Spatio-temporal scanning modality for synthesizing interferograms and digital holograms V. Bianco,* M. Paturzo, and P. Ferraro CNR Istituto Nazionale di Ottica—Sezione di Napoli, Via Campi Flegrei, 34 80078 Pozzuoli (Napoli), Italy *[email protected]
Abstract: We investigate the spatio-temporal scanning of a single-pixel row for building up synthetic interferograms or digital holograms, shifted each other of a desired phase step. This unusual recording modality exploits the object movement to synthesize interferograms with extended Field of View and improved noise contrast. We report the theoretical formulation of the synthetizing recording process and experimental evidence of various cases demonstrating quantitative phase retrieval by adopting this intrinsic phaseshifting procedure. The proposed method could be particularly suited in all cases where the object shift is an intrinsic feature of the investigated system, as e.g. in microfluidics imaging. ©2014 Optical Society of America OCIS codes: (100.3175) Interferometric imaging; (090.1995) Digital holography; (050.5080) Phase shift; (100.5070) Phase retrieval.
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1. Introduction In recent years many efforts have been spent to develop new imaging modalities for highprecision optical measurements and non-disruptive testing based on interferometric techniques [1]. Among them, Phase-Shifting Interferometry [2–5] (PSI) is one of the preferred
#214515 - $15.00 USD Received 23 Jun 2014; revised 14 Jul 2014; accepted 16 Jul 2014; published 8 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022328 | OPTICS EXPRESS 22329
analysis technique, together with Digital Holography (DH) [6–12]. The latter has the intrinsic advantage to allow flexible numerical focusing of 3D objects without any mechanical scanning process along the optical axis. On the other hand, DH suffers from low resolution due to the off-axis recording geometry used to spatially filter the zeroth diffraction order and the conjugate image. To overcome this limitation, in-line PSI has been developed, in which an in-line recording geometry is adopted to decrease the maximum spatial frequency of the interference patterns on the recording plane and a phase-shifting algorithm is used to eliminate the zeroth order noise and the conjugate image [2–11]. A common challenge to both the methods is related to the possibility to enhance the interferograms/holograms in terms of Field of View (FoV), resolution, and Signal-to-Noise Ratio (SNR). In this framework, many methods rely on scanning modalities to acquire multiple interference patterns, which, if properly combined, provide images with extended resolution and FoV, as in the case of the synthesis of hologram aperture by femtosecond-pulsed DH [13], Optical Scanning Holography (OSH), where a point-detector collects light diffused by the object and a heterodyne demodulation decodes the useful object information [14–16], or Lensless Object Scanning Holography (LOSH) [17–19]. In LOSH, a scanning of the CCD camera or an illumination angle diversity has to be provided for the scope and the recorded fringe patterns have to be properly combined in a suitable domain for generating an overall synthetic hologram or interferograms entirely containing the required information. This operation can be complex and the method performance turn out to be object spectrum dependent. Besides, object scanning and illumination angle diversity are exploited in techniques such as Ptychography and the recently developed Fourier Ptychographic Microscopy (FPM) for retrieving, by an iterative algorithm, the full complex object field from a number of intensity measures obtained with coherent and LED sources respectively [20,21]. On the other hand, Multi-Look DH [22–26] incoherently combines multiple holograms acquired in time [27,28], wavelength [29], or polarization diversity [30] to achieve remarkable noise reduction in the numerical reconstruction. Here we show and discuss a completely different imaging modality, named SpatioTemporal Scanning Interferometry (STSI), which exploits the object movement to synthesize space-time interferograms with extended FoV and improved noise contrast, starting from interferograms acquired with a linear array of sensing elements instead of a 2D array detector. CCD linear array are obviously less expensive and more compact than the standard 2D CCD sensor. Moreover they can achieve higher frame rate of acquisition. They already find different successful applications such as in ref [31]. where a bacterial colony image scanner based on linear array CCD is patented or, for example, in Linear Array Panoramic Aerial Camera [32,33]. The modality presented here can be valuable and exploitable in many fields of application but especially where a relative lateral displacement between the imaging system and the object is intrinsic in the process under investigation. In particular, the new method we propose looks particularly advantageous whenever the object movement is an intrinsic feature of the investigated system, as e.g. in the case of microfluidics [26,34–38], thus allowing to make a step toward the integration of the imaging functionalities onboard Lab-on-a-Chip platforms. On the other hand, if linear arrays made by more than a single pixel row (it is important to note that the most of commercially available liner array usually have a number of rows from ranging from 1 to 128) or a common CCD camera are available, the proposed technique can be exploited to carry out PSI with enhanced FoV, SNR and, above all, with a small subset of detector elements. Usually, in the standard PSI technique, the phase steps between three or more phase-shifted interferograms, acquired successively, are induced using piezoelectric devices in the reference arm. On the contrary, in the present work the set-up does not rely on moving parts in order to induce the required phase-shift, but we exploit the object movement to create synthetic interferograms, shifted each other of the desired phase step. As described in the “method” section, starting from an interferogram with tilted fringes, each synthetic interferogram is built from the time sequence of a single row of the CCD sensor, recorded during the object movement. The number of rows needed to obtain all the phase-shifted
#214515 - $15.00 USD Received 23 Jun 2014; revised 14 Jul 2014; accepted 16 Jul 2014; published 8 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022328 | OPTICS EXPRESS 22330
interferograms, and, therefore, the dimension of the necessary CCD sensor, depends on the accepted phase-shifting error. Indeed, the minimum phase-shift obtainable with the proposed method is inversely proportional to the period of the spatial carrier. In principle, as it is well known, the whole complex object field could be retrieved from just only one synthetic interferogram if the spatial carrier frequency is such that in the Fourier transform the conjugated orders can be separated. This means that a single pixel row suffices as detector. In the next section, we describe the more general case where multiple phase-shifted spacetime interferograms can be obtained from which the whole complex information of the object can be retrieved. Then, we show the experimental results obtained in case of different objects, a test target and some intricate shaped polymeric drops, respectively, while they are shifted by means of a linear translation stage. This work can pave the way to new solutions in the field of bio-imaging and microfluidics [26], where the sample inherently moves along the channel, so that the proposed imaging modality offers the above mentioned advantages but having no cost associated with. 2. Method In this section we describe mathematically the recording process of the STSI modality. Let
S1 ( x, y , t = t1 ) ,..., S K ( x, y, t = tK )
x = 1,..., N ; y = 1,..., M
(1)
be the stack of interferograms acquired at the instants {t1 ,..., tK } while moving the object along the x direction. In Eq. (1) K = TF , where T and F are respectively the overall acquisition time and the frame rate. If the same selected line x = x0 is extracted from the acquired stack, we can obtain the set of vectors: c 01 = S1 ( x = x0 , y , t = t1 )
c 02 = S2 ( x = x0 , y , t = t2 )
(2)
c 0 K = S K ( x = x0 , y , t = t K ) .
Sorting these vectors in a matrix, we can build a synthetic, space-time interferogram I 0 ( y , t ) . The possibility to build an extended FoV synthetic interferogram from one single line can be exploited to replace the 2D MxN sensor with a linear detector consisting of M elements, more compact and achieving higher frame rate. The process of synthesis of a space-time interferogram is sketched on the bottom of Fig. 1, where the sensing elements of the detector are highlighted with a red line as well as the extended FoV obtainable after the interferogram synthesis. Noteworthy, the spatial distribution of the noise, which is always present due to the acquisition system, changes when passing from an x − y interferogram to a y − t synthetic one. Indeed, in the space-time interferogram this turns out to be variable only
#214515 - $15.00 USD Received 23 Jun 2014; revised 14 Jul 2014; accepted 16 Jul 2014; published 8 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022328 | OPTICS EXPRESS 22331
Fig. 1. Set-up employed to build-up synthetic interferograms. BS: Beam Splitter. M: Mirror. MO: Microscope Objective. BC: Beam Combiner. MS: Moving Stage. S: Sample. D: Detector. On the bottom the process of synthesis of an extended FoV space-time interferogram is sketched. The red line indicates the active sensor elements at each acquisition in case a linear detector is employed. The interferometric fringes are reported with the blue solid lines.
along the y direction whenever the acquisition system keeps stable during the time T . As a consequence, this is easier to be estimated and filtered out in the Fourier domain. Moreover, if a 2D sensor is employed, N different synthetic interferograms are obtainable by sweeping all the lines x = xi , i = 1,..., N , so that we can build the following stack: c 01 c C = 11 cN 1
c 02 c12 cN 2
c 0 K I 0 ( y, t ) c1 K I 1 ( y , t ) = , c NK I N ( y , t )
(3)
corresponding to N − 1 different phase shifts. The minimum phase shift obtainable with the proposed method depends on the acquisition parameters as follows: ΔΨ min =
2πΔx sin ϑ p
(4)
where Δx is the pixel size along the object moving direction x , p is the pitch of the spatial carrier, and ϑ is the fringe inclination on the acquisition plane ( x, y ) with respect to the x axis. The fringe inclination ϑ also establishes the relationship between p and the pitch of the spatial carrier in the plane (y,t) , namely p ' = p cos ϑ . Hence, it results that a fringe system orthogonal to the movement direction ( ϑ = π / 2 ) does not provide an interferogram in the plane (y,t) , while fringes parallel to the x-axis ( ϑ = 0 ) allow the synthesis of space-time interferograms but cannot provide any phase-shift between them. In particular, the range of admissible angles depends on the sensor parameters as follows:
#214515 - $15.00 USD Received 23 Jun 2014; revised 14 Jul 2014; accepted 16 Jul 2014; published 8 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022328 | OPTICS EXPRESS 22332
Δy − 1 M Δy tg −1 ≤ ϑ ≤ tg N Δx Δx
(5)
The elements of the stack in Eq. (3) can be used to retrieve the complex field by a phase– shifting algorithm. Let EO ( x, y ) = EO ( x, y ) exp [iϕ ( x, y )] E R ( x, y ) = ER exp [iψ ( x, y )] ,
(6)
be the object beam and the reference beam in the acquisition plane ( x, y ) at time t = t0 , respectively. The detector (e.g. a CCD camera) records the interference pattern between the object wave and the reference wave, i.e. the signal S ( x , y , t = t0 ) = E 0 + E R = 2
2
= E0 ( x, y ) + E R2 + 2 E0 ( x, y ) ER cos ϕ ( x, y ) + ψ ( x, y ) .
(7)
In PSI both the amplitude and the phase of the object beam can be extracted tuning L times the phase of the reference beam ψ = ψ 0 + Δψ and combining the corresponding L intensity measures according to proper algorithms [2–5]. For instance, if L = 4 intensity measures are carried out selecting the phase shift values as Δψ = {0, π 2 , π ,3π 2} , we obtain: S0 = E0 2 + E R2 + 2 E0 ER cos ϕ 2 Sπ /2 = E0 + E R2 − 2 E0 E R sin ϕ , 2 2 Sπ = E0 + ER − 2 E0 E R cos ϕ 2 2 S3π /2 = E0 + E R + 2 E0 E R sin ϕ
(8)
where we put ψ 0 = 0 for the sake of simplicity. From Eq. (8) the complex object beam can be extracted: E0 =
1 [( S0 − Sπ ) + i( S3π /2 − Sπ /2 )]. 4 ER
(9)
Similarly, if three measures are carried out the complex object beam can be found as: E0 =
1+ i [( S0 − Sπ /2 ) + i( Sπ − Sπ /2 )] , 4
(10)
from which the amplitude and the phase of the object wave is obtained. Obviously, once the full complex field is retrieved, this can be numerically propagated back or forth as occurs in Digital Holography modality. This enables to acquire holograms out of the object focus plane, as in the case of cells flowing inside a channel, each of which can occupy different positions along the optical axis. In PSI the phase variation is obtained introducing a delay line in the path of the reference beam, e.g. using piezoelectric mirror devices. On the contrary, we exploit the object movement as shown in the following section. 3. Experimental results
The experimental set-up is a classical Mach-Zehnder interferometer in transmission configuration, as shown in Fig. 1. In our experiments the beam emitted by the laser source has a wavelength λ = 632nm , and a CCD camera with Δx = Δy = 4.4 μ m pixel size has been
#214515 - $15.00 USD Received 23 Jun 2014; revised 14 Jul 2014; accepted 16 Jul 2014; published 8 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022328 | OPTICS EXPRESS 22333
used to collect the pattern of interference between the object beam and the reference. A remote controlled linear stage has been used to move the object along the x direction at a proper speed, such that the object was well sampled. In order to obtain straight and parallel fringes, a Fourier configuration for the interferometer has been adopted, i.e. the object beam and the reference beam have been arranged in order to have the same curvature. As a first experiment, an interferogram of a test resolution chart has been acquired while moving the object along the x direction. Figure 2(a) shows the interferogram corresponding to a fixed position x0 of the target, where the fringe orientation is oblique with respect to the x axis. In Fig. 2(b) the synthetic interferogram is shown, which is obtained from the signal corresponding to one single column of the acquired 2D matrix (see the red line in Fig. 2(a)) at different times. Indeed, if the time diversity is exploited, this is sufficient to synthesize the image of the whole object, also enlarging the system FoV. As expected, in the synthetic interferogram the fringe orientation is perpendicular to the y axis. Figure 3 shows the creation of a synthetic interferogram from a time sequence of recordings. In this experiment, a small subset of pixel of the employed CCD camera has been used. This would also allow to maximize the sensor frame-rate, so that geometrical super-resolution in the x direction could be achieved as a result of a better sampling [39]. However, this is not the purpose of this work, so that a frame rate of 47, 3 Frames s has been selected while the target velocity was set to v = 208, 3 μ m s . Here we just want to show that a large FoV can be obtained from one single sensor line if the target movement is exploited. Again, it is worth to notice the expected change in the fringe orientation when passing from the acquired interferograms (oblique fringes) to the synthetic one. Moreover, the spatial distribution of the noise corrupting the image changes as well, becoming constant along the direction in the synthetic reconstruction. This results in an improved image contrast and, hence, a better visualization of the target. In our second experiment a pure phase object has been analyzed, namely a Polydimethylsiloxane (PDMS) drop having an intricate geometry. Figure 6(a) shows a microscope image of the drop, where it is apparent its very complex shape, with two thickness maxima on the left and the right side of the drop and a saddle point in the middle. The synthetic interferogram is shown in Fig. 4, where circular fringes, corresponding to the two thickness maxima, are well visible. Noteworthy, due to the complex morphology of the test object the fringe spacing is too small in those areas characterized by high slope, and under-sampled areas are apparent where the useful signal gets lost (see the red circled zones in Fig. 4). Synthetic interferograms are extracted by tuning the selected column x. As shown in Media 1, a change in the column selected to build the synthetic interferogram has three main consequences: A shift along the x direction of the position of the object. A shift of the horizontal fringes in the y direction. A phase shift in the object area, provided without the need of delay lines in the setup.
#214515 - $15.00 USD Received 23 Jun 2014; revised 14 Jul 2014; accepted 16 Jul 2014; published 8 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022328 | OPTICS EXPRESS 22334
Fig. 2. Interferometric imaging of a test resolution target. (a) Acquired limited FoV interferogram. (b) Extended FoV synthetic interferogram built from the red column in Fig. 2(a) while scanning the target along the x direction.
Fig. 3. Interferogram synthesis from one single detector row (red line). Extended FoV is achievable exploiting the object shift. Improved visualization is also obtained as the noise is constant along the t axis.
In particular, three synthetic holograms, if properly selected, are sufficient to implement a three step Phase Shift (PS) algorithm, in order to retrieve the quantitative information about the object optical thickness. Figure 5 shows a flow chart sketching the main processing steps of the proposed technique. Starting from the acquired interferogram stack [S1 ,...,SK ] we
#214515 - $15.00 USD Received 23 Jun 2014; revised 14 Jul 2014; accepted 16 Jul 2014; published 8 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022328 | OPTICS EXPRESS 22335
Fig. 4. (Media 1) Synthetic interferogram of a PDMS drop. The red circled zones show the areas where the fringe spacing is too small to properly sample the high slope object parts.
Fig. 5. (Media 1) Flow chart illustrating the processing blocks of the proposed technique.
extract three ensembles of vectors, namely [c 01 ,...,c 0K ] , [c11 ,...,c1K ] , [c 21 ,...,c 2K ] , respectively obtained selecting three different columns x = xi , i = 0,1, 2 . In our experiments the spacing between the most distant pixel rows employed as detectors for PS purposes is 36 pixel, a number compatible with the dimensions of commercially available linear CCD arrays. Each ensemble leads to the synthesis of a space-time interferogram, namely: I 0 (y, t) = I Δψ =0 I1 (y, t) = I Δψ =π /2
(11)
I 2 (y, t) = I Δψ =π
corresponding to three different phase shifts Δψ = [0, π 2 , π ] . These synthetic interferograms are re-aligned along the x direction so that they constitute the input of the processing block implementing Eq. (10), i.e. the three-step PS algorithm. This returns both the amplitude and the wrapped phase of the reconstructed complex wavefield E 0 , shown in Fig. 6(b) and Fig. 6(c) respectively. We applied a phase unwrapping algorithm [40] to the map shown in Fig. 6(c), in order to obtain a quantitative unwrapped phase-contrast image of the object, shown in Fig. 6(d). A pseudo-3D map of the unwrapped phase signal ϕ u ( y , t ) is
#214515 - $15.00 USD Received 23 Jun 2014; revised 14 Jul 2014; accepted 16 Jul 2014; published 8 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022328 | OPTICS EXPRESS 22336
Fig. 6. PS technique applied to a synthetic interferogram. (a) Microscope image of a complex shape PDMS drop deposited on a plane substrate. (b-c) Synthetic amplitude (b) and wrapped phase (c) of the drop, obtained applying the three step PS algorithm. (d) Unwrapped phasecontast map of the drop [40].
Fig. 7. Pseudo 3D phase-contrast map of the PDMS drop obtained unwrapping the signal reported in Fig. 6(c) [40].
#214515 - $15.00 USD Received 23 Jun 2014; revised 14 Jul 2014; accepted 16 Jul 2014; published 8 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022328 | OPTICS EXPRESS 22337
Fig. 8. (Media 2) A PDMS micro-axicon and a PDMS drop are deposited on a plane substrate. (a-b) Space-time amplitude (a) and wrapped phase (b) of the objects obtained applying the three step PS algorithm. (c) Microscope image of the micro-axicon.
reported in Fig. 7, where it is apparent the capability of the proposed method to recover the 3D shape of the object. As a further validation, we chose a phase object of particular interest, namely a PDMS micro-axicon. This is a widely studied object due to its characteristic optical properties [41,42]. Indeed, axicon lenses can change Gaussian wavefronts into Bessel beams. The extended depth of focus of a Bessel beam can be exploited in various fields, e.g. optical tweezing [43–45], optical microscopy of biological samples and cell manipulation [43–48], optical coherence tomography [49]. Besides, a small PDMS drop has been layered on the moving stage near the axicon. Figure 8(a) shows the amplitude of the complex wavefield obtained after applying the proposed technique, while the wrapped phase of both the objects present in FoV is shown in Fig. 8(b). A microscope image of the axicon (acquired at different orientation with respect to Figs. 8(a), 8(b)) is shown in Fig. 8(c). Here we chose to implement the 3-step PS algorithm (Eq. (10)) to retrieve the phase from the synthetic interferograms. However, it is worth to notice that, once the interferogram stack is acquired, synthetic interferograms can be obtained corresponding to whatever phase shift required, e.g. a 4-step PS algorithm (Eq. (9)) or the N-measures method [2–5] could be applied without further acquisitions or changes to the optical set-up. In our last experiment, we put on the moving stage a PDMS drop which took a circular shape after being cured. The corresponding synthetic interferogram is presented in Fig. 9(a). Circular fringes are well visible in the object area, whereas horizontal fringes are present in the background. The choice of any different column x=x i ,i=1,...,N provides a phase shift between the corresponding space-time interferograms, as shown in Media 2. As said above, we can select a-posteriori which PS algorithm implement to obtain the best quality phase map. In this case we applied the 4-step PS algorithm of Eq. (9) to retrieve the phase map ϕ ( y , t ) of the drop. Figure 9(b) and Fig. 10 respectively show the grayscale image and the pseudo-3D view of the unwrapped phase ϕ u ( y , t ) [40], from which the optical thickness of the object can be measured.
#214515 - $15.00 USD Received 23 Jun 2014; revised 14 Jul 2014; accepted 16 Jul 2014; published 8 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022328 | OPTICS EXPRESS 22338
Fig. 9. (a) Syntetic interferogram of a PDMS circular drop. (b) Unwrapped phase-contrast map of the object obtained after applying the four step PS formula of Eq. (9) [40].
Fig. 10. Pseudo 3D unwrapped phase-contrast map of the PDMS circular drop of Fig. 9 [40].
4. Conclusions
In this work, we have described a Spatio-Temporal Scanning Interferometry (STSI) method for building up synthetic interferograms. This possesses some intriguing features as the spacetime interferograms offer an extended FoV with respect to the classical ones. Moreover, in a space-time interferogram the noise only varies along the normal to the scanning direction, assuring an improved visualization of the object, provided that the object moves at constant velocity, and better SNR conditions for further processing, like e.g. PSI. Besides the theoretical formulation, experiments have been performed in order to show in details how we retrieve the quantitative phase maps by adopting this intrinsic phase-shifting procedure. The same configuration STSI can be exploited for improving the resolution along the scanning direction as different views of the object can provide an increased numerical aperture of the imaging system [50]. Finally, it worth to underline that, thanks to its characteristics, the proposed approach can be used in microfluidics as a simple but effective tool to obtain the phase maps of objects flowing in microfluidics channels. Acknowledgments
This work was supported by Progetto Bandiera “La Fabbrica del Futuro” in the framework of the funded project “Plastic lab-on-chips for the optical manipulation of single cells” (PLUS) and by Progetto “Smart Cities” – Aquasystem: “Procedure e tecnologie innovative per una gestione pianificata ed integrata delle risorse idriche, l'ottimizzazione energetica ed il controllo della qualità nel Ciclo Integrato delle Acque”.
#214515 - $15.00 USD Received 23 Jun 2014; revised 14 Jul 2014; accepted 16 Jul 2014; published 8 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022328 | OPTICS EXPRESS 22339
Abstract: We investigate the spatio-temporal scanning of a single-pixel row for building up synthetic interferograms or digital holograms, shifted each other of a desired phase step. This unusual recording modality exploits the object movement to synthesize interferograms with extended Field of View and improved noise contrast. We report the theoretical formulation of the synthetizing recording process and experimental evidence of various cases demonstrating quantitative phase retrieval by adopting this intrinsic phaseshifting procedure. The proposed method could be particularly suited in all cases where the object shift is an intrinsic feature of the investigated system, as e.g. in microfluidics imaging. ©2014 Optical Society of America OCIS codes: (100.3175) Interferometric imaging; (090.1995) Digital holography; (050.5080) Phase shift; (100.5070) Phase retrieval.
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1. Introduction In recent years many efforts have been spent to develop new imaging modalities for highprecision optical measurements and non-disruptive testing based on interferometric techniques [1]. Among them, Phase-Shifting Interferometry [2–5] (PSI) is one of the preferred
#214515 - $15.00 USD Received 23 Jun 2014; revised 14 Jul 2014; accepted 16 Jul 2014; published 8 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022328 | OPTICS EXPRESS 22329
analysis technique, together with Digital Holography (DH) [6–12]. The latter has the intrinsic advantage to allow flexible numerical focusing of 3D objects without any mechanical scanning process along the optical axis. On the other hand, DH suffers from low resolution due to the off-axis recording geometry used to spatially filter the zeroth diffraction order and the conjugate image. To overcome this limitation, in-line PSI has been developed, in which an in-line recording geometry is adopted to decrease the maximum spatial frequency of the interference patterns on the recording plane and a phase-shifting algorithm is used to eliminate the zeroth order noise and the conjugate image [2–11]. A common challenge to both the methods is related to the possibility to enhance the interferograms/holograms in terms of Field of View (FoV), resolution, and Signal-to-Noise Ratio (SNR). In this framework, many methods rely on scanning modalities to acquire multiple interference patterns, which, if properly combined, provide images with extended resolution and FoV, as in the case of the synthesis of hologram aperture by femtosecond-pulsed DH [13], Optical Scanning Holography (OSH), where a point-detector collects light diffused by the object and a heterodyne demodulation decodes the useful object information [14–16], or Lensless Object Scanning Holography (LOSH) [17–19]. In LOSH, a scanning of the CCD camera or an illumination angle diversity has to be provided for the scope and the recorded fringe patterns have to be properly combined in a suitable domain for generating an overall synthetic hologram or interferograms entirely containing the required information. This operation can be complex and the method performance turn out to be object spectrum dependent. Besides, object scanning and illumination angle diversity are exploited in techniques such as Ptychography and the recently developed Fourier Ptychographic Microscopy (FPM) for retrieving, by an iterative algorithm, the full complex object field from a number of intensity measures obtained with coherent and LED sources respectively [20,21]. On the other hand, Multi-Look DH [22–26] incoherently combines multiple holograms acquired in time [27,28], wavelength [29], or polarization diversity [30] to achieve remarkable noise reduction in the numerical reconstruction. Here we show and discuss a completely different imaging modality, named SpatioTemporal Scanning Interferometry (STSI), which exploits the object movement to synthesize space-time interferograms with extended FoV and improved noise contrast, starting from interferograms acquired with a linear array of sensing elements instead of a 2D array detector. CCD linear array are obviously less expensive and more compact than the standard 2D CCD sensor. Moreover they can achieve higher frame rate of acquisition. They already find different successful applications such as in ref [31]. where a bacterial colony image scanner based on linear array CCD is patented or, for example, in Linear Array Panoramic Aerial Camera [32,33]. The modality presented here can be valuable and exploitable in many fields of application but especially where a relative lateral displacement between the imaging system and the object is intrinsic in the process under investigation. In particular, the new method we propose looks particularly advantageous whenever the object movement is an intrinsic feature of the investigated system, as e.g. in the case of microfluidics [26,34–38], thus allowing to make a step toward the integration of the imaging functionalities onboard Lab-on-a-Chip platforms. On the other hand, if linear arrays made by more than a single pixel row (it is important to note that the most of commercially available liner array usually have a number of rows from ranging from 1 to 128) or a common CCD camera are available, the proposed technique can be exploited to carry out PSI with enhanced FoV, SNR and, above all, with a small subset of detector elements. Usually, in the standard PSI technique, the phase steps between three or more phase-shifted interferograms, acquired successively, are induced using piezoelectric devices in the reference arm. On the contrary, in the present work the set-up does not rely on moving parts in order to induce the required phase-shift, but we exploit the object movement to create synthetic interferograms, shifted each other of the desired phase step. As described in the “method” section, starting from an interferogram with tilted fringes, each synthetic interferogram is built from the time sequence of a single row of the CCD sensor, recorded during the object movement. The number of rows needed to obtain all the phase-shifted
#214515 - $15.00 USD Received 23 Jun 2014; revised 14 Jul 2014; accepted 16 Jul 2014; published 8 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022328 | OPTICS EXPRESS 22330
interferograms, and, therefore, the dimension of the necessary CCD sensor, depends on the accepted phase-shifting error. Indeed, the minimum phase-shift obtainable with the proposed method is inversely proportional to the period of the spatial carrier. In principle, as it is well known, the whole complex object field could be retrieved from just only one synthetic interferogram if the spatial carrier frequency is such that in the Fourier transform the conjugated orders can be separated. This means that a single pixel row suffices as detector. In the next section, we describe the more general case where multiple phase-shifted spacetime interferograms can be obtained from which the whole complex information of the object can be retrieved. Then, we show the experimental results obtained in case of different objects, a test target and some intricate shaped polymeric drops, respectively, while they are shifted by means of a linear translation stage. This work can pave the way to new solutions in the field of bio-imaging and microfluidics [26], where the sample inherently moves along the channel, so that the proposed imaging modality offers the above mentioned advantages but having no cost associated with. 2. Method In this section we describe mathematically the recording process of the STSI modality. Let
S1 ( x, y , t = t1 ) ,..., S K ( x, y, t = tK )
x = 1,..., N ; y = 1,..., M
(1)
be the stack of interferograms acquired at the instants {t1 ,..., tK } while moving the object along the x direction. In Eq. (1) K = TF , where T and F are respectively the overall acquisition time and the frame rate. If the same selected line x = x0 is extracted from the acquired stack, we can obtain the set of vectors: c 01 = S1 ( x = x0 , y , t = t1 )
c 02 = S2 ( x = x0 , y , t = t2 )
(2)
c 0 K = S K ( x = x0 , y , t = t K ) .
Sorting these vectors in a matrix, we can build a synthetic, space-time interferogram I 0 ( y , t ) . The possibility to build an extended FoV synthetic interferogram from one single line can be exploited to replace the 2D MxN sensor with a linear detector consisting of M elements, more compact and achieving higher frame rate. The process of synthesis of a space-time interferogram is sketched on the bottom of Fig. 1, where the sensing elements of the detector are highlighted with a red line as well as the extended FoV obtainable after the interferogram synthesis. Noteworthy, the spatial distribution of the noise, which is always present due to the acquisition system, changes when passing from an x − y interferogram to a y − t synthetic one. Indeed, in the space-time interferogram this turns out to be variable only
#214515 - $15.00 USD Received 23 Jun 2014; revised 14 Jul 2014; accepted 16 Jul 2014; published 8 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022328 | OPTICS EXPRESS 22331
Fig. 1. Set-up employed to build-up synthetic interferograms. BS: Beam Splitter. M: Mirror. MO: Microscope Objective. BC: Beam Combiner. MS: Moving Stage. S: Sample. D: Detector. On the bottom the process of synthesis of an extended FoV space-time interferogram is sketched. The red line indicates the active sensor elements at each acquisition in case a linear detector is employed. The interferometric fringes are reported with the blue solid lines.
along the y direction whenever the acquisition system keeps stable during the time T . As a consequence, this is easier to be estimated and filtered out in the Fourier domain. Moreover, if a 2D sensor is employed, N different synthetic interferograms are obtainable by sweeping all the lines x = xi , i = 1,..., N , so that we can build the following stack: c 01 c C = 11 cN 1
c 02 c12 cN 2
c 0 K I 0 ( y, t ) c1 K I 1 ( y , t ) = , c NK I N ( y , t )
(3)
corresponding to N − 1 different phase shifts. The minimum phase shift obtainable with the proposed method depends on the acquisition parameters as follows: ΔΨ min =
2πΔx sin ϑ p
(4)
where Δx is the pixel size along the object moving direction x , p is the pitch of the spatial carrier, and ϑ is the fringe inclination on the acquisition plane ( x, y ) with respect to the x axis. The fringe inclination ϑ also establishes the relationship between p and the pitch of the spatial carrier in the plane (y,t) , namely p ' = p cos ϑ . Hence, it results that a fringe system orthogonal to the movement direction ( ϑ = π / 2 ) does not provide an interferogram in the plane (y,t) , while fringes parallel to the x-axis ( ϑ = 0 ) allow the synthesis of space-time interferograms but cannot provide any phase-shift between them. In particular, the range of admissible angles depends on the sensor parameters as follows:
#214515 - $15.00 USD Received 23 Jun 2014; revised 14 Jul 2014; accepted 16 Jul 2014; published 8 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022328 | OPTICS EXPRESS 22332
Δy − 1 M Δy tg −1 ≤ ϑ ≤ tg N Δx Δx
(5)
The elements of the stack in Eq. (3) can be used to retrieve the complex field by a phase– shifting algorithm. Let EO ( x, y ) = EO ( x, y ) exp [iϕ ( x, y )] E R ( x, y ) = ER exp [iψ ( x, y )] ,
(6)
be the object beam and the reference beam in the acquisition plane ( x, y ) at time t = t0 , respectively. The detector (e.g. a CCD camera) records the interference pattern between the object wave and the reference wave, i.e. the signal S ( x , y , t = t0 ) = E 0 + E R = 2
2
= E0 ( x, y ) + E R2 + 2 E0 ( x, y ) ER cos ϕ ( x, y ) + ψ ( x, y ) .
(7)
In PSI both the amplitude and the phase of the object beam can be extracted tuning L times the phase of the reference beam ψ = ψ 0 + Δψ and combining the corresponding L intensity measures according to proper algorithms [2–5]. For instance, if L = 4 intensity measures are carried out selecting the phase shift values as Δψ = {0, π 2 , π ,3π 2} , we obtain: S0 = E0 2 + E R2 + 2 E0 ER cos ϕ 2 Sπ /2 = E0 + E R2 − 2 E0 E R sin ϕ , 2 2 Sπ = E0 + ER − 2 E0 E R cos ϕ 2 2 S3π /2 = E0 + E R + 2 E0 E R sin ϕ
(8)
where we put ψ 0 = 0 for the sake of simplicity. From Eq. (8) the complex object beam can be extracted: E0 =
1 [( S0 − Sπ ) + i( S3π /2 − Sπ /2 )]. 4 ER
(9)
Similarly, if three measures are carried out the complex object beam can be found as: E0 =
1+ i [( S0 − Sπ /2 ) + i( Sπ − Sπ /2 )] , 4
(10)
from which the amplitude and the phase of the object wave is obtained. Obviously, once the full complex field is retrieved, this can be numerically propagated back or forth as occurs in Digital Holography modality. This enables to acquire holograms out of the object focus plane, as in the case of cells flowing inside a channel, each of which can occupy different positions along the optical axis. In PSI the phase variation is obtained introducing a delay line in the path of the reference beam, e.g. using piezoelectric mirror devices. On the contrary, we exploit the object movement as shown in the following section. 3. Experimental results
The experimental set-up is a classical Mach-Zehnder interferometer in transmission configuration, as shown in Fig. 1. In our experiments the beam emitted by the laser source has a wavelength λ = 632nm , and a CCD camera with Δx = Δy = 4.4 μ m pixel size has been
#214515 - $15.00 USD Received 23 Jun 2014; revised 14 Jul 2014; accepted 16 Jul 2014; published 8 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022328 | OPTICS EXPRESS 22333
used to collect the pattern of interference between the object beam and the reference. A remote controlled linear stage has been used to move the object along the x direction at a proper speed, such that the object was well sampled. In order to obtain straight and parallel fringes, a Fourier configuration for the interferometer has been adopted, i.e. the object beam and the reference beam have been arranged in order to have the same curvature. As a first experiment, an interferogram of a test resolution chart has been acquired while moving the object along the x direction. Figure 2(a) shows the interferogram corresponding to a fixed position x0 of the target, where the fringe orientation is oblique with respect to the x axis. In Fig. 2(b) the synthetic interferogram is shown, which is obtained from the signal corresponding to one single column of the acquired 2D matrix (see the red line in Fig. 2(a)) at different times. Indeed, if the time diversity is exploited, this is sufficient to synthesize the image of the whole object, also enlarging the system FoV. As expected, in the synthetic interferogram the fringe orientation is perpendicular to the y axis. Figure 3 shows the creation of a synthetic interferogram from a time sequence of recordings. In this experiment, a small subset of pixel of the employed CCD camera has been used. This would also allow to maximize the sensor frame-rate, so that geometrical super-resolution in the x direction could be achieved as a result of a better sampling [39]. However, this is not the purpose of this work, so that a frame rate of 47, 3 Frames s has been selected while the target velocity was set to v = 208, 3 μ m s . Here we just want to show that a large FoV can be obtained from one single sensor line if the target movement is exploited. Again, it is worth to notice the expected change in the fringe orientation when passing from the acquired interferograms (oblique fringes) to the synthetic one. Moreover, the spatial distribution of the noise corrupting the image changes as well, becoming constant along the direction in the synthetic reconstruction. This results in an improved image contrast and, hence, a better visualization of the target. In our second experiment a pure phase object has been analyzed, namely a Polydimethylsiloxane (PDMS) drop having an intricate geometry. Figure 6(a) shows a microscope image of the drop, where it is apparent its very complex shape, with two thickness maxima on the left and the right side of the drop and a saddle point in the middle. The synthetic interferogram is shown in Fig. 4, where circular fringes, corresponding to the two thickness maxima, are well visible. Noteworthy, due to the complex morphology of the test object the fringe spacing is too small in those areas characterized by high slope, and under-sampled areas are apparent where the useful signal gets lost (see the red circled zones in Fig. 4). Synthetic interferograms are extracted by tuning the selected column x. As shown in Media 1, a change in the column selected to build the synthetic interferogram has three main consequences: A shift along the x direction of the position of the object. A shift of the horizontal fringes in the y direction. A phase shift in the object area, provided without the need of delay lines in the setup.
#214515 - $15.00 USD Received 23 Jun 2014; revised 14 Jul 2014; accepted 16 Jul 2014; published 8 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022328 | OPTICS EXPRESS 22334
Fig. 2. Interferometric imaging of a test resolution target. (a) Acquired limited FoV interferogram. (b) Extended FoV synthetic interferogram built from the red column in Fig. 2(a) while scanning the target along the x direction.
Fig. 3. Interferogram synthesis from one single detector row (red line). Extended FoV is achievable exploiting the object shift. Improved visualization is also obtained as the noise is constant along the t axis.
In particular, three synthetic holograms, if properly selected, are sufficient to implement a three step Phase Shift (PS) algorithm, in order to retrieve the quantitative information about the object optical thickness. Figure 5 shows a flow chart sketching the main processing steps of the proposed technique. Starting from the acquired interferogram stack [S1 ,...,SK ] we
#214515 - $15.00 USD Received 23 Jun 2014; revised 14 Jul 2014; accepted 16 Jul 2014; published 8 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022328 | OPTICS EXPRESS 22335
Fig. 4. (Media 1) Synthetic interferogram of a PDMS drop. The red circled zones show the areas where the fringe spacing is too small to properly sample the high slope object parts.
Fig. 5. (Media 1) Flow chart illustrating the processing blocks of the proposed technique.
extract three ensembles of vectors, namely [c 01 ,...,c 0K ] , [c11 ,...,c1K ] , [c 21 ,...,c 2K ] , respectively obtained selecting three different columns x = xi , i = 0,1, 2 . In our experiments the spacing between the most distant pixel rows employed as detectors for PS purposes is 36 pixel, a number compatible with the dimensions of commercially available linear CCD arrays. Each ensemble leads to the synthesis of a space-time interferogram, namely: I 0 (y, t) = I Δψ =0 I1 (y, t) = I Δψ =π /2
(11)
I 2 (y, t) = I Δψ =π
corresponding to three different phase shifts Δψ = [0, π 2 , π ] . These synthetic interferograms are re-aligned along the x direction so that they constitute the input of the processing block implementing Eq. (10), i.e. the three-step PS algorithm. This returns both the amplitude and the wrapped phase of the reconstructed complex wavefield E 0 , shown in Fig. 6(b) and Fig. 6(c) respectively. We applied a phase unwrapping algorithm [40] to the map shown in Fig. 6(c), in order to obtain a quantitative unwrapped phase-contrast image of the object, shown in Fig. 6(d). A pseudo-3D map of the unwrapped phase signal ϕ u ( y , t ) is
#214515 - $15.00 USD Received 23 Jun 2014; revised 14 Jul 2014; accepted 16 Jul 2014; published 8 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022328 | OPTICS EXPRESS 22336
Fig. 6. PS technique applied to a synthetic interferogram. (a) Microscope image of a complex shape PDMS drop deposited on a plane substrate. (b-c) Synthetic amplitude (b) and wrapped phase (c) of the drop, obtained applying the three step PS algorithm. (d) Unwrapped phasecontast map of the drop [40].
Fig. 7. Pseudo 3D phase-contrast map of the PDMS drop obtained unwrapping the signal reported in Fig. 6(c) [40].
#214515 - $15.00 USD Received 23 Jun 2014; revised 14 Jul 2014; accepted 16 Jul 2014; published 8 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022328 | OPTICS EXPRESS 22337
Fig. 8. (Media 2) A PDMS micro-axicon and a PDMS drop are deposited on a plane substrate. (a-b) Space-time amplitude (a) and wrapped phase (b) of the objects obtained applying the three step PS algorithm. (c) Microscope image of the micro-axicon.
reported in Fig. 7, where it is apparent the capability of the proposed method to recover the 3D shape of the object. As a further validation, we chose a phase object of particular interest, namely a PDMS micro-axicon. This is a widely studied object due to its characteristic optical properties [41,42]. Indeed, axicon lenses can change Gaussian wavefronts into Bessel beams. The extended depth of focus of a Bessel beam can be exploited in various fields, e.g. optical tweezing [43–45], optical microscopy of biological samples and cell manipulation [43–48], optical coherence tomography [49]. Besides, a small PDMS drop has been layered on the moving stage near the axicon. Figure 8(a) shows the amplitude of the complex wavefield obtained after applying the proposed technique, while the wrapped phase of both the objects present in FoV is shown in Fig. 8(b). A microscope image of the axicon (acquired at different orientation with respect to Figs. 8(a), 8(b)) is shown in Fig. 8(c). Here we chose to implement the 3-step PS algorithm (Eq. (10)) to retrieve the phase from the synthetic interferograms. However, it is worth to notice that, once the interferogram stack is acquired, synthetic interferograms can be obtained corresponding to whatever phase shift required, e.g. a 4-step PS algorithm (Eq. (9)) or the N-measures method [2–5] could be applied without further acquisitions or changes to the optical set-up. In our last experiment, we put on the moving stage a PDMS drop which took a circular shape after being cured. The corresponding synthetic interferogram is presented in Fig. 9(a). Circular fringes are well visible in the object area, whereas horizontal fringes are present in the background. The choice of any different column x=x i ,i=1,...,N provides a phase shift between the corresponding space-time interferograms, as shown in Media 2. As said above, we can select a-posteriori which PS algorithm implement to obtain the best quality phase map. In this case we applied the 4-step PS algorithm of Eq. (9) to retrieve the phase map ϕ ( y , t ) of the drop. Figure 9(b) and Fig. 10 respectively show the grayscale image and the pseudo-3D view of the unwrapped phase ϕ u ( y , t ) [40], from which the optical thickness of the object can be measured.
#214515 - $15.00 USD Received 23 Jun 2014; revised 14 Jul 2014; accepted 16 Jul 2014; published 8 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022328 | OPTICS EXPRESS 22338
Fig. 9. (a) Syntetic interferogram of a PDMS circular drop. (b) Unwrapped phase-contrast map of the object obtained after applying the four step PS formula of Eq. (9) [40].
Fig. 10. Pseudo 3D unwrapped phase-contrast map of the PDMS circular drop of Fig. 9 [40].
4. Conclusions
In this work, we have described a Spatio-Temporal Scanning Interferometry (STSI) method for building up synthetic interferograms. This possesses some intriguing features as the spacetime interferograms offer an extended FoV with respect to the classical ones. Moreover, in a space-time interferogram the noise only varies along the normal to the scanning direction, assuring an improved visualization of the object, provided that the object moves at constant velocity, and better SNR conditions for further processing, like e.g. PSI. Besides the theoretical formulation, experiments have been performed in order to show in details how we retrieve the quantitative phase maps by adopting this intrinsic phase-shifting procedure. The same configuration STSI can be exploited for improving the resolution along the scanning direction as different views of the object can provide an increased numerical aperture of the imaging system [50]. Finally, it worth to underline that, thanks to its characteristics, the proposed approach can be used in microfluidics as a simple but effective tool to obtain the phase maps of objects flowing in microfluidics channels. Acknowledgments
This work was supported by Progetto Bandiera “La Fabbrica del Futuro” in the framework of the funded project “Plastic lab-on-chips for the optical manipulation of single cells” (PLUS) and by Progetto “Smart Cities” – Aquasystem: “Procedure e tecnologie innovative per una gestione pianificata ed integrata delle risorse idriche, l'ottimizzazione energetica ed il controllo della qualità nel Ciclo Integrato delle Acque”.
#214515 - $15.00 USD Received 23 Jun 2014; revised 14 Jul 2014; accepted 16 Jul 2014; published 8 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022328 | OPTICS EXPRESS 22339
