2}

〈η[j, k]〉 = η_m must be incorporated, upon which it can be shown that

〈 ℜ {[E [m, n]]}^{2} 〉 = 〈 {(\sum_{j k}^{M - 1} \frac{η [j, k]}{M^{2}} \cos (\frac{2 π}{M} (J m + K n)))}^{2} 〉 = \frac{σ_{I}^{2}}{M^{4}} \sum_{j k}^{M - 1} \cos^{2} (\frac{2 π}{M} (J m + K n))

By employing the trigonometric identity cos2α=2cos² α−1 to evaluate the remaining sum, we obtain

〈 ℜ {[E [m, n]]}^{2} 〉 = \frac{σ_{I}^{2}}{2 M^{2}}

. Denoting the variance of R[m, n] and E[m, n] as

σ_{ℜ R}^{2}

and

σ_{ℜ E}^{2}

, respectively, we obtain

σ_{ℜ R}^{2} = σ_{ℜ E}^{2} = 〈 ℜ {[E [m, n]]}^{2} 〉 - {〈 ℜ [E [m, n]] 〉}^{2} = \frac{σ_{I}^{2}}{2 M^{2}}

It can be easily shown, e.g. by replacing cosines with sines in all steps above, that the variance of the imaginary part of the reconstruction is equal to that of the real part,

σ_{ℑ R}^{2} = σ_{ℜ R}^{2}

. We see that the variances are independent of the pixel index (m, n).

2.2. HERALDO with linear polynomial multiplication

In this section and the one after, we will be considering the simplest form of HERALDO scheme, namely an object accompanied by a line reference. In particular, the reference is aligned in vertical direction. Therefore, in the Fourier domain we must multiply the diffraction pattern with a linear polynomial of the form iΔK, where Δ is the interval between adjacent coordinates in Fourier domain (i.e. spatial frequencies). The reconstruction R[m, n] is then

R [m, n] = IFFT [i Δ K I [j, k]] = R_{0} [m, n] + \frac{i Δ}{M^{2}} \sum_{j k}^{M - 1} K η [j, k] \exp (\frac{2 π i}{M} (J m + K n))

whose real part is

ℜ [R [m, n]] = ℜ [R_{0} [m, n]] - \frac{Δ}{M^{2}} \sum_{j k}^{M - 1} K η [j, k] \sin (\frac{2 π i}{M} (J m + K n))

We may identify the second term in the right hand side as the noise in the real part of the reconstruction and denote it as ℜ[E[m, n]]. As it was in the previous subsection, the first and second terms are uncorrelated. This allows us to calculate just the variance of ℜ[E[m, n]]. The mean value of ℜ[E[m, n]] can be shown to vanish (see appendix B).

Upon calculating the mean square of ℜ[E[m, n]] (see appendix B) followed by the substitution $k_{c} = \frac{M - 1}{2}$ , the variances of the noise in the reconstruction are found to be

σ_{ℜ R}^{2} = 〈 ℜ {[E [m, n]]}^{2} 〉 - {〈 ℜ [E [m, n]] 〉}^{2} = \frac{π^{2} σ_{I}^{2}}{6 M^{4}} (M^{2} - 1) \approx \frac{π^{2} σ_{I}^{2}}{6 M^{2}}

where the last approximate sign is based on the fact that in most practices the number of pixels M in the camera used for recording is much bigger than 1. In addition, the variance of the imaginary part of the reconstruction is equal to that of the real part,

σ_{ℑ R}^{2} = σ_{ℜ R}^{2}

. The noise variances in the reconstruction are also position-independent like they are in the case of FTH.

2.3. HERALDO with approximate derivative of the autocorrelation

In the previous section, we calculated the noise in HERALDO scheme when the reconstruction is obtained through the use of polynomial multiplication of the diffraction pattern. Here, we present the noise calculation for the same HERALDO arrangement as the one before but with the reconstruction obtained through the derivative of the autocorrelation. Denoting the noisy and the noise-free autocorrelations as A[m, n] and A₀[m, n], respectively, we write

\begin{array}{l} A [m, n] = IFFT [I [j, k]] = IFFT [I_{0} [j, k] + η [j, k]] \\ = A_{0} [m, n] + \frac{1}{M^{2}} \sum_{j k}^{M - 1} η [j, k] \exp (\frac{2 π i}{M} (J m + K n)) \end{array}

The reconstruction is obtained by taking the approximate first derivative of A[m, n]. This is realized by taking the difference between adjacent elements in A[m, n] in the direction of the slit, which is vertical in this example. Therefore we will be calculating

R [m, n] = A [m, n + 1] - A [m, n]

with the real part ℜ[R[m, n]] = ℜ[A[m, n + 1]] − ℜ[A[m, n]].

At this point, the calculation of variances can be significantly simplified by realizing that ℜ[A[m, n+1]] and ℜ[A[m, n]] are uncorrelated, as can be checked by calculating the covariance between them (see Eq. (5)). Thus, according to the sum law of random variables the variance of ℜ[R[m, n]] is equal to the sum of the variance of ℜ[A[m, n + 1]] and of ℜ[A[m, n]]. We note that these variances are actually identical and are equal to Eq. (8). This analysis leads to the conclusion that the variance due to a uniform, random noise in HERALDO line reference with approximate derivative of the autocorrelation is just twice as big as that of FTH

σ_{ℜ R}^{2} = \frac{σ_{I}^{2}}{2 M^{2}} + \frac{σ_{I}^{2}}{2 M^{2}} = \frac{σ_{I}^{2}}{M^{2}}

which is different from Eq. (11) by a factor of π²/6 ≈ 1.6. This difference is caused by the fact that the Fourier theorem on the derivatives of a function does not exactly hold true in the discrete realm. The interested readers are referred to appendix C for the mathematical proof of this statement. As before, the symmetry between the real and imaginary parts can also be expected, that is

σ_{ℑ R}^{2} = σ_{ℜ R}^{2}

. Using the same argument as that discussed just above, one can easily see that the noise variance in the reconstruction of HERALDO with rectangular reference using double differentiation in horizontal and vertical directions is

2 σ_{I}^{2} / M^{2}

for both the real and imaginary parts. At this point, we would like to put forward that when reconstructing HERALDO rectangular reference using polynomial multiplication, the mean value of the noise in the reconstruction does not vanish (unlike in the preceding cases in which it vanishes identically across the object plane). This non-zero average noise is what we see as a broad, cross-shaped feature in Fig. 1(i). Therefore, upon averaging the reconstructions from each corners of the rectangle or from different acquisition events, this feature will not be canceled out.

3. Comparison on the noises in FTH and HERALDO

We have seen that, while HERALDO with linear polynomial multiplication and with the differentiation of the autocorrelation use the same diffraction patterns as the input, the two methods responds differently to the presence of signal-independent uniform noise. We may understand this effect by noticing that when the diffraction pattern is multiplied with the required polynomial, pixels near the edge of the array which are dominated by noise, get enhanced more than those around the central pixel. Thus, we might say that the biggest contribution to the noise in the reconstruction comes from the noisy pixels around the edges of the diffraction pattern. We also see that, regardless of which computational method being used, HERALDO schemes inherently have stronger response to a uniform noise than FTH. In the next section, we shall see that upon considering shot-noise, which is signal dependent, the total noise in HERALDO with differentiation of the autocorrelation can be smaller in variance than that in FTH. The result of the preceding calculations for the noise properties in the reconstruction is summarized in Table 1. Note that up to this point, we have not restricted the noise to have certain probability distribution.

Table 1. Comparison of the mean and variance of noise in the reconstruction in the case of FTH and HERALDO line reference.

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Therefore, the preceding results will apply for any kinds of probability distribution as long as it is uniform throughout the pixel array in the diffraction pattern. Fig. 2 shows HERALDO reconstructions of the line and rectangular references using differentiation of the autocorrelation, instead of polynomial multiplication, from the same diffraction patterns as in Fig. 1(e) and 1(h). Comparing Fig. 2(a) and 1(f), although not directly apparent, the former has a lower noise level than the latter. Moreover, the central vertical line in Fig. 2(a) does not extend until the edge of the pixel array. The most apparent SNR improvement is seen with the rectangular reference (Fig. 2(b)), where the objects look much cleaner and also the very noisy cross-shaped area present in Fig. 1(i) disappears. This noisy cross-shaped area, however, might not completely disappear even when the differentiation method is being used with the rectangular reference. This occurs especially in a situation where the noise near one of the camera edges is much stronger than the noise in other places, e.g. due to stray lights or non-uniformity in the pixel array.

Fig. 2 Reconstruction of HERALDO sample by differentiation of the autocorrelation. (a) Reconstruction from the diffraction pattern in Fig. 1(e). (b) Reconstruction from the diffraction pattern in Fig. 1(h). A significant improvement of the SNR can be seen when comparing Fig. 2(b) with Fig. 1(i).

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4. Calculation of the total noise

In real experiments, there are various types of noise apart from the uniform one considered prior to this point. One of these is the shot-noise which is an inherent property of light, i.e. this noise cannot be eliminated. A detailed treatment of the effect of shot noise in HERALDO can be found in [19], therefore we will simply adopt the related equations. Shot noise has Poisson probability distribution with the mean and standard deviation depending on the signal. This means, in order to incorporate this type of noise, one must first determine the signal. For this reason, we establish the signal to be the noise-free diffraction intensity I₀[j, k] and not the noisy one I[j, k]. This choice is justified by the fact that shot noise is a quantum property of light, irrespective of any other sources of noise which might be present during the detection process. This means, even under the condition of zero device noise, shot noise will still be present upon the light detection event. In the presence of shot noise, denoted by ϒ[j, k], as well as the uniform noise η[j, k] considered above, the noisy diffraction pattern will look like

I [j, k] = I_{0} [j, k] + η [j, k] + ϒ [j, k]

where 〈ϒ[j, k]〉 = 0 and 〈ϒ[j, k]ϒ[j′, k′]〉 = I₀[j, k]δ_jj′ δ_kk_′.

Here, we present the measure of noise as a ratio between the energy of the noise-free reconstruction and the total noise variance (real plus imaginary parts in the reconstruction), both of which are contained in 〈|R[m, n]|²〉, the mean square value of the reconstruction

\begin{array}{l} 〈 {| R [m, n] |}^{2} 〉 = 〈 ℜ {[R [m, n]]}^{2} 〉 + 〈 ℑ {[R [m, n]]}^{2} 〉 \\ = (σ_{ℜ R}^{2} + {〈 ℜ [R [m, n]] 〉}^{2}) + (σ_{ℑ R}^{2} + {〈 ℑ [R [m, n]] 〉}^{2}) \end{array}

First, we will work out the noise expression for the case of HERALDO with approximate derivative of the autocorrelation. In this case, it can be shown (see appendix D) that the sum of the two noise variances in Eq. (16) simplifies to

σ_{ℜ R}^{2} + σ_{ℑ R}^{2} = \frac{1}{M^{2}} (2 σ_{l}^{2} + \sum_{m n}^{M - 1} {| O_{0} [m, n + 1] - O_{0} [m, n] |}^{2})

where O₀[m, n] is the noise-free object.

Now, the second and fourth terms in Eq. (16) is equal to the noise-free reconstruction of the main sample, 〈ℜ[R[m, n]]〉² + 〈ℑ[R[m, n]]〉² = 〈ℜ[R₀[m, n]]〉² + 〈ℑ[R₀[m, n]]〉², which we denote by S[m, n]. Furthermore, suppose that all separation conditions are met. Defining SNR as the ratio between the energy of the noise-free reconstruction, |S[m, n]|², and the total noise variance, $σ_{ℜ R}^{2} + σ_{ℑ R}^{2}$ , inside a rectangular area with sides P and Q pixels wide which encloses S[m, n], we write

\begin{array}{l} S N R = \frac{\sum_{(P Q)} {| S [m, n] |}^{2}}{\sum_{(P Q)} (σ_{ℜ R}^{2} + σ_{ℑ R}^{2})} \\ = \frac{M^{2}}{P Q} \frac{\sum_{(P Q)} {| S [m, n] |}^{2}}{2 σ_{l}^{2} + \sum_{m n}^{M - 1} | O_{0} [m, n + 1] - O_{0} {[m, n] |}^{2}} \end{array}

where we have made use the fact that the total noise is independent of the pixel

σ_{ℜ R}^{2} + σ_{ℑ R}^{2}

index in the object domain. It can be easily seen that the same measure of total noise for FTH can be obtained by the substitution

2 σ_{l}^{2} \to σ_{l}^{2}

{| O_{0} [m, n + 1] - O_{0} [m, n] |}^{2} \to {| O_{0} [m, n] |}^{2}

Therefore

S N R = \frac{M^{2}}{P Q} \frac{\sum_{(P Q)} {| S [m, n] |}^{2}}{2 σ_{l}^{2} + \sum_{m n}^{M - 1} {| O_{0} [m, n] |}^{2}}

While for the case of HERALDO with linear polynomial multiplication,

S N R = \frac{M^{2}}{P Q} \frac{\sum_{(P Q)} {| S [m, n] |}^{2}}{(π^{2} 2 σ_{I}^{2}) / 3 + \sum_{m n}^{M - 1} {| O_{0} [m, n + 1] - O_{0} [m, n] |}^{2}}

From Eq. (18), (20), and (21), it is obvious that the SNR can be improved by reducing the variance of signal-independent, uniform noise. This can be realized by using cooled and shielded detector. Fig. 3(a) displays the behavior of the SNR as given by Eq. (18), (20), and (21) as a function of variance

σ_{I}^{2}

for a fixed intensity equal to 2 (arbitrary unit). It can be seen that at small variances, the SNR of HERALDO line reference is higher than FTH and vice versa when it increases beyond certain value. Thus, we may conclude that in the situation where the uniform noise is very small, HERALDO is a better option than FTH.

Fig. 3 The SNR as a function of (a) variance and (b) the incoming intensity for FTH (Eq. (20)), HERALDO line reference with polynomial multiplication (Eq. (21)), and HERALDO line reference with approximate derivative of the autocorrelation (Eq. (18)).

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For binary object and uniform illumination such as those considered in this manuscript, it’s possible to express SNR in Eq. (18), (20), and (21) in terms of the incoming intensity. The graphical representations of these equations are shown in Fig. 3(b) with $σ_{l}^{2} = 832$ , P = Q = 80 (these choices of value enclose one reconstruction completely), and M = 1024. The same as before, the pinhole in FTH and the line in HERALDO line reference are of one pixel size. The case of FTH initially has the highest SNR for low intensities. This can be understood from table 1 that the variance for the uniform noise is lowest for FTH, such that for low intensities the total noise in FTH is smaller than in HERALDO. As the intensity increases, the contribution from shot noise begins to dominate. The shot noise in FTH increases faster with intensity than it does in both implementations of HERALDO. This is so because in HERALDO, pixels close to the edges have the most contribution in the reconstruction while at the same time the variances of shot noise in places having low signal are the smallest. Also, notice that HERALDO with linear polynomial multiplication crosses the curve of FTH at a higher intensity than HERALDO with the derivative of the autocorrelation does. The insight we can get from Fig. 3(b) is that in the low photon flux regime, FTH gives better SNR than HERALDO even if the reference pinhole is just 1 pixel large and thus the resolution, which depends on the reference size, is the same as in the HERALDO cases. On the other hand, if either a high photon flux or a low noise detector are available HERALDO can be better than FTH both in resolution and in SNR.

Before drawing the conclusions, we numerically analyze the dependence of resolution and SNR on the dimensions of the reference structures which can in principle be chosen by the experimentalist. The results are summarized in Fig. 4. Here the resolutions are defined through the 10%–90% distance along a fixed vertical and horizontal edge in the “IAP” aperture. The uniform noise variance is set to be $σ_{l}^{2} = 832$ and the illumination amplitude is equal to 0.4. From Fig. 4(a), it can be seen that in the case of FTH as the pinhole radius increases, the SNR improves as expected. On the other hand both horizontal and vertical resolutions get worse in the same manner. In the case of HERALDO line reference (Fig. 4(b)), the SNR improves but the horizontal resolution deteriorates as the slit becomes wider. On the other hand, the vertical resolution remains unchanged because this resolution is only determined by the steepness of the edge of the line aperture [19].

Fig. 4 The evolution of the resolution and SNR as (a) the pinhole radius in FTH, (b) the slit length in HERALDO line reference, and (c) the length of the rectangle in HERALDO rectangular reference are increased. In (a)–(c), the vertical and horizontal resolutions are denoted by the × and + markers, respectively (a∗ marker is actually an overlapping + and ×). (d) the threshold resolution as a function of the incoming field amplitude ( $\sqrt{I}$ ).

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In contrast to the line reference (Fig. 4(b)), in rectangular reference (Fig. 4(c)) both resolutions have equal value (1 pixel) because the two derivatives in the directions of the sides of the rectangle result in a delta-like structure at the corners having equal horizontal and vertical dimensions. In terms of noise, the SNR in rectangular HERALDO is lower (SNR = 1.1) than the minimum SNR for HERALDO line reference. The reason of this low SNR for HERALDO rectangular reference is that the feature with which the sample is cross-correlated consists of only one pixel (resulting from the double derivatives in the horizontal and vertical directions), in contrast to the line HERALDO where the result of a single derivative along the line yields a feature whose width is equal to that of the original line but with the length of only one pixel. At the same time, the double derivative required by rectangular reference necessarily increases the uniform noise contribution (see the discussion following Eq. (14)). The preceding result can be used as one of the considerations by the experimentalists to decide which type of reference is suitable according to the performance of their experimental setup. For example, if one can afford to realize an imaging system with sufficiently high photon flux and spatial coherence, then using a rectangular reference is a good choice because, despite having low SNR, its resolution is invariably good for any rectangle area while for FTH, the resolution is always limited by the smallest, practically realizable pinhole diameter. However, in the photon flux-limited regime where image quality becomes critical FTH is a better option for its high SNR.

Finally, we analyze, for the particular arrangements being considered, how photon flux of the source (intensity) influences the achievable resolution. Again, we set the value for $σ_{I}^{2}$ to be 832. In order to vary the resolution, the reference size is changed such that the SNR is kept at a certain threshold value (the SNR is held constant because our goal is to study the behavior between resolution and intensity). The result is shown in Fig. 4(d). In this plot, for a given amplitude $\sqrt{I}$ the corresponding value in the vertical axis is determined by varying the pinhole radius (for FTH), the slit width (for line HERALDO), and the length of the rectangle (rectangle HERALDO) until a value where the SNR is approximately equal to 0.1 (this value corresponds to the condition where the reconstruction is barely visible due to the noise). The resolution is again calculated from the 10%–90% distance along an edge in the reconstruction. The plot shows that, for FTH and HERALDO line reference, the threshold resolution decreases (i.e. resolution gets better) in an asymptotic way as the incoming intensity increases. On the other hand, the plot for HERALDO rectangular reference is presented as a single point because the SNR and resolution are independent of the area of the reference. This means there can only be one value of amplitude which gives SNR = 0.1, which turns out to be $\sqrt{I} = 0.22$ , regardless of the reference area. If we compare FTH and HERALDO line reference from Fig. 4(d), in order to achieve 2 pixels of resolution at SNR = 0.1 and $σ_{I}^{2} = 832$ , minimum amplitudes of $\sqrt{I} = 0.068$ and $\sqrt{I} = 0.115$ for FTH and HERALDO line reference respectively are required. Thus, if the detector noise is dominant, it can be concluded that the same resolution and SNR can be achieved with FTH by a light source with lower photon flux.

5. Conclusion and outlook

This study provides a comparison of the noise response between various holographic imaging techniques. Our numerical study, taking shot noise and a detector noise floor into account, unveiled that:

HERALDO is more sensitive than FTH to a uniformly distributed noise floor which is typical to CCD cameras. The sensitivity of HERALDO against this type of noise is higher for two-dimensional references, e.g. rectangle, compared to one-dimensional (linear) references. In addition, the signal-to-noise ratio can be significantly reduced by using the correct implementation of the necessary differentiation steps for object reconstruction.
Another way for improving SNR is to suppress the signal-independent, uniform noise and when the variance of this noise is sufficiently small HERALDO provides a higher SNR than FTH.
At low photon flux when the detector noise is dominant, FTH is the optimum choice since it can yield a higher SNR even with a reasonably small pinhole. On the other hand, at high photon flux where shot noise dominates, HERALDO is more beneficial both in SNR and resolution because the resolution in HERALDO in the direction of the derivatives is solely determined by the steepness of the edge, compared to FTH where the pinhole size determines the resolution in all directions.

Thus, extended references are not always the optimum choice for holographic imaging schemes. In many applications, for example in time-resolved pump-probe experiments with limited photon flux from femto-slice synchrotrons [2,27] or table-top HHG sources [28], conventional FTH will provide a much better SNR than HERALDO. This will also be the case for single-shot imaging at laser-driven X-ray sources [3]. As reported in [22,29], the SNR and resolution obtained from HERALDO can be enhanced by implementing the phase retrieval algorithm using the reconstruction of HERALDO to determine the support area. In the future, our findings will allow for an optimum choice of the holographic imaging scheme to be applied. This will enable high resolution imaging of nanoscale objects with highest spatial and temporal resolution with implications on solid state physics, material science, chemistry, and many other fields.

Appendices

A. Appendix 1

Throughout the derivations presented in this manuscript, we will quite often come across series of the form

\sum_{k = 0}^{M - 1} \exp (\frac{i 2 π n}{M} (k - k_{c}))

for n = 0, 1, 2, …, M − 1 and k_c is a constant. In particular n = 0 corresponds to the central pixel in the object plane. The reconstructions which we are interested in are of course located some distance away from the center in order to avoid overlaps. Since, it’s these reconstructions which we will focus on, we can rule out the case of n = 0 in the above series. With n = 0 excluded, we can use the usual formula for geometric series to evaluate the sum

\sum_{k = 0}^{M - 1} \exp (\frac{i 2 π n}{M} (k - k_{c})) = \exp (- \frac{i 2 π n}{M} k_{c}) \frac{M (\exp (2 π n) - 1)}{\exp (2 π n / M) - 1} = 0

B. Appendix 2

The average value of ℜ[E[m, n]] is

〈 ℜ [E [m, n]] 〉 = - \frac{η_{m} Δ}{M^{2}} \sum_{j k}^{M - 1} K \sin (\frac{2 π}{M} (J m + K n)) = 0

By making use of the trigonometric identity sin(α + β) = sin α cos β + cos α sin β, it can be seen that the above sum does vanish because the sum over j takes the form of the real and imaginary parts of Eq. (23) (see appendix A). At this point we would like to point out that the reason the mean of ℜ[E[m, n]] being zero is because the sum over j vanishes. In the case of rectangular reference, however, the diffraction pattern is multiplied with JK and the sum over j will no longer be zero. 〈ℜ[E[m, n]]〉 will then have certain value which may depend on (m, n). This is why in Fig. 1(i) there are strong intensity regions which cross the pixel array in both vertical and horizontal through the center.

The mean square of ℜ[E[m, n]] is

〈 ℜ {[E [m, n]]}^{2} 〉 = 〈 {(- \frac{Δ}{M^{2}} \sum_{j k}^{M - 1} K η [j, k] \sin (\frac{2 π}{M} (J m + K n)))}^{2} 〉 = - \frac{Δ 2 σ_{I}^{2}}{2 M^{4}} \sum_{j k}^{M - 1} K^{2}

In arriving at the last expression, Eq. (23), 〈η[j, k]〉 = η_m, and

〈 η [j, k] η [j^{'}, k^{'}] 〉 = σ_{I}^{2} δ_{j j^{'}} δ_{k k^{'}} + η_{m}^{2}

have been used. The last expression contains quadratic and arithmetic series whose formula are [26]

\sum_{k = 1}^{M} k^{2} = \frac{1}{6} M (M + 1) (2 M + 1)

\sum_{k = 1}^{M} k = \frac{1}{2} M (M + 1)

We note that Δ is determined by the pixel pitch in the camera and is also related to the pixel pitch in the object plane. For convenience, we choose Δ such that the pitch in the object domain is equal to unit length, this leads to Δ = 2π/M.

C. Appendix 3

The analysis may be started by rewriting Eq. (13) as

R [m, n] = R_{0} [m, n] + \frac{1}{M^{2}} \sum_{j k}^{M - 1} \sum_{p = 1}^{\infty} η [j, k] \exp (\frac{2 π i}{M} (K n + J m)) \frac{{(2 π i K / M)}^{p}}{p!}

with

R_{0} [m, n] = \frac{1}{M^{2}} \sum_{j k}^{M - 1} \sum_{p = 1}^{\infty} I_{0} [j, k] \exp (\frac{2 π i}{M} (K n + J m)) \frac{{(2 π i K / M)}^{p}}{p!}

where we have made the substitution

\exp (\frac{2 π i}{M} (K n + K + J m)) - \exp (\frac{2 π i}{M} (K n + J m)) = \exp (\frac{2 π i}{M} (K n + J m)) \sum_{p = 1}^{\infty} \frac{{(2 π i K / M)}^{p}}{p!}

The infinite sum on the right is the Taylor expansion of exp(2πiK/M)−1. In the triple summation

\sum_{j k}^{M - 1}

\sum_{p = 1}^{\infty}

above, for some small values of K (or equivalently for some k’s in the vicinity of k_c), the power terms (2πiK/M)^p/p! can be well approximated by retaining only the first term p = 1 (this is because with M very big, K/M is very small for some small values of K). As K departs from 0, the contribution from the higher powers of (2πiK/M)^p/p! is becoming more and more significant. In reality, typical diffraction patterns have one property that they drop really fast as one moves away from the central peak, that is, I₀[j, k] diminishes very quickly at some small K. Using this argument, we can approximate the equation for R₀[m, n] above as

R_{0} [m, n] \approx \frac{1}{M^{2}} \frac{2 π i}{M} \sum_{j k}^{M - 1} K I_{0} [j, k] \exp (\frac{2 π i}{M} (K n + J m))

from which we can also identify that 2π/M = Δ. Based on the above equation, one can say that HERALDO with linear polynomial multiplication and with differentiation of the autocorrelation yield reasonably close results in the absence of signal-independent uniform noise. But this is not true when this kind of noise is present because η[j, k] has a constant variance until the edge of the pixel array and consequently, the approximation of taking only the first term in the Taylor expansion of exp(2πiK/M) − 1 will not hold anymore.

D. Appendix 4

Using the results from the previous subsections that 〈ℜ[E[m, n]]〉 = 〈ℑ[E[m, n]]〉 = 0 and the property 〈ϒ[j, k]〉 = 0, it can be seen that 〈ℜ[R[m, n]]〉 = 〈ℜ[R₀[m, n]]〉 and 〈ℑ[R[m, n]]〉 = 〈ℑ[R₀[m, n]]〉. The variances $σ_{ℜ R}^{2}$ and $σ_{ℑ R}^{2}$ now also contain the contribution from the shot noise. To calculate these variances, we note that IFFT[I₀[j, k] + η[j, k]] and IFFT[ϒ[j, k]] are uncorrelated, as can be checked by calculating the covariance between them. We can then again use the sum law for variances. We denote the variances of the real and imaginary parts of IFFT[I₀[j, k] + η[j, k]] as $σ_{ℜ R}^{2} (η)$ and $σ_{ℑ R}^{2} (η)$ and of IFFT[ϒ[j, k]] as $σ_{ℑ R}^{2} (ϒ)$ and $σ_{ℑ R}^{2} (ϒ)$ . So,

σ_{ℜ R}^{2} + σ_{ℑ R}^{2} = σ_{ℜ R}^{2} (η) + σ_{ℜ R}^{2} (ϒ) + σ_{ℑ R}^{2} (η) + σ_{ℑ R}^{2} (ϒ)

We have calculated

σ_{ℜ R}^{2} (η)

and

σ_{ℑ R}^{2} (η)

for three kinds of holographic imaging schemes. In the case of HERALDO with approximate derivative of the autocorrelation,

σ_{ℜ R}^{2} (η) = σ_{ℑ R}^{2} (η) = σ_{l}^{2} M^{2}

(see Eq. (14)). As for

σ_{ℜ R}^{2} (ϒ)

and

σ_{ℑ R}^{2} (ϒ)

, we will take the result derived in [19],

σ_{ℜ R}^{2} (ϒ) + σ_{ℑ R}^{2} (ϒ) = Δ^{2} \sum_{j k}^{M - 1} K^{2} I_{0} [j, k] = Δ^{2} \sum_{j k}^{M - 1} {| K U_{0} [j, k] |}^{2}

with U₀[j, k] is the noise-free Fourier field. The above expression was derived in the HERALDO scheme with linear polynomial multiplication. However, since the mean square of ϒ[j, k] is equal to the diffraction pattern which drops quite fast in the vicinity of the central fringe and so does |U₀[m, n]|, the relation O₀[m, n + 1] − O₀[m, n] ≈ IFFT[iΔKU₀[j, k]] where O₀[m, n] is the noise-free object, has a high degree of validity. Making use of Parseval’s theorem, we can then write

σ_{ℜ R}^{2} (ϒ) + σ_{ℑ R}^{2} (ϒ) = \frac{1}{M^{2}} \sum_{m n}^{M - 1} {| O_{0} [m, n + 1] - O_{0} [m, n] |}^{2}

Putting everything together into Eq. (31), we get

σ_{ℜ R}^{2} + σ_{ℑ R}^{2} = \frac{1}{M^{2}} (2 σ_{l}^{2} + \sum_{m n}^{M - 1} {| O_{0} [m, n + 1] - O_{0} [m, n] |}^{2})

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Noise property in the reconstruction	FTH	HERALDO with pol. multiplication	HERALDO with approx. derivative
Mean (ℜ/ℑ)	0	0	0
Variance (ℜ/ℑ)	$σ_{l}^{2} / (2 M^{2})$	$π^{2} σ_{I}^{2} / (6 M^{2})$	$σ_{l}^{2} / M^{2}$

Influence of detector noise in holographic imaging with limited photon flux

Abstract

1. Introduction

2. Response of FTH and HERALDO to uniform noise

2.1. FTH

2.2. HERALDO with linear polynomial multiplication

2.3. HERALDO with approximate derivative of the autocorrelation

3. Comparison on the noises in FTH and HERALDO

4. Calculation of the total noise

5. Conclusion and outlook

Appendices

A. Appendix 1

B. Appendix 2

C. Appendix 3

D. Appendix 4

References and links

Cited By

Figures (4)

Tables (1)

Equations (37)

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