This Letter presents an in-line digital holographic system that can provide full amplitude and phase ... In-line and off-axis digital holography are now used in a.
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Complex field recovering from in-line digital holography Pascal Picart1,2,* and Mokrane Malek1 1
LUNAM Université, Université du Maine, CNRS UMR 6613, LAUM, Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France 2
École Nationale Supérieure d’Ingénieurs du Mans, rue Aristote, 72085 Le Mans Cedex 9, France *Corresponding author: pascal.picart@univ‑lemans.fr Received March 5, 2013; revised July 25, 2013; accepted July 25, 2013; posted July 30, 2013 (Doc. ID 186385); published August 20, 2013
This Letter presents an in-line digital holographic system that can provide full amplitude and phase reconstruction without any reference wave, with a single recorded hologram. This major capability is obtained by using a coherent mixing between several object waves generated by a pure spatial phase modulation. A scaling parameter permits us to reconstruct the phase without any unwrapping. The capability of the method to provide quantitative phase contrast measurement and numerical refocusing is demonstrated through experimental results. © 2013 Optical Society of America OCIS codes: (090.0090) Holography; (090.1995) Digital holography; (100.3010) Image reconstruction techniques. http://dx.doi.org/10.1364/OL.38.003230
In-line and off-axis digital holography are now used in a wide range of domains, such as fluid mechanics [1,2], microscopy imaging [3–6], and phase imaging [7,8]. In the basic in-line holographic configuration, the reference and object waves are parallel (no tilt) and the path lengths of both waves are equal. For this reason, a pure in-line holographic arrangement is only able to provide an amplitude image of the object. Therefore, it cannot be used to measure the phase contrast of any transparent object. For a multiple object recovering, such as particles [1], this method suffers from the presence of twin images during the reconstruction. In off-axis digital holography, the reference wave is shaped to provide a spatial separation in the reconstruction plane or in the Fourier plane of the hologram (slight tilt) [2,3,5,8]. The use of an independent reference wave induces sensitivity to external perturbations, such as vibrations, temperature changes, etc., and leads to an increase in the setup complexity. So as to simplify the setup, the in-line configuration is well adapted, but the faculty for the phase contrast recovering has to be developed. In order to overcome such limitations, this Letter proposes (i) to simplify the recording setup by eliminating the reference wave and (ii) to measure the phase contrast of the object. This approach is based on the use of a pure spatial phase modulation so as to produce duplications of the incoming diffracted wave. Since no reference wave is included in the setup, such a method could be qualified as reference-free digital holography or self-referenced digital holography. The principle is based on a pure spatial phase modulation that produces diffraction orders, along which the incident object wave is propagated. Figure 1 illustrates the propagation scheme. The incident laser beam is diffracted by the object, and then it impacts a spatial phase modulation localized behind the object plane. This incident wave is thus diffracted in many directions by the modulation, and the propagated diffraction orders are interfering at a distance δz from the modulation plane. The sensor records the digital hologram produced by the coherent superposition of these diffraction orders. The recording plane is at distance d0 from the object plane. Note Ar0 is the complex object wave front in p the object plane, and Or AO r expiφO ri − 1 is the wave front diffracted at the distance d0 from the 0146-9592/13/173230-03$15.00/0
object plane (r is the vector of the Cartesian coordinates fx; yg in the plane perpendicular to the z direction). Due to the spatial phase modulation, multiple wave fronts are diffracted in different directions. For the nth diffracted order, the propagation direction is given by the wave vector kn 2π∕λen , where en is the unit vector of the propagation direction. Then, the wave front propagated along each diffraction order and impacting the recording plane can be written as Or; kn AO r − rn expikn · r iφO r − rn :
(1)
In Eq. (1), rn is the spatial shift produced by propagation along distance δz from the modulation plane and in the direction of unit vector en . We note that P is the number of propagated wave fronts, having a wave vector kn (n f1; …; Pg), and the recorded hologram can be written according to
Hr A2O r ( 2R
n P X
A2O r − rn
n1 P −1 X P X
) Or; kn O r; km :
(2)
n0 mn1
The digital hologram includes useful diffraction orders whose phase is given by argfOr; kn O r; km g and expressed as
Fig. 1. Basic scheme for the principle of the reference-free digital holographic setup. © 2013 Optical Society of America
September 1, 2013 / Vol. 38, No. 17 / OPTICS LETTERS
Δφnm r kn − km · r φO r − rn − φO r − rm ≅ kn − km · r jsnm j
∂φO r : ∂r · enm
(3)
In Eq. (3), enm is the unit vector of vector snm rn − rm . Equation (3) means that the digital hologram includes diffraction orders that have a phase related to the spatial derivative of the object phase, along an axis given by r · enm . Furthermore, the hologram includes a spatial phase modulation related to the scalar product kn − km · r. In the Fourier plane of the digital hologram, the diffraction orders are separated from the zero-order diffraction and localized by the spatial frequency vector kn − km ∕2π. So they can be spatially filtered [3]. From Eq. (2), the zero order in the Fourier plane is proportional to 1 Σn exp2iπu · rn . If the spatial phase modulation produces symmetric diffraction orders, then the sum of the exponential is equal to zero. So the object amplitude AO r is retrieved from the filtering of the zero order in the Fourier spectrum. After extracting each useful order, the term jsnm j∂φO ∕∂r · enm is extracted from the argument of the inverse Fourier transform of the filtered hologram spectrum. We note ∂φO ∕∂r · enm ∂φO ∕∂xk , jsnm j αk λδz∕p, and Dk 1∕αk × jsnm j∂φO ∕∂r · enm , with k varying from 1 to Q, Q being the number of really independent axis r · enm amongst the set of useful orders included in the hologram (p is the modulation period in both the horizontal and vertical directions). Then, spatial integration is carried out to obtain a quantity Ψ, the numerical method being based on the weighted leastsquare criterion [9], as λδz 1 φ FT−1 Ψ p O 2iπ
"P Q
# ~k D β u k k k1 : P Q 2 k1 β k uk
(4)
In Eq. (4), FT means Fourier transform, βk is the weight ~ k is the applied to Dk , uk is the dual variable of xk , and D Fourier transform of Dk . Note that the scaling coefficient jsnm j depends on the ratio δz∕p and that it can be adjusted so that the measured phase differences do not need to be unwrapped prior to the reconstruction (in case that they exceed 2π jumps). In the case in which phase singularities are present (i.e., light reflected from rough surfaces or scattering media), alternative reconstruction algorithms are to be expected [10,11]. The phase of the object wave can be obtained according to φO p × Ψ∕λδz. From the full object complex field recovered at the sensor plane, backpropagation to the image plane is made possible, and a focused image can be obtained. The numerical refocusing can be performed with the angular spectrum transfer function method [6]. The algorithm uses a double fast Fourier transform by applying A FT−1 fFTfAO expiφO g × Gg, with G being the Fresnel angular spectrum transfer function defined as Gu; v exp−iπd0 u2 v2 , with u; v the spatial frequencies. The experimental proof of principle is provided through the experimental setup described in Fig. 2. A cw laser at λ 532 nm is shaped to produce a collimated beam to illuminate the object placed in front of
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the microscope objective (with tube lens L1 ). The spatial light modulator [(SLM) holoeye] is based upon reflective liquid crystal on silicon [(LCOS) 1920 × 1080 pixels and pitch 8 μm]. The sensor includes 1344 × 1024 pixels with 6.45 μm pitch and records the digital holograms. The spatial phase modulation required to generate the reference-free hologram is programmed to be a symmetric two-dimensional phase grating. Figure 2 shows a zoom of the pure phase modulation image applied to the SLM. The gray levels are previously calibrated to provide a linear phase shift from 0 to 2π at the used wavelength (532 nm). From the two-dimensional phase grating, five waves are impacting the sensor plane: four waves diffracted from the modulation and the wave reflected from the SLM. These five waves are coherently mixed and give the digital hologram. With four symmetrically diffracted beams, there are six useful orders included in the digital hologram, from which we can extract four independent derivative axes (Q 4). The derivative axes are the horizontal x and vertical y axes, and two axes oriented at π∕4 and −π∕4 from the fx; yg axis. The distance sk between the four orders is λδz∕p in both p the horizontal and vertical axes (α1 α2 1) and 2λδz∕p p in the axes oriented at π∕4 and −π∕4 α3 α4 2. The spatial frequencies carrying the useful order are related to 1∕p. The 4f optical system produces an image of the SLM plane near the sensor area at distance δz. The full system (MO L1 L2 L3 ) produces an in-focus image at the sensor area (d0 0). An out-of-focus image (d0 ≠ 0) can also be produced by adjusting the position of the object in front of the microscope objective. In the Fourier spectrum of the hologram, the mean spatial frequencies of the four useful spectral contributions were estimated at f42.1; 0g mm−1 for the x-derivative spectrum, f0; 42.1g mm−1 for the y-derivative, and f42.1; 42.1g mm−1 for the π∕4 oriented axis. Thus, the pitch of the spatial phase modulation is estimated to be p 1∕42.1 23.8 μm. The orders are filtered through a binary circular mask 42.1 mm−1 in diameter [3], corresponding to the available spatial bandwidth between the useful orders. Since the transverse magnification of the system is about 11.8 (due to MO and L1 ), the spatial resolution in the reconstructed object is estimated at 2 μm and is related to the pitch of the phase modulation.
Fig. 2. Setup for the in-line holographic recording with a pure spatial phase modulation (MO, microscope objective; L1 , tube lens; and SLM, spatial light modulator).
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Fig. 3. (a) Estimated phase map φO and (b) comparison between profiles calculated from Ψ (red line) and measured using a Dektak profilometer.
The phase retrieving capability of the proposed method is demonstrated through the use of a pure phase object. This object was realized in a BK7 glass plate (refractive index n 1.51947 at λ 532 nm) by lithography and etching to engrave square plots and rails at its surface. The height of the plots was 1.4 μm, the width of each plot is 263 μm, and the width of each rail is about 45 μm. A profile of the engraved structure was measured using a Dektak contact profilometer. In the setup, distance δz was adjusted to δz 1.18 mm. The surface of the object was focused onto the sensor area (d0 0). The signal-to-noise ratio of each derivative map is estimated according to [12]. These estimations are used as weights in Eq. (4) to calculate Ψ. It was measured that σ 1 0.055 rad, σ 2 0.054 rad, σ 3 0.071 rad, and σ 4 0.079 rad. So, noise is slightly greater in the π∕4 oriented axis than in the x–y ones. The y axis exhibits the lowest noise; weighting the derivatives is carried out by considering β2 1, β1 σ 2 ∕σ 1 0.9818, β3 σ 2 ∕σ 3 0.7606, and β4 σ 2 ∕σ 4 0.6835. Then, the physical height of the object was calculated by h p × Ψ∕2πδzn − 1). Profiles obtained along the estimated surface height and the Dektak profile can be compared (note that the Dektak profile was obtained in a region that was not exactly the same as for the holographic measurement). Figure 3(a) shows the object phase φO p × Ψ∕λδz estimated from the experimental data. Thanks to the scaling parameter δz∕p, the reconstruction process leads to a continuous phase map, for which no unwrapping is necessary. After rescaling the phase and computing the surface height, profiles can be obtained and compared in Fig. 3(b). As can be seen in Fig. 3(b), the comparison between the estimated height profile and that obtained using the Dektak is quite satisfactory. Then, the object was slightly defocused from its initial position, so that the final image given by the optical system was localized nearly at 48 mm from the sensor plane (reconstruction distance is then d0 −48 mm). After calculating Ψ and rescaling to get phase φO , the complex field is backpropagated to the image plane so as to get a focused image. Figures 4(a) and 4(b) show the object amplitude and phase φO , at the sensor plane. Figure 4(c) shows profiles (along red line) of (optically) in-focus and (numerically) refocused phase contrast after numerical refocusing using the angular spectrum transfer function. In order to estimate the noise stability of the setup, the phase contrast was measured with a time series of 100
Fig. 4. (a) Out-of-focus image amplitude and (b) phase φO at CCD plane. (c) Profiles of focused and refocused phase contrast image. (d) Noise probability density function.
holograms over 60 s. The spatiotemporal phase noise distribution is then calculated by subtracting the phase at every recorded instant to the initial phase, and converted into physical units. Figure 4(d) shows the noise probability density function, and the standard deviation is estimated to σ 10 nm. This noise is due to the SLM instabilities and can be reduced by averaging independent measurements. In conclusion, this Letter presents an alternative reference-free recording system for digital holography. The setup is considerably simplified compared to off-axis configurations. The capability of the method to provide quantitative phase imaging without any unwrapping and numerical refocusing has been demonstrated. The author thanks Prof. N. Yaakoubi from LAUM for helpful discussions. This research is supported under Grant Agreement No. 2010 10262 by the Région Pays de la Loire, France. References 1. D. Lebrun, D. Allano, L. Méès, F. Walle, F. Corbin, R. Boucheron, and D. Fréchou, Appl. Opt. 50, H1 (2011). 2. J.-M. Desse, P. Picart, and P. Tankam, Opt. Express 16, 5471 (2008). 3. E. Cuche, P. Marquet, and C. Depeursinge, Appl. Opt. 39, 4070 (2000). 4. J. Garcia-Sucerquia, Opt. Lett. 37, 1724 (2012). 5. P. Ferraro, S. Grilli, D. Alfieri, S. De Nicola, A. Finizio, G. Pierattini, B. Javidi, G. Coppola, and V. Striano, Opt. Express 13, 6738 (2005). 6. C. Mann, L. Yu, C.-M. Lo, and M. Kim, Opt. Express 13, 8693 (2005). 7. B. Bhaduri, H. Pham, M. Mir, and G. Popescu, Opt. Lett. 37, 1094 (2012). 8. P. Girshovitz and N. T. Shaked, Opt. Express 21, 5701 (2013). 9. S. Velghe, J. Primot, N. Guérineau, M. Cohen, and B. Wattellier, Opt. Lett. 30, 245 (2005). 10. C. Falldorf, J. Opt. Soc. Am. A 28, 1636 (2011). 11. C. Falldorf, C. von Kopylow, and R. B. Bergmann, Proc. SPIE 8413, 84130Q (2012). 12. P. Picart, R. Mercier, M. Lamare, and J.-M. Breteau, Meas. Sci. Technol. 12, 1311 (2001).
