1. Introduction

Due to the properties of X rays, such as higher penetrating power and shorter wavelengths compared to visible light, the field of X-ray microscopy has attracted a lot of research and has a wide range of applications in various fields of study [1–3]. As a result, many different types of techniques have been developed that achieve micron and nano-level spatial resolution [4–6]. A well known type of X-ray imaging is standard projectional radiography with hard X-rays ($> 10$ keV) due to its simplicity and common use in the medical field [7,8]. This technique is also commonly used in X-ray microscopy, including the slightly adjusted form of projection-based phase contrast imaging [9]. The resolution of this projection is closely dependent on how small of a source is used, meaning a “perfect,” magnified image can be obtained with a “perfect” point source. With the help of the state-of-the-art techniques, focusing the X-ray beam down to 30 nm has been achieved; producing a projection with an equivalent resolution [10]. Unlike lens-based imaging mechanisms, this technique cannot take advantage of the wave nature of radiation to achieve diffraction-limited point-to-point imaging resolution.

Focusing optics such as imaging lenses play a critical role in producing a diffraction-limited point-to-point correlation between the object plane and the image plane, forming a magnified or demagnified image of the object [11–13]. If desired, an additional lens system is then able to map the primary image onto a secondary image plane for further magnification or demagnification; notably making an optical microscope possible. While lens-based optical microscopes are commonly used to obtain high-resolution images, the relatively longer wavelengths of visible light limits this to $\sim$200 nm resolution [11,12]. Unfortunately, unlike visible light, X rays experience a refractive index close to 1 for many materials such as those used in traditional imaging lenses, making such materials useless for focusing the X-ray beam [7,8]. To overcome this, some alternative focusing X-ray optics have been developed such as compound refractive lenses, focusing mirrors, and zone plates [14–16]. While zone plates and focusing mirrors are typically limited to soft X rays ($< 10$ keV), compound refractive lenses can be designed for hard X rays ($> 10$ keV). Unlike projectional radiography, which is a projection, or “shadow,” of the X rays that pass through the object, focusing X-ray optics produce high-resolution images by taking full advantage of the short wavelength of the X rays used. Most notably, this has been demonstrated by producing 10 nm resolution images with soft X rays [16]. We would expect even higher resolution with higher energy levels but limitations in the fabrication of the focusing X-ray optics are the limiting factor. Even as advances are made in focusing X-ray optics, it is difficult for classic imaging devices such as these to obtain magnified, high-resolution images of the interior structure of a thicker object. Recording the resulting image on a photodetector array relies on magnification produced by these focusing X-ray optics or an optical lens following the scintillator. Due to the fact that object planes further away from the focal plane do not achieve as much magnification, the magnification may not be enough for a sensor array to capture the desired resolution.

For the sake of comparison, it is worth noting that other X-ray microscopy techniques exist. This mainly includes coherent X-ray diffraction imaging (CXDI) and a similar technique of X-ray ptychography, both of which can theoretically provide diffraction-limited resolution without the need of a lens [6,17,18]. These techniques utilize the measurement of a diffraction pattern following the object that then requires post processing to recover the image. Similar to these techniques is lensless Fourier-transform imaging which also requires post processing [19]. Projection-based X-ray phase contrast imaging was already mentioned, but the larger, encompassing field of phase contrast microscopy is well studied [20,21].

In this article, we discuss the physics of two-photon ghost imaging to produce a diffraction-limited, lensless, image-forming correlation between the image plane and object plane, for which X rays allow imaging of the internal structure of the object [22–25]. Due to the diffraction-limited nature of this imaging process, sub-nanometer resolution can theoretically be achieved but will be limited by physical constraints of the imaging system. One key example of this is the fact that the ghost imaging process itself does not provide any magnification to allow for photodetectors (pixels of a CCD or CMOS) with relatively significant physical size to capture the full resolving power of the imaging process. To account for this, we propose the help of a secondary imaging device, either focusing X-ray optics or a scintillator paired with a visible-light lens assembly. With secondary imaging, the primary lensless ghost image can be mapped onto a secondary image plane with significant magnification to be resolvable by a standard CCD or CMOS. Although the addition of secondary imaging will introduce its own restrictions on resolution (among other fabrication restrictions) compared to X-ray ghost imaging on its own, we may label this setup as an X-ray ghost microscope as it produces a magnified, high-resolution image of the internal structure of an object. One interesting result of the X-ray ghost microscope is that the secondary imaging system is focused on the primary ghost image rather than the physical object. This removes the common constraint encountered in classic X-ray microscopy with focusing X-ray optics: in order to get high magnification from the setup it is desired to have the object as close to the focal point as possible, but the short focal lengths of focusing X-ray optics makes it impossible to do this for the internal structure of larger (thicker) objects [16].

To avoid confusion with other discussions of ghost imaging, it is worth discussing the different types of ghost imaging that have been developed. Visible-light ghost imaging was first demonstrated in 1995 using two-photon interference from entangled photon pairs through the measurement of the second-order coherence function [26]. In addition, this has been demonstrated recently with entangled X-ray photons [27]. 10 years after the initial, 1995 demonstration, it was demonstrated that two-photon interference of randomly created and randomly paired photons in a thermal state can produce a similar point-to-point image-forming correlation without the need of an imaging lens, commonly called lensless ghost imaging [22–25]. Interestingly, the first lensless ghost imaging of thermal light was observed in the secondary ghost imaging plane with the help of a lens, providing the inspiration for this paper. An unfolded schematic diagram of the 2005 ghost imaging experiment of thermal light is illustrated in Fig. 1. X-ray ghost imaging has now been demonstrated multiple times [28–35]; however, most demonstrations did not take advantage of two-photon interference to produce an image. The most commonly used technique for current X-ray ghost imaging is to introduce a spatially varied material in the X-ray beam to produce an artificial “speckle” pattern instead of quantum speckles inherently present in the beam through two-photon interference. This results in a classical speckle-to-speckle ghost imaging with resolution dependent on the artificial “speckle” size as opposed to being diffraction-limited.

 

Fig. 1. Experimentally achieved with a beam splitter, this “unfolded” schematic of the 2005 thermal light ghost imaging demonstration of Valencia et al. helps depict the symmetry of the two-photon amplitudes. The one-to-one lensless ghost image and the magnified secondary ghost image are results of two-photon interference. Of the many two-photon amplitudes, such as the red and blue two-photon amplitudes of the random $m$th and $n$th pair of photons shown here, each superpose constructively at a corresponding $\vec {\rho }_{i}$ to from an image.

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Classified according to their experimental setup and working mechanism, we can group the two classes of ghost imaging: (I) The observed ghost image is produced from a natural point-to-point image forming correlation that is the result of two-photon interference. This class of ghost imaging follows the mechanism of the entangled state ghost imaging of Pittman et al. [26] and the thermal light ghost imaging of Valencia et al. [22]. With thermal light ghost imaging, the point-to-point image-forming correlation is able to achieve a diffraction-limited spatial resolution of $\lambda / \Delta \theta _s$, where $\Delta \theta _s$ is the angular diameter of the light source, and also has the potential to be turbulence-free [13,36–38]. Pelliccia et al. have used type (I) ghost imaging in their first demonstration of X-ray ghost imaging [29]. Alternatively, (II) the observed ghost image is produced from a classical speckle-to-speckle correlation shown with visible light [39–41] and with a variety of X-ray sources [30,32–35]. Here, a ghost image of the object is obtained from the coincidences between two sets of identical, artificial “speckles” formed by either spatially correlated laser beams or an aperture mask following the light source to produce shadows and bright spots distributed on the object plane and on the ghost image (detector) plane. Included as a classical speckle-to-speckle correlation is computational ghost imaging which removes the need of a beam splitter by measuring the object plane and ghost image plane at separate times with an identical speckle pattern [28,31,42]. These artificial speckles are different from the two-photon interference that produces a point-to-point image-forming correlation and, correspondingly, this type of ghost imaging has a resolution dependent on the size of the classically formed speckles. Through the correlation measurement, this type of ghost imaging observes a projection or a shadow of the object, comparable to how classic X-ray imaging technology is a projection of the object. Both classes of ghost imaging use the measurement of intensity fluctuation correlation to obtain an image, utilizing either changes in the artificial speckle distribution over time or fluctuations resulting from two-photon interference.

The X-ray ghost microscope studied in this article belongs to type (I) ghost imaging via two-photon interference. As with lensless ghost imaging, the spatial resolution of X-ray ghost imaging before magnification is determined by $\lambda / \Delta \theta _s$, where $\lambda$ is the wavelength and $\Delta \theta _s$ is the angular diameter of the X-ray source. This unique characteristic of lensless ghost imaging makes the resolving potential of an X-ray ghost imaging a point of interest. As an example, ignoring potential physical constraints, using a high-energy ($>$20 keV) X-ray source with a relatively large angular diameter may produce a one-to-one (no magnification or demagnification) lensless ghost image with sub-nanometer resolution. It should be noted that achieving higher resolution with a source with a larger angular diameter is the opposite of what is desired for projectional X-ray imaging, phase contrast imaging, and CXDI which all see an increase in resolution when using a smaller, point-like source, such as modern X-ray tubes and synchrotron X-ray sources [7,43]. Type (I) ghost imaging is still possible with these small sources, but the resolution will be reduced. When expecting a ghost image with even sub-micrometer resolution, the image becomes unresolvable by any state-of-the-art CCD or CMOS detector arrays. However, this primary ghost image can be magnified significantly onto a secondary ghost image plane through either focusing X-ray optics or a scintillator-lens system, making sub-micron (potentially even 10 nm [16]) sized features resolvable by a CCD or CMOS with micrometer pixels. Importantly, the secondary ghost image is produced directly by the point-to-point image-forming correlation between the object plane, $\vec {\rho }_o$, and the secondary ghost image plane, $\vec {\rho }_i$, where $\vec {\rho }_o$ and $\vec {\rho }_i$ represent the transverse coordinates of the object plane and image plane, respectively. An additional benefit of two-photon interference is that this X-ray ghost microscope can be set up such that it is insensitive to any rapid phase variations along the optical path due to random changes in composition, density, length, index of refraction, or medium vibration, namely “turbulence-free” [36–38]. Overcoming vibrations is especially important for the extremely high resolution imaging obtainable with the X-ray ghost microscope.

This article is organized in four sections: Sec. 2 we introduce the concept of two-photon interference induced intensity fluctuation correlation, Sec. 3 we show a point-to-point image-forming correlation for producing a lensless ghost image, Sec. 4 we extend this ghost image to a secondary image using either focusing X-ray optics or a scintillator-lens system, and Sec. 5 we discuss how to obtain observable intensity fluctuation correlation when using an X-ray source with a broad spectrum.

2. Two-photon interference induced X-ray intensity fluctuation correlation

Recall, in 1905, Einstein introduced a granularity to radiation, abandoning the continuum interpretation of Maxwell [44]. This led to a microscopic picture of radiation and a statistical view of light, including X rays. In Einstein’s picture, a light source consists of many point-like sub-sources, each of which emit their own subfields. Originally labeled in German by “strahlenbündel,” which translates to “bundle of ray” in English, we now label these bundles of light as subfields or the quantum term of photons. For a thermal light source, these subfields (photons) are emitted in a random manner such that the $m$th subfield emitted from the $m$th point-like sub-source may propagate in all possible directions with a random initial phase. It has been shown that an effective wavefunction can be defined from the quantum theory of optical coherence [45,46] to specify the space-time behavior of a photon [13,47]. The effective wavefunction of a photon in a thermal state is mathematically the same function as Einstein’s subfield. In Einstein’s picture, the radiation measured at coordinate $(\boldsymbol{r}, t)$ is the result of a superposition of a large number of subfields,

The X-ray ghost microscope studied in this article utilizes the two-photon induced intensity fluctuation correlation $\langle \Delta I(\boldsymbol{r}_1, t_1) \Delta I(\boldsymbol{r}_2, t_2) \rangle$ for type (I) ghost imaging. Although the expectation or ensemble average of the intensity fluctuation measured by a single detector $D_j$, $j = 1, 2$, is zero, $\langle \Delta I(\boldsymbol{r}_j, t_j) \rangle = \langle \sum _{m\neq n} E^*_m(\boldsymbol{r}_j, t_j) E_n(\boldsymbol{r}_j, t_j) \rangle = 0$, the expectation or ensemble average of the correlation of the intensity fluctuations measured by $D_1$ and $D_2$, jointly, may not equal zero,

3. Lensless ghost imaging with X-rays

To better visualize the X-ray ghost microscope it is best to start with the working mechanism of X-ray lensless ghost imaging. This simple experimental setup, schematically illustrated in Fig. 2, consists of an X-ray beam splitter [48, 49] that divides the X-ray beam from a disk-like source into two beams. A 2D array of X-ray detectors, $D_1$, is placed in beam-one at a selected plane of $z_1 = z_{gi}$ in the Fresnel near-field. Following the output of beam-two, we place an object followed by a bucket X-ray photodetector, $D_2$, which collects all X rays transmitted from the object. The Green’s function that propagates a subfield from point $(\rho_0,z_0,t_0)$ to point $(\rho, z, t)$ is thus

 

Fig. 2. X-ray ghost imaging. A beam splitter (most likely a crystal aligned to utilize Laue diffraction which would not provide 90$^{\circ }$ separation as depicted) creates two paths for the beam, one directed passing through the object and followed by a bucket detector, and the second directed at a 2D photodetection array (CCD or CMOS). The array is placed on the desired ghost image plane, $z_1 = z_{gi}$, corresponding to different “slices”, or cross sections inside the object: $z_{gi} = z_o, z_{gi}' = z_o', z_{gi}'' = z_o'',$ etc.

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Unless extreme magnification or point-like pixels are used, we can approximate this point-to-spot correlation (somb-function) as a point-to-point correlation (delta-function) between $\vec {\rho }_{gi}$ and $\vec {\rho }_o$ to simplify the math. This point-to-point correlation, or ghost image forming function, identifies a unique internal plane of the object, $z_{gi} = z_o$, with the X-ray ghost image observed from the X-ray intensity fluctuation, or photon number fluctuation, correlation. With the point-to-point approximation and $z_{gi} = z_o$, we may approximate the X-ray intensity fluctuation correlation as

This result also indicates the reproduction of a 2D X-ray ghost image of the internal transverse cross section of $z_o = z_{gi} = z_1$. In other words, the longitudinal position of the 2D X-ray detector array, $z_1$, selected an object plane, $z_o = z_1$, of the internal structure of the object to image. Interestingly, when the 2D X-ray detector array is linearly scanned from $z_1$ to $z'_1$ to $z''_1$ along the optical axis, the selected object plane will be changed from $z_o = z_1$ to $z'_o = z'_1$ to $z''_o = z''_1$, respectively, as illustrated in Fig. 2. The imaging resolution along the $z$-axis is the coherence time of the light, similar to optical coherence tomography [56]. By scanning the position of the 2D X-ray detector array, a set of slices of the internal structure of the object can be grouped together to form a 3D ghost image of the object with sub-nanometer resolution. This differs from traditional X-ray computerized tomography (CT) imaging [57] and ghost tomography (GT) demonstrated by Kingston et al. [34] which rely on rotating the object or revolving the detectors around the object 360$^{\circ }$ [58, 59].

4. X-ray ghost microscope

Unfortunately, the sub-nanometer resolution of lensless ghost imaging is unresolvable by any state-of-the-art 2D photodetector array, such as CCD or CMOS sensors which may have micrometer sized pixels, unless photodetector array technology ever advances to sub-nanometer-sized pixels. In order to resolve the two-photon X-ray ghost image with a standard 2D photodetector array, significant magnification is necessary.

4.1 Configuration I

To magnify the X-ray ghost image to a secondary ghost image plane we may follow the optical setup demonstrated by Valencia et al. in 2005 for visible light ghost imaging. The schematic design of this standard X-ray ghost microscope matches what an optical setup would be and is shown in Fig. 3. For this configuration of the X-ray ghost microscope, focusing X-ray optics such as compound refractive lenses, X-ray zone plates, etc. can be used to produce a magnified secondary ghost image [14–16]. Note that focusing X-ray optics may limit the usable photon energies of the X-ray source unless the technology advances to accommodate higher photon energies. As with the primary ghost image, this secondary ghost image is observable from the measurement of the intensity fluctuation correlation. To confirm the nonlocal point-to-point correlation between the object plane, $z_o$, and the secondary image plane of the lens, $z_i$ (where $D_1$ will be placed, $z_1 = z_i$), we adjust Eq. (9) to include the lens system,

 

Fig. 3. The X-ray ghost microscope. Focusing X-ray optics are used to produce a magnified secondary ghost image prior to the 2D photodetection array which allows for a high resolution image of the object to be resolvable by a standard CCD or CMOS. Different primary ghost image planes correspond to different “slices” of the object. By scanning the lens and $D_1$ along the optical axis, we select new $z_{gi} = z_o = d$ planes to be in focus, obtaining a magnified 3D image of the internal structure of the object.

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Due to the high resolution of the primary ghost image, assuming the use of a source with a large angular diameter, the result of the magnified secondary ghost image has identical resolution to an image from a classical X-ray microscope that was obtained with the focusing X-ray optics. Currently, further advancing the fabrication of focusing X-ray optics is a difficult dilemma to overcome (with 10 nm resolution when using a zone plate being state-of-the-art [16]), but as the the fabrication capabilities advance, the potential of this ghost microscope setup will advance as well. In addition to matching the resolution of simply using the focusing X-ray optics to image the object itself, one major benefit to the ghost microscope setup is that the X-ray optics are imaging the ghost image plane, which isn’t physically present, as opposed to imaging the physical object (hence the name “ghost” microscope). Imaging small or thin objects with a classical X-ray microscope may not be an issue, but often it may be desirable to image the deeper internal structure (e.g. a bone) of a thicker object (e.g. a body) for which the focusing X-ray optics can not physically be close enough to the internal structure to image it with proper magnification and resolution. This would not be an issue for the X-ray ghost microscope as the physical object is placed “nonlocally” on the other path following the beam splitter and is simply followed by a bucket detector.

4.2 Configuration II

If the use of X-ray optics is not possible or not preferred, one can place a scintillator on the primary ghost image plane, $z_{gi}$ to convert it into the visible spectrum, allowing it to then be magnified by a visible-light lens system onto a secondary image plane. This is inspired by many current X-ray imaging setups that use scintillator and visible light lenses. Unlike configuration I which has limitations from the focusing X-ray optics, this design is more suitable for higher energy X-ray imaging, such as $\geq 20$ keV, depending on the scintillator used. The schematic design of the adjusted X-ray microscope is shown in Fig. 4. Unlike the standard X-ray ghost microscope which has a clear path for the two-photon amplitudes from the light source to the detectors, here it is not clear that the scintillator-lens system preserves the result of the two-photon interference, and thus the secondary ghost image. To understand how the secondary ghost image is preserved, we can say that the scintillator essentially “detects” the X rays, thus establishing the presence of the X-ray ghost image on the scintillator plane; although, prior to correlation, this plane is simply a distribution of quantum speckles from interfering photon pairs. The scintillator converts these X-ray fluctuations to the visible spectrum, $\Delta I_{\textrm {X}}(\vec {\rho }_{gi}, z_{gi}) \approx \Delta I_{\textrm {v}}(\vec {\rho }_{gi}, z_{gi})$. The lens system then images this distribution of fluctuations and produces a diffraction-limited magnified image of them on the image plane,

 

Fig. 4. The adjusted X-ray ghost microscope. Nearly identical to the standard X-ray ghost microscope, but now a scintillator is placed on the ghost image plane to convert the X-ray ghost image into the visible spectrum. Now a lens (or lens system) operating in the visible spectrum produces a magnified secondary ghost image.

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One factor that may aid in the high resolving capabilities of the X-ray ghost microscope is the optical stability of two-photon interference. It has been shown that ghost imaging and other two-photon interference phenomena can be set up to achieve turbulence-free measurements [36–38,51]. This is achieved when the superposed two-photon amplitudes experience the same turbulence and medium vibrations along their optical paths, meaning any composition, density, length, refractive index, or medium vibration induced random phase variations along the optical paths do not have any effect on each individual two-photon interference. While optical turbulence may not be prominent in the X-ray regime, this technique also includes small vibrations of the object or detectors that would become more likely to blur a classical image at the resolutions discussed.

5. Achieving observable two-photon interference with broad-spectrum X-ray sources

The intensity fluctuation correlation utilized for the X-ray ghost microscope is directly related to the second-order correlation function, $G^{(2)}(\boldsymbol{r}_1, t_1; \boldsymbol{r}_2, t_2)$, of X-rays in a thermal state [13,45–47]. This second-order correlation function follows a quantum description of light, so we use the quantized notation of photon numbers, $n(\boldsymbol{r}_j, t_j) \propto I(\boldsymbol{r}_j, t_j)$ and $\Delta n(\boldsymbol{r}_j, t_j) \propto \Delta I(\boldsymbol{r}_j, t_j)$ for $j = 1, 2$. Up to this point, we have assumed perfect temporal correlation to allow us to emphasize the spatial portion of this function. Now we will do the opposite and focus on the temporal aspect of the measurement. Similar to ghost imaging with sunlight, we have to face the problem caused by the extremely broad spectrum of X-ray sources and relatively slow detectors. Due to the limited ability of the photodetectors in determining the registration time of a photoelectron, the response time of the photodetectors and the associated electronics will affect the measurement of $G^{(2)}(t_{1}-t_{2})$, where $t_j$ is the registration time of photodetectors $D_1$ and $D_2$. For instance, due to the slow response time of the photodetectors, relatively speaking, the measured second-order correlation may have a much wider temporal width and much smaller amplitude. Note that this also holds true with the relatively slow decay time of the scintillator in configuration II. We may characterize this uncertainty as a response function of the photodetector, $D(\tilde {t}_j - t_j)$, where $t_j$ is the photon annihilation time and $\tilde {t}_j$ is the electronic registration time. Thus, the joint photodetection measurement of $D_1$ and $D_2$ can be treated as a convolution between the response functions and the second-order correlation function $G^{(2)}(t_{1} - t_{2})$,

The above result for slow detectors may not be a problem in the measurement of entangled states; however, it may affect the measurement of a $G^{(2)}$ function for thermal or pseudo-thermal light significantly. The $G^{(2)}$ function of thermal or pseudo-thermal field has two terms, the product of two measured mean photon numbers (trivial part) and the photon number fluctuation correlation (nontrivial part) [13],

How does one distinguish the relatively small, photon number fluctuations correlation part of the second-order correlation of a thermal field with short coherence time $\tau _0$? The obvious approach is to use fast photodetectors with $t_c \sim \tau _0$. However, even if faster, individual photodetectors are used instead of a full detector array, the state-of-the-art technology has not been able to produce photodetectors fast enough to achieve this for broadband light sources yet. One realistic approach that could utilize current photodetector technology is using a pulsed thermal or pseudo-thermal radiation source with pulse width of $\tau _p$ and recording a single pulse per frame. In this case, the time integral in Eq. (20) shall have similar effects on the first term of $G^{(2)}$ (product of mean photon numbers) as it had in Eq. (21),

Based on the above discussions, we may conclude that a short pulsed X-ray source with large angular diameter is a better solution for X-ray ghost imaging and the X-ray ghost microscope. Although synchrotrons have been chosen for many X-ray imaging techniques (projectional radiography, phase-contrast radiography, and CXDI) along with various other X-ray experiments in which point-like sources are desired, synchrotron X-ray sources are not necessarily the ideal candidate for the X-ray ghost microscope setups because higher-resolution is expected from a larger angular diameter X-ray source. An X-ray source with a large diameter and short pulsed emission is currently in development at Los Alamos National Laboratory and other institutions.

6. Summary

In summary, we have analyzed the working mechanism of the X-ray ghost microscope. By applying Einstein’s granularity picture of radiation, which includes X rays, we found that two-photon interference is able to produce an image-forming correlation that forms a lensless X-ray ghost image with high spatial resolution, but no magnification. Utilizing a high-energy ($>$ 20 keV) source with a relatively large angular diameter, a ghost image with sub-nanometer resolution could be achieved but is not obtainable with current photodetector array technology. Either with the addition of focusing X-ray optics or the help of a scintillator and an optical lens system, the primary ghost image can be mapped onto a secondary image plane with significant magnification and reduced (but still high) resolution. Because the primary ghost image is along a different beam than the physical object, the focusing X-ray optics or scintillator can be placed anywhere along the beam, allowing for high-resolution, magnified imaging of the deep internal structure of an object without direct physical constraints. Linearly scanning the secondary imaging setup along the propagation axis, a 3D image can be obtained with Linear Ghost Tomography. We have also found that a short-pulsed X-ray source and measuring a single pulse per frame is preferred for observing the two-photon interference induced intensity fluctuation correlation of X rays due to the wide bandwidth of the source. Even with measuring a single pulse per frame, in order to achieve a higher degree of second-order correlation it is preferred to have longer coherence time, $\tau _0$, that is more comparable to the pulse width, $\tau _p$. In addition to this, the desire for a large angular diameter makes synchrotron X-ray sources less favorable as the resolution of the primary ghost image would be greatly reduced. As sensor arrays and X-ray sources see technological advancements, this X-ray ghost microscope will open up new capabilities that would be of interest to the fields of physics, materials science, and biological imaging.