Figure 58 – uploaded by Florian B Soulard

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Figure 53: object phase correction at the object plane The phase appears to be wrapped, but this is not due to the surface itself, it is due to the illuminating beam. As the diffracted light from this chip was very low, the concentratec laser beam was shone onto the chip surface (coming directly from the output port of the laser, before the reference beam expansion and collimation). This beam is not a perfec plane wave, instead the laser section tends to expand slightly after a few meters. Thus 11 can be considered as a spherical wave, and the object phase can be corrected by spherical field matrix. This is illustrated in figure 53. Additionally, a plane fit was performed because a phase jump appeared in the middle of the chip surface. The corrected phase map is in the range [-2;+z], and this range corresponds to height values between 0 and 632.8 nm (one wavelength). It does not seem to undergo phase jumps on the IC surface, which means that the depths variations are within one wavelength. Not all the chip’s surface is visible, because the laser beam was concentrated on the right part of the chip, and this side diffracted more light than the left side. The corrected phase map is in the range |—73+70 | , and this range corresponds to height — Figure 53 object phase correction at the object plane The phase appears to be wrapped, but this is not due to the surface itself, it is due to the illuminating beam. As the diffracted light from this chip was very low, the concentratec laser beam was shone onto the chip surface (coming directly from the output port of the laser, before the reference beam expansion and collimation). This beam is not a perfec plane wave, instead the laser section tends to expand slightly after a few meters. Thus 11 can be considered as a spherical wave, and the object phase can be corrected by spherical field matrix. This is illustrated in figure 53. Additionally, a plane fit was performed because a phase jump appeared in the middle of the chip surface. The corrected phase map is in the range [-2;+z], and this range corresponds to height values between 0 and 632.8 nm (one wavelength). It does not seem to undergo phase jumps on the IC surface, which means that the depths variations are within one wavelength. Not all the chip’s surface is visible, because the laser beam was concentrated on the right part of the chip, and this side diffracted more light than the left side. The corrected phase map is in the range |—73+70 | , and this range corresponds to height

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Department of Physics — Rome III University santarsiero@fis.uniroma3.it

Light is a transverse electromagnetic wave, composed of an electric wave and a magnetic wave whose planes of oscillation are perpendicular to each other. The oscillations of the waves make a right angle with the local direction of propagation, given by the wave vectork . Assuming that the light in use is linearly polarized, monochromatic and propagating in free space, we can describe the electromagnetic wave only by the variations of one of its components (for example, the electric field). The use of complex exponential notation simplifies further calculations, and it enables an easy access to the phase and the amplitude of the wave in numerical calculations.

The modulus of the wave vector is defined from the wavelength i: Two kinds of waves are to be considered: plane waves and spherical waves. A plane wave is theoretically extending towards infinity, and all surfaces of constant phase are planar and parallel. The amplitude remains constant with propagation. In practice a plane wave is obtained by collimating a divergent or convergent beam with a lens. However, as the resulting plane wave is not extending infinitely, it can be considered as a plane wave on a small extent only (if distance is too big, diffraction is not negligible anymore).

plane of observation appear sharp. The focus plane must be adjusted for a clear visualization of other parts of the object. Points in the plane of focus make sharp points on the photographic plate or film (green dots), while points out of focus lead to bigger, blurry zones (red dots, focused out of the plate). As there is no information recorded about the depth of the scene, there is no way to recover the out-of-focus points. The use of thin lenses with a far field object allows for a bigger depth of focus (within which the points remain sharp), however in any case the information about the 3D shape is lost. What is recorded is a perspective view of the object.

Digital recording of holograms enables to reduce stability problems related to the exposure time, in particular due to vibrations, since the holograms can be recorded within a fraction of a second. Still, a high degree of coherence is a critical requirement. The phase shifting technique presented in the chapter 5 requires a high stability too, because the recording of several holograms with different phase shifts takes some time.

The optical reconstruction of the object is performed by shining a laser light through the holographic plate, with the same properties as the original reference beam (wavelength, geometrical configuration). The diffracted light reconstructs the 3D shape of the object with high fidelity, as well as a twin image. It is possible to simulate the diffraction phenomenon on a computer for a digital hologram, as it is described in chapter 2. In this case, the field is propagated and calculated on several planes in space, to reconstruct the object on different depths (simulating the way that we refocus a microscope objective on different planes, for example). From equation 9) we can see that the recording process is not perfect: the cosine function loses the information about the sign of the phase difference. This will lead to two reconstructed waves. The virtual image located behind the plate is the continuation of the original field from the object. It is the term V, (x, y) Ve (x, y) in equation 8). The real image is located in front of the plate, and light converges to form a second image. It is the term with the complex conjugate of the object field Vj (x, y)V, (x,y) in equation 8. In off-axis holography these images can be well separated however in an in-line set- up they are mixed together, and this is a major problem as it will be seen further. Moreover two other terms are present in the interference equation: /, (x, y) (responsible for the DC-term or undiffracted light) and J, (x, y) which is the intensity

$Light diffracting on dust present in the optical set-up and at the output port of the laser gives a grainy profile (corresponding to high spatial frequencies present in the beam). The laser light in input is filtered with a spatial filter (composed of a microscope objective and a pinhole), removing the noise in the incoming light and giving a smooth light in output. This beam is eventually collimated before entering the interferometer. The use of a plane wave as the reference wave is handy for the numerical reconstruction, as a wave of constant phase (normal to the plate) and constant amplitude can be used.$

Light diffracting on dust present in the optical set-up and at the output port of the laser gives a grainy profile (corresponding to high spatial frequencies present in the beam). The laser light in input is filtered with a spatial filter (composed of a microscope objective and a pinhole), removing the noise in the incoming light and giving a smooth light in output. This beam is eventually collimated before entering the interferometer. The use of a plane wave as the reference wave is handy for the numerical reconstruction, as a wave of constant phase (normal to the plate) and constant amplitude can be used.

Figure 9: the optical set-up (scattered light from the object)

Figure 10: adding fields of spherical waves (only the real part of the complex field is shown) The field at a given distance d is the sum of all the fields from each point of the hologram at that distance.

As it can be seen on figure 10, the field from a point in the hologram at distance d does not change in shape during the scanning process. Only the amplitude changes. modulated by the transparency of the corresponding hologram point, but the field is just translated across the reconstructed plane since planes are parallel. This is similar to standard image filtering, where a (small) filter matrix is moved across an image tc process the image points, summing pixel neighbours and saving the value in the central pixel. This is a convolution, and one can now consider the same process for holographic reconstruction, as the convolution between the whole hologram and a filter matrix, being the field of a point at distance d. In order to process all the points of the hologram for each point of the reconstructed plane, this filter matrix must be two times bigger than the hologram.

The alternative calculation is illustrated in figure 12. Of course all the fields involved are complex fields (modulus and phase), however only the square of the modulus (intensity) is represented for the hologram and its reconstruction, and the real part for the filter matrix. Jumps in the phase are due to the fact that phase is wrapped. When doing the pointwise multiplication between the two fields, modulus is multiplied and phase in the exponential part is added. 2.2. Decomposition in plane waves

Figure 13: an optical field as a sum of plane waves (shown in x direction only) the same as the reconstructing wave — the wavelength of the diffracted light remains the same as the wavelength of the incident light.

Figure 14: the plane wave propagation method The next figure summarizes the plane waves method in a graphical way.

All the plane waves are multiplied by the factor e*”’”“ and the angular spectrum is found at the plane at distance d. The field at the reconstructed plane is obtained by performing the inverse Fourier transform on the angular spectrum: of propagation is along the z axis, normal to the plane of the hologram. The added phase

As long as the optical path difference between two adjacent samples is less than half a wavelength, the phase difference remains below z and it is possible to record the pattern. This comes from the spherical wave equation:

Figure 16: absolute minimum distance between the object and the CCD sensor

Let us assume a round object of diameter W (same size in x and y), centred on the CCD sensor. The new distance between the vertical projection of the edge of the object and the furthest edge of the sensor is L’.

Figure 19: longitudinal resolution On the other hand, the longitudinal resolution of the system is much lower. This is due to the reduced numerical aperture (the CCD sensor is small with respect to the object distance). If we consider an object point at a reconstruction distance d, the light is focused with a very narrow angle. The maximum angle is for the minimum reconstruction distance dmin: @,,,, =4.1°, corresponding to a maximum numerical aperture of NA =nsin(@)=0.07.We want to find the minimum distance x where the beam size remains below the pixel size (for a typical reconstruction with a magnification factor of 1, the reconstructed frame pixel size is the same as the CCD sensor pixel size). It will appear as the same point across following frames within this distance (outside it will spread on adjacent pixels as a blurry remaining of the out of focus point). aperture of NA=nsin(@)=0.07.We want to find the minimum distance x where the

Figure 21: magnification of the reconstructed object

Figure 20: a hologram as a set of lenses

The first reconstructing functions were called on the command line, but a graphical user interface (GUI) was then made in order to facilitate the selection of a hologram, the reconstruction depths list entry, the output folder for saving calculated frames, etc. The main window is shown below.

Figure 23: efficient way of calculating the field from a point source on a plane (real part displayed)

Figure 24: convolution function flowchart This is summarized in the following diagram:

Figure 25: plane waves propagation function flowchart The algorithm in the code (Appendix 5) is the one shown below. The plane waves propagation is implemented according to the method in chapter 2. Some extra conditional statements have been added in order to link the output frames with the required dimensions and magnification entered in the selectArea window (see next sub-chapter). In a similar way to the convolution function, it is enough to calculate once the Fourier transform of the hologram, and to process the angular spectrum for all the required depths. Thus the FFT is called only at the first call of the function, and the angular spectrum of the hologram is kept in memory between successive calls.

The frequency plane in output of the FFT function is indexed in the following way (for an even length of samples, as it is the case for powers of 2): We can notice that there are horizontal and vertical symmetries. As the IFFT functiot takes an angular spectrum in the same manner, the fftshift function was not used to shif the DC value to the centre (it would have been redundant to apply it twice). The propagation coefficients are calculated according to this organization in the frequencies The m factor in equation 17 makes use of p* and q? (getting rid of the frequency sign) consequently it is enough to calculate only one quarter of the propagation coefficient: and to replicate it according to the symmetries. The phase coefficients matrix i calculated once at the first call of the function, and corresponding propagation factor: are calculated for every iteration from this coefficients matrix and the frame depth.

The aim of the selectArea screen is to enable to select a zone of reconstruction (in x and y, whereas the depths selection is done on the main GUI). The possibility to load a previously calculated frame into the plate limits is added to help making the selection. Also a magnification factor is introduced, which can be alternatively entered by the end- user as a desired frame sampling. While dragging the selection area, two dedicated “constraint” functions check and correct the location of the centre along with the width and height in pixels, to make sure the centre always falls on a pixel (odd dimensions) or between two pixels (even dimensions). This is required by the previous function (planeWavesPropagation) to crop or pad the matrix with zeros, and this process is done with reference to the frame centre and the frame resolution (pixels in x and y).

A custom parallel function was finally implemented to overcome these shortcomings. It is using the same principle as the multicore package from Markus Buehren: parameters and communication between the workers are done by writing files in a folder common to all the participating workers. A graphical interface enables the end-user to select the common folder for each instance, and provides messages concerning the computation.

Figure 28: Calculation time versus number of slave processes A faster processing speed is obtained, as can be seen on figure 28. However, to achieve this result, considerable changes in the way the reconstruction functions are called by the main GUI were necessary. The main drawback is still the same: four instances of Matlab must be run in parallel on the multicore computer, wasting memory. In one of the instances the main GUI is launched, in the other instances the DHWorker GUI is running, waiting for “orders” to appear in the common folder. The code for DHWorker is on the CD (Appendix 5).

Figure 27: DH Worker GUI, launched along the main GUI for parallel calculation

In the ideal case, the processing time could be divided by a factor up to 4 if the computation is perfectly parallelized. In the actual case, an improvement of 2.8 is reached, which is rather good if we consider the competition between the processes to write the propagated fields on the local disk.

Figure 32: reconstruction of the USAF 1951 test target

The halo is still not completely removed, and this is due to the fact that the plane reference wave is not perfectly smooth (small dust in the optics, reflections on the beam splitter and the collimating lens...). Also the reference beam intensity is not constant, it is more like a Gaussian profile since it is coming from an expanded laser beam. By recording the intensity of this field alone, and withdrawing it from the hologram, a better “cleaning” of the image is obtained. This is the second term in equation 9.

Figure 58 – uploaded by Florian B Soulard

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