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1Centro de Investigaciones en Óptica, A.C. Loma del Bosque 115, C.P. 37150, León, Guanajuato, México. *Corresponding author: [email protected] Received ...

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High topographical accuracy by optical shot noise reduction in digital holographic microscopy Miguel León-Rodríguez,1,* Ramón Rodríguez-Vera,1 Juan A. Rayas,1 and Sergio Calixto1 1

Centro de Investigaciones en Óptica, A.C. Loma del Bosque 115, C.P. 37150, León, Guanajuato, México *Corresponding author: [email protected] Received September 1, 2011; revised December 13, 2011; accepted December 16, 2011; posted December 19, 2011 (Doc. ID 153705); published March 16, 2012

In this work, we present a new method to reduce the shot noise in phase imaging of digital holograms. A spatial averaging process of phase images reconstructed at different reconstruction distances is performed, with the reconstruction distance range being specified by the numerical focus depth of the optical system. An improved phase image is attained with a 50% shot noise reduction. We use the integral of the angular spectrum as a reconstruction method to obtain a single-object complex amplitude that is needed to perform our proposal. We also show the corresponding simulations and experimental results. The topography of a homemade TiO2 stepwise of 100 nm high was measured and compared with the atomic force microscope results. © 2012 Optical Society of America OCIS codes: 090.1995, 030.4280, 110.0180, 110.3010, 120.5050, 120.6660.

1. INTRODUCTION Digital holographic microscopy (DHM) has become a novel tool for the study of microscopic samples, mainly yielding quantitative information of the wavefront transmitted or reflected through three-dimensional (3D) objects, and attaining only subwavelength accuracy along the axial direction in phase images, due to their interferometric nature. Lateral resolution is limited by diffraction, as it happens in classic microscopes. The method is based on the acquisition of a hologram formed by an object wave that passes through a microscope objective (MO), and then interferes with a reference wave [1,2]. The complex amplitude of the object is recovered when the hologram is reilluminated by a digital replica of the reference wave, which allows us to have quantitative measurements of phase and amplitude images of the specimen [3]. This kind of microscope has been quickly accepted for use in metrology, in its diverse fields. Some of those applications are in the analysis and characterization of microelectromechanical systems (MEMs) and micro-optical systems (MOEMs) [4–7], the study of biological samples [8–11], and in the measurement of phenomena in microfluids [12], where the reconstruction distance, a unique characteristic of digital holography, plays an important role. There are important challenges when one inspects test objects with microscopes, such as a limited depth of field, the introduction of aberrations caused by the use of optical elements inside the arrangement, and the optical noise and parasitic interferences generated by multiple reflections, among others. DHM is not an exception. Aberrations are introduced when using a MO, and there is also a limited depth of field. Although some of these challenges have already been dealt with, limitations still exist in obtaining measurements with the precision demanded by different current applications. For example, Ferraro et al. [13] tried to extend the focus depth for the analysis of MEMs using different reconstruction distances of a digital hologram. On the other hand, not only there 1084-7529/12/040498-09$15.00/0

are numerical methods to correct aberrations of the wavefront, as proposed by Colomb et al. and Di et al. [14,15], where a priori knowledge of the recording conditions of digital holograms and specific areas needs to be considered, but also hardware compensation methods. These include a reference conjugated hologram (RCH) that it is based on recording, in a flat area near the specimen, a hologram that will contain the aberrations of the system without contributions from the object. Once one has the reconstruction of the complex field of the RCH and the sample object, the phase images are subtracted from each other [16]. Noise reduction in DHM is one of the most important challenges. The lower the phase noise, the smaller will be the measurement error. Different methods to reduce noise in digital holography have been applied, mainly to eliminate speckle noise. Kang used different angles of illumination to obtain multiple holograms, either by rotating the object or the illumination source. He obtained an improved image through the process of averaging the reconstructed images [17]. Baumbach et al. [18] recorded multiple holograms to different displacements of the CCD camera to perform an averaging procedure. Rong et al. [19] varied the polarization angle of the reference wave to obtain different holograms in each case, and then he applied an averaging procedure to get an improved image. Charrière et al. [20] applied a method to reduce the shot noise of digital holograms that consisted of averaging multiple holograms recorded within a certain time, so they could have an improved phase image. The methods previously described have the disadvantage of recording multiple holograms at times defined by the speed of the capturing system (CCD, personal computer, frame grabber) and the processing time, making them inappropriate to a certain degree for very high-speed transitory phenomena. To reduce the noise due to multiple reflections or parasitic interferences, low partial coherence sources of illumination are usually used, such as laser diodes or light emitting diodes (LEDs) [21–23], whose main disadvantage is their inability © 2012 Optical Society of America

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to reconstruct the complex amplitude from the object for larger reconstruction distances [24]. In this article, we show digital holograms with shot noise when they were recorded with a CCD camera. The illumination source used in the optical setup was a commercial LED. The experimental results were in agreement with the simulations. We also show a different method to reduce the shot noise of a phase image coming from a single complex amplitude. The proposal is based on the averaging of reconstructed images at different reconstruction distances within the range determined by the focus depth. With this, we ensure an improved image without quality diminution, where a noise reduction of 50% is achieved. For comparison with a reference measurement, our system was calibrated and made traceable to the results from an atomic force microscope (AFM).

2. EXPERIMENTAL CONFIGURATION A. LED Characteristics To reduce parasitic interferences and multiple reflections, one uses a partially spatial coherent source as an illuminating source. The described investigation was carried out using a commercial ultrabrilliant LED of 5 mm in diameter, giving out 3 W of electric power, with emission in the red spectrum range, 120° of viewing angle, and 120 lm of luminous flux (data provided by the supplier). However, the LED underwent testing to more accurately determine its emission spectrum. We used an i1Pro spectrophotometer, with a spectral range from 380 to 730 nm, which was calibrated under the standards of the National Center of Metrology in Mexico, traceable to the National Institute of Standards and Technology. The peak wavelength (λ) measured was of 630 nm, and a full width at half-maximum (Δλ) of its spectrum was 24 nm. Figure 1 shows the typical normalized spectrum of the LED that was used. This device was operated at a voltage of v  2.4 V and a current of I  1 A at room temperature. Then, with the spectral data mentioned above (λ2 ∕Δλ), a coherence length of 16.5 μm is obtained [25]. B. Reconstruction of the Hologram The experimental setup used in the present study was a modified Mach–Zehnder interferometer, as shown in Fig. 2; the light source was the LED described in Subsection 2.A. When the beam is incident on the adjustable diaphragm (D), with a diameter of 300 μm, spatial coherence is increased. Then, a secondary source of partial coherence is created by the spatial filter (SP) with an adjustable diaphragm to limit the source

Fig. 2. (Color online) Schematic of the microscope implemented on a Mach–Zehnder interferometer with a spatially partial coherent source for digital holographic microscopy. Abbreviations are defined in text.

size [21]. Lens L images D at the plane of the sample specimen (S) and on the compensating plate (CP) when the beam is divided by the beam splitter (BS1). The light transmitted by the specimen (S) is collected by a microscope objective (MO1) of 10× and numerical aperture (NA) of 0.25, which forms object wave O. The object wave interferes with a reference wave R when the light is collected by a 10× 0.24 NA microscope objective of (MO2) to produce an intensity hologram I H . The intensity hologram is recorded by a PixelinkTM digital camera (complementary metal-oxide semiconductor, 1280 × 1024 pixels, 8 bits, black and white, with a pixel size of 6.7 μm × 6.7 μm). To achieve interference between the two beams, it is necessary that the optical path length be the same along the two arms of the interferometer; thus, a compensation plate (CP) was inserted. Also, mirror M1 was mounted on a one-directional displacement stage with step resolution of 1.25 μm. Mirror M2 was mounted on a piezoelectric transducer (PZT) to implement the phase-shifting technique. The distribution of intensity Ix; y at the CCD sensor plane is formed by the interference of the object wave Ox; y and the reference wave Rx; y as follows: Ix; y  jOx; y  Rx; yj2  O2 x; y  R2 x; y  Ox; yR x; y  O x; yRx; y;

(1)

where the first two terms are the DC terms, and the last ones represent the real and the virtual images, while  denotes the conjugated complex. Mirror M2 is mounted on a PZT to calculate the phase of the initial object using the phase-shifting technique, and to eliminate the DC terms and the virtual image of Eq. (1). The 2π phase module is calculated from four π∕2 phase-shifted images I 1 − I 4 with the four frame algorithm [26]:  ϕ0 x; y  tan−1 Fig. 1. (Color online) Normalized spectrum distribution of the light intensity emitted by the commercial ultrabrilliant LED.

 I 4 x; y; 3π∕2 − I 2 x; y; π∕2 : I 1 x; y; 0 − I 3 x; y; π

(2)

The amplitude of the optical field A0 x; y is calculated as the square root of the object’s intensity distribution; that is, the

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object intensity can be obtained by blocking the reference beam and recording only the beam object in the CCD. As a result, the object complex amplitude, U a x; y  A0 x; y expiϕ0 x; y  ϕx; y;

(3)

of the object wave is determined at the recording x; y plane, where ϕx; y is the phase aberration term. In order to compensate phase aberrations induced by misalignment of the optical setup and MOs, we applied the so-called RCH method. Once we applied the phase-shifting technique without the presence of the test object, we obtained the phase aberration term ϕx; y, which can be subtracted from Eq. (3) to retain only the object complex amplitude: U 0 x; y  A0 x; y × expiϕ0 x; y  ϕx; y − ϕx; y: (4) Now, the angular spectrum (AS) method can be used to calculate the object complex amplitude at any other plane x0 ; y0  in order to refocus it:  Ux0 ; y0   I −1

expikd1 − αλ − βλ1∕2  ×IU 0 x; yα;β

 x0 ;y0 

(5)

;

where Ux0 ; y0  is the object complex amplitude focused, k  2π∕λ is the wave vector, d is the reconstruction distance, λ is the wavelength, x0 ; y0  are the spatial variables, α; β are the spatial frequencies, and I denotes a two-dimensional continuous Fourier transformation. The discrete form of Eq. (5) is written as  UmΔξ; nΔη  FFT−1

expikd1 − λr 2 − λs2  ×FFTU 0 k; lr;s

 m;n

;

(6)

where FFT is the fast Fourier transform operator, Δξ and Δη are the sampling intervals at the observation plane (pixel size), and r, s, m, and n are integers (−N∕2 ≤ m; n ≤ N∕2). C. Depth of Focus of the System A limitation in microscopy, of which DHM is no exception, is a limited depth of focus (DOF). High magnifications are achievable for investigating micro-objects, but the paradigm is that, the higher required magnification is, the lower the DOF is. As the geometrical DOF of an imaging system is related to the sampling distance, this DOF is expressed as a function of the sampling distance and the NA of the MO: DOF 

Δξ ; M 2 NA

where λ is the wavelength,Δϕ is the phase step, and n − n0  is the index difference between the specimen’s material and air.

3. SIMULATION OF HOLOGRAMS A. Parameters for the Simulations The simulations presented in this study are based on the work done by Charrière et al. [20], with some differences as described below. Two plane waves of equal intensities have been considered to interfere in a DHM in-line configuration with a magnifications of 10 and 20. In order to suppress the DC term and the virtual image, we apply the phase-shifting technique mentioned in Subsection 2.B. For all simulations, 600 × 600 pixel holograms have been considered, along with a square pixel size of 6.7 μm, and an ultrabrilliant LED with a wave peak of 630 nm. As for the reconstruction method, we use AS with a reconstruction distance of 30 μm. The evaluation area for all simulations is 300 × 300 pixels in the central zone to prevent the influence of border effects due to the discontinuity introduced by the windowing of the hologram when it is processed by the FFT calculation and the apodization function applied to the hologram. We have considered the standard deviation (STD) as a measure of the phase accuracy in all sections. B. Analysis of the Proposal to Reduce Shot Noise in Phase Images Shot noise depends only on optical power, and it follows Poisson statistics [27]. Figure 3 illustrates the STD of the reconstructed phase images as a function of the optical level. This optical level was expressed through the average number of photons per pixel for simulated holograms with shot noise. One can note that the simulated shot noise causes a phase STD of 0.2542 deg, corresponding to an average number of 8000 photons per pixel, being in agreement with Charrière et al.’s results. Now we can say that higher intensity corresponds to lower shot noise. A way to virtually increase the number of photons is by carrying out an averaging of the reconstructed phase images in order to get an improved phase image. In order to reduce the shot noise of the reconstructed phase image, we suggest performing a spatial averaging process of the reconstructed phase images that are obtained from different reconstruction distances within the system’s DOF. These reconstructed images are uncorrelated with each other at

(7)

where M is the magnification. DHM has as a unique feature in that is possible to refocus the object complex amplitude at any plane within the maximum refocus distance. Also, it is possible to refocus the complex amplitude at different distances within the DOF (ΔDOF). Another important factor is the physical thickness of the specimen, which is given by the phase to specimen height formula h  λΔϕ∕2π∕n − n0 ;

(8)

Fig. 3. (Color online) STD of the reconstructed images using the AS reconstruction method as a function of the optical power.

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specific reconstruction distances, and computed from the same complex amplitude. By stating the hypothesis that these reconstructed images are uncorrelated with similar STDc , one can write the following for the STD of a typical averaged image σ x : 1 σ x  p STDc ; C

(9)

where C is the number of images to average. If four images are averaged with similar STDs, one concludes from Eq. (9) that, theoretically, noise reduction is 50%. The correlation coefficient between these reconstructed images can be expressed as [28] CC p;q  

PN PN PN PN m

n

m

n I

p m;n−hI p iI q m;n−hI q i

I p m;n−hI p i2

PN PN m

n

I q m;n−hI q i2

Fig. 5. (Color online) Effect of averaging on a series of simulated holograms for a mean number of photons per pixel of 100, 500, 4000, and 8000: STD of the performed phase image as a function of the number of phase images C used in the averaging procedure.

1∕2 ; (10)

images from a single complex amplitude. Complex amplitude can be obtained with a single digital hologram in an off-axis configuration or, as in this case, an in-line configuration. This method allows faster recording than temporal averaging, where a lot of digital holograms have to be recorded. In order to determine whether the reconstruction distance difference shows any change, a last simulation was done with a magnification of 20. Figure 6 shows this change when there was a difference in reconstruction distance of Δd  0.6 μm. These phase images are uncorrelated. This result allows us to conclude that higher magnification is required; lower Δd can be managed in order to get uncorrelated phase images. The reconstructed phase images were obtained from the same simulated holograms with an average number of 8000 photons per pixel.

where CC p;q ∈ −1; 1, p and q are the two images to be correlated, hI p i is the average irradiance of image p, and hI q i is the average irradiance of image q. Figure 4 shows the correlation coefficient of the reconstructed phase images as a function of the reconstruction distance. One can see that, when there is a reconstruction distance difference of Δd  2 μm from one reconstructed phase image to another, these phase images are uncorrelated. The reconstructed images were computed from the same simulated hologram with an average number of 8000 photons per pixel. For example, if we average reconstructed phase images at 30, 32, 34, and 36 μm reconstruction distances, then p shot noise reduction will be 50% (1∕ 4  0.5), in concordance with Eq. (9). This simulation validates the hypothesis that uncorrelated phase images are obtained from the AS reconstruction method at specific reconstruction distances using a single complex amplitude described in Eq. (4). Four series of holograms with shot noise have been simulated, with different illumination levels: 100, 500, 4000, and 8000 mean photons per pixel. Figure 5 depicts the STD of the reconstructed phase image as a function of the number of phase images C used in the averaging procedure. In agreement with Eq. (9), STD reduction obeys the C −1∕2 law. We can conclude from this behavior that the averaging process proposed is valid for different illumination levels. This averaging technique allows us to reduce the shot noise effect in phase

In this part, the experimental results of the recorded hologram’s 1280 × 1024 pixels are presented. All the zones of interest are clearly depicted and defined in text and images. We take central zones (800 × 800 pixels) of the reconstructed images to prevent the influence of border effects described in Subsection 3.A. The camera has an electronic shutter, which enables it to reduce the exposure time down to 40 μs, and an electronic gain that is adjustable from 0 to 17 dB in 14 increments. To compensate the system aberrations, we use the RCH in all the reconstructed images. It is complicated to get an average number of 8000 photons per pixel (a camera’s full well capacity) with an

Fig. 4. (Color online) Correlation coefficients between reconstructed phase images at different reconstruction distances from the same complex amplitude.

Fig. 6. (Color online) Correlation coefficients between reconstructed phase images at different reconstruction distances from the same complex amplitude with a magnification of 20.

4. EXPERIMENTAL RESULTS

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LED illumination source in the setup presented, even with the maximum integration time and no electrical gain. In order to know the number of photons impinging on a specific pixel of the detector, the optical power of the intensity was measured with a Mod 515 powermeter and an 818 series Newport photodetector. First, a comparison between a blank experimental hologram and the simulated hologram results is shown. The recorded illumination intensity of the blank holograms was 6.7 × 10−5 W∕cm2 , corresponding to an average number of 5100 of photons per pixel, with 38 ms of integration time and no electrical gain in the camera settings. We show the reduction of system aberrations when the RCH method is used. Figure 7(a) presents a phase image that is reconstructed without RCH correction. A DST  12 deg is computed in the area enclosed by the black square. On the other hand, Fig. 7(b) shows the phase image after the RCH method was applied, with DST  0.7 deg in the area enclosed by the black square. We apply the procedure defined in Subsection 3.B to confirm the noncorrelation among phase images reconstructed at different distances. The reconstructed phase images were obtained of the recorded experimental holograms. The graph in Fig. 8 shows this justification. When there is a Δd  2 μm from one reconstructed phase image to another obtained from the same complex amplitude, a noncorrelation exists between the images. These results validate our proposal. With the Δd  2 μm calculated, we can average C images. In Fig. 9(a), the phase image is shown without any averaging. The image was reconstructed with d  30 μm, where the DST

Fig. 7. Reconstructed phase image from an experimentally recorded hologram without any specimen in the system. (a) Phase image without aberration correction where DST  12.6 deg is measured, delimited by the black square. (b) Phase image with aberration correction using the RCH method, where DST  0.7 deg is measured, delimited by the black square.

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Fig. 8. (Color online) Correlation coefficient between reconstructed phase images at different reconstruction distances from the same complex amplitude, which comes from real holograms with an average number of photons per pixel of 5100.

measured in the area defined by the black square is 0.69 deg. On the other hand, in Fig. 9(b), a phase image is shown after applying the proposed averaging procedure with 10 phase images, where the DST is 0.231 deg in the area defined by the black square. With these results, we verify a prospective decrease of 68.4% in noise. Figure 10(a) shows this decrease of shot noise for the marked profile as a white line in the phase images in Fig. 9. As has already been mentioned, the numbers of photons can be virtually increased by applying the averaging of reconstructed phase images. In Fig. 10(b), the DST of the reconstructed phase image is presented as a function of the number of phase images C used in the averaging procedure. This figure shows a comparison between simulation results with an

Fig. 9. Shot noise reduction through proposed averaging technique of experimental holograms. (a) Single reconstructed phase image with a phase STD of 0.693 deg. (b) Ten averaged phase images reconstructed with a Δd  2 μm to each other with a phase DST of 0.231 deg.

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Fig. 10. (Color online) Effect of phase DST reduction occurs when the proposed averaging method is applied. (a) Measurement of phase DST of the images in Fig. 9, where the profile is indicated by a white line. One can clearly see a diminution of DST when 10 phase images are averaged. (b) Measurement of phase DST as a function of the number of phase images, where a behavior of C −1∕2 is shown in the DST reduction.

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Fig. 11. Illustration of the experimentally determined DOF. (a) One of the recorded intensity holograms from an Edmund NBS 1963A resolution card and the profile taken to measure the DOF. (b) Evolution of intensity determined on a profile line when reconstruction distance is increased.

average number of 5100 photons per pixel and the experimental reconstruction results. Experimental values of the phase STD are larger than the values coming from simulations. One can see an offset of about 0.37 deg at the beginning, due to the presence of sources of noise other than shot noise, quantum efficiency, and small optical defects in the optical components. One can see that, if the number of averaged images increases, the offset also decreases. As we expected, a reduction of phase DST was obtained in an almost perfect behavior of C −1∕2 . Noise sources should be speckle and parasitic defects due to multiple reflections provoked by defects of the optical elements in the interferometer. In addition, noise related to the quantum efficiency is determined by the camera specifications given by the manufacturer as signal-to-noise ratio. Both speckle and multiple reflections are reduced greatly by using low spatial and temporal coherence source, as that used in our experimental setup. The noise related to the quantum efficiency is decreased by using a zero gain and high integration time. But the integration time during hologram recording is not the highest and the camera’s well capacity is not reached due to low illumination, therefore, the noise related to the quantum efficiency of the CCD detector is the main factor, which is the difference between the experimental and simulated results. A. Decrease of Shot Noise in Amplitude Images A principal limitation in the proposal is limited DOF. As we have already seen in Subsection 2.C, DOF is related to the sampling distance and NA of the optical system. In the system

Fig. 12. Reconstructed focused amplitude images. (a) Reconstructed focused amplitude image without averaging. (b) Reconstructed amplitude image when the averaging process is performed with four focused amplitude images.

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Fig. 13. (Color online) Comparison between profiles measured along the white lines defined in Figs. 12(a), 12(b).

described, the theoretical DOF is of about 0.268 μm. However, in the experimental reconstruction, DOF is higher than would have been expected from Eq. (7), due to the fact that the spatial resolution introduced by the optical setup is limited [24]. To experimentally illustrate the determination of DOF, in Fig. 11(b) we have plotted the intensity evolution with the reconstruction distance on a line profile from the recorded hologram shown in Fig. 11(a), where the profile zone is marked with a black line. The starting image is defocused by 40 μm. After zooming on the focus zone, we conclude that the DOF is 9 μm. The object used to record the intensity hologram was an Edmund NBS 1963A resolution card, where the zone of interest corresponds to 18 double lines per millimeter. The reconstruction distance was 15 μm. As the Δd calculated in the previous section was 2 μm, then we can average four images to carry out what we propose. These images have to be reconstructed at a distance within the DOF to guarantee that image quality is not affected.

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Figure 12 shows the improvement of the image when the proposed method is employed (the averaging of four reconstructed amplitude images at different reconstruction distances inside the DOF). This evaluation is through DST in the area defined by the white square in each image. We calculated a DST  1.57 gray levels (GLs) for an image focused without averaging [Fig. 12(a)], and DST  0.869 GL for the proposal [Fig. 13(b)]. So we can say that the averaging process also performs the amplitude image. For illustrating that the amplitude image improvement and quality are not affected when the averaging process proposed is performed, we have plotted a line profile marked by the white lines in Fig. 12. Figure 13 shows this plot and the comparison between the focused amplitude image without averaging and the improved amplitude image. One clearly notes, from this profile comparison, a little difference in the transition edges where the improved image (dashed curve) almost follows the one focus image transitions edges (solid curve). On the other hand, we can note clearly the improvement on the top and bottom areas from these profiles. B. Decrease of Shot Noise in Phase Images First, a 100 nm stepwise specimen made of TiO2 thin film, with a refraction index of 1.82 for a wavelength of 632.8 nm, was used as a phase calibrating gauge. The sample was made at home using a Balzer B-510 vapor deposition machine. To ensure a real and accurate measurement reference, the test sample was analyzed with a Digital Instruments 3100 AFM, with a maximum scan area of 100 μm × 100 μm × 7 μm. However, the following results contain an object-dependent “bias,” although the measurements should be interpreted in terms of noise reduction, not absolute values converging to zero. Figure 14(a) shows the reconstructed phase image of the

Fig. 14. Topographic measurement of the TiO2 stepwise specimen. (a) One reconstructed phase image. (b) Improved phase image when the averaging process is performed. (c) Corresponding zoomed area inside the white dashed rectangle in (b). (d) Numerical data extracted from AFM.

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stepwise, where the reconstruction distance was 10 μm. The STD measured in the zone enclosed by the black square is 3.44 nm. Every TiO2 step sample phase image shown in this section is inverted to render topographic phase images. We applied the proposed averaging of four reconstructed phase images at Δd  2 μm reconstruction distance inside the DOF. Figure 14(b) shows the improved phase image where the reduction of DST is of about 1.24 nm (corresponding to 0.57 deg of STD reduction). One can notice a difference with the predicted value of the experimental results, where the STD reduction was 0.35 deg [Fig. 10(b)]. We can say that the higher than expected SDT reduction is mainly due to a lower intensity recording than the blank recorded holograms. This lower intensity is due to the glass plate thickness located in the arms of the interferometer (4.7 mm glass of each). There are some other causes that generate noise in the phase image, such as the quantum efficiency of CCD and small optical defects in the optical components. The noise generated by these causes is also reduced by some percentage. The evaluation zone is marked in Fig. 14(b) by the white solid square. We extracted a measured profile and the topographic sample surface. Figure 14(d) shows the topographic measurement done by AFM. These data were extracted and corrected numerically. We corrected horizontal and vertical tilt mainly (all the calculations have been made in a MATLAB environment). Figure 14(c) shows the similar zone where the comparing information was taken of the improved phase image by zooming in on the white dashed rectangle of Fig. 14(b). Figure 15 presents the profile comparison extracted from the white lines in Figs. 14(c) and 14(d). These profiles show their respective measured heights to compare them. In this figure, a difference between both graphs can be noticed. The bigger difference is principally in the transition edges, due to divergences in the lateral resolution between the microscopes. As the lateral resolution of the AFM (the lateral space between taken samples named sampling rate is of 200 nm) is bigger than the lateral resolution of the DHM (sampling rate of 670 nm), the transition edge from DHM is softer than AFM. Also, we can note a phase map improvement in the top and bottom areas compared against the topographic measurement made with the AFM (blue solid curve), due to what was commented above. DHM should be capable of a bigger lateral resolution than achieved in this announcement by using a MO with the highest NA. This makes our proposal an alternative method comparable with AFM. In addition, the DHM method has some advantages over AFM. One of them is the time to get the topographic measurement; while our proposal takes just

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Fig. 16. (Color online) 3D TiO2 step images. (a) Topographic measurement done by AFM, (b) topographic measurement enhanced through the averaging process done by DHM.

seconds to perform the topographic measurement, the AFM takes several minutes to complete this task. Another advantage is the flexibility to put under test a bigger sample area than in the AFM method. A last important advantage of our proposal is that it is cheaper in its implementation that the AFM. Finally, Figs. 16(a) and 16(b) present the 3D topographic surface of the TiO2 step from Figs. 14(d) and 14(c), respectively. Figure 16(a) corresponds to the data provided by the AFM. The smaller sampling rate commented on above has a better ability to detect defects in the sample. Figure 16(b) corresponds to an improved topographic data obtained by DHM.

5. CONCLUSION

Fig. 15. (Color online) Comparison between profiles measured along the white lines defined on Figs. 14(c), and 14(d).

The phase information has great importance in DHM for the analysis and characterization of 3D objects, such as biological samples and micro-optical systems. In this study we have shown a different way to decrease the shot noise in DHM. We have shown the existence of shot noise in digital holograms when using a commercial LED as an illumination source. A way to demonstrate this fact is when the experimental results obtained are in agreement with the simulations. We also show a proposal that is based on the averaging process of reconstructed images at different reconstruction distances within the range determined by the focus depth. We obtained an improved phase image without quality diminution, in which a noise reduction of 50% was achieved, and that was

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in agreement with the value predicted by simulations. In addition, the averaging process does not demand the recording of a number of complex amplitudes, which is an advantage affecting the time of computation. So we have shown axial topographic measurements in agreement with the measurements made with a standard AFM and mentioned some advantages of our proposed method over the AFM method.

ACKNOWLEDGMENTS The authors recognize the financial support provided by the Council of Science and Technology of the Guanajuato State (CONCYTEG) under grant No. 09-04-K662-055-A03. Also, M. Leon recognizes a scholarship from the National Council of Science and Technology (CONACyT).

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