1. Introduction

Ghost imaging is a technique that nonlocally produces an image of an object. In this technique, the output of a (partially) coherent light source is split into two: one beam hits the unknown object after the fixed distance D and the total transmitted photon flux is recorded; the intensity distribution of the second beam is captured by a 2-D camera after the same optical path length D. The spatial information of the object is coded in the measured 2-D speckle pattern correlated with the total transmitted flux. The first successful ghost imaging experiment, conducted in 1995, relied on entangled photon pairs generated by spontaneous parametric down conversion (SPDC) [1]. Following this, Bennink et al. [2] presented an experimental demonstration of ghost imaging using a pseudothermal light source.

Ghost imaging using pseudothermal light sources has been extensively studied in recent years [3–6]. Because of the experimental robustness and simplicity of this technique compared with those using entangled photon sources, it is suitable for imaging applications. However, pseudothermal ghost imaging has the disadvantage of relatively low resolution at high visibility conditions of the speckle structure. Many strategies are available that can significantly improve the visibility in pseudothermal ghost imaging, such as higher-order ghost imaging [7–9]. Recent studies [10–12] have shown that high-order intensity correlation can more significantly improve the visibility or resolution of the image than low-order intensity correlation. Further, the visibility can be improved using other methods such as intensity fluctuation-fluctuation correlation of chaotic-thermal light [13,14], especially in noisy environments. Using the speckle manipulation method to enhance the visibility has gained more interest in recent years. For example, increasing the size of the speckle [15–20] produced from a pseudo-thermal light source can generate high quality reconstruction. In contrast, decreasing the speckle size results in higher resolution, but with reduced visibility [21–23]. Thus, it is clear that a tradeoff between resolution and visibility is inevitable for high-quality imaging. The influence of detector noise on ghost imaging was studied recently [24]. However, this study did not consider the influence of detector noise on visibility with object band width below 200 µm. Under these macroscopic conditions, the visibility increases with the square root of the speckle size. For macroscopic objects, the transmitted signal intensity is much higher than the detector noise, which can be safely ignored. In microscopic ghost imaging, where the object size and speckle size are comparable, the noise level must be considered because the small transmitting area results in low signal in the detector, whereas the noise level remains constant.

In this study, we report the limitations of the measured contrast-to-noise ratio (CNR) in microscopic ghost imaging using a pseudothermal light source. In our experiments, we controlled the speckle size using an aperture to regulate the beam size of the incoming pseudothermal light [15]. A scaling of the CNR different from that of previous experiments was observed with the increase in speckle size. After the expected initial increase in the CNR, a sudden drop was observed, if the microscopic objects were imaged. To study these observations, we developed a numerical model and obtained an upper limit for the speckle size without impairing the CNR based on the noise level of the photodiode. Thus, high-quality ghost imaging microscopy can be realized only if the speckle size is carefully adjusted for a given object. This research greatly impacts nanoscale microscopic imaging, especially high-resolution XUV/X-ray imaging, where ghost imaging can be a potential alternative for low dose imaging of biological samples to avoid radiation damage [25].

2. Experimental results and discussion

2.1 Methods

Pseudothermal ghost imaging relies on correlating a measured 2-D intensity pattern (reference path) to a transmission scalar value (bucket value) of an object, illuminated by the same speckle pattern (object path). By varying the speckle pattern over time and simultaneously measuring the transmitted light through the object, a statistical reconstruction of the image is possible. Therefore, the measured 2-D pattern is weighted with the corresponding bucket value in every iteration and is added to the ghost image. Thus, the quality of reconstruction depends directly on the number of iterations and the correlation between the pattern and bucket value. Disturbing the optical elements in one of the beam paths results in a decorrelation of the intensity pattern and the bucket value.

In addition, the algorithm used in this work, which is published in [18,26], considers the nonlinear background and fluctuations of the light source, given by:

To estimate the spatial resolution of the retrieved ghost image, the slanted-edge method was used. An area of the retrieved ghost image was chosen such that it is equally covered by signal and background (Fig. 1). The pixel values parallel to the edge of the objects were averaged to a line-out perpendicular to the edge because of the noisy image. By fitting an error function ${f_{err}}(x )= A\int_0^x {{e^{ - B{\tau ^2}}}d\tau }$, with fitting parameters A and B to the measured line-out and applying the 10% to 90% criterion, a reliable resolution R can be calculated.

 

Fig. 1. Resolution measurement of a ghost image. Because of the noisy image, all line-outs must be calculated by averaging the pixels parallel to the edge of the objects. By fitting the line-out with an error function and applying the 10% to 90% criterion, the resolution R can be retrieved.

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2.2 Experimental setup

The experimental setup to study the scaling of the CNR with speckle is shown in Fig. 2(a). A Helium-Neon laser (λ = 632.8 nm) and a rotating diffuser were used to generate a pseudo-thermal light beam. The rotating diffuser is in the focal plane of the lens of focal length f = 150 mm. From $\delta {x_0} = {{\pi \lambda f} \mathord{\left/ {\vphantom {{\pi \lambda f} {{\omega_0}}}} \right.} {{\omega _0}}}$ [27,28], different average speckle sizes δx₀ can be obtained by changing the beam waist (ω₀), by widening the aperture from 0.5 mm to 2.25 mm. The generated speckle field is bifurcated by a beam splitter into two spatially correlated beams, namely, transmitted (reference path) and reflected (object path) beams. The object beam hits the object (µ), which is followed by an integration of the transmitted light by a photodiode, thereby producing the bucket signal. The laser power required to reach the photodiode is in the order of a few microwatts. It must be noted that the reference beam does not interact with the object, and its speckle pattern is recorded using a CCD camera, whose specifications are 1200×1600 pixels and an effective pixel size of 5.86 µm × 5.86 µm << δx₀. Ghost imaging with randomly distributed fields is possible only if the propagation lengths of the two beam paths are the same (beam splitter to object and beam splitter to CCD camera), to conserve correlations. Here, the error in propagation lengths between the beam paths is in the order of a few millimeters.

 

Fig. 2. (a) Schematic of the microscopic ghost imaging experimental setup with variable speckles size generation (aperture + diffusor). (b) Objects used; because of the laser-cutting fabrication process, the object sizes must be validated by conventional microscopy.

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The µ-shaped transmitting objects were laser-cut from a 100-µm-thick black-coated aluminum foil to ensure stability and full absorption at the outer areas. Figure 2(b) shows the designed objects and the corresponding results obtained from laser cutting. The widths of the vertical bar of letter “µ” in objects 1, 2, and 3 are 120 µm, 60 µm, and 30 µm, respectively. Correspondingly, the areas of objects 1, 2, and 3 are calculated as ∼180 × 10³ µm², ∼77 × 10³ µm², and ∼22 × 10³ µm², respectively.

2.3 Experimental results and discussion

For every object, the speckle size was increased from 17 µm to 63 µm in 8 steps, followed by reconstruction with 10000 iterations under the same conditions. Starting with the macroscopic object 1, the retrieved speckle size dependent CNR was found to be in good agreement with [15–20], which indicates a growth of the CNR proportional to $\sqrt {\delta {x_0}}$. By reducing the size of the object into the microscopic regime, where the speckle size became comparable to the smallest features of the object, the CNR scaling exhibited a sudden change. Using object 2, with features in the speckle size range, it was observed that the CNR showed an increasing trend for small speckle sizes until it reached a maximum, following which it dropped continuously, as shown in Fig. 3. To our knowledge, the observed scaling depicted in Fig. 3 has never been reported in the literature. This indicates, for the first time, that an optimal speckle size exists in microscopic ghost imaging, using which the maximum CNR can be achieved for the given object size. Moreover, this scaling is easily observed from the ghost images represented by false color, in the inset of Fig. 3.

 

Fig. 3. Exemplary retrieved CNR vs. speckle size for object 2. After a maximum at ∼45 µm speckle size, the CNR begins dropping.

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It must be noted that the retrieved ghost images that are clearly visible to the naked eye are not identical to the images with the highest CNR, because the definitions of visibility and CNR are different. Visibility is a measure of image quality based on the total signal-to-background noise difference, whereas the CNR considers the full image noise (given by Eq. 2). Therefore, images with the highest visibility and highest CNR are observed for a different speckle size.

To clearly understand and generalize the influence of speckle size on the CNR of the retrieved ghost images for an object of a given size, we evaluated the CNR as a function of the number of speckles N_obj falling on the object. We defined ${N_{\textrm{obj}}} = {{{A_{obj}}} \mathord{\left/ {\vphantom {{{A_{obj}}} {{A_{spe}}}}} \right.} {{A_{spe}}}}$ as the ratio of the transparent area of the object to the average speckle size, which is related to the speckle size by $\delta {x_0} = 2 \times {({{{{A_{spe}}} \mathord{\left/ {\vphantom {{{A_{spe}}} \pi }} \right.} \pi }} )^{{1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}}}$. In Fig. 4 (black lines), we summarize the dependence of the CNR on N_obj for the three above-mentioned samples. In Fig. 4(a), for the largest object, the retrieved CNR immediately starts decreasing if N_obj is higher than a minimum value of approximately 60. The N_obj value cannot be reduced further by increasing the speckle size (by reducing the aperture size) because of the resulting low intensity of the speckle field. However, A_obj can be decreased to obtain a smaller N_obj, by inserting samples 2 and 3, whose transparent area is smaller. As shown in Fig. 4(b), the maximum CNR is observed at a lower N_obj of 53; for N_obj above this threshold, the CNR of object 2 decreases with the increase in N_obj. However, in Fig. 4(c), the CNR of object 3 increases until N_obj remains below 55. This clearly indicates that the CNR does not always increase with the increase in speckle size. The best CNR of an object can be achieved if the N_obj is in the range 53–55 in our setup, indicating that the CNR can be quantitatively optimized for microscopic ghost imaging.

 

Fig. 4. CNR as a function of the number N_obj of the speckles falling onto the object 1 and the test slit 1. (a) object 2 and test slit 2; (b) object 3 and test slit 3; (c) The resolutions (green bars) of the retrieved ghost images of the “µ” shaped objects increase with higher N_obj. Note: ghost images of an object are imaged in the same dynamic range.

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Further, Fig. 4 shows that the mean CNR of object 3 is lower than that of objects 1 and 2. This can be attributed to the following causes: the CNR decreases rapidly with increasing speckle size for microscopic objects because the bucket intensity decreases and nears the detector noise level. Therefore, the bucket signal for weighting the speckle pattern in the image retrieving algorithm is no longer correct because it is now the sum of the true bucket signal and is no longer a negligible noise signal. Comparing the results with different N_obj, the following conclusion was derived: the non-linear scaling of the retrieved CNR strongly depends on the ratio between object size and speckle size of the pseudothermal light for the same number of independent iterations.

In addition to the CNR, the resolution of the retrieved images was evaluated. The trade-off between resolution and CNR mentioned previously is shown in Fig. 4. Because of the increase in resolution with smaller speckle sizes, which is independent of the dimension of the objects, the trade-off must only be considered for macroscopic objects, as shown in Fig. 4. On the contrary, in the case of microscopic objects, because of the limited detection capabilities of the system, decreasing speckle size results in higher CNR and higher resolution.

As seen from the measurements (Fig. 4) and the simulation (Fig. 5), the resolution converges to a value that is limited by the pixel size of the image detector. The blurred sharp edge of the object in the image is the result of a convolution between the edge and the average Gaussian speckle function. However, reducing the average speckle size below the pixel size does not improve the resolution. Further, the estimated resolution varies slightly for different measurements because of the finite number of recorded speckle fields. Thus, in Fig. 4 we observe for certain speckle sizes a deviation from the overall trend which can be considered as outliers. The overall trend agrees very well with expected behavior as shown in Fig. 5.

 

Fig. 5. Results of the noise dependent CNR simulation. (a) Retrieved ghost images as a function of detector noise and number of speckles falling onto the object N_obj; (b) Calculated CNR of the ghost images as a function of N_obj for different relative noise levels. In Fig. 5(a) and 5(b) the same colors are used for the same RNL. With the increasing detector noise, the CNR starts to drop for larger speckle sizes, marked with the dashed white boxes in the reconstructions (maximum of CNR) in (a). For negative RNL, the CNR follows a growth in a square root scaling, whereas with increasing noise, the CNR starts to follow a concave function. The resolution which is marked by green dashed line, increases with the number of speckles corresponding to the experiment.

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To verify that our observations are very general, i.e., independent of the specific shape of the sample, the measurement was repeated with a different sample with the same A_obj. To create an alternative object with an adjustable open area, we used two perpendicular aligned slits forming a rectangular transparent window. The width of the vertically aligned slit was adjustable, whereas the horizontally aligned slit had a fixed width of 150 µm. Thus, the total transmission of our object was easily adjusted and artifacts caused by small irregularities of the laser-cut “µ” shape were avoided. For comparison, we set the width of the first variable slit to realize the same transparent area for objects 1, 2, and 3, which was approximately 1200, 517, and 147 µm, respectively. Figure 4 (blue lines) shows the experimental results of the CNR for the test slits. For comparison, we show the ghost images obtained for different slits which were retrieved by 10000 independent iterations of the speckle field. The agreement of the estimated CNR vs. N_obj for the two completely different shaped objects with the same transparent area was remarkably good. The slight discrepancy observed maybe because of some uncertainties in the estimated area of the “µ” and/or small errors in setting the slit width. However, our results rule out the possibility that this behavior is a result of a specific structure of the object.

2.4 Simulation results

To validate the experimentally observed sudden drops in CNR at a certain speckle size, a numerical model was developed to simulate the observed behavior. This model can independently vary the relevant parameters over a wide range to identify their role in the observed scaling of the CNR for large speckle sizes. First, we implemented a function to numerically generate random speckle fields. In the random phase field, Ф, of the required dimension N × N all the signals have the same amplitude to simulate a transparent phase-only diffusor as that used in the experiment. The Fourier transform of the input field, $\widetilde \phi$ is multiplied with a Gaussian filter function $g({\overrightarrow r } )= \exp [{ - ({{{\overrightarrow r }^2}} )/2{\sigma^2}} ]$, where $\overrightarrow r$ is the radial vector origin from the image center. Note that the parameter σ is proportional to the inverse of FWHM of the speckle size. Calculating the back-propagation via an inverse Fourier transform yields a complex speckle field Ф_s at the position of the object and camera for each iteration of the algorithm. Here, the Gaussian filter function in the far-field works in the same way as a circular transmission function in the near-field and was chosen based on existing standards. In the simulation, we varied the speckle size in the range 5–100 µm to agree with the experimental parameters. For the object path, Ф_s was multiplied with the object transmission function and further propagated to the single-pixel detector. Here, the individual pixels were threshold-filtered, i.e., only the signals that are above the adjustable noise level were summed, resulting in the bucket value. To quantify the detector noise relative to the measured intensities, we introduced the relative noise level (RNL), which is defined as follows:

Figure 5 shows the reconstruction figures for 10 different RNLs ranging from –0.11% to 1.08% and 25 different speckle sizes. A negative RNL does not affect the CNR as concluded from similar reconstructions shown in Fig. 5(a) and the CNR evolution in Fig. 5(b); this follows the theoretically predicted square root law [15–20]. If the RNL is positive, CNR increases with the speckle size up to a certain value before it starts to drop, as observed in the experiments. This behavior can be attributed to the decorrelation between the bucket value, which is strongly modified by the noise, and the reference field whose noise contribution is negligible. Incoming low-signal fields, with intensities below the noise level, are detected with the same bucket value, resulting in a comparable weighting during the reconstruction. The algorithm cannot distinguish different low-intensity fields, which leads to an accentuation of the fields containing no object information, resulting in an incoherent stacking of the measured fields. As a result, the reconstruction is blurred and the CNR drops. With the increase in noise level, this effect becomes increasingly prominent for fields with smaller speckle sizes, resulting in a shift of the maximum CNR to smaller speckle sizes. The same effect can be attained by choosing smaller objects. The lower transmission reduces the margin between typical signal levels, and the noise background of the detector results in a drop of the CNR. By comparing the simulation results with the experimental results described above, three different regimes are identified for a given combination of speckle size–object: macroscopic objects (object 1, Fig. 4(a)) exhibit features larger than the speckle size resulting in a growth of the CNR for larger speckles. The transition from the macroscopic to the microscopic regime (object 2, Fig. 4(b)) is observed for an object with features in the range of the speckle size. Here, the detector noise is considered, resulting in the CNR drop. Microscopic objects (object 3, Fig. 4(c)) are characterized by a monotonic decrease of the CNR with larger speckles because of the influence of the detector noise in the whole range.

3. Conclusions

We analyzed the role of speckle size, object size, and detector noise on the maximum achievable CNR and resolution in microscopic ghost imaging. From our experiments, we obtained the scaling laws for the achievable CNR and resolution of the retrieved ghost images, which strongly depends on the ratio of the object size and the speckle size of the pseudothermal light for the same number of independent iterations. The same CNR scaling was observed for two different shaped objects with the same transmission, namely, rectangular aperture and a more complex structure, indicating, that the CNR is shape independent. For all samples, the highest resolution was observed at the smallest speckle sizes, which results in a trade-off between resolution and CNR for macroscopic objects, whereas microscopic objects show a common growth of these quantities. The resolution itself is limited by the pixel size of the imaging device and is comparable to the resolution achieved with shadow grams of macroscopic objects.

An analysis of our simulation unambiguously identified the noise level of the photodiode as the major limitation for the growth of the CNR for large speckle sizes. To the best of our knowledge, the CNR behavior is observed for the first time in the field of microscopic ghost imaging. When imaging small objects with low transmission, the detector noise must be considered. The novel CNR scaling found in this research will be of great importance in future experiments on microscopic ghost imaging in regulating the speckle sizes to obtain the best visibility and resolution of reconstructions.