1. Introduction

In the coherent diffraction imaging (CDI), over-sampled diffracted intensity from an object is used to retrieve the phase of diffraction signal using an iterative algorithm. Experimentally measured diffraction amplitude and the retrieved phase are put together and Fourier-transformed back to real space to reconstruct an image or an electron density map of an object. The oversampling of the diffraction amplitude and the phase retrieval algorithms are two key elements of the CDI techniques. Sayre [1] was the first to recognize the relation between the Shannon sampling and the Bragg’s law. Bates [2] stated that one needs to over-sample the diffraction intensity twice as fine as the Bragg frequency in each dimension to satisfy the Shannon’s criterion, since the autocorrelation of an object is twice as large as the object itself. Later, Miao [3] showed numerically that it is enough to sample diffraction intensity points more than the number of unknown variables for reconstructing even 2D and 3D images, which was also verified experimentally [4].

Gerchberg and Saxton(G-S) [5] and Fienup [6,7] worked extensively on the phase retrieval algorithms. G-S established the use of iterations between the real and the reciprocal space with repeatedly imposing the known information called constraint in each domain. The constraint in the reciprocal domain is the experimentally measured diffraction intensity. One obtains an initial real space image by Fourier transforming the measured diffraction intensity with an arbitrary initial phase. A real space constraint, typically zero-density requirement outside the object boundary called an object support, is then applied to the real space image obtained previously. This image is again inverse Fourier-transformed to yield a diffraction intensity and a phase. In the second iteration, the diffraction intensity is replaced by the measured intensity and the phase obtained in the first iteration is used. Cyclic application of the constraints in the two domains for few hundred iterations leads to convergence to a correct image. Fienup added a feedback feature to the G-S algorithm, which speeds up the convergence greatly. Subsequently, Fienup and co-workers worked on phase-retrieval stagnation problems, reconstructing a complex valued object, and finding tighter upper bounds on the support using the autocorrelation [8–10]. Miao et. al. were successful in reconstructing lithographed characters from soft x-ray diffraction pattern [11], which initiated substantial amount of research in both coherent diffraction imaging [12–18].

In this study, we have used the ‘shrink-wrap’ algorithm [15] for image reconstruction. Unlike other iterative phase retrieval algorithms, this algorithm estimates the object support from the measured intensity by first estimating the support from an autocorrelation function which is updated in the subsequent iterations by the boundary of the object in the corresponding iteration. The advantage of this algorithm is that one needs not to know the object boundary independently.

In most CDI experiments, the diffraction amplitude was obtained in a transmission geometry or in a Bragg geometry. In the transmission geometry, a probing light beam transmits through a sample to yield a diffraction pattern. To measure this diffraction pattern with sufficient intensity, a sample should be partially transparent to the employed light source. It is, however, impractical to measure the diffraction signal of samples grown on typical substrates in the transmission geometry due to the absorption by the substrate. In most applications, substrate thickness is much larger than the attenuation length of probing light. On the other hand, the Bragg geometry, where the diffraction profile near a Bragg reflection peak is measured, is only available to periodic crystalline samples [12]. In this paper, we present a method which is simple but can overcome the above mentioned limitations associated with the transmission geometry. We gather diffracted signal around a specularly reflected beam of a sample on a substrate in reflection geometry in the Fraunhofer regime. The intensity pattern is then “phased” by applying an iterative phase retrieval algorithm to reconstruct the phase and the amplitude of the wavefield at the surface of a sample.

There have been a few reports on the phase retrieval carried out in the reflection geometry. Taguchi et. al. measured the back reflected light from a grid pattern of alternating bars and spaces with pitch of 10 μm, and retrieved the phase of the reflected signal applying a design model as an initial input and the known sample diameter as a constraint for the object [19,20]. Applying known illumination pattern as a support constraint, Fienup reconstructed a coherent image using a reflected far-field laser speckle intensity from an object [21]. However, this requires an illumination pattern in favor of the phase retrieval. Different from these previous reports on the reflection-based diffraction imaging experiments, we report a reconstruction of the phase map and the surface image from a reflection pattern using a He-Ne laser beam without relying on any a priori knowledge of either the object or the support constraint. The support constraint was determined by the autocorrelation function obtained experimentally, and the image was reconstructed starting from a random initial phase. We have also elaborated on the phase image of the wave field of the reflected light at the sample surface.

2. Theoretical considerations

The principle of the diffraction imaging in the reflection geometry is as follows. When a coherent plane wave is incident on a sample with an angle ${\theta }_i$ with respect to the average surface normal as illustrated in Fig. 1 , the reflected exit wave field right at the surface can be expressed as,

 

Fig. 1 Schematic of the reflection geometry used in this experiment.

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The wave field arriving at a point $(X,Y)$away from the specularly reflected beam in the detector plane which is tilted with respect to the average surface plane by ${\theta }_i$ can be predicted by applying the Huygens-Fresnel principle,

The wave field at the detector position $(X\cos {\theta }_i,Y)$ is a Fourier transform of the reflectance r(x,y) multiplied by the phase factor $e^{i\varphi (x,y)}$ _, where $\varphi (x,y)=(4\pi /\lambda )z(x,y)\cos {\theta }_i$, which retains the information of the surface height profile. We note that the detector coordinate in the x direction is scaled by an oblique factor $\cos {\theta }_i.$

The magnitude of the wave field $\left|\psi (X,Y)\right|$ given in Eq. (3) can in principle be measured by placing a detector far away from the sample in the Fraunhofer regime. In practice, however, one may place a convex lens close to the sample position in the Fresnel regime which brings all the rays in a particular direction to a single point on the back focal plane of lens [22]. By placing a detector at the focus of the lens, one readily obtains the Fraunhofer pattern. The usage of a lens serves two purposes: one is to measure the Fraunhofer pattern at a practically manageable distance from the sample and the other is to cover a large solid angle of the scattered field. In the case of short wavelength x-rays, the Fraunhofer condition is readily satisfied within a few meters and it is not necessary to use such a lens.

This measured amplitude$\left|\psi (X,Y)\right|$constitutes only half of the total information required for reconstructing the exit wave field, $e^{-i\varphi (x,y)}r(x,y)$. The other half information, the phase of the diffracted field, cannot be measured, and has to be recovered from the measured intensity using a phase retrieval oversampling algorithm. Eq. (3) suggests that the reconstructed field is the exit wave field at the sample surface plane specified by the (x,y) coordinate where an appropriate physical support constraint is applied.

The ‘shrink-wrap’ algorithm [15] used in this study does not require the knowledge of the object support a priori. As described previously, this algorithm estimates the initial object support from the autocorrelation function which is obtained by Fourier-transforming the measured diffraction intensity. Being roughly twice the size of the object, this first estimate provides a rather loose constraint of the object. During iterations, the support is updated periodically by tracing the boundary of the object in the corresponding iteration. As the iterations proceed, the support approaches towards the true support of the object. Except for the estimate of the object support, all other real and reciprocal space operations are the same as in a typical hybrid input-and-output (HIO) algorithm. We monitor the progress of image reconstruction using the real space image error, which is the measure of the total density remaining outside the support.

3. Experimental configuration

Figure 2 shows the experimental set up used in this study schematically. A 2 mW He-Ne laser source with a wavelength of 632.8 nm was used as a light source. A 500 µm aperture was used to select the central portion of the laser beam and a diaphragm aperture is placed to block all the higher lobes of the Airy pattern from the aperture allowing only the central bright disc to the sample. The size of the resultant Airy disc of the beam defining aperture at the sample position was 2.5 mm. Neutral density filters were used at the downstream of the aperture to adjust the intensity of the illuminating beam. Samples were placed at 90 cm downstream of the aperture. Placing the sample at a distance much larger than the Rayleigh distance, 2a ²/λ [23], where a is the diameter of the beam defining aperture, we ensured that a sample be positioned well in the Fraunhofer regime of the aperture diffraction.

 

Fig. 2 Schematic diagram of the experimental setup. A He-Ne laser beam passing through a 500µm aperture illuminates the sample defined by another 500µm aperture. The reflected light is collected by a biconvex lens that focuses the beam on to a CCD. A microscope image of the first test sample is also illustrated.

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Another aperture with a diameter of 500 µm was attached on to the sample to limit the total size of the beam and also to ensure the zero density constraint outside the aperture. The outside of the aperture was coated by dull, black colored oxide in an un-brushed form to minimize the reflection from it. Without this sample aperture, the Gaussian beam arising from the beam defining aperture slowly goes to zero and the boundary becomes blurred, which is not desirable for applying the oversampling algorithms that require a sharp boundary. The incident beam thus defined becomes much smaller than the dimension of the Airy disc, and it can be approximated as a plane wave. A well defined plane wave illumination on to a sample is required to ensure the exact Fourier transform relation between the object exit wave and the intensity distribution at the focal plane of the lens placed close to the sample.

Beam reflected from the sample was collected by the lens, which has a diameter of 50 mm. The Fourier transform of the sample exit wave was formed at the back focal plane of the lens where the CCD was placed. The focal length f of the lens was 116 mm. A Princeton instrument, PI-SX back illuminated 16 bit CCD which has 1340×1300 pixels was used to record the diffraction patterns. Each pixel is 20 µm in size. Using the CCD, the linear dimension of the field of view in real space given by$\lambda f/\mathrm{\Delta }q$ was 3.67 mm, where $\mathrm{\Delta }q$ is the smallest interval in the Fourier space specified by the pixel size. Therefore, for a sample size of 500 µm, the achieved oversampling ratio was about 7.3. The reconstruction resolution in the real space, estimated by $\lambda f/N\mathrm{\Delta }q$where N is the number of pixels, was about 3.45 µm.

4. Results and discussions

4.1 Image reconstruction of an array of periodic Au bars on Si

The first test sample was an array of periodic gold bars patterned on a Si wafer with 20 µm width and 40 µm period as illustrated in Fig. 2. The pattern was prepared by UV-lithography. After developing the patterned photo-resist, we deposited a ~5 nm thick Ti adhesion layer and a ~50 nm thick Au layer.

The reflection patterns at various incident angles are shown in Fig. 3(a-c) . The periodic nature of the sample is well illustrated in the Bragg peaks. Due to the limited dynamic range of the detector, the pixels in the intense central region are saturated under exposures over 70 ms. On the other hand, exposures as much as 1500 ms were required to collect the weak signal near the edge of the CCD. We acquired the data under six different exposures of 30, 70, 200, 400, 800 and 1500 ms. A diffraction pattern was generated by selecting unsaturated areas with statistically meaningful signal in those six shots and patching them after scaling by the exposure time. The set of exposures was turned out to be the best combination to cover the reciprocal space shown in the figures without saturating any pixel. As the incident angle was increased, the beam illuminating the sample forms a larger footprint along the horizontal x direction. We observed that the noise level was increased at high incident angles, which obstructs the reconstruction.

 

Fig. 3 Reflection pattern obtained at 15°, 30°, 45° incident angles respectively. Each of the diffraction patterns were generated by combining 30, 70, 200, 400, 800, 1500 msec, exposures. As the incident angle is increased, the diffraction pattern is elongated in the Q_x direction.

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The image reconstruction was realized by applying a phase retrieval algorithm to the measured diffraction data. We adopted the shrink-wrap algorithm discussed before. In the first iteration, the object support was set to the boundary of the autocorrelation function defined at the intensity level at 1% of its maximum. Then, the HIO algorithm with a feedback parameter β of 0.9 was applied. At every twentieth iteration following the first iteration, the support was replaced by the boundary of the calculated real space object in the iteration. To estimate the boundary, we first smoothed the object by multiplying a Gaussian blurring function. The boundary was obtained by truncating the smoothed object at the 10% of its maximum value. The support was updated until the 1000^th iteration where it traces the boundary of the true object reasonably well. We treated the exit wave field as a complex quantity and did not apply any positivity constraint which is often assumed for a real object.

As described by Shapiro [24], we adopted a procedure of generating a reconstructed image by averaging large numbers of fluctuating reconstructions, which was found to be effective in obtaining a consistent image. The logic behind the averaging is that during averaging the true signals are reinforced while the noise signals are gradually diminished. After initial 1000 iterations, we obtained the reconstructed amplitude and phase at every 50 additional iterations and averaged over such 11 iterates. The reconstructed amplitude and phase images from the exit wave obtained by the procedure discussed above are illustrated in Fig. 4 . They are reconstructed from the reflected intensities measured with 15°, 30° and 45° incident angle. To take care of the effect of the oblique factor on the reconstructed image at different incident angles, for all the reconstructed images, the scale in the horizontal direction was multiplied by the oblique factor $1/(\cos {\theta }_i)$ as discussed in Eq. (3). The reconstructed amplitude and the phase images are quite similar to the sample image obtained by an optical microscope shown in Fig. 2. Thus, we believe that we have demonstrated that it is possible to realize the surface image reconstruction, which is the amplitude image, using only the reflection intensity profile.

 

Fig. 4 Reconstructed amplitudes (a,b,c) and phases (d,e,f) at 15°, 30°, 45° incident angles. The images are obtained using shrinkwrap+HIO algorithm. Each image was generated by adding 11 different reconstructed amplitude/phase reconstructions starting from a random initial phase value and multiplied by the support constraint. Images are multiplied with the oblique factor of 1/cosθ_i on to x scale.

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The reconstruction provides not only the magnitude of the exit wave field which corresponds to the surface image but also the phase of the wave field. The phase of the exit wave, $Arg[r(x,y)]\varphi (x,y)=Arg[r(x,y)](4\pi /\lambda )z(x,y)\cos {\theta }_i$as shown in Eq. (3), retains the depth information, z(x,y) of the surface as well as the phase change during the reflection, Arg[r(x,y)]. If one knows the surface height profile or the phase change during reflection, the other value can be easily obtained from the phase map. For example, there is no phase change during reflection from typical dielectric materials, and the phase of the exit wave determines the surface height profile directly. In the case of absorbing materials such as metals, Arg[r(x,y)] is finite and is also a function of the incident angle which can be calculated for a known material. In principle, it should also be possible to extract Arg[r(x,y)] and the height profile from the phases of the exit wave at two or more incident angles.

A three dimensional illustration of the phase map obtained from the reconstruction of the 15 degree data is shown in Fig. 5(a) . We extracted a line profile perpendicular to the gold bars in the phase map. The standard deviation in the phase values obtained from 11 iterations was assigned to the error in the phase determination. The phase profile was fit to a series of step functions whose edges are smoothened by an error function. The smoothening is to take the finite lateral resolution into consideration. Fig. 5(b) shows the result of the fit together with the line profile of the phase value extracted from the phase map. Although it is difficult to assign an exact phase value at the gold or silicon regions due to the limited lateral resolution, the phase map in principle provides the phase change during reflection or the depth profile if one or the other can be determined independently. As a result of the fit, the phase difference between the top and the bottom of the structure and the error in its determination can be estimated. The contrast in the phase map tends to decrease with increasing the incident angle due to the noise increase in the diffraction profile. Figure 5(c) illustrates the ratio of the phase error to the phase difference as a function of the incident angle. The phase error sharply increases to 20% of the contrast when the angle of the incidence is above 35 degrees. For the current system, quantitative phase determination is reliable with the angle of incident below 35 degrees. Further improvement of the noise reduction is required to increase the incident angle.

 

Fig. 5 (a) 3D rendering of the reconstructed phase map for the data obtained at an incident angle of 15°. In this 3D plotting, the phase value outside the central 70 pixels (in radius) was set to zero to illustrate the variation in phase value clearly. (b) Non-linear least squares fit for the line profile across center of phase map for the 15° incidence. The line profiles were fit to a series of step functions whose edges are smoothened by an error function. (c) Ratio of the phase error to the phase difference at the top and bottom layers.

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Since the variation of the phase is less than π as shown in Fig. 5(b), we have also tried to apply additional constraint requiring positivity on the imaginary part of the exit wave field in the shrink wrap algorithm. The iterations converge about twice as fast as those without this constraint, which illustrates the effectiveness of the constraint as reported previously [3]. The quality of the final reconstruction, however, did not improve, and the image quality remained similar even with the positivity constraint.

4.2 Non periodic, isolated sample on an anti-reflection coated window

The second test sample was an isolated number ‘2’ patterned on an anti-reflection (AR) coated window. The importance of testing an isolated sample is that it is not desirable to place an aperture onto samples such as sensitive biological specimen or samples of irregular shapes. By adopting an AR-coated window as a sample holder, the background signal reflected from the holder can be minimized.

The test sample was prepared on an AR-coated float glass substrate. A number ‘2’ on a negative resolution bar was selected and used as a mask for the photolithography. The usual photolithography procedure of spin coating photo-resist, mask alignment, UV exposure, and developing was followed by e-beam deposition of the chromium layer. The patterned number ‘2’ sample has a dimension of 300×200 μm² with line width of 30 μm. A microscopic image of the sample is shown in the Fig. 6 .

 

Fig. 6 Optical microscope image of a Cr letter patterned on an AR coated substrate using UV lithography.

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Different from the previous set-up, no sample aperture was used and the specimen was placed at 450 mm downstream of the beam defining aperture. Taking into account the oversampling condition, the maximum possible sample size for a lens of focal length f = 53 mm and a detector with a pixel size a =12 μm was 1.39 mm. We used a Dalsa 1M30P CCD (1024x1024 pixels, dynamic range of 12bits) for this measurement. Sample was mounted on a XY stage and introduced into the beam, with 3 different angles, in such a way that angle between the incident beam direction to the sample surface normal is 15°, 30° and 45°.

The optimum condition for the exposure time was 31 ms, and 100 shots were taken to improve the statistics. Measured diffraction patterns at various incident angles are shown in Figs. 7(a) -7(c). The diffraction intensity is concentrated mostly on the central region except for the two diffraction lines that correspond to the two straight edges in the sample. Diffraction pattern is elongated in the horizontal direction as the incident angle is increased. Two artifacts were observed in the diffraction patterns which were caused by the technical problems of the CCD. One is the horizontal streak which is more pronounced in the left part of the specular reflected beam. We think that this was caused by the incomplete flush of the charge when the chip is exposed to light continuously even with internal shutter closed. The other was the non-uniform read out of the CCD. Albeit with these artifacts, the image was successfully reconstructed, although it could have affected the quantitative value of the reconstructed wave field.

 

Fig. 7 Measured diffraction patterns (a,b,c) and the reconstructed amplitudes (d,e,f) and phases (g,h,i) from the sample ‘2’on an anti-reflection coated substrate at 15°, 30°, and 45° incident angles. As the angle of incidence is increased, the diffraction pattern is elongated in the Q_x direction. Plotted reconstructed image size is 301x301 pixels. Image is multiplied by the support. We also multiplied the reconstructed image by the oblique factor 1/cosθ_i in the x direction.

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Image reconstruction was performed again using the shrink-wrap method based on the HIO algorithm, which is most appropriate for this reconstruction, since we do not use an aperture on to sample to define the support. Shrink-wrap algorithm was initially run for 1000 iterations, and the amplitude of the wavefield was saved after every additional 50 iterations for 10 times. The reconstructed amplitude images at 15°, 30° and 45° of incidence were shown in Figs. 7(d)-7(f) respectively. Reconstructed image was multiplied by the support obtained after 1500 iterations. The images shown in the figure were the averages of 11 such reconstructions. The lateral reconstruction resolution in this case was 3.49 μm. When the angle between the sample surface and the incident beam is different from normal, the foot print of the beam was elongated in the incident direction, and the reflection from substrate increases, which degraded the quality of the reconstructed image.

The sample can be approximated as an amplitude object since the reflection amplitude from the AR window was negligible and the phase of the exit wave in this area is not well defined. The amplitude outside the number ‘2’ is close to zero and the phase in this region is meaningless. Consistently, the phase map illustrated in Figs. 7(g)-7(i) shows that the phase inside the number ‘2’ is relatively uniform but it is not well-defined outside it. The only meaningful information is the amplitude from the isolated number ‘2’, and one may treat the system as a real object. Therefore, the real object approximation can be applied for the image reconstruction. We also note that the amplitude and phase reconstruction was much easier for this aperiodic surface than the periodic surface discussed in section 4.1.

5. Conclusion

We have demonstrated that it is possible to reconstruct the surface image using a reflection signal from the surface. Without a priori knowledge, both the amplitude and the phase of the reflected exit wave field of a sample were obtained successfully. The amplitude corresponds to the conventional surface image and the phase image contains information related to the phase change during reflection and the surface height profile. We have also reconstructed an image of an isolated sample which can be treated as an amplitude object using the reflection signal. The reflection based coherent diffraction imaging will be especially useful in the applications of soft x-rays with extremely short penetration length to most material systems. We expect that the proposed method can also be adapted to hard x-ray reflection-based microscopy in the grazing incident geometry.